Thesis (pdf) - Swinburne University of Technology

Bose-Einstein Condensation
in Micro-Potentials
for Atom Interferometry
Dipl. Phys. Falk Scharnberg
A thesis submitted for the degree of
Doctor of Philosophy at
Swinburne University of Technology
Melbourne, Australia
Hamburg, March 06, 2006

Abstract
Interferometry with atoms is a young discipline of physics. The first interfer-
ometers were realised in the early 1990s, using light pulses as beamsplitters
in momentum space. Recent developments in atom optics have led to new
proposals for interferometers, where the splitting takes place spatially. Mag-
netic and optical traps are both suited for this kind of interferometer if a well
defined and highly controllable trap is realised. In this thesis results from two
experiments that in principle allow the creation of such traps are presented.
After reviewing the principles and techniques of atom optics necessary for
the understanding of this thesis, a theoretical discussion about the spatial
single atom interferometer in a double well potential follows. It is shown that
within reasonable limits the system can be reduced to two levels and solved
by the Bloch equations. Using the realistic case of a not perfectly symmetric
double well potential allows an understanding of the physics behind such an
interferometer: how phase is accumulated and how localisation of the atoms
leads to the loss of the interferometric signal.
Then two experiments are presented. One experiment was newly built up as
part of this thesis: it uses a novel hybrid “atom chip” with a combination of a
magneto-optical film and current-carrying structures to produce the magnetic
trapping potentials. This experiment allowed 5·10 8 87 Rb atoms to be captured
in a magneto-optical trap, where the atom-chip’s surface acts as a mirror. The
atoms were then transferred into a purely magnetic trap which was created by
iii

current carrying structures on the chip. From there on, RF radiation enforced
evaporative cooling, so that a quantum degenerate Bose-Einstein condensate
of up to 10 5 atoms was created. It was also shown that the magneto-optical
film is able to hold and trap the atoms.
In the second experiment atoms are trapped in the spatially varying inten-
sity of light of 1.03 µm wavelength. This experiment used an existing set-up as
a basis and was modified and improved for this new project. This experiment
starts with 10 9 87 Rb atoms in a 6-beam magneto-optical trap. After examining
and optimising the loading process, 1.5 · 10 5 atoms could be loaded directly
from the Magneto-optical trap into the optical dipole potential of two crossed
laser beams at 1.03 µm. Evaporative cooling was demonstrated though the
phase transition to quantum degeneracy was not reached.
iv

Acknowledgements
It is very hard for me to order my acknowledgements as there are so many
people whom I would like to thank. Fortunately, the first address of my thanks
is not a question at all: I would like to thank my supervisor Prof. Peter
Hannaford for all that he has done. Without exaggeration I would like to call
him a luminary, not only as a physicist and especially spectroscopist, but also
as a supervisor who has a keen and generous eye for the needs of all of his
students.
And now it becomes difficult. I will thank location-wise, and start with
Swinburne. Here someone that deserves more thanks than I could count is
Shannon Whitlock. He joined after my first year and worked on the experi-
ment during my stays in Hannover. Without him this experiment would never
have become such a success. I enjoyed working with him, be it in the lab or
discussing the atom interferometer theory. He also proved to be invaluable
during my writing up in Germany, being my eyes in the lab and answering all
the things I forgot to write down and now couldn’t look up quickly. Further-
more I want to thank the project leader of the atom chip experiment Prof.
Andrei Sidorov for his constant endeavours to support me and promote the
experiment. I enjoyed working with him in the lab. Dr. Brenton Hall I would
like to thank for the fun in the lab and the expertise he brought and shared
with us students. Someone who also taught me quite a few tricks in the lab
was David Gough. I want to thank him that he made it possible that I could
v

hear and speak German in my Hamburg accent in Queensland. I also want to
thank all others who are part of the atom optics group at Swinburne, mainly
Dr. Alexander Akulshin and Prof. Russell McLean, for helping out whenever
I needed a tool, advice or just another hand. Mark Kivinen I want to thank
“in triplicate” for his great workshop work and Sharon Jesson I want to thank
for all the fights with the administration that I didn’t have to fight.
Now, more private thanks go to my friends Craig Lincoln and Ruth Plathe.
It was great fun sharing a flat with Craig, and I could always count on Ruth
to brighten my mood with her jokes. I also thank Heath who taught me a lot
about the Australian culture and way-of-life. There are more people to thank:
Dru, Craig and Grant, Jürgen and Holger, Grainne, Wayne (Rowlands), Wayne
and Bob, Saeed and probably a dozen more who I should mention. I thank all
of you for making my time overseas so enjoyable.
On the Hannover side, the first thanks are reserved for Prof. Gerhard Birkl
and Prof. Wolfgang Ertmer. They allowed me to work with them in Gerhard’s
group as an exchange student, and made it possible for me to split my work
between Australia and Germany.
I would like to thank all my colleagues of the group A6 in Hannover. First
Dr. Rainer Dumke, who introduced me to the system there and whose dry
humour you can’t forget. Next is André Lengwenus with whom I enjoyed
working during my first time in Hannover: you are doing better work than
you admit to yourself. Dr. Tobias Müther I would like to thank not only for
his vast knowledge on everything and the discussions we had, but also for his
work in the lab, although our overlap there was rather small. Without him,
the successes of the experiment would be unthinkable. I would also like to
thank Johanna Nes and Anna-Lena Gehrmann, the first “women only” crew I
have ever seen “man” an experiment.
Very special thanks go to Dr. Michael Volk. Just like I had my flat mate
vi

Chapter 0: Acknowledgements
in the neighbouring lab at Swinburne, I had a friend and former flat mate
at the neighbouring optical table in Hannover. I enjoyed the smoking and
later the non-smoking breaks we had together. Furthermore I want to thank
Dr. Norbert Herschbach and Dr. Peter Spoden for their encouraging words
whenever I felt a bit disheartened, and Sascha Drenkelforth. And of course
thanks go to all other group members who I didn’t name here; I wish you all
the best for the future.
I also thank my parents for all they have done and for all their support.
Even though this is now the end of my list, the most important person in my
life has not been thanked yet.
Nicole, I love you. I have learnt that I never want to be away from you so
far and for so long.
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viii

Declaration
I, Falk Scharnberg, declare that to the best of my knowledge, this thesis con-
tains no material which has been submitted to another university for the award
of any other degree, previously published or written by another person except
where due reference is made in the text. Where the work is based on joint
research or publications, the relative contributions of the respective workers or
authors are disclosed.
ix
Falk Scharnberg,
Hamburg, March 06, 2006

Publications by the Candidate
• A. I. Sidorov, R. J. McLean, F. Scharnberg, D. S. Gough, T. J. Davis, B.
J. Sexton, G. I. Opat and P. Hannaford. Permanent-magnet microstruc-
tures for atom optics. Act. Phys. Pol. B 33, 2137-2155 (2002).
• J. Y. Wang, S. Whitlock, F. Scharnberg, D. S. Gough, A. I. Sidorov,
R. J. McLean and P. Hannaford. Perpendicularly magnetized, grooved
GdTbFeCo microstructures for atom optics. J. Phys. D: Appl. Phys. 38,
4015-4020 (2005). Included as appendix A.3.
• B. V. Hall, S. Whitlock, F. Scharnberg, P. Hannaford and A.I. Sidorov.
Bose-Einstein condensates on a permament magnetic film atom chip. In:
Laser Spectroscopy; Proceedings of ICOLS 2005, E. A. Hinds, A. Fer-
guson and E. Riis (Editors), page 275-282 (World Scientific, Singapore,
2005).
• B. V. Hall, S. Whitlock, F. Scharnberg, P. Hannaford and A.I. Sidorov.
A permanent magnetic film atom chip for Bose-Einstein condensation.
J. Phys. B: At. Mol. Opt. Phys. 39, 27-36 (2006). Included as appendix
A.2.
• A. I. Sidorov, B. J. Dalton, S. Whitlock and F. Scharnberg. The asym-
metric double-well potential for single-atom interferometry. Phys. Rev.
A 74, 023612 (1-9) (2006). Included as appendix A.1.
xi

xii

Contents
Abstract iii
Acknowledgements v
Declaration ix
Publications by the Candidate xi
1 Introduction 1
2 Theoretical Background 13
2.1 Atoms and Electromagnetic Fields . . . . . . . . . . . . . . . . . 14
2.1.1 Dressed state model . . . . . . . . . . . . . . . . . . . . 15
2.1.2 Absorption and emission of photons . . . . . . . . . . . . 18
2.1.3 Detection of atoms . . . . . . . . . . . . . . . . . . . . . 21
2.1.4 Trapping of atoms in light fields . . . . . . . . . . . . . . 23
2.1.5 Cooling atoms with light . . . . . . . . . . . . . . . . . . 24
2.2 Atoms and Magnetic Fields . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Magnetic trapping . . . . . . . . . . . . . . . . . . . . . 28
2.2.2 Permanent magnets . . . . . . . . . . . . . . . . . . . . . 34
2.3 Evaporative Cooling . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 A simple model . . . . . . . . . . . . . . . . . . . . . . . 36
xiii

CONTENTS
2.4 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . 39
2.5 The Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . 41
2.5.1 Mirror MOT . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.6 Atom Interferometry . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6.1 Atom interferometry with symmetric double well poten-
tials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 The Asymmetric Double Well 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Two Mode Approximation and the Bloch Equations . . . . . . . 56
3.2.1 Two mode approximation . . . . . . . . . . . . . . . . . 56
3.2.2 Bloch equations . . . . . . . . . . . . . . . . . . . . . . . 59
3.3 The Results of the Model . . . . . . . . . . . . . . . . . . . . . . 65
3.3.1 General results . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.2 Comparison with experimental data . . . . . . . . . . . . 74
3.3.3 CARP: Coherent Adiabatic Readout Process . . . . . . . 76
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4 The Permanent Magnetic Chip Experiment: Apparatus 81
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 The Laser Systems . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2.1 Main laser . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2.2 Repumping laser . . . . . . . . . . . . . . . . . . . . . . 87
4.2.3 Optical pumping laser . . . . . . . . . . . . . . . . . . . 88
4.2.4 Optical paths on the experiment table . . . . . . . . . . 90
4.3 The Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3.1 Experiment chamber . . . . . . . . . . . . . . . . . . . . 94
4.3.2 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . 95
xiv

CONTENTS
4.3.3 Procedure to reach UHV . . . . . . . . . . . . . . . . . . 96
4.4 The Magnetic Field Coils . . . . . . . . . . . . . . . . . . . . . . 98
4.4.1 Offset coils . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4.2 Quadrupole coils . . . . . . . . . . . . . . . . . . . . . . 100
4.4.3 Bias field coils . . . . . . . . . . . . . . . . . . . . . . . . 100
4.5 The Atom Chip . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.5.1 Overall design . . . . . . . . . . . . . . . . . . . . . . . . 101
4.5.2 Current-carrying wires . . . . . . . . . . . . . . . . . . . 103
4.5.3 Magneto-optical film . . . . . . . . . . . . . . . . . . . . 105
5 The Permanent Magnetic Chip Experiment: Results 109
5.1 Overview and Timing Sequence . . . . . . . . . . . . . . . . . . 109
5.2 The Magneto-Optical Traps . . . . . . . . . . . . . . . . . . . . 111
5.3 The Wire Magnetic Trap . . . . . . . . . . . . . . . . . . . . . . 124
5.3.1 Evaporation and BEC . . . . . . . . . . . . . . . . . . . 135
5.4 The Permanent Magnetic Trap . . . . . . . . . . . . . . . . . . 145
6 The All-Optical BEC Experiment: Apparatus 153
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2 The Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . 156
6.3 The Diode Laser Systems for Magneto-Optical Trapping . . . . 157
6.3.1 Main laser . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3.2 Repumping laser . . . . . . . . . . . . . . . . . . . . . . 159
6.3.3 Chirp lasers . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.4 The Dipole Trap . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.5 Detection of Atoms . . . . . . . . . . . . . . . . . . . . . . . . . 162
7 The All-Optical BEC Experiment: Results 165
xv

CONTENTS
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.2 The Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . 166
7.3 Loading the Dipole Trap . . . . . . . . . . . . . . . . . . . . . . 168
7.4 Evaporative Cooling in the Dipole Trap . . . . . . . . . . . . . . 174
8 Summary and Outlook 179
8.1 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . 179
8.2 Outlook and Future . . . . . . . . . . . . . . . . . . . . . . . . . 181
A Reprints of selected Publications by the Candidate 185
A.1 Asymmetric double-well potential for single-atom interferometry 186
A.2 A permanent magnetic film atom chip for Bose-Einstein con-
densation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
A.3 Perpendicularly magnetized, grooved GdTbFeCo microstructures
for atom optics . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
B Determining the Temperature from One Image 199
C The Atom: 87 Rb 201
D Technical Details of the Coils for the Permanent Magnetic
Chip Experiment 203
E List of Equipment 205
xvi

Chapter 1
Introduction
One revolution in physics which has had a big influence on everyday life was
the discovery by Planck in 1900 of the quantisation of action, which led to
quantum mechanics. In slightly more than one human lifetime this has led to
the discovery of semi-conductors and their use in computation, and to lasers
and their use in communication, medicine and many other fields. Even more
impressive has been the impact of this discovery on physics and the understand-
ing of Nature itself. Here, one of the most controversially discussed concepts
is the often appearing duality. One manifestation was labelled “wave-particle-
dualism”, as quantum mechanics allows classical waves (like electromagnetic
waves) to behave like particles [Ein05] and, vice versa, particles to behave like
waves [de 24]. The term is put in inverted commas as it will be later explained
that this “dualism” is not an “exclusive or” in the boolean sense. The wave-
like behaviour usually is connected to the particle’s ability to interfere with
itself, while the particle-like behaviour of a wave usually is connected to the
wave’s ability to scatter, to transfer momentum by that scattering and finally
to be localisable.
At first, this duality was explained by and contributed to another duality:
the fundamental impossibility to measure both position and momentum with
1

infinite accuracy [Hei27, Boh27]. Later it was shown that these two dualisms
are not related to each other but both are fundamental in quantum mechan-
ics [Scu91, Eng96]. Indeed the problem to measure wave-like or particle-like
attributes is related closely to the number-phase uncertainty.
This can be easily seen in terms of interferometric concepts. Once it is
known which way is taken in the interferometer’s arms (knowledge of the num-
bers) we lose the interference (knowledge of the phase) [Bru96, D¨98]. These
experiments showed that imperfect knowledge of which way leads to a decrease
in the contrast of the interference fringes to a correlated degree, and not to a
total destruction of the fringes .
Even before this interpretation of the wave-particle dualism, other concepts
of classical waves were reformulated into quantum mechanics, such as the co-
herent state 1 [Gla63], which corresponds to a monochromatic plane wave and
is the state with a minimum uncertainty in both phase and number. This the-
ory was first formulated for photons, where we have no number conservation.
When dealing with atoms, we find the difficulty that the overall number of
atoms usually is not only constant but also very well defined. In a single cloud
of atoms, we can in principle measure the exact number of atoms with zero
uncertainty. Such a state is called a Fock state or number state and has no
defined phase (see for example [Nol94]).
This raises the question: if we can measure the exact number of atoms,
how can we still create states with a defined phase for interferometry? How
else can we talk about the wave-like properties of atoms but by the ability to
use them in interference experiments?
The answer to this question is given in the field of “atom optics”. The name
itself implies atoms, and thus countable particles, with optics, the discipline of
2005.
1 The quantum theory of coherence was awarded half of the Nobel prize for physics in
2

Chapter 1: Introduction
physics that totally relies on the wave-like properties of its subjects, and indeed
this is the field where the wave character of the atoms becomes important. Of
course, when talking of wave character, one must be able to parameterise the
atoms by a wavelength. This is done by the so-called ‘de Broglie’ wavelength,
named after the scientist who first postulated the wave-like behaviour [de 24].
This wavelength is the ratio of Planck’s action quantum h and the momentum
of the particle p, and thus is inversely proportional to the square root of the
temperature T of a slowly moving particle:
λdB = h
p ∝
�
1
T
This means, the colder an ensemble of atoms is, the longer is the wavelength
of the particles. Considering Rubidium atoms, at room temperature the wave-
length is 27 pm, while for very low temperatures like 1 µK the corresponding
wavelength is in the optical range of 467 nm. Reducing the temperature by
another factor of 100 to 10 nK leads to a ten-fold increase in the wavelength
to 4.7 µm, a wavelength that is very much larger than the ‘size’ of the atom
in the particle picture.
These temperatures are interesting for a physicist, as in one case the mo-
mentum of an atom is comparable to the momentum of an interacting photon,
the spectroscopist’s main tool. Then the measurement process is not negligi-
ble anymore. The case of even lower temperatures is interesting in itself, as it
seems to be a contradiction - or at least hard to imagine - that a wavelength
can be much longer than the carrier wave’s extension.
Reaching these very low temperatures is not trivial though. Today, tem-
peratures in the region of a few µK or less can only be achieved using dilute
gases. With the introduction of the magneto-optical trap (MOT) [Raa87] and
the understanding of the mechanisms of how to cool atoms with the help of
laser radiation [H¨75, Win75, Dal89], it became possible to create samples of
3

several million up to several hundred million atoms at these very low tem-
peratures. These break-throughs were rewarded with the award of the Nobel
prize for physics to Steven Chu, William Phillips and Claude Cohen-Tannoudji
in 1997. Having such cold atoms in such numbers allowed the creation of
several types of atom interferometers, which the field of metrology has bene-
fited from. These interferometers ranged from devices to measure the Earth’s
gravity [Pet99] and rotation [Gus97] to atomic clocks, which in principle are
interferometers only in the time domain [Rus98, Wil02], and to interferome-
ters that were used to measure atomic properties [Eks95] or natural constants
[Gup02]. All of these interferometers work with free atoms, either falling or
accelerated from the trap in which they were cooled, and the beamsplitters
needed for interferometry are created by light pulses and the change of the
atomic momentum on absorption of a photon [Kas91]. With the thermal mo-
tion and kinetic energy being greatly reduced, it became possible to trap and
confine large numbers of atoms with rather weak forces, stemming for example
from the interaction between a magnetic field and the magnetic moment of
an atom or from the interaction of the induced atomic electric dipole-moment
with an electro-magnetic field. With these magnetic and optical traps even
more versatile atom interferometer set-ups are possible.
Two main implementations of atom interferometers have been proposed: in
the first example, an otherwise stationary trap is split into two, held there and
recombined [Hin01]. The second example works with confined atoms that pass
through wave guides and beam splitters. Only recently it has become possible
to create these time dependent (in the first case) or spatially dependent (in the
second case) potentials. Here either micron-sized optics and lenses are used to
create optical traps which can be split and recombined [Dum02b] or waveguides
are constructed [Dum02a]. For magnetic traps, the proposal [Wei95, Thy99]
and realisation [Rei99, Ott01] of the so-called “atom chip” made the tailoring
4

Chapter 1: Introduction
of micron-sized potentials possible. Here micron-sized current-carrying wires
on a chip, as known from microelectronics, induce the trapping fields. Both
the time dependent and spatially dependent interferometer types are obtain-
able with magnetic trapping or confinement. Switchable waveguides [M¨01],
beam splitters [Cas00] and the splitting and recombining of traps [Est05] have
all been demonstrated. It has to be noted that splitting and merging with
macroscopic traps was also achieved [Tho02, Tie02], but atom chips allow
more precise control and thus are favoured.
Another break-through in the field of atom optics, which may prove very
useful for atom interferometry, was the creation of Bose-Einstein Condensates
(BEC) from dilute gases of alkali atoms [And95, Dav95, Bra95]. The existence
of this new state of matter was first postulated by S. N. Bose and A. Einstein in
1925 [Ein25], and the experimental proof of this phase transition was realised
70 years later, resulting in the award of the 2001 Nobel prize in Physics to E.
Cornell, C. Wieman and W. Ketterle. In simple words, the phase transition
from a thermal cloud of atoms to the condensate occurs when the de Broglie
wavelength of the atoms is larger than the mean distance between them. The
waves then overlap and become coherent. In the end, the waves of all atoms
oscillate in phase and a macroscopic coherent matter state is obtained. This
transition was found for dilute gases of atoms with temperatures of a few hun-
dred nanokelvin, as explained above. Shortly after the experimental realisation
of a BEC, theoreticians began investigating its behaviour in double-well po-
tentials, which are traps that are split into two [Mil97, Sme97, Jav97]. These
results were then used to examine the feasibility of interferometry with a BEC
[Men01]. An early result was that the splitting of the BEC will have a thresh-
old. At this point, it will become virtually impossible for atoms to tunnel from
one well to the other and the condensate “fragments” [Spe99]. This will fix
the atom number in each well, and the coherence between the two wells will be
5

lost—making any meaningful interferometry impossible after this point when
the atoms are localised in the left or right well. This collapse into separate
BECs with undefined relative phase was experimentally seen not only for two
wells [Shi05] but also for an array of traps [Gre02a, Gre02b]. These results
seem to indicate that the splitting of the BEC has to stay in the so-called
Josephson regime to be useful for interferometry [Jav86, Sme97, Sak02], or
that the whole process of splitting, phase evolution and recombining for the
measurement has to be done on short timescales to maintain the coherence
between the wells. The coherence between the wells will only survive as long
as the atom numbers are undefined. One way to achieve this is to split only
so far that some tunnelling is still allowed, so that the atom numbers of the
wells fluctuate and are not defined, like in the the above-mentioned Josephson
regime. Another way to keep the coherence is to split the single atoms between
the wells. As long as localisation can be prevented, the atoms in the two wells
then are coherent. However, a BEC consists of interacting particles, and these
interactions can be interpreted as measurements of the location of the atoms.
This limits the coherence time, but there are ways to reduce the interaction
strength by so-called Feshbach resonances [Mar02, Web03, Shi04].
Theoretical work is continuing on this topic, examining the splitting pro-
cess more closely [Bor04] for weakly interacting atoms and ways are being re-
searched on how to increase the sensitivity of such an interferometer [Neg04].
Other proposals use the BEC only as a means to create atoms in the ground
state of the potential well. Atoms that do not interact with each other, because
they have a scattering length of zero, will overcome some of the problems of
the interferometers proposed with interacting BECs. These then in principle
work like multiple single atom interferometers [Dud03], where the single atom
interferometers are the ones proposed for atom chips [Hin01, H¨01c]. Other
6

Chapter 1: Introduction
proposals use the BEC and then use techniques that are beamsplitters in mo-
mentum space rather than in real space [Pou02].
A working interferometer using a BEC and the splitting and recombining
of the trap has been experimentally demonstrated [Shi04, Jo07]. In [Shi04]
a limit on the coherence time between the wells was found, after which the
relative phase became undefined. Another similar experiment has shown this
limitation as well, by having random positions of the interference fringes from
one experimental run to another [Shi05]. The traps in these two experiments
are comparable for the important parameters; the main difference is that the
splitting in the second experiment takes about 40 times the coherence time
that was determined in the first experiment. How such an increase can affect
a single atom double well device and destroy any interferometric signal is
covered in chapter 3 of this thesis. A BEC of interacting particles has even
more possible effects that lead to localisation and the loss of a defined phase
relation. Nevertheless the chapter presented here (Chapter 3) can already
give an overall idea and lead to a new “gut feeling” of suitable timescales
for experiments. In another experiment, a Michelson interferometer using a
magnetic wave guide has also been demonstrated [Wan05b]. Here optical pulses
acted as beam splitters in momentum space, in exactly the same way as the
interferometers with cold atoms in free space which were described above.
This thesis is organised in the following way. Chapter 2 gives information
about the theoretical background that is needed to understand all subsequent
chapters. This background is well known and can be found in textbooks for
graduate students, for example [Met99, Mes90, Mey91, Pet02]. In later sections
of the chapter recent developments play a stronger role and the respective
publications are discussed there. The chapter contains no new material.
Chapter 3 deals with a theoretical analysis of the single atom interferome-
ter, and early results of this work have been published by the author [Sid06].
7

Here a model is presented which in contrast to most publications [Hin01, H¨01c]
incorporates an unavoidable experimental effect: the imperfect symmetry of
the double well potential. A two mode approximation is developed for this
case. The analogy with all two-level systems allows the use of the Bloch equa-
tions. Their validity was checked with the numerical results of the full system,
performed by S. Whitlock [Whi04] using the XMDS software developed at the
University of Queensland. The simplicity of the Bloch model allows simula-
tions on a much faster scale and application to many related problems. The
presented two mode model is well known [All75, Mey91]. To the author’s
knowledge the application to this kind of interferometer is new. The inter-
pretation of the model using the Bloch sphere and the solutions of the Bloch
equations give further insight into the physics that occurs in such an interfer-
ometer, identifying the main process as Larmor precessions. It is shown that
an asymmetry is crucial and needed for an interferometer to work, and should
be incorporated in any model, as it gives rise to the phase shift that the in-
terferometer measures. The model also shows how experimental imperfections
that translate into asymmetries in the potential can be overcome, clarifying
a common misconception among experimental physicists about adiabatic pro-
cesses and their role in atom interferometry. It also explains where and how
the change from phase difference to measurable number difference is mani-
fested. Although the model strictly is applicable to single atoms only, the
main results have to be kept in mind for BEC-based interferometers. A/Prof.
B. Dalton is acknowledged for correcting a factor in the Bloch equations. Two
main limits of the model are identified by comparing the two mode model
with the full numerical analysis. A new way to read out a double well atom
interferometer was inspired by questions of Prof. P. Drummond of the Uni-
versity of Queensland and is based on an idea of Prof. A. Sidorov. A related
read out process was later successfully implemented in a BEC-based double
8

Chapter 1: Introduction
well system [Hal07b] on the permanent magnetic chip described in chapters 4
and 5. The model is kept simple, so that different experimental parameters
can be implemented easily. Even though it works with single atoms in double
well, we find a surprisingly close match with the experimental data for a BEC
in a double well [Jo07]. These two experimental publications emphasise the
importance of this chapter, although it was mainly intended not as a tool for
prediction, but to give a deeper understanding of the actual physical processes
in double well potentials. Further applications to double well problems outside
of atom optics are briefly discussed. Although the author is not first author
on a publication that reports the early results [Sid06], the work on which this
publication is based was mainly undertaken by him.
Chapters 4 and 5 contain the description of the technical details and the
results of the experiment which the author worked on at Swinburne University
of Technology. Chapter 4 contains a full description of the experimental set-up
of the permanent magnetic film atom-chip experiment. Here the experimental
apparatus including the laser systems, the vacuum chamber and the novel
hybrid atom chip with both current-carrying wires and a permanent magnetic
film are presented and characterised. This work was done mainly in association
with S. Whitlock and Dr. B. Hall. In the early stages this part of the work was
supported by D. Gough and Prof. A. Sidorov. Results have been published
covering the magnetic film and its properties [Wan05a] and experimental data
with some technical details of the set-up [Hal06]. The TbGdFeCo magneto-
optical film was fabricated by J. Wang, S. Whitlock and D. Gough. As the
setting up of this unique experiment is an integral part of this thesis, the set-up
and the individual parts are described in some detail.
After the characterisation of the experimental set-up, the analysis of the
data taken with the new permanent-magnetic-film atom-chip is presented in
Chapter 5. With the help of this chip, atoms were first collected and cooled in a
9

mirror MOT and then transferred into a current-carrying wire magnetic trap,
in which the atomic sample was further cooled by allowing the atoms with
highest kinetic energy to “evaporate” from the trap [Hes86, Mas88]. With
this cooling, the phase transition to a quantum degenerate BEC was achieved.
Then, with the current in the wires turned off, trapping of ultracold atoms
in the field of the permanent magnetic structure was successfully achieved
and these results are presented at the end of this chapter. A BEC was then
successfully realised on a similar permanent magnetic structure [Hal05, Hal06]
while the author was in Hannover working on the all-optical BEC experiment
described in Chapters 6 and 7. The measurements in this chapter were taken
together with S. Whitlock and Dr. B. Hall, and have been published in [Hal06].
Chapters 6 and 7 present the experimental set-up and the results of the
experiment at the University of Hannover, where the author participated in
the experiment for limited periods. A tri-national network grew out of this
collaboration. In Chapter 6 the set-up of the all-optical BEC experiment is
described. This apparatus was set up and first used by Dr. F. Buchkremer
for the coherence investigations in optical waveguides and traps [Buc01]. The
following experiment implemented miniaturised optical elements like arrays
of micron-sized lenses to demonstrate their in-principle use for quantum in-
formation and atom interferometry [Dum03a]. The author first joined the
experiment at that stage. An existing experimental set-up was modified and
improved, and the author was involved in experiments and measurements with
the aim of producing an all-optical BEC. This work was done together with Dr.
R. Dumke in the very beginning. Most of the changes were implemented after
Dr. Dumke had left and then optimised together with A. Lengwenus [Len04].
While the author was absent, the dipole laser system that is now used to trap
the atoms for evaporation was set up by Dr. T. Müther, J. Nes and A.-L.
Gehrmann. A description of the full apparatus can be found in their theses
10

Chapter 1: Introduction
[M¨05, Geh05]. In this thesis the description is kept short, and the interested
reader is referred to the above-mentioned theses for more detailed descriptions.
In Chapter 7 the results from the all-optical BEC experiment are presented.
As a non-resident researcher the author worked with several colleagues over
the time. The collection and pre-cooling of the atomic sample in a MOT
is characterised first. This work was done together with A. Lengwenus and
has also been published in his Diploma thesis [Len04]. The results from the
optical trapping, the optimised loading of atoms into this trap and the evap-
orative cooling were achieved together with Dr. T. Müther, J. Nes and A.-L.
Gehrmann and published in their respective theses [M¨05, Geh05].
The thesis ends with a summary and discussion and an outlook for possible
future experiments.
11

Chapter 2
Theoretical Background
In the branch of atomic physics called “Atom Optics” we deal with cold and
ultra-cold atoms. To exploit the wave like properties of atoms [de 24] the
thermal de Broglie wavelength of the atoms λdB defined by
λdB = h/p = h/ � 3m · kBT (2.1)
has to be increased by cooling down the atoms while confining them in a trap.
The mechanisms of how to cool and how to trap these atoms are thus of main
interest. To understand these mechanisms, one has to understand the principal
interactions between the atoms and the fields that are used to trap and cool
the atoms. Using light and magnetic fields for these purposes, we are in the
fortunate position that this theory is well understood.
This chapter will introduce the general theory and emphasize and explain
certain points in more detail, which are needed for a full understanding of the
experiments that are the main part of this thesis. This chapter starts with
an explanation of the interaction between an atom and electromagnetic waves.
Here we need to differ between two main interactions: one that allows the
trapping of atoms, and one that allows us to cool and to detect our atomic
sample. The detection of atoms will also be addressed here. The next section
13

2.1. Atoms and Electromagnetic Fields
covers the interaction of the atoms with constant magnetic fields, which are
another means to trap the atoms. It will then be explained how one can cool
the atoms to quantum degeneracy; here the removal of energy from the sample
is coupled to the removal of particles. A combination of magnetic interaction
and optical forces leads to the magneto-optical trap, which is discussed in the
next section. The experiments discussed in this thesis have slightly different
arrangements for this trap, so both are presented here. The chapter ends with
a section on a main application of cold atoms and atom optics: using the wave
like properties of cold atoms to perform interferometry.
2.1 Atoms and Electromagnetic Fields
The interaction between atoms and electromagnetic fields is a topic which can
be treated in a nearly arbitrarily detailed fashion [CT97, CT92]. Although,
to understand the basic concepts, some assumptions can be made to simplify
the problem and not lose too much information. If the spectral linewidth of
the light is small enough so that only two atomic energy levels are coupled
by the light, then other atomic energy levels which do not contribute to the
problem can be ignored, reducing the complexity [Mey91]. Further, one can
ignore the quantised nature of the light, assuming classical fields that obey
Maxwells equations: if the light is coherent and of sufficient intensity, then
the light field can be well approximated by a classical field [Gla63]. In the
experiments described here, we work with laser light that fulfills both these
conditions. The first simplification will be used to explain how light can create
a conservative potential that can be used to trap atoms. Both simplifications
are needed for the simple model of absorption and scattering explained in the
following section. This model in turn is needed to understand the mechanisms
of cooling atoms optically [Win79].
14

2.1.1 The dressed state model
Chapter 2: Theoretical Background
To describe the situation of an atom in a light field correctly, one needs to
treat the atom, the light field, and their interaction quantum mechanically.
This leads to the so-called dressed states model [Dal85], which will be used
later (section 2.1.4) to explain how atoms can be trapped with light. In the
following, that derivation will be outlined.
If the light is monochromatic, in the sense that the width of the frequency
distribution of the light is small compared to the energy difference between
atomic levels, the atomic polarisability can be calculated from an ansatz that
treats the atom as a two-level quantum system. The ground state is denoted
by |g〉, the only excited state by |e〉. The Hamiltonian of this atomic system
is then
HA = ¯hω0|e〉〈e| (2.2)
where the energy of the ground state has been set to zero and the energy
difference between the levels is ¯hω0. If the light has frequency ωL = ω0 + ∆, it
can be described by the Hamiltonian
HL = ¯hωL(a † a + 1
) (2.3)
2
where a † and a are the creation and annihilation operators, and ˆn = a † a is the
number operator with eigenvalue n, the number of photons in the field. It is
then said that the light is detuned by ∆ with respect to the transition.
To include the interaction of atom and light, we need a third operator
VAL = − � d · � E(�r) (2.4)
Here � d is the operator of the induced electric dipole moment of the atom and
�E(�r) is the operator of the electric field strength at position �r.
15

2.1. Atoms and Electromagnetic Fields
The overall Hamiltonian of the interacting atom-light system is thus
H = HA + HL + VAL
= ¯hω0|e〉〈e| + ¯hωL(a † a + 1
2 ) − � d · � E(�r) (2.5)
We now apply the rotating wave approximation, only allowing transitions be-
tween the nearly degenerated states |g, n〉 and |e, n−1〉. There are cases where
this approximation is not justified. This will be addressed in equation (2.13)
later this section. For now we can assume it to hold. The transition matrix
element is then
〈e, n − 1|VAL|g, n〉 = ¯hΩR
2
Here ΩR = d · E/¯h is the Rabi frequency.
(2.6)
The interaction lifts the degeneracy, so that the eigenstates of Hamiltonian
(2.5) are linear combinations of the eigenstates of the uncoupled system |{g, e}, {n, n−
1}〉
|1, n〉 = cos θ|e, n − 1〉 − sin θ|g, n〉
|2, n〉 = sin θ|e, n − 1〉 + cos θ|g, n〉 (2.7)
Here we use the generalised Rabi frequency
to define the angle θ by
Ω =
cos 2θ = − ∆
Ω
�
∆ 2 + Ω 2 R
and sin 2θ = ΩR
Ω
(2.8)
(2.9)
The eigenvalues of the energy of these dressed states are not degenerate any-
more (see also Fig. 2.1):
E1,n = (n + 1)¯hωL + ¯h
(Ω − ∆)
2
E2,n = (n + 1)¯hωL − ¯h
(Ω + ∆)
2
(2.10)
16

Chapter 2: Theoretical Background
The difference in the energies ±¯hΩ due to the applied light field is called the
ac-Stark shift. In the limit of small frequencies ΩR, the Ω in equation (2.8) can
be expanded around ΩR/(4|∆|). This results in energy shifts of ¯hΩ 2 R /(4|∆|)
of the ground and excited state. For negative detunings ∆ < 0 (red detuned
light), the energy of the ground state is lowered by this amount, while the
energy of the excited state is raised by the same amount. In this limit we have
θ → 0, and the dressed states become identical to their respective unperturbed
eigenstates.
Ε
e, n
g, n+1
e, n-1
g, n
h∆
h∆
ω
h 0
Figure 2.1: Energy of the system atom-light without interaction between
the atomic states and the light field (left side) and including the interaction
(“dressed states”, right side). The notation is explained in the text. The cou-
pling increases the energy gap of the doublet states and in case of resonance
will cause an “avoided crossing”.
hΩ
hΩ
1, n
2, n
1, n-1
2, n-1
The Rabi frequency of a light field is a function of its intensity,
Ω 2 R = 1
2 Γ2 · I
17
I0
(2.11)

2.1. Atoms and Electromagnetic Fields
where Γ is the linewidth of the atomic transition and I0 = ¯hΓω3 0
12πc 2 the saturation
intensity. We can now calculate the change of the energy between the two
levels, as a function of intensity and detuning, and get
∆E = U = 3πc2
2ω3 Γ ·
0
I
∆
∝ I
∆
(2.12)
It needs to be remembered that the potential depth scales with U ∝ I/∆,
proportional to the intensity and inversely proportional to the detuning.
In the above, the rotating wave approximation was used when evaluating
the Hamiltonian of equation (2.5). This approximation is generally valid if the
difference between the frequency of the atomic transition ω0 and of a photon ωL
is small. In the correct quantum mechanical treatment, a second term appears
which does not contain the difference but the sum of the frequencies in the
denominator. For similar frequencies, especially in the optical regime, the
difference term dominates over the sum term. In the case of our experiment,
the transition has a wavelength of 780 nm ( 87 Rb D2) while the laser light has a
nearly 300 nm longer wavelength. Here the influence of the second term with
the sum is no longer negligible. Instead, we find a corrected energy shift
�
�
Γ Γ
+ · I(�r) (2.13)
∆E(�r) = 3πc2
2ω 3 0
ω0 − ωL
ω0 + ωL
The scattering rate which will be introduced later in equation (2.27) also needs
to be corrected in the same way and the term Γ/∆ be replaced by the sum.
In our case neglecting the counter rotating term leads to an approximation of
the potential depth that is more than 10% too small, while the scattering rate
in the rotating wave approximation results in a value that is nearly 30% too
small.
2.1.2 Absorption and emission of photons
A simple way to describe the absorption and emission of photons by atoms is
the density matrix formalism. The quantum mechanical state Ψ of the atomic
18

system is expanded into
Ψ = �
i=1,2
Chapter 2: Theoretical Background
ciφi
where the coefficients ci are complex and normalise the state.
(2.14)
A pure state is defined by the density matrix ρ = |Ψ〉〈Ψ|. The time evolution of
the density matrix for a Hamiltonian H is given by the von Neumann equation:
i¯h dρ
dt
= [H, ρ] (2.15)
As we restrict ourselves to an atom with two levels only, with |φ1〉 = |g〉
the ground state and |φ2〉 = |e〉 the excited state, the density matrix can then
easily be written down
ρ =
⎛
⎝ ρee ρeg
ρge ρgg
⎞
⎠ =
⎛
⎝ |ce| 2 cec ∗ g
cgc ∗ e |cg| 2
⎞
⎠ (2.16)
Normalisation requires |cg| 2 +|ce| 2 = ρgg +ρee = 1. We can now insert equation
(2.16) into equation (2.15). To include spontaneous emission of photons by the
atoms, we include a finite lifetime Γ −1 of the excited state. The time evolution
of the two level system can now be written down as [All75]:
dρgg
dt = +Γρee + i
2 (Ω∗R ˜ρeg − ΩR ˜ρge)
dρee
dt = −Γρee + i
2 (Ω∗R ˜ρge − ΩR ˜ρeg)
d˜ρge
dt
d˜ρeg
dt
� �
Γ
= − + i∆
2
� �
Γ
= − + i∆
2
˜ρge + i
2 Ω∗ R(ρee − ρgg)
˜ρeg + i
2 ΩR(ρgg − ρee) (2.17)
Here ˜ρij = ρije −i∆t , with ∆ being the detuning. The Rabi frequency ΩR is
defined by equation (2.6).
Equations (2.17) are called the optical Bloch equations. Using the inversion
w = ρgg − ρee, the normalisation and the fact that ρeg = ρ ∗ ge, we can simplify
19

2.1. Atoms and Electromagnetic Fields
them to
dw
dt = Γ(1 − w) − i(ΩRρ ∗ eg − Ω ∗ Rρeg)
dρeg
dt
= −
� Γ
2
�
− i∆ ρeg + iΩR
w (2.18)
2
The steady state solutions of these equations are retrieved by setting the time
derivatives to zero:
w =
ρeg =
1
1 + s
iΩR
2(Γ/2 − i∆)(1 + s)
Here, we have introduced the saturation parameter s which is given by
s =
|ΩR| 2
2|Γ/2 − i∆| 2 = |ΩR| 2 /2
Γ2 /4 + ∆2 (2.19)
(2.20)
(2.21)
The saturation intensity I0 for a transition of wavelength λ into an excited
state of lifetime τ = Γ −1 is
I0 = πhc
3λ 3 τ
The saturation parameter s0 for zero detuning, ∆ = 0, is
2 |ΩR|
s0 = 2
Γ2 I
=
Using these definitions equation (2.21) simplifies to
s =
I0
s0
1 + (2∆/Γ) 2
(2.22)
(2.23)
(2.24)
From equation (2.19) we see that with increasing saturation s, the steady
state changes from a high occupation of the ground state (w = 1) to an
equal occupation of both states w = 0 in the limit of infinite saturation. The
occupation probability for the excited state is given by a Lorentzian
ρee = 1
(1 − w) =
2
s
2(1 + s) =
20
s0/2
1 + s0 + (2∆/Γ) 2
(2.25)

Chapter 2: Theoretical Background
In the steady state, the decay and excitation of the higher state are equal,
and the decay rate Γ is known. We can now determine the rate of scattering
processes Γsc for this case:
Γsc = Γ · ρee = 1 Γs0
2 1 + s0 + (2∆/Γ) 2
(2.26)
The intensity dependence of the scattering rate is implicit in the saturation
parameter s0. Substituting the s0 yields
Γsc = 3πc2
2¯hω 3 0
� Γ
∆
� 2
· I ∝ I
∆ 2
(2.27)
We see that the rate of photons scattered by the atoms depends linearly on
the intensity of the illuminating light and is proportional to the reciprocal of
the square of the detuning.
2.1.3 Detection of atoms by fluorescence and absorption
From equation (2.27) we can calculate the scattered light power of a single
atom to by multiplying the scattering rate with the energy of a single photon
of the resonance frequency ω0
PA = ¯hω0 · 3πc2
2¯hω 3 0
� �2 Γ
· I (2.28)
∆
For N atoms, the total scattered power is thus Pt = N · PA. If this is not
spatially resolved but imaged into a photodiode, this photodiode will detect a
power PPD
PPD = Pt · πr2
∝ N (2.29)
4πa2 Here, r denotes the radius of the lense used for imaging, and a is the distance
between the atomic cloud and this lense. A single photodiode is sufficient to
measure the atom number.
A better signal to noise ratio in the detection of atoms is possible when a low
intensity resonant beam shines though the atomic sample and then is detected.
21

2.1. Atoms and Electromagnetic Fields
It is important that the signal does not contain spontaneously emitted photons.
This is the case when the spatial angle that is gathered by the imaging optics
is much smaller than 4π.We will consider only a single pixel of a CCD camera
here, the same argument then also holds for central column density absorption
measurements onto a single photodiode.
We consider an area A inside the atomic cloud, and an initial intensity of
light IA illuminating this area. Due to the absorption, this intensity is reduced
by ∆I = −PA/A when passing the cloud. Here PA is the scattered power per
atom, see equation (2.28). The measurable is the intensity IA(NA) after NA
absorptions, NA is the number of atoms in a column through the cloud with
area A. For a laser frequency of ωL and any number of atoms N, we receive
the rate equation
dIA(N)
dN
= ¯hωLΓ
2A
IA(N)/I0
1 + IA(N)/I0 + (2∆/Γ) 2
(2.30)
This can be solved for the number of atoms as a function of the attenuated
intensityIA by substituting N by NA
NA = 2AI0
¯hωLγ
��
1 +
� � �
2 � �
2∆ IA(0)
· ln
+
Γ IA(NA)
IA(0)
�
− IA(NA)
I0
(2.31)
For a weak absorption IA ≪ I0 the last difference term can be neglected. It
has to be noted that the polarisation of the absorbed beam influences the
saturation intensity I0. Linear polarised light does not optically pump the
atoms into just one magnetic substate, so that more than one sublevel can
contribute to the transition strength, and the saturation intensity is larger
than for circular polarised light. For a camera, the overall number of atoms
in the cloud is then given by summing over all pixels. An extensive overview
on the detection and probing of Bose-Einstein condensates can be found in
[Ket99].
22

2.1.4 Trapping of atoms in light fields
Chapter 2: Theoretical Background
A spatial varying intensity of light I(�r) leads to a trap if the potential has a
local minimum. For red detuned light this is reached by a local maximum of
the intensity, while for blue detuned light (∆ > 0) the atoms can be trapped in
a local minimum. Figure 2.2 schematically shows how such a trap works by a
spatially dependent shift of the energy levels of the atoms. In the experiment
described here (chapters 6, 7) we work with red detuned light only. Thus,
trapping occurs in local maxima of the intensity. For this, a single focused
beam is already sufficient, with the radial confinement given by the waist of
the beam and the axial confinement due to the Rayleigh range. This set-up
was used in the first experimental realisation of an optical dipole trap [Chu86].
Figure 2.2: A spatially dependent intensity of light can be used to trap atoms.
The intensity of a red detuned Gaussian beam is causing a trapping potential
by the ac-Stark effect. In high intensity areas the atomic ground state is
shifted to lower energies while in low intensity areas it remains unperturbed
and remains at a relatively higher energy.
Two crossed beams with perpendicular polarisation and same focal char-
acteristics have the advantage that the confinement is much more isotropic.
23

2.1. Atoms and Electromagnetic Fields
The confinement in each axis is given by the waist of the beams when their
foci overlap. The potential is twice as deep as that of a single beam of equal
power, but usually the crossed beam is created from the two beams split off
one single beam and their intensity is only half as big as the intensity of the
beam they originate from. Also, one has to note that the particles need a lower
energy than the trap depth to escape the trap. The beams themselves have a
finite intensity outside the crossing region and the atoms can use these arms
as escape routes. This leads to an effective trap depth that is half the absolute
value, or equal to the trap depth that a single uncrossed beam of the same
power as one of the crossed beams would have.
In the experiments that use dipole traps to evaporate atoms to create a
BEC, crossed beam configurations are used. The experimental set-ups differ in
the lasers used [Bar01], how the confinement is created by two beams [Web03,
Kin05]. A common problem in evaporation with optical traps is that ramping
down the intensity also reduces the gradient of the trap: the focal width of
the light which gives the radial width of the trap remains unchanged. A less
deep trap with the same radius is shallower, the gradient and curvature of the
trap do not stay constant over the forced evaporation but decrease. For a good
overview on different trapping designs, see [Gri00].
2.1.5 Cooling atoms with light
Doppler cooling
As the photons not only carry an energy E = ¯hω = hc/λ but also a momentum
�p = ¯h � k (2.32)
where � k is the wavevector with absolute value k = | � k| in the propagation
direction of the light, it was proposed by [H¨75, Win75] to use light for the
deceleration and cooling of atoms.
24

Chapter 2: Theoretical Background
During each scattering event, both energy and momentum have to be con-
served. The force that an atom experiences in a light field is the product of
the scattering rate and the momentum that is transferred to the atom in each
scattering process.
�F = ¯h Γ
2
s0
1 + s0 + (2∆/Γ) 2 · � k (2.33)
To cool an atom, we need to reduce the thermal motion. This can be
done by the force of the spontaneous scattering. Each spatial direction can be
cooled by one pair of beams if we use light with a negative detuning ∆ < 0 and
irradiate the atom with a pair of counterpropagating beams of same intensity
and detuning.
Consider an atom moving with a velocity �v due to the thermal motion. The
Doppler effect will shift the frequency of the light of the counterpropagating
beam to a different, smaller detuning than the rest frame detuning ∆, so
the effective detuning the atom experiences is ∆ − �v · � k. Under the same
conditions the atom sees a higher detuning from the light propagating in the
same direction. The probability to absorb a photon is different for each beam:
it is higher for the counter propagating beam. After each absorption, a photon
will be emitted by the atom. This is done isotropically and after many cycles
the momentum change due to the emissions averages out. The atom will have
experienced a net momentum transfer that is directed against its propagation
and slowing it in this direction. As this is done in every spatial direction,
the undirected thermal motion is reduced and thus the atom is cooled. This
procedure has been termed “optical molasses”, as in the linear approximation
of equation (2.33) the force is linear in the velocity of the atom like viscous
damping in mechanics.
Involved with the re-emission of the photons is a heating process; so the
lowest possible temperatures that can be reached this way is the equilibrium
between the cooling and the heating process. It can be shown that the lowest
25

2.1. Atoms and Electromagnetic Fields
possible temperature is reached for a detuning of ∆ = − Γ
2
[Neu78]. This temperature is called the Doppler-limit
and for 87 Rb is TDoppler = 146 µK.
Sub-doppler cooling
TDoppler = ¯h
kB
Γ
2
or half a linewidth
(2.34)
Soon after the first experimental observation of cooling atoms by radiation,
it was observed that the temperature of the atoms was actually below the
predicted doppler limit for the temperature [Let88]. This behaviour can be
explained by the fact that the atoms are not pure two level systems [Dal89],
but have magnetic sublevels. The cooling then stems from spatially dependent
stark shifts in an optical standing wave and the optical pumping of the atoms
in the substate with less energy.
Assume a light field that is created by two counterpropagating linearly
polarised waves, with perpendicular planes of polarisation (lin⊥lin), and an
atom where the ground state has two magentic sublevels mF = ±1/2. In
the basis of circular polarisation, the light field creates two standing waves
of different polarisation σ +,− that have nodes which are separated by λ/4
from each other. The magnetic substates now are shifted depending on the
substate and the intensity and polarisation of the light. Also, σ + -polarised
light drives transitions into the mf = +1/2 state and σ − -polarised light into
the mf = −1/2 state. Now, at positions with high intensity of one polarisation,
the level that the atom is pumped into is a strongly negative shifted level. If
the atom moves further, the shifting intensity of the standing wave decreases
and the potential energy of the state increases. This results in a loss of kinetic
energy. A decrease in the intensity of one polarisation means an increase in
the intensity of the other, so the atom can undergo a new transition and again
26

Chapter 2: Theoretical Background
be pumped into a low-lying energetic state. Figure 2.3 depicts this concept.
It can be seen as the atom is moving uphill in the potential, and close to the
peak, is falling back into a valley where it starts a new climb. Because of the
similarity to the fate of the greek tragic hero Sisyphos, the cooling mechanism
was dubbed “Sisyphus cooling”.
E
λ/2
m = - 1/2
F
m = + 1/2
F
Figure 2.3: Sub-Doppler cooling: the atom continuously “moves uphill” as
transitions from the energetically higher to lower mF -state are driven.
The ratio of this cooling and the Doppler cooling can be shown to be about
2|∆|/Γ [Met99]. For larger detunings ∆ lower temperatures can be reached,
but this cooling mechanism has a smaller capture range and only works with
sufficiently cold, Doppler cooled atoms. The lowest limit here is given by the
momentum of the last emitted photon ¯hk
Trec = Erec/kB = (¯hk)2
2mkB
with m being the mass of the atom. For 87 Rb this recoil limit is 349 nK.
27
(2.35)

2.2. Atoms and Magnetic Fields
2.2 Atoms and Magnetic Fields
That the interaction between atoms and magnetic fields can expose the atoms
to a force was first shown by Otto Stern 1 and Walther Gerlach in 1922. An
atom with magnetic moment �µ in an inhomogenous field � B will experience a
force
�F = � ∇(�µ · � B) (2.36)
This force can be expressed by a potential U = −�µ · � B and � F = − � ∇U. The
potential finds its extreme values for parallel and antiparallel alignment of the
magnetic moment towards the magnetic field and at the position of a local
maximum or minimum of the field. This can be used for trapping of neutral
atoms and was first demonstrated in 1985 [Mig85]. Using magnetic fields to
confine atoms was first proposed by W. Paul [Fri51], although here a hexapole
field was used as lense and not as a trap in 3D.
2.2.1 Magnetic trapping
Unlike the case with optical traps, it is not possible to create local maxima
of the field strength with static magnetic fields [Win84]. That means we have
to trap the atoms in a local minimum. For small fields we find anomalous
Zeeman splitting, in which the magnetic moment of an atom can be expressed
by µ = −µBgF mF , with µB the Bohr magneton, gF the Landé factor and mF
the magnetic quantum number or substate of an atomic state with hyperfine
state F . For a positive Landé factor (gF > 0), this reduces the choice of
trappable atomic states to those for which the magnetic moment is counter-
aligned with the magnetic field vector. In quantum mechanics, this is achieved
by magnetic substates with positive quantum numbers mF , which then are
1 O. Stern also first showed that whole atoms possess wave-like properties observing
diffraction in a beam of He [Kna29]
28

Chapter 2: Theoretical Background
trapped in the local minimum of the magnetic field. These are called low or
weak field seeking states.
The simplest way to create a field with a minimum is a quadrupole field,
which can be created by two coils in an anti-Helmholtz configuration. This
field has a zero at its centre and from there a linear increase in the absolute
field strength. Different mF -states thus differ in their energy by an amount
proportional to the magnetic field:
∆E = gF µBB∆mF
(2.37)
The linearity of the field of this trap is a drawback though. Atoms moving
towards the centre have their magnetic moment counter-aligned with the di-
rection of the field vectors. When they pass through the field’s zero crossing,
the atom sees a reversed direction of the field and accordingly experiences a
force that expels it from the centre. This has been called a spin flip or a Ma-
jorana spin flip, although it is not the spin that changes, but the direction of
the magnetic field. In fact, if the spin and thus the magnetic moment is able
to change with the field then the atom will not be lost.
Another simple design creates a quadrupole trap in two dimensions. A
current I is running through an infinitely long wire, and a homogeneous bias
field B0 is applied perpendicular to the wire. There is one point r0 ∝ I/B0
where the field of the wire Bw ∝ I/r is just canceled by the bias field. This
field can be well approximated by a quadrupole field. If we now add a second
bias field along the wire, the overall field never crosses zero, although it has a
field minimum at the same position as before. Here the atomic spin can now
adiabatically follow the changing field direction and thus Majorana spin flips
are suppressed. Such a configuration is generally called a Ioffe-Pritchard (IP)
configuration [Pri83].
29

2.2. Atoms and Magnetic Fields
Miniaturised traps: Atom chips
A single wire and bias field producing a magnetic quadrupole field was used
in 1933 to study spin flips in an atomic beam [Fri33]. Experiments with free
standing wires and magnetic bias fields were successful in deflecting, guid-
ing and trapping atoms [Row96, For98b, For00]. It was proposed to use
microfabricated current-carrying wires [Wei95], much like the conductors in
micro-electronics, to create the trapping magnetic potential. This device
was called an atom chip. Soon after the proposal the trapping of atoms
[Rei99, Fol00, For00] and the use of such a chip as a magnetic mirror for atoms
[Lau99b] were experimentally demonstrated. The use of atom chips proved to
be a breakthrough technology in evaporative cooling and Bose-Einstein con-
densation [Ott01, H¨01a]. Traps created by these chips in general have a much
higher gradient than “macroscopic” traps. This increases the collision rate
and reduces the time needed equilibrating the atomic sample and in this way
reduces the time needed for the whole evaporation process. A shorter time
relaxes the conditions on the lifetime of the trap and the surrounding vacuum.
Recently chips with slightly larger structures showed the same benefits in fast
and ‘easy’ condensation [Sch03, Val04].
The field of an infinitely long wire with current I at a distance �r with an
external bias field � B0 is
�B = µ0
�I × �r
2π r2 + � B0 (2.38)
We choose the component of the external field that is perpendicular to the
wire and reduce the problem to this dimension, where we put the origin of the
axis into the wire. Then we have
B = µ0 I
− B⊥0
(2.39)
2π r
This field has a zero at the position r0 = µ0
2π
30
I
B⊥0
, and for the gradient of this

magnetic field at the trap position we find
�
∂B �
�
∂r � = −
r=r0
µ0 B
2π
2 ⊥0
I
In the Gaussian cgs system this is simply
�
∂B �
� =
∂r
1 B
2
2 ⊥0
I [cgs]
� r=r0, [cgs]
Chapter 2: Theoretical Background
(2.40)
(2.41)
Of course, an infinite wire is not realisable and does not allow confinement
in the direction of the wire. Bent wires can create the offset field � B0 and
create the confining field. Here two major designs have been considered. In
the first design the wire is bent into a U-shaped form, where the connecting
parts are bent away from the ‘infinite’ part in the same direction. The overall
field of this wire alone is a quadrupole field in all three dimensions, the field
components of the side wires cancelling themselves at the trap centre. This
trap’s centre is not directly above the wire. It is shifted slightly away from the
wires in the dimension of the bent wires, see Fig. 2.4.
A more symmetric arrangement is created by bending the wire in a Z-
shape. This creates a harmonic trap, as the fields from the side wires add
constructively at the trap’s centre. The field of this trap for a current I, from
the centre of the ‘infinite’ central wire bar with length wx, perpendicular to
the chip’s plane can be calculated to:
⎛
z ·
⎜
�B(z) = I · ⎜
⎝
2wy
(w 2 x/4+z 2 ) √ w 2 x/4+z 2 +w 2 y
− 1
z · wx √
w2 x /4+z2 0
⎞
⎟
⎠ + � B0
(2.42)
In this thesis the Gaussian cgs system is used when covering magnetic fields,
as for applications in atom optics it yields convenient numbers. The field is
expressed in Cartesian coordinates, where the x-direction is chosen along the
wire and the z-axis is perpendicular to the chip’s plane. The terms wx and wy
31

2.2. Atoms and Magnetic Fields
give the extensions of the wire in the labelled direction, where it is assumed
that the bent parts of the wire have the same length wy. The finite width of
the wire is still neglected here; the correction terms can be found in [Rei02].
A wire configuration in an “H”-shape now allows one to produce a U-shape
or a Z-shape, depending on the arms the current passes through [Fol00]. The
different different trapping potentials for a single wire, a U-shaped and a Z-
shaped wire with their respective bias fields are depicted in Fig. 2.4, taken
from an article with an extensive overview on atom chips [Fol02].
Figure 2.4: Magnetic fields and currents (top) and corresponding potentials
(bottom) of different possible wire configurations for trapping atoms: (a) a
single wire with a quadrupole field, (b) a U-shaped wire with homogenous bias
field, (c) a Z-shaped wre with homogenous bais field. Taken from [Fol02]
Using typical parameters (I = 30 A, B0 = 60 G, wy = 4 · wx = 10 mm),
we expect our trap to be about 0.8 mm away from the wire with trapping
frequencies of ν ≈ 300 Hz (see Figure 2.5).
It is possible to use more than one wire to create trapping potentials. If
more than one parallel wire is used, then it is possible to create configurations
where the magnetic potential has more than one minimum (a double well
potential) [Hin01, Est05], waveguides for atoms [Thy99], or traps where the
32

absolute magnetic field / G
50
45
40
35
30
25
20
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
distance from wire / mm
Chapter 2: Theoretical Background
Figure 2.5: Magnetic field of a Z-shaped wire plus bias field with parameters
close to the experimental parameters. The field is given as a function from the
distance from the wire perpendicular to the plane defined by the Z-structure
and calculated by equation (2.42).
wires create their own bias field [Ott01]. Using bent wires can also create IP-
type traps without external fields [Wei95] or even complicated structures like
conveyor belts for atoms [H¨01b], waveguides [Lea02] or beamsplitters [Cas00,
M¨01]. A new generation of chips has structures to capture and evaporate the
atoms, and structures which create the field for the experiment [Zim04]. When
these structures create a periodic potential with or without perturbations,
they allow us to investigate problems of solid-state physics with a mesoscopic
number of atoms.
In general one can state that atom chips simplify the creation of a BEC. The
high gradients allow very efficient evaporation, relaxing the requirements on
the vacuum by reducing the evaporation and rethermalisation times. Current-
carrying wires allow complicated structures and fields, such as atom guides,
beam splitters and switches, which are all promising candidates to become
parts of an atom interferometer on a chip [Shi05, Wan05b]. Several overview
articles on the topic have been published [Rei02, Fol02, For03].
33

2.2. Atoms and Magnetic Fields
2.2.2 Permanent magnets
Instead of using current-carrying wires on a chip to induce magnetic fields,
it is possible to use permanent magnetic structures instead. Among the first
applications of permanent magnetic structures in atom optics was a periodic
array of magnets [Roa95, Sid96, Lau99a], which has a field strength that decays
exponentially with distance from the surface. This decaying field can then be
used as a mirror, similar to evanescent light [Bal87]. The materials used in
mirror experiments were magneto-optical ferrimagnetic films like TbGdFeCo,
alloys like CoCr [Sid02b] and even computer hard drives [Lev03b], audio tape
[Roa95] or video tape [Ros00].
When using periodic arrays like the above, the magnetic fields of the indi-
vidual elements are summed. We now consider a single magnet of thickness
h uniformly magnetised in perpendicular direction, which means it has its
magnetisation axis along its height and perpendicular to its plane. Then, the
magnetic field � Bfilm produced by this magnet will be the same as the field of
a current Ieff that propagates along the edges of the magnet. The strength of
this effective current depends on the magnetisation MR and the thickness h
of the element, Ieff = hMR. The magnitude of the field and the field gradient
near the edge of the magnetic element is then like the field of a single wire (see
equation (2.39) )
Bfilm = µ0 hMR
2π z
∂
∂z Bfilm = − µ0
2π
hMR
z 2
(2.43)
where z is the distance from the magnetic element and much larger than the
thickness h. As the edge of a magnet can be seen as a current through a wire,
it was proposed to implement atom traps by replacing the current-carrying
wires by permanent magnetic structures [Dav99a, Dav99b, Dav99c, Sid02b].
Here it was proposed to use parallel slabs of magnets, so that a single slab
34

Chapter 2: Theoretical Background
would be equivalent to a double-wire structure, and two slabs would create the
field of four wires. These waveguides and even 2D magneto-optical trapping
was demonstrated [Ven02]. Even though these magnetic fields are permanent,
it would still be possible to load them from the outside by using bias fields
to compensate the barriers that otherwise confine the atoms once they are
trapped [Sid02b]. Careful design of perpendicular slabs can lead to a 2D
array of traps for quantum computation [Gha06], similar to dipole traps using
microstructured optics [Bir01, Dum03b].
In our experiment, we use the edge of a ferrimagnetic magneto-optical film
(TbGdFeCo) sandwich structure [Wan05a] as the equivalent of a single straight
wire. Other experiments use a layered structure of a different magneto-optical
film (Pt-Co), with a corner cut out of their slab (this is equivalent to a Z-
shaped wire) [Eri04], or a comparatively thick layer of Fe-Pt film [Xin04].
In the latter experiment the permanent magnetic material has the form of a
capital F, for which the resulting field creates a self-biased Ioffe-Pritchard type
trap [Bar05]. Recently, the first Bose-Einstein condensate was formed in a trap
from a permanent magnetic chip [Sin05].
2.3 Evaporative Cooling
Optical cooling methods are limited by the lowest temperatures that can be
achieved. High density samples can show strong absorption, leading to an
anisotropic cooling, and have a higher chance of reabsorbing photons emitted
by cooled atoms, which results in a higher heating rate. For Bose-Einstein
condensation, it is important to reach a high phase space density: that is
very low temperatures (a narrow distribution in momentum space) and high
densities (a narrow distribution in space). Both of these factors are directly
limited when cooling with radiation.
35

2.3. Evaporative Cooling
The light field of a laser is a highly ordered system and can be considered
as a coolant into which the sample is immersed; the emission of photons into
different modes adds entropy and transfers energy from the atoms into the
light field. Instead of having the sample immersed in a cooling medium, a way
to further cool the sample is to let parts of the sample carry away an amount
of entropy and energy that exceeds the mean value of the sample. This of
course reduces the number of particles in the cloud. For this the trap depth is
lowered, so that particles with high energies can escape the trap. Because it
can be compared to hot water, where the highest energy molecules evaporate
and leave a cooler sample, this method has been called evaporative cooling
[Hes86, Mas88].
2.3.1 A simple model
A simple model of evaporative cooling is presented now, following the treat-
ments of [Met99] and [Arn99]. This model allows us to introduce some quan-
titative measures to characterise the cooling process. It highly simplifies the
actual processes: ergodic behaviour of the atoms is assumed, as is classical
thermodynamics without quantum influences. Only s-wave scattering is taken
into account and only with a constant cross section for elastic scattering. Fur-
ther, full evaporation is assumed and the rethermalisation is much faster than
the cooling rate. For the experiment here the model applies sufficiently well,
but one must be aware of its limitations.
We assume our trapping potential to be of the form
U(�r) = �
i=1,2,3
ci
�
�
�
�
xi
ai
�
�
�
�
si
(2.44)
where ai is the characteristic length of the trap in the xi direction, and si gives
the power scaling for each axis. It can be shown that the volume V of the
36

Chapter 2: Theoretical Background
atomic cloud is then proportional to a power of the temperature T
V ∝ T ξ
ξ = � 1
si
(2.45)
(2.46)
and the dependency on the trap geometry is reduced to a single parameter ξ
[Met99, Arn99]. For a linear trap in 3 dimensions, si = 1, we get ξ = 3; for a
harmonic trap in 3D with si = 2 it is ξ = 3/2.
We now assume N atoms of temperature T in this trap which we assume
to be infinitely high initially. We reduce the trap depth to a value of ηkBT ,
so η is a truncation parameter. This will let some atoms escape the trap, we
will label the ratio of the remaining atoms N ′ and the initial number N with
ν ≡ N ′ /N. Now we can define a measure for the temperature decrease from
T to T ′
γ ≡ log(T ′ /T )
log(N ′ /N) = log(T ′ /T )
log ν
(2.47)
From this we can deduce power laws for the thermodynamic quantities.
The primed values are the ones after the reduction of the trap height. They
are:
T ′ = T · ν γ
N ′ = N · ν
V ′ = V · ν γξ
(2.48)
Additional important quantities when evaporating atoms from a trap are the
atom density n = N/V of the remaining cloud and the phase space density
(PSD) ρ which scales with the atom density and the cube of the thermal de
Broglie wavelength:
ρ = nλ 3 dB
37
(2.49)

2.3. Evaporative Cooling
The change of these two quantities can also be expressed in terms of ν, γ and
ξ:
n ′ = n · ν 1−γξ
ρ ′ = ρ · ν 1−γ(ξ+3/2)
(2.50)
One can now show [Met99, Arn99] from the density of states in an external
potential that the fraction of atoms remaining in the trap after decreasing the
trap depth to ηkBT becomes
ν =
N ′
N
= γ(ξ + 3/2, η)
Γ(ξ + 3/2)
(2.51)
where γ(ξ + 3/2, η) is the lower incomplete Gamma-function, with Γ(ξ + 3/2)
being the corresponding complete Gamma-function.
The dynamics of the evaporation process are covered in [Pet02]. The key
results are that the decay times for evaporation τev and other losses that do
not change the temperature of the atoms, τloss, need to be comparable for the
optimal choice of the threshold energy at which atoms can leave the trap. It
is also shown that for evaporation trap potentials with larger values of the
parameter ξ are preferable, which means linear traps are better than harmonic
ones. Most important in experiments is that the time needed to rethermalise
the atoms, and thus the elastic scattering time τel, decreases as the evaporation
proceeds. The scattering rate scales as the atomic density times the thermal
velocity ∝ T 1/2 . To achieve runaway evaporation one needs
d
d log N log τel = γ ·
τloss
τev + τloss
giving a stringent requirement for successful evaporation.
· (ξ − 1
) − 1 > 0 (2.52)
2
Of course, this model can not be applied directly to optical traps, as there
it is not only the trap height, but also the trap’s curvature and gradient that
change during the evaporation ramping. Thus, the characteristic length in
38

Chapter 2: Theoretical Background
equation (2.44) is not constant. With the failure of this approximation the
model as presented here becomes invalid. The problem of evaporating in an
optical trap was treated in [O’H01], where a constant η and one case of ramping
down the potential was assumed. Still, one can use the parameters as defined
here for the evaporative cooling in optical traps to get a rough idea of the
processes.
2.4 Bose-Einstein Condensation
The main reason why such an effort is made to reach ultra-cold temperatures
is the condensation of atoms into a Bose-Einstein condensate (BEC) [Ein25,
And95, Dav95, Bra95]. In this state, all atoms occupy the same quantum state
of the trap, the ground state. One then has a macroscopic and coherent matter
wave, a tool which promises to be of great use in atom interferometry.
Bosonic particles do not follow the Maxwell-Boltzmann distribution. In-
stead quantum effects lead to the Bose-Einstein distribution, so that for a
sample at temperature T N(E) particles will be found at an energy E:
N(E) =
1
e (E−µ)/kBT − 1
(2.53)
The chemical potential µ vanishes for photons, leading to Planck’s equation
for black body radiation. This was the historical problem which S. Bose solved
and A. Einstein extended for quantized energies. A new state of matter was
predicted by this distribution. For very low temperatures, a phase transition
will occur. Then the lowest possible state, the ground state, would gain a
sudden and large increase in its population. The threshold condition for this
phase transition is related to the phase space density, using the same notation
as above:
ρ = nλ 3 dB ≥ 2.612 (2.54)
39

2.4. Bose-Einstein Condensation
This is the critical value for a three dimensional box. In a 3D harmonic po-
tential the threshold is less than half that value [Pet02, Arn99], although older
sources claim it is independent of the actual trap geometry [Bag87]. Once the
critical PSD is reached, the other atoms will condense into that region. A more
pictorial description of this condition is that the wavefunctions of the atoms
must overlap each other, so that they would then constructively add up to one
large coherent wave function.
As these atoms are interacting with each other, the description of the whole
system is not trivial. Fortunately, in many cases one can use standard approxi-
mation methods. Using the Hartree-Fock approximation allows one to describe
the interaction of a single particle with many others by estimating a mean field,
which averages the influences of all other N particles into one additional non-
linear potential term. When this additional non-linear term is added to the
Schrödinger equation, one gets the Gross-Pitaevskii equation [Gro61, Pit61]
for the wavefunction Φ of the condensate:
�
− ¯h2
�
2
∆ + V + NVNL|Φ| · Φ = i¯h
2m ∂
Φ (2.55)
∂t
The term N · VNL · |Φ| 2 describing the interaction between the atoms is depen-
dent on the scattering of the atoms: it is assumed that only s-wave scatter-
ing takes place between the atoms with scattering length a0, so that VNL =
4π¯h 2 a0/m. For a BEC this assumption in general is valid. The scattering
length of Rubidium can be taken as constant over the range of parameters
that are used in the experiments here. It is positive, leading to a repulsive in-
teraction between the atoms. Magnetic fields and optical fields can be used to
tune the scattering length using resonances between atomic states and molec-
ular states, so-called Feshbach resonances. By tuning the respective field, so
that the scattering length vanishes, it is possible to create a coherent mat-
ter wave of non-interacting particles. This simplifies the description, as the
40

Chapter 2: Theoretical Background
non-linear term vanishes and the whole sample can be described by the single
particle Schrödinger equation. Especially for atom interferometry in double
well potentials this can be advantageous (see section 2.6.1 and chapter 3).
Unfortunately for the isotope of 87 Rb the Feshbach resonances are not easily
accessible [Mar02, The04].
For an extensive description of this model with a detailed assessment of
when the assumptions and approximations made are valid, see for example
[Pet02, Arn99]. An overview suited for experimental physicists covering the
route towards Bose-Einstein condensation, detection of condensates, early re-
sults and an extensive list of literature can be found in [Ket99].
2.5 The Magneto-Optical Trap
Both the optical and the magnetic traps described in the earlier sections have
one common drawback: their depths allow us to trap only atoms that are
already sufficiently cold. If one tried to fill these traps from an atomic reservoir
at room temperature, the number of captured atoms would be negligible. The
method of choice to overcome this is to trap and pre-cool the atoms in a
magneto-optical trap (MOT) [Raa87], a combination of a quadrupole magnetic
field and a three dimensional optical molasses with circular polarisation. The
absolute magnetic field increases linearly from its zero point in each direction,
| � B| ∝ |�x|. If we consider a single transition |J = 0〉 → |J ′ = 1〉, we will
find a Zeeman splitting of the magnetic sublevels of ∆E = gJµBBmJ ′. This
Zeeman splitting due to the magnetic fields provides a natural quantisation
axis radially along the � B field, with mJ ′ being a “good” quantum number for
our purposes. This quantisation only depends on the magnetic field, and due to
its radial symmetry there are no angular dependencies. Electronic transitions
with mJ ′ = 0 are called π transitions, transitions with mJ ′ = ±1 are called
41

2.5. The Magneto-Optical Trap
σ ± transitions. We will consider circularly polarised light. This polarisation
is independent of the magnetic field. It is always defined in relation to the
direction of propagation. One can define this by the projection of the photon’s
spin onto its direction of propagation. This is called the helicity and possible
values are ±¯h. Another common way to define the polarisation is to describe
the evolution of the electrical field vector along its propagation: with the light
coming towards the observer, it is called right-handed circular (RHC) when
the vector rotates clockwise and left-handed circular (LHC) when the vector
rotates anti-clockwise. Positive helicity is the same as LHC [Bra86].
We now illuminate the atoms with counterpropagating light that is po-
larised to drive transitions from the ground state to the excited state with
mJ ′ = −1, called σ− light. Of course, the light that is σ − on one side of
the trap centre will change to σ + once it has passed the centre point: the
handedness of the light does not change, but the direction of the magnetic
field reverses. So, at each point we find not only the cooling and trapping σ −
light, but also σ + light. Fortunately, the Zeeman effect shifts the energy of the
mJ ′ = +1 state upwards, so that the detuning is further increased and absorp-
tion of this light is not probable. This scheme is depicted in Figure 2.6. We
can thus say that the incoming light has to have the same circular polarisation,
which translates to one beam pair (passing through the magnetic field coils)
with one handedness, while the other two pairs have the opposite handedness.
Up to the point where the sum of the Zeeman shift and the thermal Doppler
shift of the atoms are less than the red detuning of the light this results in a
force that drives the atoms back to the centre. When the detuning is large
compared to these shifts, the force can be written as a sum of a spatially and
a velocity dependent part:
�F = −β�v − κ�r (2.56)
This force leads to a damped harmonic oscillation of the atoms with a spring
42

negative
helicity,
RHC
LHC
I coils
positive
helicity,
LHC
left handed circular
LHC
right
handed
circular
Chapter 2: Theoretical Background
E
hνLaser
m =+1
J
m =0
J
m =0
J
J=1
m =-1
J
Figure 2.6: Schematic view on a magneto-optical trap with helicities of the light
for a chosen direction of the current in the quadrupole coils (left). Influence
of the magnetic field on the energy of the atomic sublevels, increasing the
probability to absorb a photon from the σ − polarised beam (right).
constant κ. For the standard MOT situation the motion is overdamped. For a
MOT with Rb, one finds temperatures that are below the Doppler limit. The
cooling mechanism is not the Sisyphus cooling described above, but due to
non adiabatic passage through polarisation gradients, also called σ + σ − -cooling
[Dal89].
2.5.1 The mirror MOT
In the standard design of a MOT the optical molasses is created by three
pairs of counterpropagating beams. The whole design is symmetric around the
centre point, and thus to a plane through the centre point. This symmetry can
be used to further simplify the set-up. If a mirror is placed such that it stands
at an angle of 45 degrees to the coils that create the quadrupole field, then only
43
σ +
σ −
J=0
r

2.5. The Magneto-Optical Trap
two pairs of light beams are needed [Rei99]. The left side of Figure 2.7 shows
a sketch of this. Here one pair of beams passes through without reflection and
LHC
I coils
positive
helicity,
left handed
circular
Mirror
LHC
negative
helicity,
right handed
circular
LHC
σ −
σ +
LHC
Mirror
Figure 2.7: Schematic view on a mirror MOT with helicities of the light (left).
Sketch of the change of polarisation and handedness of the light close to the
surface (right)
this does not differ from the description above: as stated, the light changes its
polarisation from σ − to σ + upon passing through the field’s zero point while
it keeps its handedness. The other pair is made up of two beams that are
reflected into each other. One of these beams initially points with the field
lines of the magnetic field; the other beam has a � k vector that points against
the field. They are chosen with different handedness so that with the different
relative alignment to the magnetic field their circular polarisation is thus the
same. Upon reflection, only the handedness of the light changes. The relative
alignment of � k vector and magnetic field changes once before the reflection, and
once after the reflection: the light leaving the trap has the opposite circular
polarisation. Figure 2.7, right, explains these changes graphically.
σ −
RHC
σ +
Instead of using external coils to create the quadrupole field, it is possible
to use current-carrying structures on the mirror holder (see section 2.2.1).
A U-shaped wire then creates the magnetic field [Rei99, Fol00]. This MOT
44
RHC

Chapter 2: Theoretical Background
typically has a higher field gradient, and is closer to the surface of the chip.
The terms U-wire MOT and compressed MOT will be used synonymously.
The possibility of creating first a mirror-MOT with external coils for a large
trapping volume and high atom numbers, which is then changed to a U-wire
MOT, has some important benefits. The U-wire MOT allows easy alignment
of the atomic cloud with the chip regarding the relative position to the wire
and the matching of the cloud size to the potential that the Z-wire will create
for magnetic trapping. Also, the switching times of the quadrupole coils is
replaced by the switching time of the bent wire, which is significantly shorter.
Furthermore, the fabrication of these chips uses standard technologies, making
these chips rather easily accessible [Lev03a]. These reasons are partly why
atom chips and mirror MOTs have become so successful and part of many
atom optics experiments [Sch03, Wil04, Val04].
2.6 Atom Interferometry
Interferometry with atoms is an important branch of today’s physics, and
one of the two most promising future main applications 2 of cold atoms and
atom optics, just like interferometry is an important part of optics. The pos-
sibility to confine atoms in free space and to cool them to low temperatures
opens the door towards interferometry with heavy particles. Here we will con-
centrate on interferometry with trapped atoms, though many schemes exist
which use cold atoms that are not confined during the interferometric process
[Gus97, Pet99, Stu03]. This is closely related to the question of using internal
or external [H¨01b] degrees of freedom for the interference. As a rule of thumb,
the proposals using internal degrees use unconfined atoms, and usually the
2 The other possible application is quantum information processing. How closely these
two are related can be seen for example in [Dum03a], [M¨05] and [Vol05].
45

2.6. Atom Interferometry
beamsplitting process is done by momentum transfer due to the process of
manipulating these degrees of freedom [Kas91]. The exception to this rule
are proposals that use state selective potentials. The use of external degrees
of freedom for interferometry is usually connected to using trapped or guided
atoms, where the these degrees then are quantised. The condensation of atoms
into a BEC and with the trapping of atoms in microscopic potentials further in-
creased the interest in atom interferometry. The production of cold atoms and
BEC in microtraps on atom chips [Fol02, H¨01a, Ott01] and in micro-optical
systems [Dum02a, Dum02b] has stimulated a great interest towards novel im-
plementations of atom interferometers [Hin01, H¨01b, And02] that are based
on the use of double well potentials. Now, a large number of coherent atoms, a
macroscopic wavefunction, can be manipulated with very precise control. Only
recently chip-based atom interferometers [Sch05b, Est05, Wan05b, Shi05] and
an interferometer using an optical double well trap [Shi04] were demonstrated.
These interferometers used cold atom clouds and BECs. A good overview of
the whole field of interferometry with atoms is given in the book of [Ber96].
2.6.1 Atom interferometry with symmetric double well
potentials
The double well potential in quantum mechanics has for a long time been ex-
amined by theoretical physicists. The first studies were in the field of solid
state physics [Set81, Wei87] or inspired by effects that that were known from
this field like Josephson oscillations [Jav86, Jac96]. With the advent of the
Bose-Einstein condensate in atomic physics [And95, Dav95, Bra95], interfer-
ence of a single BEC distributed over both wells and between BECs located
in a well each was studied [Cas97, And97]. These experiments showed that
the phase between two independent condensates was unpredictable, which was
46

Chapter 2: Theoretical Background
explained by the fact that the BECs in this case are Fock states with a fixed
number of atoms and as such their relative phase is undefined [Jav96, R¨97].
A phase between two BECs is always measurable, but it will be different and
unpredictable from shot to shot.
Double well atom interferometers are suited to be realised on atom chips
[Jo07]. Precise control over the splitting and merging processes becomes possi-
ble on a sub-micron scale when microfabricated structures are used. Splitting
[Cas00] and merging [H¨01c] of cold atomic clouds was demonstrated soon after
the first atom chips became available, and interference of BECs split and re-
combined in double wells [Shi04, Shi05] or in a chip based waveguide [Wan05b]
has been demonstrated. The use of a BEC in interferometry can enhance the
precision of measurements by a factor of √ N[Kas02]. At the moment, phase
diffusion [Jav97] due to localisation of atoms is a major limitation [Shi04].
Double well interferometry with single atoms will allow longer measurement
time and larger splitting distances. In these regards single atom interferometry
has an advantage. On the other hand, by the statistical nature of quantum
mechanics, a single measurement of a single atom will not yield information
about the phase. A way to overcome this is to repeat the experiment, or run
identical experiments at the same time. Even more promising is a BEC of
non-interacting particles. Then the phase diffusion due to mean field effects
can be neglected and we look at N single atoms in a single trap.
Two schemes have been proposed to split a potential into a double well us-
ing the technology of atom chips [Hin01, H¨01c]. These schemes use a three step
approach. The starting point is an atom in the ground state of a symmetric
potential. In the first step this potential is split adiabatically into a symmetric
double well. Then a phase difference between the parts of the atom in each
well is then induced. The double well is then adiabatically recombined into the
original potential, and the population of the excited state measures the effect
47

2.6. Atom Interferometry
of the spatially-asymmetric potential. This can be viewed as a Mach-Zehnder
interferometer where the two separated wells give rise to different optical paths.
The three main steps are shown schematically in Figure 2.8.
Figure 2.8: Schematic view on the interferometer process. An atom is prepared
in the ground state of a potential (left). This potential is then split, so that for
a symmetric the potential the ground state of that split potential populates
both wells. Then a phase shift between the wells is introduced (centre). Upon
merging the two wells again, this leads to a population in the first excited state
of the merged potential. Solid line: potential, dashed: ground state, dotted:
first excited state wavefunction
The concept of adiabaticity needs to be addressed further. We start with
the atom in the lowest energy eigenstate |φ0〉 of the single well. We can sup-
press transitions to excited states by choosing the time scale for splitting and
recombination stages much longer than the inverse energy gap between the
relevant states. The energy gap between the ground state energy E0 and E2,
the energy of the second excited state, always exceeds the gap between the
ground state energy and the first excited state energy E1. We can adiabat-
ically isolate the two lowest energy eigenstates (|φ0〉 and |φ1〉) from higher
excited states (|φ2〉, |φ3〉, etc) by a suitable choice of the time scales:
h · (E1 − E0) −1 ≫ Tadiabatic isolation ≫ h · (E2 − E0) −1
(2.57)
In the symmetric case, the states |φ2n〉 and |φ2n+1〉 become degenerate for large
splittings, then the left hand side of inequality (2.57) becomes infinite and can
48

Chapter 2: Theoretical Background
be neglected [H¨01c]. In this case, one often changes from the energy basis to
a localised basis, where
|L〉 = 1
√ 2 (|φ0〉 + |φ1〉)
|R〉 = 1
√ 2 (|φ0〉 − |φ1〉) (2.58)
The basis vectors |L, R〉 indicate whether the atom is localised in the left or
the right well. This can be seen from the central figure of in Figure 2.8: the
energetic states |φ0,1〉 differ by a phase factor of −1 in their respective right
well population. In the sum of these states, only the left well population is
non-vanishing. In subtracting the states, this phase factor is canceled and only
the right well is populated.
This allows us to apply a two-mode approximation, using the two lowest
energy eigenstates of the describing Hamiltonian. The theory of the inter-
ferometer is then simpler and allows us to introduce a the two level Bloch
equations. These can be visualised by the motion of the Bloch vector on the
Bloch sphere and simplifies the interpretation of the physics. This is done in
the following chapter, where the need for a symmetric potential is dropped
and the more general case of an asymmetric double well potential is discussed.
49

2.6. Atom Interferometry
50

Chapter 3
A single particle in an
asymmetric double well
potential
This chapter addresses one problem that atom interferometric experiments
have to deal with. The problem addressed here is an imperfection that is
unavoidable in experimental set-ups and actually needed for an interferometer:
the imbalance in the potentials of the two arms of a single particle, double
well interferometer. Because this imbalance is needed to give rise to the phase
difference, it will be explained how it has to be introduced and how it can
destroy the interferometer if this is not done properly. Although only Mach-
Zehnder-like interferometers will be specifically addressed, like in section 2.6.1,
the results of this chapter can easily be generalized to other situations.
A known approach is to express the single atom in a double well system by
the localised wavefunctions of the left and right well as in equation (2.58). This
requires the truncation to the lowest two states of the symmetric double well,
which is a well known method when treating double well problems [Gri98].
51

Localised wavefunctions were also used to examine a time dependent symmet-
ric double well [H¨01c] and time independent asymmetric double wells [Hu00].
The problem of non-adiabatic transitions in two level systems was first raised
in 1927 by F. Hund [Hun27] and treated in 1932 individually by L.D. Landau,
C. Zener and E.G.C. Stückelberg [Lan32, Zen32, Stu32]. Their work is now
known as the Landau-Zener-Stückelberg (LZS) theory and is concerned with
exact solutions for special forms of potential curves. For a so-called Landau-
Zener (LZ) type of curve crossing, the slopes of the potential curves have the
same sign. In terms of the double well potential, the LZ model has a linearly
varying asymmetry and a constant underlying symmetric potential. For an
extensive overview of the topic and several exact analytical solutions of special
potentials, see [Nak02].
In our case of the asymmetric double well potential, we also deal with a
LZ-type of curve crossing, except that the potentials do not cross but asymp-
totically become degenerate. More importantly the use of the LZS theory is
impossible in our case, as neither of the two important parameters is constant;
both are time varying. In addition, neither of them can be neglected over the
full range of our interest. Indeed, we are most interested in the range where
both parameters are of the same order of magnitude. So, instead of offer-
ing exact or approximation solutions for special cases [Bam81, Rob82], here
the emphasis is on the interpretation and understanding of the physics. That
this understanding is needed can be seen by the the examples in the field of
atom interferometry where the influence of asymmetries in experimental set-
ups seems to be underestimated or ignored, see for example footnote [24] of
[Shi05], where a difference between the trap frequencies of the wells of 12%
was mentioned but the effect of that asymmetry on the coherence between the
separated BECs was neglected.
The behaviour of a BEC in a symmetric double well system has been the
52

Chapter 3: The Asymmetric Double Well
subject of theoretical research [Jav97, Mil97, Sti02b, Tho02, Sak02], includ-
ing analytical investigations using a two mode approximation [Jav99, Cor99,
Spe99, Men01, Sti02a, Mah02]. The influence of non-adiabaticity of the split-
ting of a BEC has also been examined [Bor04, Mel03] during the time the
model presented here has been developed. Mellish et al. use a two mode ap-
proach and “dressed state” basis states like in solid state physics to describe
the loading of a BEC into an optical lattice [Mel03]. Smerzi et al. have exam-
ined a BEC in an asymmetric double well, using two coupled Gross-Pitaevskii
equations in the far split regime only, without overlap between the wavefunc-
tions of the different wells [Sme97]. During the time the model here has been
developed, Sakellari et al. have studied Josephson effects of BECs in double
wells by introducing a time dependent linear asymmetry and using a two mode
approach [Sak04].
Single particles in double well systems have also been the subject of theo-
retical research. The use of degenerate isomers of molecules as double well sys-
tems for detection of very small energy differences has been proposed [Har78],
and later the theoretical justification for the sensitivity of double well systems
has been studied [Sim85]. The multi-state dynamics of magnetically trapped
atoms with a double well formed by the avoided crossing of atomic spin states
and RF radiation have been studied [Vit97], based on previous work with
time independent fields, by Cook and Shore [Coo79]. Double wells and the
influence of pulsed coupling have been examined from a LZS point of view
[Bam81, Rob82]. Tunnelling processes, their resonances and how a modula-
tion of the potential changes these resonances has been studied [Gro91]. A
good overview on two mode models and double well systems with an emphasis
on these tunnelling processes is presented in [Gri98].
This chapter begins with the introduction of the model system and Hamil-
tonian and how the imbalance is introduced. This system is kept in a most
53

3.1. Introduction
general form, so that it is easily applicable to other similar situations. The sec-
ond section deals with the main approximation made to simplify the model:
the system is restricted to the lowest two levels leading to the well known
Bloch equations. These equations allow easy interpretation of the different
parameters and their roles in the interferometric process. Some obvious lim-
itations of the model will be discussed here as well. This is then followed by
the numerical results of this model and compared with the numerical solutions
of the full Hamiltonian and experimental data of a BEC based interferometer
[Jo07]. This part of the chapter includes a novel way to read out an atom in-
terferometer, similar to that which has been used for a BEC based double well
experiment in our group [Hal07b]. The last section of this chapter summarizes
the insights of the model and its limitations. A short list of similar physical
problems where this model can be used is presented. The results of the first
sections of the chapter have been published [Sid06]; some later results were
obtained after the contents of the publication had been decided and for that
reason only have not been included in the paper. The author withdrew from
his position as first author on that publication after agreement could not be
reached with some of the co-authors.
3.1 Introduction
We investigate the dynamics of a single particle due to a time dependent, one
dimensional Hamiltonian ˆ H(t) [Sid06]. Any Hamiltonian can be expanded into
a sum of one part with even symmetry and one part with odd symmetry. The
Hamiltonian thus can be written as the sum of its symmetric part ˆ H0(t) and an
antisymmetric part ˆ Vas. For the symmetric potential ˆ V0(x, t) a rather general
54

form is chosen [Pul03]:
ˆH = ˆ H0 + ˆ Vas
∂2 Chapter 3: The Asymmetric Double Well
ˆH0 = − 1
2 ∂x2 + ˆ V0(x, t) (3.1)
� �
ˆV0(x, t) = 1 + β(t) − x2
� �
2
1/2
(3.2)
2
ˆVas = Vasˆx (3.3)
The Hamiltonian is normalised by dividing by ¯hω0, and the variables are
rescaled by
x = x′
a0
t = ω0 · t ′
V =
a0 =
V ′
¯hω0
�
¯h
mω0
(3.4)
where the primed quantities are the variables of the unscaled Hamiltonian.
Dimensionless harmonic oscillator units are used. The normalised asymmetric
part of the potential ˆ Vas is linear in the spatial variable x and constant in time.
The eigenvectors of ˆ H(t) will be labelled |φi〉 with eigenvalues Ei. The
eigenvectors of ˆ H0(t) are |σi〉 and their eigenvalues are Eis, where for both
cases i = 0, 1, 2 . . .. All eigenvalues and eigenvectors are time dependent. The
eigenvectors of the symmetric Hamiltonian have defined symmetry. A notation
common in the field of solid states physics [Hu00] calls the symmetric ground
state |S〉 = |σ0〉 and the first excited and thus lowest anti-symmetric state
|AS〉 = |σ1〉.
The shape of the potential is characterised by a time dependent splitting
parameter β(t). For β = 0 it is a single well that is nearly quartic. The
splitting is realised by increasing this parameter to a large value. For β ≫ 0
55

3.2. Two Mode Approximation and the Bloch Equations
the double well potential appears with two nearly harmonic wells. The distance
between the minima is 2 √ 2β. In the following β(t) will change linearly or will
be constant. We are allowed to choose our eigenvectors |φi〉, |σi〉 to be real, as
a one dimensional problem is considered.
3.2 Two Mode Approximation and the Bloch
Equations
3.2.1 The two mode approximation
The two mode approximation is now applied. The eigenstates of the total
Hamiltonian ˆ H(t) do not possess a defined symmetry. We will use a standard
matrix mechanics approach [Mes90] in the basis of localised atoms, the so-
called left, right basis |L, R〉 (L-R basis). The basis vectors are defined as
|L〉 = 1
√ 2 (|σ0〉 + |σ1〉)
|R〉 = 1
√ 2 (|σ0〉 − |σ1〉) (3.5)
For the split potential, β ≫ 0, the states |L, R〉 describe atoms that are lo-
calised in the left or right well. For the unsplit potential the states |L, R〉
correspond to higher probability amplitudes at the sides of the single well. In
the left, right basis, the Hamiltonian takes the form
ˆHL,R = 1
⎡
⎛
⎣(E0 + E1) · ˆ1 −
2
56
⎝ Vasym
∆0
∆0 −Vasym
⎞⎤
⎠⎦
(3.6)

Chapter 3: The Asymmetric Double Well
Here ˆ1 is the unity matrix, the columns and rows are ordered L-R and the
following definitions are used:
∆0 = E1s − E0s (3.7)
Vasym = 〈R| ˆ Vas|R〉 − 〈L| ˆ Vas|L〉 (3.8)
= −〈σ0| ˆ Vas|σ1〉 − 〈σ1| ˆ Vas|σ0〉
All of these quantities are real. The energy gap between the eigenstates of the
symmetric Hamiltonian |σ0,1〉 is given by ∆0. In solid state physics this term
sometimes is denoted by ∆SAS as the energy difference between |S〉 = |σ0〉 and
|AS〉 = |σ1〉. The quantity Vasym is a measure of the asymmetry between the
wells and would be zero for a symmetric Hamiltonian. These two quantities
are related to the energy gap of the total Hamiltonian ˆ H(t), which we call ∆
following a notation used in [Hu00], by:
∆ 2 = ∆ 2 0 + V 2
asym = (E1 − E0) 2
(3.9)
The dependence of ∆, Vasym and ∆0 on the splitting parameter β(t) is shown
in Fig. 3.1. The energies were calculated numerically for the symmetric and
asymmetric potential using a Numerov recursion method, and the difference
was calculated from equation (3.9). All quantities are expressed in the natural
oscillator units of equations (3.4), so that they can easily be rescaled with
the actual experimental parameters. The antisymmetric contribution roughly
follows a square root behaviour, but deviates for low splitting values. The
square root can be explained thus. The definition of the symmetric splitting
potential, equation (3.2), establishes a square root relation between β (linear)
and x (squared). The asymmetry is defined by equation (3.8) and is linear in
the distance between the wells. This means it has a square root dependence
on β. The overlap between the states |L, R〉 is suspected to be the reason
for the deviation from the square root. The symmetric part shows a decay
57

3.2. Two Mode Approximation and the Bloch Equations
proportional to e−(β−δ)2, like the falling slope of a Gaussian displaced by δ.
A square root with a quartic root correction and a Gaussian had been fitted
to the points that were directly determined from the stationary Schrödinger
equation, and were used in the later calculations of the model.
Energy in hν
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 1 2 3 4 5 6 7
splitting parameter β
Figure 3.1: Energy difference ∆ between ground and first excited state (solid
line) for ˆ Vas = 0.02x, as a function of the splitting parameter β. Dotted line:
symmetric fraction ∆0; dashed: asymmetric fraction Vasym.
The energy eigenvectors can be expressed in terms of the L,R basis, and
are given by
|φ0〉 = 1
√ 2 ( √ 1 + V |L〉 + √ 1 − V |R〉)
|φ1〉 = 1
√ 2 ( √ 1 − V |L〉 − √ 1 + V |R〉) (3.10)
where the asymmetry is now represented by the scaled quantity
V = Vasym
∆
(3.11)
The degree of asymmetry is described by V , which is related to the angle θ
used for similar situations in the field of solid state physics, so that V=tan θ or
58

Chapter 3: The Asymmetric Double Well
V=cot θ. This θ also appears in equations (2.7) and (2.9) of the dressed states
model in section 2.1.1. It has to be pointed out that for a nearly symmetric
(unsplit) trap, we still have
|L〉 = 1
√ 2 (|φ0〉 + |φ1〉)
|R〉 = 1
√ 2 (|φ0〉 − |φ1〉) (3.12)
but for a far split trap, when V ≈ 1, the left and right well states can be
expressed by
|L〉 = |φ0〉
|R〉 = −|φ1〉 (3.13)
When looking at the eigenfunctions |φ0,1〉 for different splittings β, we can
see why a two mode approximation should be sufficient to describe the inter-
ferometer. Figure 3.2 shows these eigenfunctions for three values of β and a
fixed asymmetry. We see how over a small increase in the splitting distance
the wave functions change from more or less symmetric functions to fully lo-
calised eigenfunctions. This has been called the “flea on the elephant” effect
[Sim85], when a very small disturbance (for Rb in a trap of frequency 1 kHz,
the asymmetry only needs to be about 1/36 of gravity to reach our model value
of 0.02·x) has a huge effect on the system [Har78]. This is the main reason why
one should not consider symmetric potentials as described in section 2.6.1.
3.2.2 The Bloch equations
When dealing with the dynamics of a two level system, like in section 2.1, one
can always describe the system by the Bloch equations, as all two level prob-
lems have been shown to be equivalent [Fey57] and can be directly mapped
onto the special unitary group SU(2). Another way to represent such a prob-
lem is by rotating vectors on a unit sphere, the so-called Bloch sphere. This
59

3.2. Two Mode Approximation and the Bloch Equations
ψ
1
0.5
0
−0.5
(a) β=1
−1
−6 −4 −2 0
x
2 4 6
20
15
10
5
0
V(x)
ψ
1
0.5
0
−0.5
(b) β=2.5
−1
−6 −4 −2 0
x
2 4 6
20
15
10
5
0
V(x)
ψ
1
0.5
0
−0.5
(c) β=5
−1
−6 −4 −2 0
x
2 4 6
Figure 3.2: Instantaneous energy eigenfunctions of the ground state (dashed-
dotted line) and the first excited state (dotted line) for different splittings β =
(a) 1, (b) 2.5,(c) 5. The solid line is the asymmetric potential. The asymmetry
is ˆ Vas = 0.02ˆx
allows a graphical representation of the different states and the parameter V ,
which were introduced in the previous paragraphs. A projection onto two di-
mensions of this sphere is shown in Fig. 3.3. As our physical system is one
dimensional and we can choose our eigenfunctions to be real, this projection is
a good means to visualise the problem and understand the underlying physics.
In [All75, Mey91] extensive treatments of the two level problem are given,
including the derivation of the Bloch equations. It was pointed out by [Dal04]
that these derivations [All75] rely on time independent basis vectors. In the
case of time dependent basis vectors these derivations are valid if and only
if there exists a set of basis vectors that are real and with opposite symme-
try. Fortunately these two restrictions are trivially fulfilled in any two level
problem: any Hamiltonian can be expanded as a sum of even and odd func-
tions, and as basis vectors the eigenvectors of the even part of the Hamiltonian
are used. The eigenfunctions of this symmetric Hamiltonian always have de-
fined symmetry. All two level systems are one dimensional or have degenerate
energy levels. Degeneracy happens through dimensions in the Hamiltonian
being symmetric. The one dimensionality of the problem allows us to choose
the phase of these eigenvectors as zero so that the eigenvectors are real. Thus,
60
20
15
10
5
0
V(x)

R
V
σ 0
Ψ
Chapter 3: The Asymmetric Double Well
φ 1
Figure 3.3: Projection of the Bloch sphere, with energy eigenstates |φ0,1〉, and
different basis states |L, R〉, |σ0,1〉. The situation is an incomplete split of the
traps, V ≈ 1/2. The state |Ψ〉 is closer to |φ0〉 than to |σ0〉, so the splitting
is nearly adiabatic. The projection onto the |L, R〉 axis shows how the atomic
wavefunction is overbalanced to the left well.
it is valid to use the derivation as presented in [All75]. We also find the first
restriction to our approximation. It will not hold when the ground or first
excited state of the asymmetric potential is a linear combination with a signif-
icant contribution from the second or higher excited states of the symmetric
basis. This can happen for example when the asymmetry and splitting are too
large, so that the asymmetric first excited state becomes localised in the same
well as the ground state.
φ 0
We can now apply the standard treatment of the two level problem to our
system: any state vector can be written as
σ1
|Ψ〉 = cL|L〉 + cR|R〉 (3.14)
61
L

3.2. Two Mode Approximation and the Bloch Equations
where the complex amplitudes fulfil |cL| 2 + |cR| 2 = 1. The state vector |Ψ〉
fulfils the normalised time-dependent Schrödinger equation
where the Hamiltonian is given by equation (3.6).
i ∂
∂t |Ψ(t)〉 = ˆ H(t)|Ψ(t)〉 (3.15)
We identify the components of the Bloch vector −→ Y as
Yx = cLc ∗ R + c ∗ LcR = Re(cLc ∗ R)
Yy = i · (c ∗ LcR − cLc ∗ R) = Im(cLc ∗ R)
Yz = |cR| 2 − |cL| 2 ,
The Bloch equations can be obtained from the Schrödinger equation (3.15)
when the state vector is substituted using equation (3.14). The Bloch equa-
tions 1 are then given by
˙
Yx= −Yy · Vasym
˙
Yy= Yx · Vasym +Yz · ∆0
˙
Yz= −Yy · ∆0
(3.16)
and can be solved numerically as rate equations using the Runge-Kutta algo-
rithm. These equations are analogous to those for the case of a two level atom
driven by a single mode laser field. The symmetric transition frequency ∆0
is equivalent to the Rabi frequency, while the asymmetry frequency Vasym is
equivalent to the detuning. The component Yz is a measure of the imbalance
of the atomic population between the wells (Yz = 0 means balanced popula-
tion), and in the far split case it is a measure of the excitation amplitude of
the first vibrational state. The component Yx contains information about the
symmetry of the atomic wave function: for Yx = −1 the state is symmetric,
while the antisymmetric state is given by Yx = 1. It is thus a measure of the
excitation of the vibrational state in the unsplit trap.
1 I thank B. Dalton for correcting a factor in these equations [Dal04].
62

Chapter 3: The Asymmetric Double Well
The Bloch equations can be expressed as a vector product
d −→ Y
dt = −→ Ω × −→ Y , (3.17)
where the torque vector is −→ Ω = (−∆0, 0, Vasym). This form of the vector with
‘Rabi frequency’ ∆0 and ‘detuning’ Vasym also explains the definition of the
‘generalised Rabi frequency’ ∆ in equation (3.9). In the case of atoms in RF
or laser fields, the Rabi frequency is determined by the external driving field.
This driving field usually is approximated by the rotating wave approximation
(RWA). In our case, no such approximation is needed; the Rabi frequency is
fully determined by the symmetric energy gap. It is noted that no relaxation
terms appear in the Hamiltonian and therefore no decay to the ground state
or other states is taken into account.
A high potential barrier suppresses tunnelling between the wells (∆0 → 0).
In the symmetric case (Vasym = 0) the left and right well states become de-
generate. This is equivalent to zero detuning or resonance. Thus, in an inter-
ferometric process, we do not observe Rabi oscillations (like the Stückelberg
oscillations seen in the group of R. Grimm [Chi05a] on a system of ultracold Cs
molecules [Chi05b]), but instead we observe Larmor precessions. The principal
difference between these two is shown in the schematic diagram of Fig. 3.4.
Here the spectral flow of the levels used for interferometry is depicted. The
thick lines show the evolution of the coupled levels, or dressed states, during
the beam splitting (or trap splitting in our case), the free evolution or phase
accumulation, and the final beam recombining (in our case the recombining of
the trap). In the Rabi case (Fig. 3.4 left) we see the uncoupled states (dotted
lines) cross so that the previously higher excited state becomes the state with
lower energy. The coupling between the states leads to an avoided crossing.
The Larmor case (Fig. 3.4 right) is different; here we see the uncoupled states
degenerate, and only the coupling lifts this degeneracy, leading to a split of
63

3.2. Two Mode Approximation and the Bloch Equations
Energy
Time
Figure 3.4: Schematic view of the spectral flow of the coupled (thick line)
and uncoupled (dotted line) energy levels for an interferometer using Rabi
oscillations (left) and Larmor precession (right), as functions of time. The plots
show the full interferometer process including splitting, holding and merging.
The double well interferometer is of the Larmor kind with coupled levels.
the two states of the interferometer. In a sense, the Larmor interferometer is
a special case of the Rabi interferometer: it just stops at the crossing point of
the uncoupled levels and stays there during the phase evolution.
Energy
We are interested in the excitation probability after the interferometric
process. Thus, when we assume an arbitrary state |Ψ〉 = cL|L〉 + cR|R〉, we
want to calculate |〈φ1||Ψ〉| 2 . We find:
|〈φ1|||Ψ〉| 2 = |〈φ1|(cL|L〉 + cR|R〉)| 2
= 1/2((|cL| 2 + |cR| 2 ) + V (|cR| 2 − |cL| 2 )
− √ 1 − V 2 (cLc ∗ R + c ∗ LcR)
= 1
2 (1 + Yz · V − Yx · √ 1 − V 2 ) (3.18)
The result depends on the parameter V . It is thus easy to probe two of the
three Bloch components by measuring in a merged trap (V ≈ 0) or in a split
trap (V ≈ 1).
In the recombined trap, the influence of the asymmetry is small, ∆ ≈ ∆0
64
Time

Chapter 3: The Asymmetric Double Well
and V ≪ 1. In the far split trap the asymmetry dominates: ∆ ≈ Vasym and
V ≈ 1. Equation (3.18) then reduces to
|〈φ1|||Ψ〉| 2 = 1 − Yx for 2
|〈φ1|||Ψ〉| 2 = 1 + Yz for 2
the unsplit case
the split case
(3.19)
Especially the second equation is of interest. The consequences of that equation
are discussed in section 3.3.3.
3.3 The Results of the Model
3.3.1 General results: the interferometer and the im-
pact of adiabaticity
In this section the results of the rate equation (3.16) will be presented. These
are then compared with the results of the numerical integration of the Schrödinger
equation without the two mode approximation, which were calculated in our
group by S. Whitlock [Whi04] using the eXtensible Multi-Dimensional Sim-
ulator (XMDS) software developed at the University of Queensland 2 . This
comparison will show another limitation of the approximation and a good
agreement apart from this, with a considerably shorter calculation time. The
effects will be addressed and the physical meaning will be explained. Unless
otherwise noted, the asymmetry will be ˆ Vas = 0.02ˆx and any ramps of the
splitting β will be linear.
The rate equations can be used to examine the important parameters. For
the symmetric case, it is important not to excite the atom into the second ex-
cited state. Thus, experimentalists might be tempted to extend the splitting
time when they encounter problems in balancing the atomic population in the
traps. Figure 3.5 shows the behaviour that is to be expected in this case. Here
2 For more information see: http://www.xmds.org
65

3.3. The Results of the Model
V; Y z ; |〈 φ 1 (x) | Ψ 〉| 2
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0 20 40 60 80 100 120 140 160 180 200
Figure 3.5: Dynamic evolution of the parameters V (dashed), the probability
for excitation of the first excited state |〈φ1||Ψ〉| 2 (dotted) and the Bloch vector
component Yz (solid line) for a splitting time of Ts = 200 and a final splitting
β = 3.5. This figure shows an adiabatic splitting, with nearly no excitation
of the atom in the potential well. After the split, the atom is localised in the
energetically lower well.
the splitting time was Ts = 200 for a final splitting of β = 3.5. The plot shows
the evolution of the parameters V (dashed line), the excitation of the first
excited state (dotted line) and the parameter Yz (solid line), which contains
information about the balance between the wells during the splitting. The ex-
citation shows us, that for this splitting time and parameter values, the overall
evolution is adiabatic. Of course, this leads to the population accumulating in
the lower well (Yz = −1 corresponds to all of the atom in the lower well). With
the atom localised in one well, this cannot be used as an interferometer, as we
have intrinsic which-way information [Eng96]. Thus, when doing experiments
with double wells one has to take care that the splitting time is adiabatic for
the transition |φ0〉 → |φ2〉 but non adiabatic for |φ0〉 → |φ1〉.
T s
If we keep this mind, we see how the interferometer works. This is shown in
66

Chapter 3: The Asymmetric Double Well
Fig. 3.6 for the same splitting, holding and recombining times Ts,h,r = 20 and
a final splitting of β = 12.5. Here the figure shows the component Yx (solid
V; Y x ; |〈 φ 1 (x) | Ψ 〉| 2
1
0.5
0
−0.5
−1
0 10 20 30 40 50 60
total time T
Figure 3.6: Dynamic evolution of the parameters V (dashed), the probability
for excitation of the first excited state |〈φ1||Ψ〉| 2 (dotted) and the Bloch vector
component Yx (solid line) for equal times of Ts,h,r = 20 and a final β = 12.5.
This figure shows a full interferometer process with splitting, phase evolution
and merging. During the non adiabatic split the atom is excited into a balanced
superposition of the ground and first excited state. In the holding stage, the
excitation of the atom does not change in the potential well, although the
phase between the population in the left and right well oscillates. This is the
well known Larmor precession. When the trap is recombined non adiabatically
this phase difference gives rise to a population difference in the energy levels.
line), the measure of the symmetry and thus of the inversion of the energetic
states in the unsplit trap. As the Bloch vector is normalised, | � Y | 2 = 1, our
knowledge of Yz ≈ 0 3 allows us to directly relate Yy to Yx. As the splitting is
much larger than before, the critical time for splitting and merging is less than
3 In the limit of small times, non adiabatic splitting is the same as the sudden approxi-
mation [Sch65] which keeps the shape and thus the symmetry of the wavefunction.
67

3.3. The Results of the Model
the actual time of the ramps. During a time T of about 5, the parameter V
goes from zero to one (and in the same amount of time it goes back to zero
later in the merging process). After this time we see that the component Yx
precesses around the revolution axis. The precession frequency is constant for
the time between 20 and 40; before and after that the precession frequency
changes slightly as the trap is still being split/merged and thus the driving
potential difference between the wells changes. A good interferometer works
when both paths are equally occupied [D¨98]; so in our case we need a balanced
distribution between the left and right well. We know from equations (3.13)
that this means we need a state corresponding to |Ψ〉 ≈ 1
√ 2 (|φ0〉 + |φ1〉). From
Equation (3.18) we also know that the excitation does not change during the
phase evolution stage. A change in excitation in the split trap would result
in a change of the population difference between the wells, which is unwanted
and, due to the high tunnelling barrier, also impossible. The plot, Fig. 3.6,
shows this behaviour clearly: the dotted line depicts the excitation probability.
It stays constant at about 1
2
when the parameter V changes.
during the phase accumulation, and changes only
During the interferometric process, a phase difference between the localised
wavefunctions is accumulated. This illustrates the fact that we look at a
precession (Larmor) and not an oscillation (Rabi). Only during the splitting
and merging is this phase difference translated into an excitation.
These results were compared with the results from the above mentioned
full numerical integration of the Schrödinger equation of S. Whitlock [Whi04].
Figure 3.7 shows the excitation probability after a complete interferometer cy-
cle, from left to right with different splitting and recombination times Ts,r =
5, 20, 200, as a function of the holding time, with a final β = 12.5. This value
for β was chosen as it translates into a splitting of the two wells of ten times
the ground state sizea0. The points are the results of the full numerical sim-
68

|〈 φ 1 (x) | Ψ 〉| 2
1
0.5
0
0 10 20 30 T 40
h
a)
|〈 φ 1 (x) | Ψ 〉| 2
1
0.5
Chapter 3: The Asymmetric Double Well
0
0 10 20 30 T 40
h
b)
|〈 φ 1 (x) | Ψ 〉| 2
1
0.5
0
0 10 20 30 T 40
h
Figure 3.7: Excitation probability for different splitting and recombining times
Ts,r = (a) 5, (b) 20, (c) 200 as a function of holding time Th. Solid line: Bloch
model results; points: results from full numerical simulation [Whi04]. (a): A
too short a splitting/recombining time leads to non adiabatic excitation to
higher states, causing a deviation between the two-mode-approximation and
the full analysis. (b): for longer times the agreement is very good. (c): A
too long a splitting/recombining time leads to adiabatic splitting and merging
with reduced fringe size. The different lines here are from different fits to the
asymmetric component Vasym (see text). Errors in this fit lead to phase errors
that accumulate during a prolonged splitting and merging.
ulation; the lines are the results of the two mode approximation. For short
splitting/recombining times Ts,r = 5, we see a deviation. While the Bloch
model reaches unity values in the excitation probability, this is not the case
for the numeric calculations which incorporate more levels. This is due to non
adiabatic transitions to higher energy levels which the Bloch model ignores.
Indeed for even smaller times the signal from the full numeric calculation shows
an obvious modulation on the signal with a higher frequency, and the results
of the Bloch model act as an envelope. For splitting/recombining times of
20, both models show remarkable agreement. In Fig. 3.7 (c), the signal of
the excitation reaches only values of |〈φ1||Ψ〉| 2 ≈ 1/2. This can be attributed
to the onset of adiabaticity in the transition |φ0〉 → |φ1〉. This results in an
unbalanced distribution of the atom between the wells, which in turn leads to
69
c)

3.3. The Results of the Model
a reduction in the final signal. It can be seen as intrinsic which-way informa-
tion when the atom preferentially populates one well, and this destroys the
interference [Eng96]. The dashed and dotted line in Figure 3.7 (c) use differ-
ent fits for the antisymmetric part: one is a square root without the quartic
root correction, and the other is a natural logarithm. It is notable that the
model is robust with regard to the maximum signal, but is fragile regarding
the phase of the signal. This error is accumulated during the splitting and
merging phases, when 0 < V < 1, as a result of the fact that Vasym �= 0 in a
merged trap because of the overlap between |L〉 and |R〉. Any error will scale
with the splitting/recombining time and thus it appears dominant when these
times become longer, and a small relative error can add up to a large absolute
error.
As the quality of the interferometric signal will depend on how symmetric
the atomic population is distributed between the wells, we examine the popu-
lation of both wells after the splitting process only. We define a filling factor
F as a measure of the symmetry, requiring F = 0 if the atom is fully localised
in one well and F = 1 if it is evenly distributed. With |Ψ〉 = a · |φ0〉 + b · |φ1〉,
we define F as
F = 2 · |a| · |b| (3.20)
This fulfils our requirements. The decay of the filling factor is shown in Fig.
3.8, for a splitting of β = 12.5 and different asymmetries. Split times range
from Ts= 0.5 to Ts = 1000. The results of the Bloch model (dotted line) and the
full numerics (solid line) are in good agreement for times Ts > 20. For smaller
times, the two mode approximation of the Bloch model fails, and excitations
into higher modes take place. In the case of the high asymmetry, Fig. 3.8
(d), the two mode approximation reaches its limit: for higher asymmetries,
the adiabatic isolation to two levels begins to fail (see equation (2.57)). Then
both lowest states of the system appear in the same well especially for small
70

Chapter 3: The Asymmetric Double Well
values of the splitting parameter β(t). This fact of adiabatic following is not
to be confused with the decay into the ground state caused by a relaxation
term in the Hamiltonian. Most of the results up to this point have also been
published in [Sid06]; the following results are exclusive to this thesis.
F
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10 0
10 1
Figure 3.8: Decay of the filling factor F as a function of splitting time; Bloch
model: dotted line; full numerical simulation: solid line [Whi04]. Asymmetries
ˆVas = 0.01ˆx (a), 0.02ˆx (b), 0.05ˆx (c), 0.1ˆx (d). The failure of adiabatic isolation
to two levels is apparent for short times. Part (d) shows how large asymmetries
also limit the Bloch model.
The behaviour of the filling factor F translates into a reduced fringe ampli-
tude. Using only the Bloch model, the maximum fringe size, max(|〈φ1|||Ψ〉| 2 ),
was calculated as a function of the same splitting and recombining time, for
different asymmetries, for a maximum splitting of β = 12.5. The results for
some asymmetries are shown in Fig. 3.9. The solid lines are the results of the
Bloch model; the dashed lines are fits of exponential functions of the type
d)
10 2
c)
max(|〈φ1|||Ψ〉| 2 ) = C · e κTs,r (3.21)
where κ is a negative decay rate, and the factor C can be used to determine
71
a)
b)
T s
10 3

3.3. The Results of the Model
max( |〈 Ψ| φ 1 〉| 2 )
10 1
10 0
10 −1
10 −2
10 −3
10 −4
10
0 100 200 300 400 500 600 700 800 900 1000
−5
Figure 3.9: Decay of the interferometric signal (on a logarithmic scale) versus
splitting/recombining time for different asymmetries: ˆ Vas = 0.01ˆx (a), 0.02ˆx
(b), 0.1ˆx (c). Solid line: Bloch model results; dashed line: fitted simple expo-
nentials. Part (c) is at the edge of the applicability of the model.
the time where the fit crosses the unit signal line. This time can be used as
a characteristic timescale, which should not be exceeded in order to retain
good visibility of the fringes and thus a working interferometer. As we can
see from Figure 3.9, both the decay rate and the time for unity maximum
signal are functions of the asymmetry Vas, where ˆ Vas = Vasˆx. If we plot these
dependences, we obtain Figure 3.10.
T s,r
On the left of Figure 3.10, we see the time where the exponential fit crosses
the unity signal threshold as a function of the asymmetry. The circles are the
results from the Bloch model, and the line is a fitted function 4 . The results
for Vas = 0 are included, which corresponds to no decay (κ = 0) and an
infinitely large unity signal threshold time. We see that the dependence of
the threshold time on the applied asymmetry Vas is inverse to a high degree.
We relate the imperfections to the problem of setting the interval of the data
4 Time for maximum unity signal = (−0.006 + 0.382 · Vas) −1
72
(c)
(b)
(a)

unity maximum signal
800
700
600
500
400
300
200
100
0
0 0.02 0.04 0.06 0.08 0.1 0.12
V as
Chapter 3: The Asymmetric Double Well
0
−0.01
−0.02
−0.03
−0.04
−0.05
−0.06
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Figure 3.10: Time for maximum unity signal (left) and decay rate κ (right) of
the signal as functions of the applied asymmetry Vas, with fits as explained in
the text. The model seems limited to asymmetries that do not exceed 10% of
the trapping frequency during the splitting and merging. The triangle in the
right figure is the result for an asymmetry that exceeds this limiting value.
to which the exponential function is fitted. Especially for large asymmetries,
when the model begins to break down, the signal shows some modulation (see
Fig. 3.9 (c)) which makes a decision difficult on which points to include in the
fitting process. Even though this was decided “by eye”, the data shows good
agreement with the reciprocal behaviour.
decay rate κ
On the right of Fig. 3.10 we see the decay rate κ as a function of the
asymmetry. The circles and the triangle are results from the fit to the Bloch
model. The line is a linear fit 5 , enforcing the results for the symmetric trap
but ignoring the two points with Vas ≥ 0.1. Here we see the breakdown of the
model due to large asymmetries more clearly. The model fails as the ground
and first excited state both occupy the same well for a non negligible time of
the splitting process. Thus, the deviation from the linear behaviour can be
5 κ = −0.403 · Vas
73
V as

3.3. The Results of the Model
used to check an experimental set-up (assuming it has a variable asymmetry)
and determine the maximum asymmetry allowed during the splitting process.
3.3.2 Addendum: Comparison with experimental data
During the period amendments were being made to this thesis, the results
of an atom interferometer using a BEC in an asymmetric double well were
published in the group of W. Ketterle [Jo07]. The experiment described in
that publication differs from the situation in our model only by the fact that
the experiment used a BEC while our model is based on a single atom. One
of the key results of the publication is an observed decay in the fringe contrast
with increasing merging time. The authors state that the “dependence [...]
on the recombination time allows [...] to speculate” about the causes of this
decay. “Furthermore, it is not clear during what fraction of the ramp time of
the [...] recombination time the effective merging [...] occurs. [...] Another
open question is what the rate of phase evolution is at the moment of the
merger.” [Jo07].
Even though the situations of the experiment (uses a BEC) and the model
(which uses a single atom) are not identical using either a BEC or a single
atom, the experimental parameters given in [Jo07] can be changed into the
model’s harmonic oscillator units by use of Equation 3.4. The separation of
the wells of d ∼ 6 µm corresponds to a splitting parameter of β = 39.05. The
experiment showed an interference that oscillated with a frequency of 500 Hz,
corresponding to an asymmetry between the wells of Vasym = 0.0283 when fully
split. The results of the experiment and the model can now be compared. This
is shown in Fig. 3.11. The results of the model have to be seen as preliminary:
the experiment’s far splitting of the wells leads to a large potential difference
between the wells compared to the case for which the model was used before.
74

enormalised interference fringe contrast
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
Chapter 3: The Asymmetric Double Well
0.6
0 10 20 30 40 50 60 70
merging time in ms
Figure 3.11: Comparison of the experimental data of [Jo07] with the predic-
tions of our model. The experimental data is shown by the squares. The
results of the model (crosses interpolated by the thick line) are taken from
three individual simulations for a splitting time of 75 ms with different tem-
poral resolution. The thin line shows the results of one of these simulations.
The reason for the oscillation is given in the text.
Thus, the temporal resolution in the simulations was chosen to be too low,
and a stroboscopic effect caused by the difference of the atomic interference
frequency and the sampling frequency of the calculation appears in the calcula-
tions. An attempt was made to overcome this effect by increasing the sampling
frequency. Unfortunately, even after three runs with increasing time needed
for the calculations, the beating in the calculations still appeared. The figure
shows the envelope of the results of the three calculations with different tem-
poral resolution (crosses and thick line). The result of one of these simulations
(thin line in the figure) shows the stroboscopic effect. The step-like structures
at times of ∼ 25 ms and ∼ 45 ms are clearly caused by the still insufficient
temporal resolution. The agreement between the experimental results and the
model is remarkable, especially since the experiment uses a BEC and the model
75

3.3. The Results of the Model
uses a single atom. The remaining small deviations may be explained by this
difference. Because of this good agreement, the model presented here can be
used to answer the above questions raised in the publication [Jo07].
3.3.3 CARP: a Coherent Adiabatic Readout Process for
double well atom interferometers
The results presented above offer an intriguing new possibility to read out
an atom interferometer based on a double well. While for the interferometric
process an adiabatic splitting of the asymmetric well is unwanted as it reduces
the signal, it provides a simple means to read out the energy state of the particle
in the trap. Equation (3.19) for the split trap shows that the occupation
probability for the first excited state is only dependent on the Bloch vector
component Yz. This was already identified as a measure of the imbalance of
the atomic population between the wells.
Thus, if we add an additional phase of adiabatic splitting to the interfero-
meter which so far consisted of a non adiabatic splitting, a phase evolution
and a non adiabatic recombination phase, we can read out the interferometer
by simply measuring the population of one of the wells. For this phase it is
advisable to apply the highest possible asymmetry that does not lead to the
two lowest states being located in the same well in order to reduce the required
time for adiabaticity.
If we plot the energy spectrum of the two lowest, uncoupled states (as in
Fig. 3.4), we obtain a graph that resembles a fish for the overall process. This
is shown schematically in Fig. 3.12, where the time is progressing along the
horizontal axis and the energy of the states is in the vertical axis.
A similar read out process that relies on the relative atom numbers in each
well after adiabatic splitting of the wells has recently been used in our group to
76

Energy
Chapter 3: The Asymmetric Double Well
Interferometer Read−out
Figure 3.12: Schematic spectral flow, like in Fig. 3.4, showing the two lowest,
uncoupled states for an interferometer with a simple read-out possibility: the
CARP. The times for the actual interferometer and the read-out process are
not to scale.
measure the double well asymmetry of a trapped BEC in a permanent magnet
film trap [Hal07b]. Here an experimental cycle time of 30 s and a single shot
sensitivity of 16 Hz or δg
g = 2 · 10−4 was achieved. The main source of noise
has been identified as the shot-to-shot number fluctuations in the condensate.
3.4 Summary
In this chapter it was shown how the two level Bloch equations can be applied
to an asymmetric double well system. This was done with an emphasis on the
application as a single atom interferometer, and the restrictions that an asym-
metry imposes on the splitting and merging time scales were shown. The main
advantages of this model are that the physics becomes easily understandable:
any double well interferometer works by Larmor precession. Only a certain
asymmetry can be applied, so that during the processes the ground and first
77
Time

3.4. Summary
excited states do not significantly occupy the same well. This is one limit to
the model, but it also promotes the use of traps with high trapping frequencies
(miniaturized magnetic or optical traps) as the increased steepness allows one
to deal with higher asymmetries. The only disadvantage of this model is that
higher modes are neglected. The model fails for processes that non adiabati-
cally couple these higher levels to the lowest two. For longer times, the results
of the model agree with the results of the full numerical integration.
The most important result of this model is the fact that, for a real double
well interferometer, an optimum time for splitting and merging the trap exists.
A too short a time will lead to non adiabatic transitions from the ground state
to the second excited state or higher states [H¨01c]. A too long a time will be
adiabatic in the transition between the ground state and the first excited state
and will inevitably lead to a localisation of the atom. This adiabaticity can be
useful in the read out of the interferometer, as it translates the population of
the energy states into a spatially distinguishable population, as pointed out in
section 3.3.3.
To extract information from a single atom interferometer, it is necessary to
have a statistical sample of identical experiments. This can be done either by
parallel experiments or by repeating the experiment under identical conditions,
or by a combination of both. It is possible to create arrays of several tens of
atom traps [Dum02b]. Let us assume that we want at least 100 measurements,
which means we need to run the overall experiment 10 times. If we further
assume we have double well potentials with a trap frequency of ω0
2π
= 1 kHz,
we have a characteristic time of Tchar ≈ 0.16 ms. From Figure 3.9 we see that
a linear and not-optimised splitting (and recombination) should take at least
10 · Tchar. The holding phase then is variable, while for the proposed read-out
we are limited by the maximum of the asymmetry that can be applied. Figure
3.9 proposes times of the order of magnitude 100·Tchar for a well localised atom.
78

Chapter 3: The Asymmetric Double Well
So, as a rough estimation, one can say that one run of the interferometer should
take about T = 150 · Tchar or T ≈ 24 ms for a 1 kHz trap. This corresponds to
a repetition rate of about 40 Hz ignoring preparation times. These preparation
times will be the dominant factor in the usefulness of the double well potential
as an interferometer to measure spatially or temporally varying external fields
[Hal07b]. To test fundamental quantum mechanics or in the application to
quantum computing the external conditions are either under our control or not
of interest as a measurable. Here the demand for a fast repetition of identical
interferometers is not as strong as if used as a measuring device for external
fields. The application of the double well single atom interferometer for the
purposes of quantum computing or tests of quantum mechanics is thus more
likely. For atom interferometers to measure imbalances between the wells, it
is more likely that trapped BECs are used (see [Hal07b, Jo07] and references
therein).
The model here was kept simple. It can easily be applied to a plethora
of problems, and its simplicity allows easy adaption to these problems. For
example, a proposed atomic double well interferometer on a chip can lead to
different trap frequencies when split [Hin01]. These different trap frequencies
can be translated into an asymmetry Vasym and the model here can be used.
Another direct application of this model is possible in double well problems
in solid state physics, like [Det04, Hu00, Hol04], and to macroscopic double
wells in atom optics [Tho02]. Some quantum computing proposals also rely
on double well potentials, both on the solid state side [Hol04] and the atom
optics side [Cal00, Mom03, Dum02b, Buc02]. A similar model was used to de-
scribe STIRAP-like processes and other properties of three level systems in a
triple well system [Eck04]. The model presented here neglects any interactions
between atoms. Interactions enhance the localising effect of the asymmetry,
as they would add a non-linear term to the Schrödinger equation which is de-
79

3.4. Summary
pendent on the atomic wavefunction or atomic density in the traps, similar
to the Gross-Pitaevskii equation (2.55). To include such a term in the model
would require a complex model, as the two mode approximation is question-
able for a N atom system where the first excited state is N-fold degenerate.
Even though these interactions were neglected, the read out process presented
inspired a way to read out asymmetry measurements of a double well with a
BEC [Hal07b] with high accuracy.
The only things that have to be changed to the specific problem are the
temporal flows of the matrix elements ∆0 and Vasym, according to the spectral
flow of the energy states and the ramps that are used to split or merge the
traps. These are the parameters that take the roles of the Rabi frequency and
the detuning in the more common form of the universal two level problem.
This makes it easy to compare our model to experimental results of the decay
of fringe contrast in an atom interferometer that utilizes BECs [Jo07]. Even
with the obvious difference between a single atom and a BEC the comparison
shows good agreement of the model.
80

Chapter 4
The Permanent Magnetic Chip
Experiment: Apparatus
4.1 Overview
This chapter describes the permanent magnetic chip experiment that is located
at Swinburne University of Technology, Australia. The major goal was to
achieve Bose-Einstein Condensation of 87 Rb atoms held in a magnetic trapping
field, where this field was produced by a permanent magnetic microstructure
on an atom chip. To achieve this, atoms were collected and optically precooled
by the standard methods of laser cooling and trapping. For magnetic trapping
and evaporation of the atomic ensemble to quantum degeneracy an ultra-high
vacuum (UHV) system was set up, containing the atom chip that was produced
in house. The chip was designed to trap 87 Rb atoms magnetically with a
potential that permits fast and efficient evaporative cooling towards quantum
degeneracy. This experiment was newly set up as a part of this PhD project
and it will be described and explained in detail in the following sections.
Permanent magnetic structures have a number of potential advantages over
81

4.1. Overview
current-carrying wires, but they also have a handicap: their fields are perma-
nent, and only in some situations is it possible to fully cancel the effect of these
fields when needed. This makes some standard procedures like temperature
measurements by TOF more difficult or even impossible. On the other hand
permanent magnets allow high effective currents, without any ohmic heating
and without any chance of breaking the wire. This is true especially for small
structures, which would be realisable only by very thin wires. Permanent
magnets exhibit no current noise and reduced thermally induced current fluc-
tuations in the conducting materials, which can cause spin-flip losses [Jon03].
This can lead to extraordinary low heating rates of the atoms in the trap,
which for wire-based traps can not be easily realised. With the magnetic film
trap presented here, heating rates as low as 3 nK/s were observed, while the
current-carrying wire of the set-up presented here had a heating rate of 270
nK/s [Hal07b, Hal07a]. Another remarkable advantage of permanent magnetic
structures is the ability to create true closed loops of the effective currents,
whereas for a current-carrying wire, there always has to be a connection to
the power supply. This allows the creation of potentials with a magnetic film
that are not possible with current-carrying wires. Here permanent magnetic
structures are advantageous.
Section 4.2 contains a description of the optical system, starting with the
laser systems that are used to cool, trap, manipulate and image the atoms.
This is followed by a description of the UHV apparatus and the procedure
to reach a sufficiently high vacuum. The third and fourth parts describe the
sources of the magnetic fields needed to trap the atoms. The third section
contains information on the magnetic field coils required for the complete op-
eration of the experiment. This is followed by the section describing the central
piece of the experiment, the permanent magnetic atom chip. Here the design
of the chip is introduced. This also covers the current-carrying wire structure
82

Chapter 4: The Permanent Magnetic Chip Experiment: Apparatus
and the permanent magnetic structure. A standard PC with two digital and
analogue I/O cards is used to control the experiment. Via connector boards [E
21] the AOMs (RF frequency and switching), the quadrupole and offset coils
(switched by MosFets), the Rb-dispensers (switched by MosFets), the shutters
and the CCD camera can all be addressed. This is done using the LabView
software package and a home-coded program.
4.2 The Laser Systems
This experiment uses three laser systems to provide light of the appropriate
frequencies, powers, linewidths, polarisations and beam shapes, necessary to
trap, cool, and engineer the atomic states and to image the ensemble. Each
system is based on diode laser technology with a wavelength of 780 nm and
resonant with the D2 line of 87 Rb. These laser diodes are readily available
since they are mass produced for CD players and recorders. The laser systems
are labelled according to their use in the experiment. They are:
• The main laser: a laser system based on a commercial tapered amplifier
system. It is used to produce the light needed for trapping and cooling
the atoms in the MOT and the light which is used for the absorption
imaging, resonant with the |F = 2〉 → |F ′ = 3〉 transition of the D2 line
(see Figure C.1 in the appendix C).
• The repumping laser: a commercial diode laser system. As the main
laser also excites some atoms into the |F ′ = 2〉 state, from where they
can decay into the |F = 1〉 ground state, the repumping light is tuned
to the |F = 1〉 → |F ′ = 2〉 transition. The repumping laser recycles any
population in the unwanted atomic substate |F = 1〉 back to the cooling
cycle.
83

4.2. The Laser Systems
• The optical pumping laser: a home built, grating stabilized laser
system, similar to the repumping laser. Its light is used to optically
pump the atoms into the state |F = 2, mF = 2〉 before the transfer from
the MOT to the magnetic trap.
All laser systems are located on a separate optical table, and their light is
brought to the experiment table by optical fibres. To reduce the amount of
stray light that could interfere with the experiment, the lasers are shielded by
an opaque PVC box.
4.2.1 The main laser
The main light source is a commercial tapered amplifier [E 1]. Here a low
power extended cavity laser diode, which is frequency stabilised by controling
temperature, current and the grating, is used to seed a high power tapered
amplifier. This produces a single tranverse mode, 500 mW output beam which
can be frequency tuned by 10 GHz across the D2 spectrum of Rb without
mode hops.
Frequency stabilisation follows the dither-free polarisation spectroscopy
scheme presented in [Pea02, Pet03], which has the big advantage of having
only one zero crossing in the error signal. This makes jumps to other frequen-
cies impossible, and is in this way superior to standard frequency-modulated
peak-lock techniques. On the other hand, it is not a peak-locking technique
and thus relies on high stability of laser powers and the used electronic devices
to avoid frequency drifts of the locking point.
The set-up is depicted in Fig. 4.1. Part of the light of the diode laser in Lit-
trow arrangement (the master laser) is split off after the optical diode. This
part double-passes an acousto-optical modulator (AOM) [E 2], with a centre
84

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400
AOM
Figure 4.1: The main laser system: tapered amplifier, spectroscopy of the
master laser and path of the amplified light. In this and all following sketches
lenses are depicted by double arrows. Other labels are explained in the text.
frequency of 80 MHz and a bandwidth of 30 MHz. The AOM 1 is used to
detune the light for the different tasks and is driven with frequencies ν1 ≈ 63
MHz for cooling or ν1 ≈ 55 MHz for absorption imaging. A polarising beam
splitter cube (PBS) is used to deflect the detuned light, that had its polari-
sation rotated by 90 o by double-passing a quarter-wave plate (λ/4), into the
spectroscopy branch. There it is split again: most of the light passes another
λ/4-plate and irradiates the 87 Rb atoms in a glass cell (Rb cell) as circular po-
larised pumping light. This pump beam induces a birefringence in the atomic
vapour. It is interrogated by the linearly polarised remaining amount of light,
which first passes a half-wave plate (λ/2) and then the Rb cell. The probe
400
1 The AOM output frequency fAOM was calibrated to fAOM = (525.87 · U + 1279.97) 1/2
MHz as a function of applied voltage U.
85

4.2. The Laser Systems
light is then split at another PBS (stray light is blocked by a pinhole in front
of the PBS) and collected by two photodiodes (PD).
The difference of the two signals is taken electronically and fed into a PID
regulator [E 3], which in turn drives the piezo-electric translators (PZT) of the
grating and stabilizes the master laser frequency to the |F = 2〉 → |F ′ = 3〉
transition. The internal feed forward from the PZT-driver to the driver of
the laser diode current is set to zero. Instead, for a fast correction, the error
signal is also fed into the modulation input of the current driver. The resulting
reduction in the noise is shown by the error signals and the power spectra of
these signals in Fig. 4.2. The modulation of the current as the fast branch of
the frequency stabilisation shows a significant effect in the acoustic frequencies
up to 1.5 kHz.
error signal / V
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0
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(a)
noise Amplitude, linear a.u.
0.025
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0 200 400 600 800 1000 1200 1400 1600 1800 2000
frequency / Hz
Figure 4.2: Closed loop error signals (a) and power spectra (b) of the main laser
without (dashed line) and with (solid line) current as fast correcting element.
To reduce drifts in the signal, precautions have been taken. After the de-
flection into the spectroscopy part, the light instantly passes a second, perpen-
dicular PBS. This ensures that we have well linearly polarised light. Second,
lenses are placed in front of the PDs. With their focussing they correct for
inevitable slight misalignments of the lenses in front of and after the AOM,
which in turn lead to a frequency dependent position of the beam.
86
(b)

Chapter 4: The Permanent Magnetic Chip Experiment: Apparatus
The master laser is used to seed a tapered amplifier. The amplified light
passes another AOM 2 [E 4] which runs at a frequency of ν2 ≈ 110 MHz. This
AOM has its centre frequency at 110 MHz and a bandwidth of 50 MHz. The
first order of the light is coupled into an optical fibre [E 5] [E 6], while the zeroth
order is lost. The optical fibre cleans the mode of the light and transfers it to
the experiment table. As the coupling efficiency into the fibre is very sensitive
to the position and angle of the incoming light, the AOM is used as a fast
switch to turn the light on and off. The two AOMs are set up so that they
work against each other in their frequency shifts. The light that is used in the
experiment thus has an overall detuning of 2ν1 − ν2 against the locking point,
which is close to but not exactly equal to the |F = 2〉 → |F ′ = 3〉 frequency, as
the polarisation locking scheme here does not lock to the peak of the absorption
like it does with the conventional modulated lock-in techniques. The overall
spectral width of the frequency stabilized laser light has been estimated to
≤ 1.6 MHz using a scanning etalon.
4.2.2 The repumping laser
The light source for closing the |F = 2〉 → |F ′ = 3〉 cooling cycle is produced
by a commercial diode laser system [E 7], which is a grating stabilised laser
in Littrow configuration. The output wavelength is stabilised by a Doppler-
free saturation spectroscopy using a peak-lock lock-in technique (see Fig. 4.3).
After passing the optical diode, part of the light is split into the spectroscopy
arm. Here another beamsplitter reflects a small amount of the light through
a 87 Rb-cell on a photodiode. The light transmitted by this beamsplitter is
counterpropagating this probe beam in the cell as the saturation light. The
wavelength of the light is modulated via the PZT of the grating, and the
2 This AOM was calibrated to fAOM = (997.09 · U + 2332.31) 1/2 MHz.
87

4.2. The Laser Systems
photodiode’s signal feeds the lock-in electronics [E 8]. The resulting error signal
is used to stabilise the laser directly to the |F = 1〉 → |F ′ = 2〉 transition.
TUI
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Figure 4.3: The repumping laser: extended cavity diode laser system with
spectroscopy
Most of the light of the diode laser is not used for the spectroscopy and is
coupled into an optical fibre, [E 5], and transported to the experiment table.
As there is no AOM in this set-up there is no possibility to detune or quickly
switch the light. The switching is done by mechanical shutters in other parts
of the set-up (see section 4.2.4).
4.2.3 The optical pumping laser
A home built diode laser is the light source for the optical pumping. This
laser is an extended cavity diode laser in the Littrow design, similar to the
repumping laser above. It is current and temperature stabilised [E 9]. Its
frequency stabilisation is a polarisation spectroscopy scheme like the one used
for the main laser in section 4.2.1. The set-up is depicted in Fig. 4.4.
The spectroscopy works in the same way as the spectroscopy of the main
laser, with a circularly polarised pump beam and a linearly polarised probe
88

Chapter 4: The Permanent Magnetic Chip Experiment: Apparatus
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Figure 4.4: The optical pumping laser: extended cavity diode laser, spec-
200
troscopy and an AOM to detune and switch the light.
beam which is then split onto two photodiodes. The difference of the signals
of the PDs is the error signal which is fed into a home built PID regulator.
The resulting output signal is fed to the PZT of the diode laser to stabilise
the laser to the frequency of the |F = 2〉 → |F ′ = 3〉 transition. Unlike the
PI-controling electronics for the tapered amplifier, we have no fast branch in
the correction signal working with the current of the laser diode.
The light for the experiment is deflected by a PBS. It then double-passes an
AOM, which is used to detune the light to 50 MHz above the |F = 2〉 → |F ′ =
2〉 transition. Switching the light is carried out by turning the AOM off while
still applying the frequency, so that the undiffracted light is dumped. This
allows us to switch much faster than the irradiation time of the atoms required
to optically pump the atoms, which is of the order of a few 100 µs. After
89
PD
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200
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4.2. The Laser Systems
double-passing a λ/4-plate, the light passes the PBS without deflection and
is coupled into an optical fibre [E 5] and transferred to the experiment. This
set-up allows us to tune the frequency of the light to the |F = 2〉 → |F ′ = 2〉
transition frequency and with sufficient fast switching times.
4.2.4 The optical paths on the experiment table
After the light has been transferred to the table with the vacuum chamber,
some beams of the different sources have to be overlapped and some have to
be split. For a working MOT, the light of the main laser and the repumping
laser have to be overlapped. Some of the light of the main laser has to be split
off before the addition of the repumping light for the absorption imaging. How
this is achieved is depicted in Fig. 4.5. Some beam shaping occurs in this part,
and there are mechanical shutters to fully block the light when unwanted. The
whole set-up is enclosed in a PVC box to reduce the amount of stray light that
could disturb the experiment.
The light for the MOT is prepared at a PBS, where the light of the main
laser (cooler) and the light of the repumping laser (repumper) are added after
both have passed a λ/2-plate. This light is then expanded by a factor of 10
in a telescope. A shutter [E 10] is placed in the focus of this telescope to stop
the light when trapping magnetically. This light proceeds towards the MOT.
The light for the absorption imaging is split off the cooler light before it is
combined with the repumper light. A rotatable PBS allows control over the
intensity of the absorption light for a good signal to noise ratio in the image of
the atoms. A λ/2-plate and a second PBS are used to redefine the polarisation
axis. The half-wave plate is not necessarily needed, but adds robustness to the
system by always allowing us to work in a regime of Malus’ law which is
insensitive to slight polarisation changes and mismatches. The beam is then
90

Chapter 4: The Permanent Magnetic Chip Experiment: Apparatus
1000
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light
50
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Figure 4.5: The optical paths to overlap the different beams. The set-up is on
the same optical table as the vacuum chamber and is enclosed in a PVC box.
expanded by a factor of 20. This is done to achieve a flat intensity distribution
of the light at the location of the atoms. A shutter [E 10] is placed in the focus
of this telescope and blocks the light until the actual pictures are taken. The
shutter starts to open at 1.7 ms and finishes 2.5 ms after the opening signal
arrives. It begins to close 1.5 ms and is fully closed 2.3 ms after the signal. The
times were determined using a fast photodiode [E 11] directly after the shutter,
with the beam appropriately weakened. Both opening and closing signal reach
the 3 · σ-line, which is calculated from the datapoints of the last 0.4 ms, after
2.5 ms. The results for the shutter of the MOT light are consistent with this.
91

4.2. The Laser Systems
To the light of the optical pumping laser we add a portion of the repumping
light by a PBS. This light is then expanded in a telescope, before it continues
its way to the chamber.
The final manipulation of the MOT light occurs outside the PVC enclosure.
It is shown in Fig. 4.6, together with the optical paths of the absorption and
the imaging light. These beams are sent to the sides of the chamber to an
elevator stage. The MOT light is split into 4 beams of equal power (about 25
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λ/2
PBS
λ/2
absorption
light opt. pumping
MOT light
Figure 4.6: The optical paths close to the chamber. This part of the set-up is
following the part of Fig. 4.5, but outside the PVC enclosute.
mW and diameter 3.5 cm) using a combination of λ/2-plates and a PBS. Each
beam then passes a λ/4-plate to be circularly polarised and is expanded by a
92

Chapter 4: The Permanent Magnetic Chip Experiment: Apparatus
telescope with f1 = −50 mm, f2 = 100 mm. The beams are the two pairs of
counterpropagating light needed for a surface MOT.
Both the optical pumping light and the light for the absorption imaging
are elevated to the position of the trap. Their polarisations are defined by a
PBS with following λ/4-plates (see Fig. 4.7). The optical pumping beam is
slightly tilted towards the axis given by the absorption beam and the magnetic
fields, and retroreflected on itself, while keeping its handedness after double-
passing a second λ/4-plate. The absorption imaging light passes through the
atomic cloud and is imaged on a CCD camera [E 12]. The imaging optics of
the camera consists of an achromatic doublet with f = 100 mm and allows a
resolution of 9.5 × 9.5 µm 2 per pixel. The camera itself has a pixel size of 9 × 9
µm 2 .
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CCD
camera
Figure 4.7: The optical pumping and the absorption light at the chamber. The
light is brought to height by an ‘elevator’ (see top of Fig. 4.6).
The fluorescence of the atoms, when trapped in the MOT, is focused onto
a photodiode. A confocal lens (f = 20 mm) is placed at a distance of twice its
focal length from both the atoms and the photodiode. An aperture in front of
the photodiode filters out unwanted scattered light. Infrared sensitive cameras
93

4.3. The Vacuum System
are mounted so that the atoms can be directly monitored from the bottom and
at an angle from the side, for quick online position control.
4.3 The Vacuum System
To magnetically trap atoms and to have trap lifetimes that allow evaporative
cooling towards a BEC, collisions with untrapped and hot background atoms
have to be avoided. This is done by trapping the atoms in an Ultra High
Vacuum (UHV) with a pressure of the order 10 −10 − 10 −11 Torr. To achieve
such a vacuum, care has to be taken in the design of the experiment chamber
and the pumping down of the vacuum. The chamber has to be baked out for
several days while pumping down. In the following the chamber itself will be
introduced. After that the pumps and their arrangement will be described.
This section finishes with the procedure for baking and pumping to reach the
desired UHV.
4.3.1 The experiment chamber
The chamber is shown in Fig. 4.8. It was designed by A. Sidorov [Sid02a], and
is made of electropolished steel (304SS), which has a magnetic permeability
close to the vacuum permeability, µ304SS < 1.05 × µ0. The overall dimensions
of the chamber are 9.6 × 9.6 × 4.9 cubic inches, the diagonal extension is 11”,
and the inner diameter is 6”.
The sides allow attachment of 8” (outer diameter) vacuum components
at the tapped holes. We have attached windows with anti-reflection (AR)
coating on the outside. Two of the MOT beams are passed through these
windows. In the horizontal and vertical plane, there are 4.5” flanges with
tapped holes. AR coated viewports are mounted on the horizontal flanges, for
the optical pumping and the absorption imaging. Through the top opening,
94

9.6"
Chapter 4: The Permanent Magnetic Chip Experiment: Apparatus
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Figure 4.8: The experiment chamber, front and side view. Dimensions are
given in the imperial system.
the chip and its holder are inserted and fixed. The bottom flange leads to
the vacuum pumps. The four 2.75” flanges with clear holes at 45 o have the
following attachments. The two lower flanges hold AR coated viewports which
are used for the second pair of MOT beams. One of the upper flanges holds
a cold cathode gauge [E 13] for monitoring the pressure. At the other upper
flange an electrical feedthrough (ceramaseal 8962-06-CF, 12 pin, copper, 55A)
is mounted which provides electrical connections to the Rb dispensers and the
chip wires.
4.3.2 Vacuum pumps and further vacuum system
The chamber is mounted on top of the vacuum system that includes the vac-
uum pumps. The tubes of the vacuum system are commercially available tubes
95
8"

4.3. The Vacuum System
produced from stainless steel. The main axis of this system is parallel to the
short axis of the chamber (see Fig. 4.9).
Ti: sublimation
pump
experiment
chamber
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valve
Ion pump
Turbo
pump
Figure 4.9: The vacuum system with its pumps. The experiment chamber is
mounted on top of this set-up. The diaphragm pump (see text) is omitted.
During the pumping down stage, the valve to the turbo pump [E 14] is
opened. This pump has a pumping capacity of 50 l/s. A diaphragm pump
[E 15] with ≈ 0.9 l/s is connected to it to produce a sufficiently low pressure
for the turbo pump to operate. Once the pumping down is completed and
the required pressure has been reached, the valve is closed and the pressure is
maintained by a 75 l/s ion pump [E 16]. A Titanium sublimation pump [E 17]
with four filaments can be used to temporarily further reduce the pressure.
4.3.3 Procedure to reach UHV
To reach the ultra low pressure multiple steps were needed. Before any bak-
ing, the chamber was flushed with Argon. The windows and viewports were
replaced by metal blanks, so that a higher temperature could be applied: after
about a week of baking at 300 degrees, a pressure of p < 7.5 × 10 −12 Torr
was reached (the pressure was lower than the minimum pressure that could
be measured by the cold cathode gauge). During that week, the chip was
96

Chapter 4: The Permanent Magnetic Chip Experiment: Apparatus
allowed to outgas in a second small metal chamber at 10 −9 Torr. After this
initial cleaning, the viewports were put back and the chip was placed inside
the chamber. The vacuum was then pumped down and baked out according
to Table 4.1.
Time 0 35 60.5; 67 80 96
(h) → 35 → 62; → → →
35 59 65.5 80 96 103
Pressure → → max → → →
(Torr) 3× 4× 1.7× 2.2× 8.3× 1.7
10 −7 10 −6 10 −6 10 −9 10 −11 10 −11
Temperature 20 20 140 100
( o C) → → →
special (a) (b)
140 100 20
Table 4.1: The pumping down process. Special: (a) all 4 Ti:siblimation pump
filaments were fired, the Rb-dispensers were outgased, baking was prepared,
(b) at 60.5 h the turbo pump heating was turned on; at 62 h the Ti:sublimation
pump filaments were fired; at 65.5 h turbo pump heating was turned off.
After an overall time of more than 100 hours, a final pressure of < 2×10 −11
Torr was achieved. The development of temperature and pressure are also
shown in Fig. 4.10 starting after the first 35 hours of pumping without heating.
The firing of the sublimation pump’s filaments is clearly visible by the peak at
t = 1630 min.
Before the experiment reached BEC, we had to open the vacuum once more.
Due to unknown reasons the vacuum had degenerated by roughly one order of
97

4.4. The Magnetic Field Coils
temperature / o C
200
180
160
140
120
100
80
60
40
20
0 1000 2000 3000 4000 5000 6000 7000
time / min
10−11
Figure 4.10: The pressure (solid line) and the temperature (dotted line) during
the pump down. The zero point here is set after the system was pumped out
for 35 hours.
magnitude. After the renewed baking and pumping down, the vacuum values
as stated here were reached again.
4.4 The Magnetic Field Coils
To trap the atoms, we use magnetic fields, which couple to the magnetic mo-
ment of the atoms. These fields are produced by different parts of the set-up.
To compensate for the Earth’s magnetic field and other stray fields that may
perturb the atoms, three pairs of coils surround the chamber so that for each
main spatial axis the compensation can be done independently. For the initial
loading of atoms in the mirror MOT, we have a set of large quadrupole coils.
The atoms are then transferred to a surface MOT whose field is created by a
U-shaped wire on the chip and a pair of bias coils. These are also used for
98
10 −5
10 −6
10 −7
10 −8
10 −9
10 −10
pressure / Torr

Chapter 4: The Permanent Magnetic Chip Experiment: Apparatus
magnetic trapping. The chip also carries a permanent magnetic structure that
is used to trap the atoms.
We will now describe the different sets of coils, each of which is labelled
according to its activity. The arrangement of all pairs of coils is depicted in
Fig. 4.11.
240 mm
160 mm
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Figure 4.11: External coil arrangement around the chamber. Coils with di-
ameter 160 mm are quadrupole coils, coils with diameter of 400 mm are bias
coils. The other coils are offset coils.
4.4.1 The offset coils
To compensate for unwanted magnetic fields, pairs of square coils have been
aligned along the three orthogonal axes around the chamber. Individual dc cur-
rents run through these coils continuously, reducing the influence of the Earth’s
magnetic field. This allows us to realize uniform ballistic expansion during
a molasses stage and to achieve a cold atomic ensemble with sub Doppler
99
400 mm

4.4. The Magnetic Field Coils
temperatures, reducing the temperature by a factor of ≈ 5 compared to the
temperature of the atoms in the MOT.
To these compensation coils, a second set of windings has been added
which can be addressed independently of the compensation coils, to give us
more flexibility. These fields can be switched on and off and are used during
the loading of the surface MOT and to reduce the magnetic field at the bottom
of the magnetic trap, thus increasing the trap frequency. The dimensions and
characteristics of these pairs of coils are given in table D.1 of appendix D.
4.4.2 The quadrupole coils
To create a surface MOT, the symmetry axis of a magnetic quadrupole field
has to to be aligned at 45 o to the reflecting surface [Rei99]. We achieve this
by a set of external coils in an anti-Helmholtz configuration. The dimensions
of this pair of coils is given in table D.2 in appendix D. These coils generate
a spatially large field, which is roughly matched with the beam size of the
cooling beams, so that the atoms see the MOT forces over a large volume. This
increases the capture volume and effectively the capture velocity of the MOT.
We water-cool the coils, so that they can dissipate the power without heating
themselves. The coils are driven with a current of 10 A, giving a gradient of
10 G/cm. Due to the high inductance, the switching time is around 10 ms.
As this MOT is needed to collect a large number of atoms initially only, which
are then transferred to a MOT created by the chip, the long switching time is
not a limitation to us.
4.4.3 The bias field coils
For trapping with a single wire, a homogeneous offset field has to be applied
to cancel out the wire’s field. The coils have been wound from a copper wire
100

Chapter 4: The Permanent Magnetic Chip Experiment: Apparatus
with a polyesterimide coating. The coils are wound 6 by 11 turns (width by
height), with a coil former of PVC. The maximum current is limited by the
melting point of the PVC. At 30 A with a 50:50 duty cycle, the coils reach a
temperature of 60 o C and create a field of 85 G. The coils are driven by two
power supplies (18 V, 20A) in parallel. The dimensions of this piar of coils in
Helmhotlz configuration is given in table D.3 in appendix D.
The time needed to switch off the coils is sub ms, although there is a 1.5
ms delay between the signal and the onset of the decay of the field. If this is
taken into account, the fast switching allows the imaging of ballistic expansion
and time of flight measurements. The longer time needed to switch on the
coil (≈ 20 ms) is not a problem, as the magnetic field here needs to increase
gradually when changing from the mirror MOT to the compressed MOT and
later to the magnetic trap (see chapter 5).
4.5 The Atom Chip
This section describes the main part of the experimental set-up. The atom
chip used in this experiment is novel in its combination of current-carrying and
permanent magnetic structures. It allows us to trap atoms with magnetic fields
produced by a current-carrying wire or by a permanent magnetic strip. This
section starts with a presentation of the overall arrangement of the different
parts. Then these parts are described in detail, starting with the wire structure.
The section ends with the description of the permanent magnetic film and its
properties.
4.5.1 Overall design
The atom chip and how its single parts are assembled is shown on a photo-
graph and schematically in Fig. 4.12. A copper block acts as the base piece
101

4.5. The Atom Chip
and as a heat sink. On one side of the block the chip is mounted, and the
other side is connected to a solid copper feedthrough with 19 mm diameter
(Ceramaseal, 800 A rating) and thus to the top of the vacuum chamber. This
feedthrough acts as a thermal conductor to the atmosphere and can be used as
a possible cold finger when cooled with liquid Nitrogen. Our atom sources are
Rb dispensers, as described and used in [Rap01, For98a]. They are mounted
on ceramic blocks (Macor) at the sides of the copper block. To insulate the
current-carrying structure from the copper block, a ceramic slide is glued be-
neath it. The reflective surface needed for a surface MOT is created by the
gold-coated magnetic film deposited on a glass slide on one half; the other
half is made up by a gold-coated glass slide. The adjacent glass slides have a
thickness of 300 µm each, with the long edges polished. The thickness of the
gold coating is 170 nm. The electrical connections use bare copper wires (Ø =
1.6 mm), BeCu barrel connectors, and a power feedthrough (Ceramaseal, 55
A rating, 12 pin). A publication describing the chip is available [Hal06].
Magnetic film
and coated glass
Atom chip
Insulation
Copper block
with dispensers
Figure 4.12: Photograph (left) and schematic view (right) of the overall chip
arrangement.
102

Chapter 4: The Permanent Magnetic Chip Experiment: Apparatus
4.5.2 The current-carrying wires
We now describe the current-carrying wire structure. The procedure for pro-
ducing the chip was developed by [Val04] A 0.5 mm silver foil (99.99 % purity)
was glued onto a Shapal-M ceramic base-plate (thickness 2 mm) using epoxy
Epotek H77. The structure itself consists of an “H”-like structure and two
end-wires on each side of the “H” (see Fig. 4.13).
22 mm
4 mm
6mm
Figure 4.13: Schematic view of the current-carrying wires on the chip. All
distances are from centres of the wires. White: remaining foil; grey: foil cut
away, black: holes drilled through the foil and the Macor. The ‘H’-structure
9 mm
allows the wire to be used in ‘U’- or ‘Z’-configuration.
The structure was cut into the foil using a PCB milling machine. This
is a cost effective solution which allows rapid prototyping and the in-house
production of the chip. The thick parts, leading to the connectors, have a
width of 4.7 mm for the H-wire and 4.5 mm for the end wires. The thin
wire parts have widths of 1 mm. The length of the bridge part of the “H”
is thus 4 mm long. The structure was cut into the foil with a cutting depth
of approximately 0.5 mm, in 6 consecutive cuts of about 90 µm each. The
width of the insulating grooves is 0.5 mm. The holes on two sides of the chip
103
32 mm

4.5. The Atom Chip
are 2.0 mm and 1.7 mm in diameter and used to connect the wires to the
feedthrough and to hold the chip to the copper block. This connection was
later identified to be the point of highest resistance in the system, and led
to substantial heating at the connections. This led to outgassing of the chip,
and in turn to a strongly reduced lifetime of the magnetic trap, one of the
main reasons why BEC was not reached in the initial runs. To reduce this
effect each connection was bridged by an additional thin sheet of metal from
the chip to the wire. These new, additional connections led to a decrease in
the overall resistance of chip and connectors from about 0.5 Ω to 0.32 Ω. The
wire resistance of the “Z”-wire was measured to be 4.6 mΩ while the chip
was changed, highlighting the need to measure the resistances before the set-
up is assembled and placed into the vacuum. Before the modification of the
connectors, a maximum current of 25 A could be run through the wire without
adverse effect on the lifetime. After the modifications a current of 31.1 A was
used without decreasing the lifetime of the magnetic trap. After the machining
and before the chip was mounted on the copper block, the silver was polished
using 600 grade sandpaper.
Having an “H”-shaped structure allows versatile operation. An “U”-shaped
wire (creating a quadrupole trap) and a “Z”-shape wire (for a harmonic trap)
are both possible. In the “U”-shaped regime, trap gradients of up to 40 G/cm
are possible for magneto-optical trapping limited only by the breakdown of
polarisation gradient cooling at high field gradients.
The end wires, with a separation of 9 mm, are not only used to provide a
confining field when trapping with the edge of the magnetic film, but also serve
as our RF antennas for the evaporative cooling. A frequency synthesizer [E
19] is connected to the wires to create the RF frequency. A sensing resistance
of 5 Ω is used to measure the resonance spectrum of the antenna loop. The
resonance shows a Lorentzian curve, centred at 11.8 MHz with a FWHM of
104

Chapter 4: The Permanent Magnetic Chip Experiment: Apparatus
2.9 MHz. This knowledge allows better control over the power of the radiation
during a ramping of the frequency as done in the process of evaporative cooling.
4.5.3 The magneto-optical film
The main component of the permanent magnetic strip is a ferrimagnetic magneto-
optical (MO) film. The structure is made from a TbGdFeCo / Cr sandwich
structure [Wan05a], on a glass slide, coated with gold for reflectivity. The
TbGdFeCo material has its magnetisation axis perpendicular to the film plane.
The structure is schematically shown in Fig. 4.14. The film was produced at
Swinburne University, with in-house technology that was developed for this
task. It was prepared using a commercial thin film deposition system [E 20]
with three magnetron sputtering guns and electron beam evaporation. The
sandwich consists of six alternating layers of the magneto-optical material
(Tb6Gd10Fe80Co4, ≈ 170 nm thickness) and Chromium (≈ 140 nm thickness).
This is coated by a thin (10 nm) Cr layer and a gold layer of 170 nm thickness
for reflectivity.
A hysteresis curve of the film was taken using the Magneto-Optical Kerr
Effect (MOKE) and shows a nearly square-like behaviour (Fig. 4.15). The co-
ercivity is around Hc = 4 kOe. The remanent magnetic field was determined
to be Brem = 2.8 kG, which is equivalent to an effective current of Ieff = 200
mA. The film was also analysed by SQUID measurements [Whi05, Hal06]. The
coercivity here was determined to be 2.7 kOe and the remanent magnetisation
to be 3.3 kG, in reasonable agreement with the MOKE results. The Curie
temperature of a similar film was measured by Imation to be TC ≈ 300 o C. A
similar film was produced together with samples for a magnetic mirror exper-
iment [Wan05a]. Here a single-layer TbGdFeCo/Cr film was examined for the
influence of the Cr underlayer on the grain size of the MO film using atomic
105

4.5. The Atom Chip
Cr
Au
Glass
MO
film
Figure 4.14: Schematic view of the
magnetic film, with alternating layers
of Cr and magneto-optical film on a
glass slide, with gold coating.
photodiode signal / V
0.35
0.3
0.25
0.2
0.15
0.1
0.05
−10
−8
− H c
−6
−4
−2
0
∝ B rem
applied field / kOe
Figure 4.15: Hysteresis curve of the
film by MOKE, used to determine the
magnetic polarisation Brem and the
coercivity Hc.
force microscopy (AFM). The MO film on Cr shows a smaller grain size (about
40 nm) and smoother rounder grains than a Cr or MO film alone (grain size
of 50 nm). It was found that without the Cr mid-layers, the coercivity was
reduced to less than 1.5 kOe. In addition, the remanent magnetisation and co-
ercivity deteriorated for film thicknesses larger than 200 nm. Using magnetic
force microscopy (MFM) the domain structure of the magnetised sample was
examined, showing excellent magnetic homogeneity (see Fig. 4.16). No do-
main structures are visible in the magnetised sample, limited by the resolution
of the MFM of 100 nm.
106
2
4
6
8
10

Chapter 4: The Permanent Magnetic Chip Experiment: Apparatus
Figure 4.16: Magnetic force microscope (MFM) images of the Gd10Tb6Fe80Co4
magnetic film surface, with 1 µm/scale. Left (a) the unmagnetised sample
showing magnetic domains of micron size. Right (b) the uniformly magnetised
sample showing no visible magnetic structure. Taken from [Hal06].
107

4.5. The Atom Chip
108

Chapter 5
The Permanent Magnetic Chip
Experiment: Results
5.1 Overview and Timing Sequence
In this chapter the results of the permanent magnetic chip experiment located
at Swinburne University of Technology are presented. In this overview the
steps taken to reach degeneracy of the atomic sample are outlined, with the
most important parameters and results. In the following sections, the impor-
tant steps are explained and further results are presented there.
Each experiment starts with a phase of loading atoms in the mirror magneto-
optical trap. For this, the Rb dispensers are pulsed for 12 s at 6.5 A. To allow
the vacuum to recover, this is followed by a 25 s waiting phase. During the
loading the cooling light is detuned by 18.5 MHz, and in the waiting phase it is
detuned by 15 MHz. With a gradient of about 10 G/cm from the quadrupole
coils, we trap about 5 · 10 8 atoms at about 90 µK. The atoms are then trans-
ferred to the compressed MOT (CMOT) by linearly ramping up the current in
the U-wire to 8 A and applying a bias field of about 7 G perpendicular to the
109

5.1. Overview and Timing Sequence
wire in the plane of the chip. This ramp takes 20 ms, and at the same time
the current in the quadrupole coils is reduced to zero. This is performed at a
detuning of 15 MHz. As the gradient increases to about 33 G/cm, the atomic
cloud heats up to a temperature of 140 µK, while it is compressed to less than
one quarter of its previous size. To further cool the atoms, the field gradient of
the CMOT is decreased to 11.2 G/cm, while the light is turned off and detuned
to 56 MHz in a time of 2 ms. The position of the cloud is held constant. The
sample of atoms is then optically cooled to 40 µK by illuminating with the
highly detuned light for 2 ms. This is followed by a stage of optical pumping
in which the magnetic substates of the atoms are changed to the mF = 2 state.
For this, the CMOT is turned off, while the bias field is increased to 6 G over
2.4 ms. The optical pumping is then performed with a 200 µs pulse of about
0.2 mW/cm 2 , with a frequency of about 50 MHz below the F = 2 → F ′ = 2
transition. A 500 µs pulse of repumping light of low power is also applied here.
The atoms are now prepared to be loaded into the magnetic trap created
by the wire. The bias field is increased to 19.5 G in 1 ms, and a current of 21.5
A is passed through the Z-shaped wire. Mode matching is achieved by using
the bias coils. We collect 8 · 10 7 atoms at 50 µK in this initial magnetic trap.
In a 100 ms linear ramp, this trap is adiabatically compressed to a gradient
of about 510 G/cm by further increasing the currents in the wire (to 31.1 A)
and bias coils (to a field of 56.3 G). This moves the trap closer to the chip
surface and heats the atoms to 160 µK. Further cooling is performed using
the mechanism of forced evaporative cooling by applying a RF-current to the
end-wires. The frequency is then ramped down from 20 MHz to 0.78 MHz in
10 seconds. At the same time the trap is further compressed by reducing the
current in the Z-wire and keeping the bias field constant. At the end of the
evaporation, the wire current is thus 24.9 A, leaving us with a gradient of 635
G/cm if we ignore the effect of the permanent magnetic film. At this stage
110

Chapter 5: The Permanent Magnetic Chip Experiment: Results
the transition to a Bose-Einstein condensate has taken place and we have up
to 10 5 atoms in the condensate.
As the atom cloud is close to the magnetic film, the effect of its field can
not be ignored, and for imaging the atoms with the time-of-flight technique,
we need to adiabatically move the atoms away from the chip. This is done by
a logarithmic ramp of 50 ms decreasing the bias field to 37.5 G, which moves
the atoms about 1 mm away from the surface. Imaging is performed by taking
4 images, each with a 100 µs pulse of resonant light. The first picture is taken
of the MOT before the transfer. The second is taken without illumination
during the RF-sweep as a correction for dark counts of the camera. The third
picture is the main picture. This is divided by the fourth image, which is a
background picture without atoms.
This chapter starts with the characterisation of the initial atomic ensem-
ble in the mirror magneto-optical trap, followed by a characterisation of the
compressed MOT. The next part covers the initial magnetic trapping and the
adiabatic compression of that trap. The subsequent section deals with the
cooling by RF-induced evaporation and the reaching of quantum degeneracy.
The chapter closes with some results on the trapping of atoms in the field of
the permanent magnetic film.
5.2 The Magneto-Optical Traps
The starting point of all experiments is collecting a large number of atoms at
reasonably low temperatures. This is done by trapping atoms in a mirror MOT
(see section 2.5.1). We use quadrupole coils (see section 4.4.2) to create a MOT
with a large trapping volume, which enables us to trap and cool a very large
number of atoms. From this point, the current in the coils is ramped down.
Ramping up currents in the wire structure on the chip (see section 4.5.2) and
111

5.2. The Magneto-Optical Traps
in the Helmholtz coils that are placed around the chamber (the ‘bias coils’,
see section 4.4.3) creates a new magnetic field that replaces the initial field.
Increasing the current in the bias coils moves the trap centre towards the chip’s
surface and increases the gradient of the magnetic field, creating a compressed
MOT (CMOT). The new gradients are chosen so that the atomic cloud can
be loaded into the magnetic trap with minimal losses. To further reduce these
losses, a stage of molasses-like cooling in a less steep trap is applied. A further
important step for optimal loading is to optically pump the atoms into the
mF = 2 magnetic substate.
In this section data from the mirror MOT is presented first, which can be
used to calibrate some laser parameters of the system. This is followed by the
results for the compressed MOT including further cooling. The section ends
with a description of the optical pumping as a preparation for the magnetic
trapping.
The mirror MOT
Early measurements on the atomic ensemble in the MOT were performed using
column density absorption. For the column density absorption, a weak laser
beam was scanned in frequency over the |F = 2〉 → |F ′ = 3〉 resonance of the
D2-line. Its intensity was measured after passing through the atomic cloud as
a function of the frequency. The intensity had to be weak so that this directed
beam did not destroy the MOT. The result of this measurement is shown in
Fig. 5.1. Fitting a Lorentzian to the logarithm of the normalised absorption
curve gave a maximum absorption of 96% and a full width at half maximum
(FWHM) of 9.1 MHz. The Lorentzian curve was not perfectly centred at zero
detuning, marked by the right dotted line. At a detuning of δf ≈ −16 MHz, a
small yet steep dispersive-like structure is visible. The left dotted line marks
the structure and intercepts the frequency scale. The structure appears at the
112

Chapter 5: The Permanent Magnetic Chip Experiment: Results
logarithm of normalised Intensity
0
−0.5
−1
−1.5
−2
−2.5
−3
−50 −40 −30 −20 −10 0 10 20 30 40 50
Detuning / MHz
Figure 5.1: Column density measurement of the MOT by absorption; thin line:
a Lorentzian fitted to the logarithm of the absorption signal, dotted lines are
explained in the text.
detuning of the trapping light fields from the atomic resonance and has its
source in a Λ-system where the probe and the trapping lasers couple the atom
states [Tab91]. We thus work with a trapping laser detuned by δf = 15.1 MHz,
the distance between the two dotted lines.
To evaluate the quality of the magneto-optical trap we determined the
linewidth of the |F = 2〉 → |F ′ = 3〉 transition of the D2-line absorption
spectrum by absorption imaging: our goal is to collect a large number of
atoms at a low temperature at a high density in the trap. The frequency of
the absorption light was tuned over the atomic resonance. Each picture was
taken with the MOT turned off after a loading time of 30 s and corrected for
the background. The data are depicted in Fig. 5.2. It shows the number of
absorbing atoms versus the detuning of the imaging light, and a Lorentzian was
fitted to the data. The Lorentzian has a FWHM of 8.7 MHz. This is slightly
closer to the natural linewidth of 6.1 MHz than the result of the column density
measurement. Here the MOT was turned off before each image was taken. As
the profile is Lorentzian, only the natural linewidth of the transition, collisional
113

5.2. The Magneto-Optical Traps
Number of
absorbing atoms
x 108
8
7
6
5
4
3
2
1
0
−30 −20 −10 0 10 20 30
Detuning / MHz
Figure 5.2: Integrated absorption spectrum of the MOT, plotted as number
of absorbing atoms versus detuning of the imaging light. Solid line: fitted
Lorentzian.
broadening and the linewidth of the absorption laser contribute significantly
to this result. Doppler broadening has a Gaussian form and would have led to
a Voigt profile in the absorption spectrum. In the convolution of Lorentzian
curves, the widths of the profiles add to the resulting width. The profile being
Lorentzian means that we deal with cold atoms. Allowing for the natural
linewidth only we can give an upper bound for the emission linewidth of the
main laser: ∆νMOPA < 2.6 MHz. A scanning etalon was used to measure
the spectral linewidth of the laser directly. The result here was ∆νMOPA ≤ 1.6
MHz. The difference probably stems from collisional broadening, and indicates
that we have a high atomic density in the MOT.
We can use the above measurement to measure the overall number of atoms
if we correct for the shape of the absorption probability. The above yields a
number of atoms of N = (2.2 ± 0.8) · 10 9 atoms, using equation (2.31) summed
over all camera pixels and taking the average over the atom numbers calculated
from the data for different detunings. We qualitatively find that our MOT
fulfills the requirements of collecting a large number of cold atoms at high
114

Chapter 5: The Permanent Magnetic Chip Experiment: Results
density. Typically, the trap is at a distance of 4.6 mm from the chip surface
and the atomic cloud of 5 · 10 8 atoms has a 1/e size of about 1.1 mm. The left
side of Fig. 5.3 shows the absorption image after correction for the background
and taking the logarithm of the intensity. The radial profile of this cloud is
depicted in Fig. 5.3, right, with a Gaussian curve fitted to it.
log absorption signal
3
2.5
2
1.5
1
0.5
0
0 500 1000 1500 2000 2500 3000
radial distance / µm
Figure 5.3: Picture (left) and radial profile (right) of the atoms in the mirror
MOT.
The final particle number in a MOT is determined by the ratio of its loading
and loss rates. These rates can be determined when the fluorescence signal of
the MOT is collected by a photodiode as a function of the time after turning on
the MOT and after turning off the source of atoms. A graph of the photodiode
signal when the MOT is loaded with atoms is shown in Fig. 5.4 (a). A graph
of the decaying fluorescence signal of the MOT which is not being replenished
with atoms is shown in Fig. 5.4 (b). The photodiode was calibrated with
3.2 · 10 7 atoms/V. As the maximum voltage is slightly less than 15 V, the
photodiode saturates at about 4.8 · 10 8 atoms in the trap and does not allow
the detection of higher atom numbers.
The dynamics of the atom number are fully characterised by the loading
115

5.2. The Magneto-Optical Traps
numer of atoms
4.5
4
3.5
3
2.5
2
1.5
1
0.5
x 108
5
(a)
0
0 10 20 30 40 50 60 70 80 90
time /s
4.5
4
3.5
3
2.5
2
1.5
1
0.5
x 108
5
0
0 10 20 30 40 50 60 70 80 90 100
Figure 5.4: Time dependence of the fluorescence signal of the mirror MOT,
number of atoms
(b)
time /s
(a) after being turned on, (b) after turning off the atom source.
rate RL and the loss rate γ; the atom number N(t) follows a rate equation:
Integration gives
dN(t)
dt = RL − γ · N(t)
N(t) = RL
γ · (1 − e−γt ) (5.1)
In the case of having turned off the atomic source, this simplifies to
with the solution
dN(t)
dt
= −γ · N(t)
N(t) = N(0) · e −γt . (5.2)
Using these solutions functions have been fitted to the data points. The
loading rate is given here as RL = 1.82 · 10 7 s −1 . The loss rate from the
loading data is γ1 = 0.037 s −1 , and from the decaying signal is γ2 = 0.011 s −1 .
These correspond to lifetimes of τ1 = 27 s and τ2 = 91 s, respectively. As we
use dispensers for loading the trap, switching off the source does not instantly
affect the MOT. The dispensers give rise to a high partial pressure of Rb atoms
in the vacuum. As long as these Rb atoms are still in the vicinity of the trap,
116

Chapter 5: The Permanent Magnetic Chip Experiment: Results
the MOT can use them as a background gas and replenish itself from there.
Thus, we have to expect the measured loss rate for the turning off to be lower
than for the turning on, or start the measurement after the vacuum definitely
has recovered.
To measure the temperature of an atomic ensemble we use the “time-of-
flight” (TOF) or ballistic expansion technique. Here the trapping potential
and the cooling lasers are turned off, releasing the atomic cloud which starts
to fall under gravity. It will also expand along all three spatial directions.
Using the identity between thermal and kinetic energy along one axis, we get
Ekin = Eth
1
2 mv2 = 1
2 kBT
T = m
v 2 . (5.3)
Here, the mean velocity of the ensemble has to be taken. At each time, the
atomic cloud will show a variance. This statistical term is also called the
second order central moment of a distribution and is a measure of the spread
of the distribution around the mean value. This value can either be taken
explicitly from the image taken or after fitting a Gaussian curve to the data,
where the variance σ appears in the functional term:
kB
x2
−
fGauss ∝ e 2σ2 Using the fit to a Gaussian distribution will lead to smaller values than calcu-
lating the second order moment directly, as the fitting procedure will smooth
out noise in the wings of the distribution which in the other case will add to
the moment.
As the cloud in the trap before the release at t = 0 has a variance σ0, the
image at a time ti will have a variance of (under the assumption that the cloud
117

5.2. The Magneto-Optical Traps
at all times has a Gaussian profile):
σ 2 i = σ 2 v(ti) + σ 2 0
(5.4)
If we assume the dynamic part to behave like σv(ti) = v · ti, we can substitute
after squaring and get v2 = σ2 i −σ2 0
t2 . If the initial size is unknown, the mean
i
velocity can be determined by taking two images at different times, so that
the temperature is
T = m
As we work with 87 Rb only, we can substitute and get
kB
σ2 1 − σ2 2
t2 1 − t2 . (5.5)
2
T = 10.45 nK σ2 1 − σ2 2
t2 1 − t2 , (5.6)
2
where the variances σi are measured in µm and the times ti in ms. The size
of the atomic cloud in the trap can be determined from the intercept of the
graph.
The final temperature of the atomic ensemble is determined by the detun-
ing of the trapping light. We have examined the temperature of the atoms as
a function of the detuning using TOF measurements. For each detuning the
temperature was determined by linear regression. The variance was calculated
directly from the data. The results are shown in Fig. 5.5. The maximum pos-
sible detuning here was ∆ = −36.5 MHz. This led to a minimum temperature
of 27 µK which is considerably smaller than the Doppler-limited temperature.
With the recoil limit for 87 Rb, Trec = 359 nK, we had a temperature of about
75 · Trec. The set-up was later changed to allow a larger detuning.
The compressed MOT (CMOT)
The atoms that were collected in the mirror MOT are then transferred into a
MOT where the magnetic field is created by the wire on the chip. The current
runs through the wire in a U-configuration creating a quadrupole field when a
118

Chapter 5: The Permanent Magnetic Chip Experiment: Results
Temperature / µ K
700
600
500
400
300
200
100
0
−40 −35 −30 −25 −20 −15 −10 −5 0
Detuning / MHz
Figure 5.5: Temperature of the atoms in the mirror MOT as a function of the
detuning of the trapping light. Each temperature and the error are the results
of a linear regression for time-of-flight measurements.
uniform bias field is added. The trap is moved closer to the surface while at
the same time the gradient is increased. This is done as a preparation for the
magnetic trapping, trying to keep a large number of atoms at a low temperature
in a potential that is deformed towards the potential of the magnetic trap.
Here it is described how both low temperatures and a large atom number in
the magnetically trappable substate F = 2, mF = 2 can be realised.
We observed a loss of atoms when moving our trap. This loss appeared
when our trap centre moved to about 1 mm from the surface of the atom chip.
As in this phase both gradient and position are changed, we examined the loss
mechanism more closely.
The loss of atoms from the compressed MOT was measured for different
settings. In one setting, the distance of the atomic cloud from the atom chip
surface was kept constant (d = 1.29 mm), while the gradient of the magnetic
field was varied ( ∂B
∂z
= 39 G/cm and ∂B
∂z
the other setting, the gradient was fixed at ∂B
∂z
= 77 G/cm)(see Fig. 5.6 (a)). In
= 77 G/cm and measurements
were taken at distances of d = 0.78 mm and d = 1.29 mm (shown in Fig. 5.6
119

5.2. The Magneto-Optical Traps
(b)). For the same distance with different gradients we see different loss rates.
A higher gradient causes a higher loss rate. The loss seems to be independent
of the distance, as long as the cloud does not touch the surface.
Remaining fraction
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(a)
10 1
10 2
Hold time / ms
10 3
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(b)
10 1
Hold time / ms
Figure 5.6: Examination of the main loss mechanism when the MOT was
compressed and moved towards the chip: remaining fraction of atoms in the
CMOT as a function of the hold time at: (a) same distance from the surface
of 1.29 mm and different magnetic field gradients of 39 G/cm (crosses) and
77 G/cm (dots); (b) same magnetic field gradient of 77 G/cm and different
distances from the surface of 0.78 mm (crosses) and 1.29 mm (dots). The lines
are included to guide the eye.
Remaining fraction
The influence of the permanent magnetic film also becomes noticable here.
We compared the value of the applied bias field calculated from the applied
current and the known dimensions of the coils with the value of the field from
the distance between the trap and the wire rw. When using the simple equation
for a wire with current Iw and perpendicular bias field Bb (see equation (2.39)):
Bb = µ0 Iw
2π rw
10 2
(5.7)
the values deviate at small distances from the wire. When correcting for the
magnetic field of the film, it has to be kept in mind that the wires and film are
120

Chapter 5: The Permanent Magnetic Chip Experiment: Results
separated by a small distance δR, so that the distance between the magnetic
film and the trap is R = rw − δR. For a film with a magnetisation equivalent
to a current IF we thus get:
Bb = µ0 Iw
2π rw
+ µ0 IF
2π R
(5.8)
We now have the option to fit this function while keeping one parameter fixed:
if we assume to know the distance in our experiment (δR = 0.55 mm, the
sum of the thickness of the glass slide with the film and half the thickness of
the silver foil) we obtain much better agreement between the two values of
the magnetic fields by using an equivalent current of IF = 0.44 A, which is
in rough agreement with the value for the effective current obtained from the
SQUID measurement [Hal06]. If we assume to know the effective current of
the film IF = 0.2 A from the SQUID measurement, then the data yields a
distance of δR = 0.46 mm between the magnetic film and the current-carrying
wire. Figure 5.7 shows a plot of the values of the bias magnetic field calculated
from the coil dimensions, from equation (5.7) and from equation (5.8), for a
wire current Iw = 6 A. The solid line is the result of either fitted function
including the influence of the film. The dotted line is the result of equation
(5.7) for a distance between film and wire of δR = 0.46 mm, the dash-dotted
line is the result of the same equation for δR = 0.55 mm. It is clear that for
distances smaller than 1 mm the influence of the magnetic film is no longer
negligible, but it is not clear which of the two cases is the actual case.
Typically, the compressed MOT is run with a current of Iw = 8 A in the
wire. The bias field is created by operating the bias coils (section 4.4.3) with
a field of Bb,1 = 18.7 G and a set of the smaller compensation coils in the x-
direction (section 4.4.1) with a field of Bb,2 = 11.2 G in the opposite direction.
This leads to an overall bias field of Bb ≈ 7.5 G. Two counteracting fields are
used in this way to reduce the switching time, which in our set-up is longer
121

5.2. The Magneto-Optical Traps
Applied bias field / G
20
18
16
14
12
10
8
6
4
2
0
0 0.5 1 1.5 2 2.5 3
Distance from surface / mm
Figure 5.7: Examining the influence of the magneto-optical film: the applied
bias field Bb is plotted against the distance between the trap and the surface
R. Circles are the calculated bias fields from the current and the dimensions of
the coils against the measured trap position for this current. The dash-dotted
and dotted line result from equation (5.7) for a wire trap without influence
of the magnetic film with Iw = 6 A and different distances between film and
wire. The solid line is calculated from equation (5.8) with the influence of the
film, IF = 0.44 A.
for building up a field. This longer time is avoided by the faster turn-off of the
countering field, which effectively results in a fast increase. Taking an effective
current of 0.44 A for the film, the trap emerges at R = 1.6 mm from the surface
with a gradient of ∂B
∂z
losses in the transfer to a negligible amount.
= 33 G/cm. This gradient is low enough to reduce the
For efficient loading into the magnetic trap we need both cold atoms and
atoms close to the chip surface. At distances as close to the surface as ours,
the proximity of the permanent magnetic film makes the usual mechanism of
sub-Doppler cooling impossible. Thus a compromise had to be made: cooling
with far detuned light at a lower gradient. For this, the field of the bias coils
is reduced to Bb,1 = 13.7 G over 1 ms; thus we reach an overall bias field of
122

Chapter 5: The Permanent Magnetic Chip Experiment: Results
Bb = 2.5 G. At the same time the current in the U-wire is reduced to Iw = 2.8
A. This keeps the trap at the same position, but quickly lowers the gradient to
about one third: ∂B
∂z
= 11.2 G/cm. During this switching, the light is turned
off and changed to a much higher detuning of 56 MHz. We examined the final
temperature of the atoms in the cloud as a function of both the final detuning
and the time the cooling light was applied. The results are shown in Fig. 5.8.
A short pulse of 2 ms is sufficient to cool the atoms, and as expected a larger
Temperature / µ K
150
100
50
0
0
−10
−20
−30
detuning / MHz
−40
−50
−60
12
10
8
6
4
cooling time / ms
Figure 5.8: Sub-Doppler cooling the atoms in a weak magnetic field: the
temperature of the trapped atoms against the detuning and the duration of
the cooling light. The lowest temperature is achieved with a cooling pulse of
≥ 2 ms with maximum detuning. The distance of the trap from the surface is
1.6 mm.
detuning leads to lower final temperatures. We thus apply a pulse of 2.5 ms
with light detuned by 56 MHz and reach temperatures as low as 40 µK for a
distance from the the surface of 1.6 mm.
To increase the population in the magnetically trappable atomic substate
F = 2, mF = 2, we turn off the light and the current in the wire, while
increasing the bias field to Bb = 6 G over 2.4 ms. The bias field now acts
123
2
0

5.3. The Wire Magnetic Trap
as a quantisation axis for the atomic spin and lifts the degeneracy of the
substates by the Zeeman effect. Using σ + -polarised light will move the atomic
population towards the state with the highest magnetic quantum number mF
by conservation of angular momentum. As any heating or a change in position
of the atoms is undesirable, the pumping light is retroreflected so that the net
heating is kept minimal. Further, only a low light power of 2 mW is applied
for a time of 200 µs. It was found that the number of atoms trapped by
the magnetic field could be increased by a factor of about 5 by locking the
light to the frequency of the F = 2 → F ′ = 2 transition detuned by 50 MHz
to the blue. This increase shows that before the pumping we have the five
magnetic substates very evenly populated and that our pumping works with
nearly 100% efficiency. This high efficiency is unexpected, as the detuning of
50 MHz corresponds to a Zeeman splitting between adjacent mF -states when
the magnetic field is about 71.5 G. In our case, we have a field of 6 G only,
and thus would expect the best pumping to occur at a detuning of roughly 4.5
MHz. The discrepancy remains unexplained.
5.3 The Wire Magnetic Trap
The transfer trap
Once the atoms have been transferred to the magnetically trappable state, we
have to create a magnetic potential that captures these atoms at their position.
From there the atoms can be further manipulated towards Bose degeneracy.
To trap the atoms magnetically, the bias field is ramped to Bb = 19.5
G in 1 ms, while the current in the wire is switched to Iw = 21.5 A in the
Z-configuration. This leads to a magnetic trap at the same distance of 1.6
mm from the surface with a gradient ∂B
∂z
= 88.4 G/cm. Strictly speaking, as
we use the Z-wire configuration for magnetic trapping, the gradient is not a
124

Chapter 5: The Permanent Magnetic Chip Experiment: Results
good parameter, as the trap is harmonic at the bottom, with a gradient of
zero but a finite curvature. In non-central regions of the trap, the harmonic
approximation fails (see Figure 2.5). But as all trapping parameters are given
by the current through the wire and the bias field, one can use any combination
of two parameters that only depend on Iw and Bb. The reasons why the bias
field and the gradient are chosen as parameters are explained below. Additional
bias fields of 3 G along the wire direction and perpendicular to the surface are
applied to match the field of the U-wire with the field of the Z-wire and to
further suppress Majorana spin flips. This trap holds ≈ 8 · 10 7 atoms at a
temperature of 50 µK.
In the Z-magnetic trap, the number of atoms slowly decays due to differ-
ent mechanisms. To measure these decays, a simple procedure is used: after
different times of magnetic trapping, the trap is changed back to a CMOT. If
needed, the trap is first moved back towards the CMOT position by ramping
the wire current and the bias field accordingly. Now, the fluorescence of the
recaptured atoms in the CMOT can be detected. This provides an easy means
to measure the lifetime of the magnetic trap.
To find the optimum trapping parameters, different settings were examined.
The trap depth and the compression are the important parameters. The trap
depth and its position r0 are determined by the bias field Bb and the current
through the wire Iw over Bb ∼ Iw/r0, where Bb is a direct measure of the
trap depth. The compression is determined by the gradient which scales like
∂B/∂r ∼ I/r 2 0 ∼ B 2 b /Iw. We want the trap to be as deep as possible to collect
all atoms, but we are limited by the maximum current through the wire and
the fact that our wire is separated from the chip surface, limiting the choice of
r0. Close to the surface we need a higher compression to prevent atoms from
touching the surface and being removed from the trap. On the other hand, too
high a gradient increases the chance of atoms not being able to adiabatically
125

5.3. The Wire Magnetic Trap
follow the field and leads to Majorana ‘spin flip’ losses. We thus need to find
a compromise. For this we first used two different initial numbers of atoms in
the same magnetic trap (Iw = 21.7 A, Bb = 39 G). The result is shown in Fig.
5.9 (a) on a logarithmic scale. We also varied the trap parameters. The larger
trap above was compared to a very shallow trap with a bias field of 13.7 G
and a current of 7 A and a steep deep trap with 54 G and 31.1 A. In terms of
the gradient, we thus compare 350 G/cm with 134 G/cm and 467 G/cm. The
results are shown in Fig. 5.9 (b) on a logarithmic plot.
relative atom number
1
0.5
0.3
0.2
0 5 10 15 20 25 30
time / s
(a)
1
0.5
0.3
0.1
0.04
0 5 10 15 20 25 30 35 40
Figure 5.9: Examining the lifetime of different magnetic traps: decay of the
number of trapped atoms on a logarithmic scale as a function of time. (a)
same trap parameters, different starting populations of 7.4·10 7 atoms (circles),
4.3 · 10 7 atoms (crosses). (b) different trap parameters: normal (o), shallow
relative atom number
time / s
(x), steep (+). The lines are fits of equation (5.9) to the data.
If we take into account density dependent effects, the loss rate changes from
dN/dt = −α · N to dN/dt = −α · N − β · N 2 . The loss coefficient α describes
constant losses like in radioactive decay, so the loss rate is proportional to
the number of atoms and each atom has the same probability to be lost.
Density dependent losses are described by β. The proportionality to N 2 is an
approximation for a large number of atoms N. Strictly speaking, a density
126
(b)

Chapter 5: The Permanent Magnetic Chip Experiment: Results
dependent loss is proportional to the number of atoms N and to the number
of remaining atoms N − 1 which can interact with the first atom. For large
numbers, the approximation N · (N − 1) ≈ N 2 is valid. This rate equation can
be solved analytically [Bro91], and still leads to a single exponential form with
the single particle loss rate α in the power of the exponential and the starting
particle number N0:
N(t) =
αN0
(α + βN0) · e αt − βN0
(5.9)
If we fit this function to the data for the different traps, we find the following:
first of all, the data of the first 0.5 seconds doesn’t fit to the model. During
this time, the cloud of atoms touches the surface of the chip and atoms are
lost at a high rate, which drops to zero once the cloud loses contact with the
chip. This effect is not covered by the above model, and thus the data for
times t ≤ 0.5 s was ignored. Second, from Fig. 5.9 (a) we can see that the
atom number is not the decisive parameter. The single loss parameter α is
negligible in both cases. If we look at the product of the density dependent
loss parameter and the starting number of atoms, βN0, we have a parameter
that is comparable to α. The value βN0 ≈ 0.1 s −1 is similar for both starting
atom numbers. If we now look at Fig. 5.9 (b), we can see that the steeper and
the shallower traps lose atoms at a higher rate than the intermediate trap. In
the steep trap, the two body loss rate is slightly higher. This has to be related
to the higher collision rate and thus the larger number of inelastic collisions
which cause loss. In this trap, one particle losses can’t be neglected anymore
and have to be taken into account with α ≈ 0.02 s −1 . This loss has to be
attributed to ‘spin flips’ when the atoms are unable to adiabatically follow the
changing magnetic field at the trap’s minimum. In the shallow trap the one
particle loss parameter is slightly higher than in the other traps with α ≈ 0.05
127

5.3. The Wire Magnetic Trap
s −1 , while the two body losses βN0 ≈ 1 s −1 exceed the values of the other
traps by an order of magnitude. Here the atoms are freely evaporating.
In summary, we infer that the main losses are caused by three mechanisms:
initially, a large number of atoms is lost by contact with the atom chip and
later, collisions between atoms and the atoms’ inability to adiabatically follow
the magnetic potential. In the shallow trap, the atoms can escape easier than
in the other two traps and any single collision can lead to the escape of atoms.
In the steep trap, the main loss mechanisms are ‘spin flips’ directly related to
the high curvature of the trapping potential and a slightly increased number
of inelastic collisions compared to the intermediate trap.
The compressed magnetic trap
Once we have trapped the atoms magnetically, this trap is compressed slowly
to improve the scattering characteristics of the atoms. We deepen the trap
and increase the gradient and by so doing we increase the density and the
temperature of the atomic sample. This is done ‘adiabatically’ in the sense
that the phase space density ρ does not change. In our set-up, we increase the
current to Iw = 31.1 A and the bias field to Bb = 56.25 G in a time of 100 ms.
As mentioned above, the set-up of the wire not being coincident with the
surface is a limiting factor. We examined how far the trap can be compressed
by measuring the recaptured atom number for variable bias fields at different
currents. The results are plotted in Fig. 5.10, with the atom number as a
function of the gradient. The magnetic film was taken into account, as 0.55
mm above the wire with an effective current of 0.44 A. The current-carrying
wire was taken as infinitely long with infinitesimal width. The dependence on
the trap depth given by the bias field is obvious: for the same gradient a higher
current generates a deeper trap, as it needs a higher bias field. A deeper trap
is capable of trapping more atoms. The influence of the gradient shows the
128

Chapter 5: The Permanent Magnetic Chip Experiment: Results
number of atoms / 10 7
4
3.5
3
2.5
2
1.5
1
0.5
0
200 400 600 800 1000 1200 1400 1600
Gradient / Gcm −1
Figure 5.10: Compressing the magnetic trap: atom number versus magnetic
field gradient, for wire currents of 31.1 A (crosses), 20.5 A (circles). The
optimum gradient is about 850 G/cm.
best trapping for values of around 850 G/cm. This optimum value is nearly
independent of the trap depth.
This Z-trap being of the IP-type has no zero crossing of the field. This
difference of the field’s minimum from zero can be measured by changing the
strength of the bias field along the wire direction and measuring the losses.
Once this field is scanned to a value that compensates the intrinsic offset field
of the Z-wire, a sudden decrease in the number of atoms indicates the losses
due to the spin flips which can occur in the vicinity of the field’s zero-crossing.
The data presented in Fig. 5.11 was taken with a wire current of 31.1 A
and a perpendicular bias field of 61.6 G, leading to a nominal gradient of 611
G/cm ignoring the film. The drop in the atom number at a field of about 3
G indicates that here the magnetic fields cancel each other. The wire alone
creates a bias field of 3 G which is counteracted by the field of the external
bias coils.
The increase up to 3 G is due to the slight changes in trapping depth and
gradient during the scan. The magnetic field changes the energy of the state
129

5.3. The Wire Magnetic Trap
number of atoms / 10 7
3
2.9
2.8
2.7
2.6
2.5
2.4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
bias field along wire / G
Figure 5.11: Atoms are lost from the trap as the magnetic field in the direction
of the wire cancelled out. The drop in the atom number at a bias field between
3 G and 3.5 G (arrow) is caused by Majorana ‘spin flips’, indicating the value
of the magnetic field component along the wire, induced by the Z-wire itself.
The dotted lines are to guide the eye.
|F, mF 〉 by ∆E due to the Zeeman effect (see equation (2.37)). The splitting
between the adjacent mF -levels at a field of 3 G corresponds to a frequency
f = ∆E/h = 2.1 MHz. In our case, in the compressed stage of the magnetic
trap, we apply a bias field of 1.6 G along the wire. The resulting effective
field of 1.4 G leads to an energy difference of 1.96 MHz between the trapped
|F = 2, mF = 2〉 and the untrapped |F = 2, mF = 0〉 state. This offset has to
be taken into account in the next paragraph.
Radiofrequency radiation can be used to cool the atoms by evaporation. It
can also be used to examine the ensemble by a spectroscopic means. Ramping
down the frequency to different end values fi leads to different numbers of
atoms remaining in the trap Ni. Atoms with energies higher than Ei = hfi
will be ejected from the trap and not be detected in a recapture fluorescence
measurement. Figure 5.12 shows how this number decreases when the fre-
quency of the radiation is linearly ramped down in 5 s from 25 MHz to 3.5
130

Chapter 5: The Permanent Magnetic Chip Experiment: Results
MHz. The cut-off energy is given in thermal units E ∝ kBT with correspond-
ing temperatures in the low mK regime to allow the relation to the following
calculations. The data was taken from two scans that differed slightly in the
fraction of remaining atoms
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
cut−off energy in thermal units k B ⋅mK
Figure 5.12: Atoms remaining in the trap as a function of the cut-off energy
of the RF radiation. This energy is equivalent to the trap depth and is given
in thermal units.
rate of the ramp. The trap is created by the same magnetic fields and wire
current as in the previous paragraphs. The data is corrected for an RF-offset
of 2 MHz.
If we now take the differences of the atom number ∆N and the energy for
adjacent frequency values, ∆E = h(fi+1 − fi), we obtain an estimate of the
number of atoms per unit energy interval as a function of their energy ñ(E) :
∆N
∆E
≈ dN
dE
= ñ(E) (5.10)
This can be plotted against the energy, where the lower frequency of the dif-
ference equation gives the value for the abscissa. Figure 5.13 shows such a
plot, using the data shown in Fig. 5.12. To guide the eye, two additional lines
are shown in the plot. The dotted line is the classical Maxwell-Boltzmann
131

5.3. The Wire Magnetic Trap
distribution for an ideal gas of N atoms at temperature T = 300 µK [Alo92]:
dN
dE =
2πN
(πkBT ) 3/2
√ −E
k Ee B T (5.11)
The solid line is for a Bose-Einstein distribution with temperature T = 120 µK.
With gi being the density of states, the distribution here is given by [Bec85]:
ñ(Ei) =
gi
e (µ−Ei)/(kBT ) − 1
(5.12)
Here the quantisation of the energy becomes important. Depending on the trap
geometry, gi states can be degenerate at the same energy Ei. The chemical
potential µ is the energy that is needed to add an atom to the ensemble while
keeping the volume and temperature constant. For a harmonic trap, the degree
of degeneracy depends on the number of dimensions. In a one dimensional trap,
there is no degeneracy. In a 2D trap, the energy Ei is i+1-fold degenerate, and
in the 3D case it is gi = 1/2(i + 1)(i + 2). For the plot an isotropic, harmonic
3D trap with a trapping frequency of 200 Hz was chosen. The first 100,000
levels were taken into consideration, using every 10th for the calculation. The
chemical potential µ of the Bose-Einstein distribution and the atom number of
the Maxwell-Boltzmann distribution were chosen to roughly fit the amplitude.
As the occupation was not calculated for every single level the physical meaning
of the chemical potential µ was lost anyway. The temperatures were chosen
to roughly fit the form of the data. We can see that the Maxwell-Boltzmann
distribution does not fit the actual distribution of the atoms in the trap. Using
only atom number and temperature, it can not be modified to fit both the
rising edge and the decaying tail. The Maxwell-Boltzmann distribution has
to be modified to take the confining potential into account, which is properly
done in the evaporation model of [Met99].
We can take the data points shown in Fig. 5.12 as a function of RF-
frequency νRF against atomic loss. We rescale and normalise the RF frequency
132

Chapter 5: The Permanent Magnetic Chip Experiment: Results
atom number per energy interval
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0 0.5 1 1.5
energy in thermal units k B ⋅mK
Figure 5.13: Distribution of the number of atoms in a compressed magnetic
trap per energy interval. The points are derived from the results shown in
Fig. 5.12. To guide the eye, lines have been added: the dotted line shows
the Maxwell-Boltzmann distribution of an ideal gas, the solid line shows the
Bose-Einstein distribution for a 3D isotropic harmonic trap. The energy is
expressed in thermal units.
into units of an unknown temperature. This gives us the atom loss as a function
of the truncation parameter η (used in the evaporation model, see section
2.3.1). This parameter is defined by the energy E = η · kBT at which the trap
is cut off. Thus, we get
η = E
kBT
h νRF − νoffset
=
kB T
(5.13)
where νoffset is the RF-offset frequency which is defined by the parallel bias
field. We can now use the relation of remaining atoms N ′ /N and the truncation
parameter of equation (2.51)
N ′
N
= γ(ξ + 3/2, η)
Γ(ξ + 3/2)
(5.14)
where γ(ξ + 3/2, η) is the lower incomplete Gamma-function as a fitting func-
tion. Here the data was fitted with the offset frequency and the temperature
133

5.3. The Wire Magnetic Trap
of the ensemble as free parameters, for the case of an harmonic (ξ = 3/2) and
a quadrupole (ξ = 3) trap. This is shown in Fig. 5.14. For the harmonic trap,
η
12
10
8
6
4
2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
fraction of remaining atoms
Figure 5.14: Truncation parameter η as a function of the atoms remaining in
the trap after reducing the trap depth. The results of Fig. 5.12 have been
used to calculate the parameter for the case of a harmonic trap (circles) and
a linear trap (crosses). Solid lines: fitted functions for the respective trap (see
equation (5.14)).
we obtain an offset between the trap bottom and the field zero of 2.36 MHz and
a temperature of T = 142 µK. In the case of the linear (quadrupole) trap, the
offset frequency is much smaller with 790 kHz and the temperature is slightly
smaller with T = 110 µK. When we allow the trap geometry parameter ξ as a
free parameter in the fit as well, then the results become inconsistent with all
a priori knowledge we have of the trap: with fitted results of ξ = 3.8 we would
have a less than linear rising trap. The fit then results in an offset frequency
of 1.5 Hz where we would have had to see massive losses due to Majorana
spin flips. When we compare the confidence levels of the three fits, all of them
reach values of r 2 > 0.995. Now, there are arguments for assuming a linear
trap and arguments that would propose a harmonic trap. The geometry with
a Z-shaped wire leads to a trapping potential which is harmonic at the bot-
134

Chapter 5: The Permanent Magnetic Chip Experiment: Results
tom, for low energies, while it is well approximated by a linear trap for higher
frequencies (ignoring the influence of the magnetic film) (see also Fig. 5.16).
A strong argument for the harmonicity of the trap is the offset of the trap of
2.36 MHz, which is in good agreement with the previously determined offset
from the above measurements on the bias field along the wire (Fig. 5.11).
Time-of-flight measurements of the atom cloud’s expansion have resulted in a
temperature of about 150 µK, close to the temperature we deduce from the fit
for a harmonic potential.
We see that we can not clearly specify the power-law of the trap. For
either a purely linear or for a purely harmonic trap the results are not fully
satisfying. This is closely related to the geometry of the wire which leads to two
regimes, one scaling with the square of the distance from the wire, the other
scaling linearly. From the above, we can conclude that our atomic cloud after
compression is in the middle region, with a tendency towards the harmonic
trapping regime.
After this compression stage, we typically work with 8 · 10 7 atoms at a
temperature of 160 µK. The trap is 560 µm from the chip surface, with a
gradient of ∼ 510 G/cm (ignoring the influence of the film). A bias field in the
direction of the wire of 1.6 G is applied, and the trap bottom for the atoms in
the mF = 2 state is about 2 MHz above the level for the mF = 0 state or zero
magnetic-field energy.
5.3.1 Evaporation and BEC
Once the trap has been compressed, we begin the cooling by evaporation. This
is done by applying a so-called “RF-knife”. By passing a radiofrequency cur-
rent along the end wires of the chip, we create radiation of the same frequency
at the trap. This causes atoms which have an energy that matches the energy
135

5.3. The Wire Magnetic Trap
corresponding to the applied frequency to undergo spin flip transitions into
magnetically untrapped states. In this way we can remove in a controlled way
atoms that have an energy higher than a threshold energy, leaving only low
energy atoms in the trap. Because of this, the RF-aided evaporation is a tech-
nique of forced evaporative cooling. The underlying mechanisms are explained
in section 2.3. This part is divided into two: one part covers the evaporation
process and the related important properties of the cold atoms, and the other
part shows the results once the BEC was reached.
Evaporation
Here results are presented that were obtained before the chip’s connectors were
modified, and with a vacuum an order of magnitude worse than is presented in
section 4.3. This vacuum problem was caused by the heating of the connectors
and wires. These will be compared with recent results [Whi05] of the BEC. The
results showing the BEC were taken with a substitute main laser, a commercial
high power laser diode [E 33]. The stabilisation set-up remained unchanged.
The most important parameter that characterises the atomic sample is the
phase space density ρ = n0 · λ 3 dB . Here the peak density n0 is given by the
the number of atoms N divided the volume which can be approximated by
the measured size of the atomic cloud, V = 1
√ 3 σaxial · σ
2π 2 radial , assuming a
3D gaussian distribution. The de Broglie wavelength can be calculated from
equation (2.1) once the temperature of the atomic cloud is known.
In the following, the data from the BEC includes TOF measurements on the
temperature for expansion times of 5, 10, and 15 ms. The cloud sizes σi were
determined by the intercept of the linear regression and by in-trap absorption
images. The absolute atom number N is the mean value taken from three
ballistic expansions. The data taken earlier provided no direct determination
136

Chapter 5: The Permanent Magnetic Chip Experiment: Results
of the temperature but a means to achieve this is presented in Appendix B.
Here the atomic spread σ was taken from in-trap absorption measurements.
Different ramps were used when ramping down the RF frequency. They
are shown in Fig. 5.15. The markers on the lines depict the times and the
value of the frequency at which data was taken for the following comparison.
The solid line marked with circles is the actual ramp that is now used to reach
Bose-Einstein condensation. This is a logarithmic ramp from a frequency of
20 MHz to 0.78 MHz in 10 seconds. The trap parameters also differed for the
RF frequency / MHz
20
18
16
14
12
10
8
6
4
2
0
0 2 4 6 8 10 12
time / s
Figure 5.15: The different logarithmic ramps of the radio frequency. The ramp
that leads to BEC is marked by circles.
measurements. The successful trap was run at a wire current of Iw = 31.1 A,
with bias fields of By = 1.6 G along and Bx = 54 G perpendicular to the wire.
Figure 5.16 shows the calculated magnetic field as a function of the distance
from the chip surface. Here the Z-shape and the magnetic film were taken
into account, and the finite width of the wires was neglected. The dotted
line shows a quadratic fit to the trap’s base. It leads to a trap frequency of
ν = 217 Hz. The magnetic field offset of this plot is much larger than is seen in
the experiment. This has to be explained by the calculation which takes into
account the Z-shape of the wire and the bias field along the wire, but not any
137

5.3. The Wire Magnetic Trap
other sources of fields in this direction, which can counteract these fields. The
magnetic field / G
50
45
40
35
30
25
20
0 0.2 0.4 0.6 0.8 1 1.2
distance from surface / mm
Figure 5.16: Calculated magnetic field of the trap that leads to quantum
degeneracy, plotted against the distance from the chip (solid line). Harmonic
approximation (dotted line) leads to a trap frequency of 217 Hz.
unsuccessful first attempt to reach BEC presented here used a wire current of
Iw = 19 A and a perpendicular bias field of Bx = 25 G. This leads to a trap
frequency in the harmonic approximation of ν = 100 Hz. Going back to Fig.
5.15 its RF ramp is shown by a solid line marked with crosses.
The evolution of the temperature during the evaporation process is shown
in Fig. 5.17 (a). The marking is the same as for the RF ramps in Fig. 5.15:
the trap that leads to a BEC is marked by circles. The older trap which did
not reach degeneracy is marked by crosses. Solid lines interpolate between the
points. The influence of the compression on the temperature can be seen on
this graph: the higher compressed trap has an initial temperature of nearly one
order of magnitude higher than the less steep trap. Both ramps lead to final
temperatures of the order of Tfinal ≈ 300 nK, close to the recoil temperature
of 87 Rb.
Knowing the temperatures, we can calculate the de Broglie wavelengths
using equation (2.1). These are shown in Fig. 5.17 (b), where the marking
138

temperature / µ K
10 3
10 2
10 1
10 0
(a)
Chapter 5: The Permanent Magnetic Chip Experiment: Results
10
0 5 10 15 20 25
−1
cut off frequency / MHz
atomic de Broglie wavelength / m
10 −6
10 −7
(b)
10
0 2 4 6 8 10 12 14 16 18 20
−8
cut off frequency / MHz
Figure 5.17: (a) Temperature of the atomic cloud, using the approximation
introduced in the text. (b) De Broglie wavelength of the atoms. Both graphs
are on a logarithmic scale as a function of the end value of the radio-frequency.
Circles: the trap that leads to BEC; +: the older trap. The lines are interpo-
lations between the points.
is the same as above. In the shallow, older trap we measure a slightly longer
wavelength than in the trap that leads to BEC. They are of the same order
of magnitude and with λdB ≈ 300 nm slightly shorter than the de Broglie
wavelength of the recoil temperature.
An experimental challenge, and how it can be overcome, becomes apparent
when the atomic density n0 is plotted against the cut-off frequency. Here for
the newer set of data the density was calculated once using the atomic cloud
size obtained directly from the in-trap imaging and once using the extrapo-
lated value for the size that comes from the linear regression which is used
to determine the temperature by ballistic expansion. The result of this dual
approach, together with the result of the older data, is shown in Fig. 5.18 (a),
using the same markings to depict the different measurements.
When we compare the different results, we see that the one that relies
on the in-trap imaging (circles) shows a breakdown in the density for end-
139

5.3. The Wire Magnetic Trap
atomic peak density in m −3
10 18
10 17
10 16
0 2 4 6 8 10 12 14 16 18 20
cut off frequency / MHz
(a)
phase space density
10 −1
10 −2
10 −3
10 −4
10 −5
10 −6
10 −7
0 2 4 6 8 10 12 14 16 18 20
cut off frequency / MHz
Figure 5.18: (a) Density of the atomic cloud, (b) Phase space density of the
atomic cloud, on logarithmic scales as a function of the end value of the radio-
frequency. Circles: the trap that leads to BEC using the spread of the in-trap
imaging; squares: the same trap using the spread calculated from the ballistic
expansion; +: the older trap. The lines are interpolations between the points.
frequencies smaller than 2 MHz. On the other hand, the results that use
the extrapolation from the ballistic expansion (squares) are much smaller for
higher frequencies. We can relate this to the problem of resolution, when the
trap reaches a size comparable to the resolution of the system. In our case,
for the points with the smallest frequency, we have a trap size of less than
10 pixels on the camera. Here possible errors in the focussing and imaging
will lead to a large overestimation of the atom cloud’s size, which in turn will
lead to too large a volume and thus to a systematically too small density.
It is thus better to rely on the ballistic expansion to achieve values for the
atomic spread. The problem of the breakdown of the density can also be
seen on the graph for the shallow, older trap. The threshold of the breaking
down is at a slightly smaller frequency. As this trap is less confining, this is
consistent with the above explanation. Unfortunately it is impossible to use
the different approach for this set of data, as here only in-trap images were
taken. The influence will be discussed further when the phase space densities
140
(b)

Chapter 5: The Permanent Magnetic Chip Experiment: Results
are compared. Apart from systematical errors in the imaging the small size
of the trap also leads to large statistical errors as can be seen in the error
bars. Here an absolute minimum uncertainty of one pixel to either side was
assumed together with a five percent relative uncertainty. The absolute error
dominates for small cut off frequencies in the measurements that rely on the
in-trap imaging, and is absent in the measurement that uses a linear regression
for determination of the trap size. It is favourable to use a linear regression
rather than the in-trap imaging technique for small trap sizes.
The most important figure of merit of any experiment that tries to reach
quantum degeneracy is the phase space density. Bose-Einstein condensation
requires a value of the phase space density of the order of one. The low
temperatures of the atoms have a major effect on this density, as they increase
the de Broglie wavelength which appears as a cube in the definition in equation
(2.49). The results of our experiments are shown in Fig. 5.18 (b), with the
same markings as above. We see that the trap that finally leads to BEC and
the older trap that was unsuccessful show a very similar behaviour and a large
increase in the PSD towards the end of the evaporation. The lowest frequency
of the RF radiation in Fig. 5.18 (b) is 850 kHz, which corresponds to a value
of ρ ≈ 0.03. The phase transition was observed at a frequency of 763 kHz. It
is difficult to extrapolate the PSD for that frequency from Fig. 5.18 (b), but
it is clearly very much greater than 0.03. The trap that did not lead to BEC
reached a maximum value of ρ ≈ 0.01 for a lowest frequency of about 1.1 MHz.
Fig. 5.18 (b) also shows the main reason why the older trap did not reach
quantum degeneracy. When we ignore the problems of determining the tem-
perature and the subsequent uncertainties, it appears as if the lowest cut-off
frequency of the RF knife was chosen to be too high. However, this is not the
only reason. It is also important that the trap was shallower by a factor of two.
This led to a density that was lower by roughly the same factor (Fig. 5.18, (a)).
141

5.3. The Wire Magnetic Trap
All figures here show the resulting problem with the shallow trap: the forced
evaporation started with RF frequencies of about 8 MHz. For higher frequen-
cies neither the temperature nor the phase space density change appreciably.
A lower atomic density also reduces the elastic collision rate in the trap. This
increases the time needed for rethermalisation and can be a further reason why
quantum degeneracy was not reached. The currents running through the chip
wires in the unsuccessful attempt were low enough not to heat the chip and
reduce the lifetime of the trap by outgassing. The problem of losing atoms to
the surface initially in the magnetic trap is a further reason for the failure of
these attempts.
Bose-Einstein Condensation
Figure 5.19 shows absorption images after ramping the cut-off frequency from
20 MHz down to 0.78 MHz in a logarithmic sweep over 10 seconds. The phase
transition to quantum degeneracy is clear. Here the wire current was also
ramped from 31.1 A initially down to 24.9 A. This leads to a further increase
in the trapping frequency and thus in the collision rates. A bias field of 56.25
G was applied perpendicular to the wire. A second bias field along the wire
had a strength of 1.6 G. We can see the phase transition starting at a cut-off
frequency of 763 kHz. The trap was measured to be 350 µm from the surface;
the calculated value ignoring the magnetic field of the film is 280 µm. The
trap’s gradient, also ignoring the magnetic field of the film, is 635 G/cm. In
the images shown here, the atom number goes from 9 · 10 4 atoms at 771 kHz
down to 4 · 10 4 atoms at 755 kHz. In the intermediate step we find 7.6 · 10 4
atoms.
The profiles of the atomic clouds for each end frequency are shown in
Figs. 5.20 and 5.21. Here the signal was summed over one axis and plotted
against the other. The plots show the data points and fitted functions. For
142

Chapter 5: The Permanent Magnetic Chip Experiment: Results
771 kHz 763 kHz 755 kHz
Figure 5.19: The phase transition from a thermal cloud to a BEC can be seen
by absorption images of the atomic cloud for different final radiofrequencies
(771 kHz, 763 kHz, 755 kHz). Each image shows an area of 1.26 × 1.17 mm 2 .
integrated atomic density axis in a.u.
0.16
0.15
0.14
0.13
0.12
0.11
0.1
0.09
(a)
0.08
0 0.2 0.4 0.6 0.8 1 1.2
vertical position in mm
integrated atomic density axis in a.u.
0.16
0.15
0.14
0.13
0.12
0.11
0.1
0.09
(b)
0.08
0 0.2 0.4 0.6 0.8 1 1.2
vertical position in mm
integrated atomic density axis in a.u.
0.16
0.15
0.14
0.13
0.12
0.11
0.1
0.09
(c)
0.08
0 0.2 0.4 0.6 0.8 1 1.2
vertical position in mm
Figure 5.20: Vertical profile of the atom cloud for end frequencies of (a) 771,
(b) 763 and (c) 755 kHz. Solid lines are fitted Gaussians for 771 and 763
kHz (thermal clouds). For 755 kHz the sum of a Gaussian and a quadratic
Thomas-Fermi profile (BEC) was fitted. The dotted line is the Gaussian ther-
mal contribution.
the thermal clouds, the data was fitted to a Gaussian distribution, while the
profiles of the condensed atoms were fitted to the quadratic function of the
Thomas-Fermi profile. For Fig. 5.21 (b) the phase transition is clearly visible.
The underlying Gaussian of the thermal cloud is shown by the dotted line in
that plot. Figure 5.21 (c) shows deviations and oscillations in the wings of the
Gaussian. These are also clearly visible in Fig. 5.19, 755 kHz. The structure
in the wings indicates that we are looking at a surrounding thermal cloud here,
and see a super-imposed pattern of the light being diffracted by the BEC.
143

5.3. The Wire Magnetic Trap
integrated atomic density axis in a.u.
0.18 (a)
0.16
0.14
0.12
0.1
0.08
0 0.2 0.4 0.6 0.8 1
horizontal position in mm
integrated atomic density axis in a.u.
0.18 (b)
0.16
0.14
0.12
0.1
0.08
0 0.2 0.4 0.6 0.8 1
horizontal position in mm
integrated atomic density axis in a.u.
0.18 (c)
0.16
0.14
0.12
0.1
0.08
0 0.2 0.4 0.6 0.8 1
horizontal position in mm
Figure 5.21: Horizontal profile of the atom cloud for end frequencies of (a) 771,
(b) 763 and (c) 755 kHz. The solid line is a fitted Gaussian for 771 kHz. For
763 and 755 kHz the sum of a Gaussian and a quadratic Thomas-Fermi profile
(BEC) were fitted. The dotted lines are the Gaussian thermal contribution.
There is an inconsistency in the trap bottom, which has been measured to
be about 0.5 G or a frequency of 350 kHz. Above in section 5.3 we found that
for a trap with a wire current of 31.1 A, a perpendicular bias field of 61.6 G
and a parallel bias field of 1.6 G, the trap has an offset field of 3 G, while a
calculation would lead to an offset field of 21.2 G, so that our calculation is
too high by 18 G absolute. When calculating the fields with the parameters
used for the BEC, we see that the ramping down of the current as in the
above reduces the offset field to 16 G, which is still too high by about 15.5 G.
As well, the calculation has its minimum at 200 µm from the trap, less than
the measured 350 µm. Part of the discrepancy between measurement and
calculation has to be related to the difficulties of adjusting the bias field to be
perpendicular to the wire and adjusting the edge of the magnetic film to be
parallel to the wire. Any angle between them leads to additional parallel fields.
If we incorporate this in the calculations, we see that the largest influence is
due to possible misalignments between the wire and the bias field, outweighing
the influence of the film by a factor of about 5. In the linear regime of small
angles, it contributes to roughly 1 G per degree away from the perpendicular.
If we allow an angular error between the chip and the bias field of 5 degrees,
144

Chapter 5: The Permanent Magnetic Chip Experiment: Results
we can reduce the difference between theory and measurement by 6 G. An
angular misplacement of 10 degrees would account for nearly all the observed
differences, but appears rather large and should be noticeable “by eye” in the
experimental set-up. It was later seen that the substrate with the magnetic
film moved by about 50 µm due to unevenly cured epoxy [Hal06]. Thus it is
not clear where the discrepancy originates, but it seems to be rather constant
in absolute values, and explainable with geometric displacements.
5.4 The Permanent Magnetic Trap
A film-based MOT
As a very early result of the implementation of the magnetic film, we were able
to show that the film is sufficient to create one of the basic magnetic fields for
a magneto-optical trap, similar to [Ven02]. This was done purely qualitatively,
and has to be seen as an “in principle” demonstration. Figure 5.22 shows
the fluorescence from this MOT, where the trapping field is created solely by
the film, a homogeneous bias field and the confining field from the end wires,
running anti-parallel currents to create a quadrupole potential. This picture is
an integration over 5 shots, each corrected for the background. On the top the
reflection from the chip is visible. As the camera resolution was not calibrated,
no measure of the atom number or temperature was performed. This result was
very promising. Not only were we able to trap atoms with the film, the trap
also was strong enough to be detected with fluorescence detection, even though
for signal-to-noise reasons it was integrated over several shots. The reason for
the low atom number lies in the small volume of the quadrupole potential, and
thus the velocity range for capture is small. The small number of atoms in the
film MOT was insufficient for further experiments, and highlights the need for
a large MOT from where the atoms are then transfered to the magnetic trap.
145

5.4. The Permanent Magnetic Trap
Figure 5.22: Fluorescence of atoms in a MOT created by the permanent mag-
netic film, integrated over 5 images which were corrected for the background.
To align the external coils for the large MOT and the current direction in the
wire magnetic trap, one needs to know the direction of the effective current
in the film. This knowledge can be deduced from the polarisation of the film-
MOT beams once the film MOT appears, and the direction of the current in
the wire for magnetic trapping can be matched to the direction of the effective
current of the magnetic film.
The film-based magnetic trap
The first experiments that were performed with this trap involved transfering
atoms from the “Z”-wire magnetic trap to the permanent magnetic film trap.
The trapping potential there is created by the magnetic fields of the film, a
homogenous bias field perpendicular to it and the field of the two end wires
with currents running through them in a parallel direction. It is not possible
to transfer a large number of atoms to the magnetic film trap directly without
an intermediate RF evaporation cooling stage in the wire. The reason for this
is the constant effective current of the film. While with the two degrees of
freedom of a variable wire current and bias field it is possible to change the
146

Chapter 5: The Permanent Magnetic Chip Experiment: Results
B,I
B',
d
B depth
I wire
time
trap distance
from chip
B': gradient
Figure 5.23: Schematical view of the ramps for wire current and bias field
to transfer the atoms from the wire trap to the film trap. Additionally, the
development of the trap position from the surface and its gradient are sketched.
trap depth and gradient independently, with the fixed effective current and a
variable bias field these two properties are connected. For mode matching, it
is possible to choose the bias field such that the trap based on the magnetic
film has the same gradient as the one based on the wire (for 21 A this is
approximately 100 G/cm), but the depth of the film based trap is reduced.
Without intermediate cooling, the atoms in the wire trap would be too hot to
stay in the film trap and would spill over, making the capture of a large atom
number impossible. To avoid this, a stage of RF cooling down to about 50 µK
was applied, with a ramp of 10 seconds and an end frequency of about 2 MHz.
To then transfer the atoms to the magnetic film trap, both the wire current
and the bias field were ramped down, as shown schematically in Fig. 5.23.
This led to a trap of 8 · 10 6 atoms roughly 220 µm below the surface. By
turning off the current in the end wires, a ballistic expansion along the wire
can be detected. Some results are shown in Fig. 5.24.
The density of the atomic sample was summed up perpendicular to the wire
to result in a density profile. Unfortunately, the wire axis does not completely
147

5.4. The Permanent Magnetic Trap
0 ms
8 ms
16 ms
Figure 5.24: Ballistic expansion of the atoms along the magnetic film. For
this only the current in the confining end wires was turned off. The left side
shows absorption pictures. The right side shows the atomic density profiles
in arbitrary units versus position, including a Gaussian fit. Expansion times,
from top to bottom are T = 0 ms, 8 ms, 16 ms. The size of each image is
5.9 × 0.4 mm 2 . The profiles are scaled with one mm/tick.
coincide with the camera axis. To compensate for this, a linear function was
added to the Gaussian function in the fitting process. From the Gaussian fits
we can deduce the spread σx and calculate an axial temperature along the wire
axis from this data using equation (5.6). The results are shown in Fig. 5.25.
Measuring the temperature by standard ballistic expansion is not possible in
this set-up, as the film’s inhomogenous magnetic field will push the atoms away
and not allow a measurement of the thermal spread of the atomic cloud.
When we omit the first two data points in Fig. 5.25, this procedure results
in a temperature of Tx = 54 µK. It is clear that the linear behaviour fails
for expansion times t < 8 ms. A high density effect may be the reason: the
atoms are first confined tightly in two dimensions and loosely in the third. The
barrier in the third direction is removed. The sudden energy gain for t < 8 ms
can be related to atoms that relax from the high energy states in the remaining
confined directions by collisions and convert their potential energy from one
axis to kinetic energy along another axis. Only when the density has been
sufficiently reduced the expected linear behaviour can be seen and thus we can
148
0.5
0
0.5
0
0.5
0

Chapter 5: The Permanent Magnetic Chip Experiment: Results
2 2
σ / mm
x
6
5
4
3
2
1
0
0 50 100 150 200 250 300 350 400
T 2 / ms 2
Figure 5.25: Squared atomic cloud size σ 2 x as a function of the squared ex-
pansion time T 2 , taken from the results and Gaussian fits as presented in Fig.
5.24. The solid line allows one to extract the temperature after relaxation
processes have ended and results in Tx = 54 µK.
indeed approximate the overall temperature by this one measurement. These
results were obtained before the BEC was reached and demonstrate that even
with the early experimental imperfections we were still able to cool the atoms
enough to transfer them to the magnetic film trap.
With the atoms trapped by the field of the magnetic film, we are able
to further examine the properties and characterise the film. We see the first
evidence of the so-called fragmentation of the atomic cloud [Lea03, For02,
Kra02, Jon04]. To check for this, the difference between the actual profile and
the fitted function was taken for all data points. It is depicted in Fig. 5.26,
where the differences are plotted for all six expansion times from 0 to 20 ms.
The differences between the actual profile and the fit show a common profile.
This can be explained by the finite roughness or inhomogenous magnetisation
of the film, with roughness leading to an effective current that is corrugated
by perpendicular currents [Est04, Sch05a, Wan04]. These lead to changes in
the magnetic field, which affect the density of the atoms. Positions with a
149

5.4. The Permanent Magnetic Trap
(density profile − Gaussian fit )/ a.u.
0.08
0.06
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
0 1 2 3 4 5 6
position / mm
Figure 5.26: Differences between actual profiles of the atomic cloud after bal-
listic expansion and the fitted Gaussian profiles (see Fig. 5.24) as a function
of position along the wire. The plot overlaps the differences of 6 expansion
times from 0 to 20 ms.
higher overall field will be less populated than positions with lower magnetic
field. Further investigations have later been carried out and the source of the
fragmentation has been identified as inhomogeneities in the magnetisation of
the film [Whi07].
During the writing of this thesis, evaporation and Bose-Einstein conden-
sation was achieved while trapping the atoms in the field of the permanent
magnetic film. This work has now been published [Hal05, Hal06]. As a result,
further characterisation of the properties of the film was possible. We use eqns.
(2.43) and add a bias field with components parallel, By, and perpendicular,
Bx, to the edge of the magnetic film. The trap will now appear at a distance
from the film z0 where Bx = Bfilm, while By lifts the trap bottom from zero
field. The result is a 3D harmonic trap with a radial frequency of
ωradial = µ0
2π
hM
z2 �
µBgF mF
0 mBy
(5.15)
where h and M are the thickness and magnetisation of the film, µB is the Bohr
150

Chapter 5: The Permanent Magnetic Chip Experiment: Results
magneton, gF the Landé factor, mF the magnetic quantum number and m the
atomic mass.
Increasing the homogeneous bias field by about 5% for a time of 2 to 5 ms
excited harmonic oscillations of the BEC’s radial centre of mass within the film
trap. The position of the atomic cloud was then measured after 10 ms of free
expansion over five periods of the oscillation, allowing a determination of the
trap frequency with an accuracy of about 0.1%. Additionally, the trap bottom
was measured using RF outcoupling with an accuracy of better than 1%. These
sets of data were taken as functions of the distance from the magnetic film and
allow a calculation of both the magnetic field strength and the gradient. The
results are shown in Fig. 5.27. The figure also shows the expected results
calculated from eq. (5.15) when using an effective current of hM = 0.2 A.
Figure 5.27: The magnetic field strength (a) and gradient (b) as functions of
the distance from the chip surface. Data points (circles) and predictions of
a simple model (dotted line) show good agreement. Experimental errors are
dominated by the image resolution. Taken from [Hal06].
151

5.4. The Permanent Magnetic Trap
152

Chapter 6
The All-Optical Bose-Einstein
Condensate Experiment:
Apparatus
6.1 Overview
This chapter describes the all-optical Bose-Einstein Condensate experiment at
the Institute for Quantum Optics, University of Hannover, Germany. Its goal
was to achieve Bose-Einstein condensation of 87 Rb atoms held in an optical
dipole trap, without any magnetic fields. Atoms were collected from a decel-
erated atomic beam into a magneto-optical trap, and after pre-cooling loaded
into the dipole trap. Evaporation was forced by ramping down the amplitude
of the trapping light. The apparatus used for this experiment was first set up
to examine the coherence of 85 Rb atoms in miniaturised trapping and guiding
potentials [Buc01]. Following this project, the apparatus was used to analyze
microstructured optical elements for use in atom optics and quantum informa-
tion processing [Dum03a]. The success of this work stimulated the interest in
153

6.1. Overview
using a degenerate atomic ensemble in these micro-optical potentials for atom
interferometry. To achieve this, the experimental set-up had to be changed to
a more suitable isotope ( 87 Rb) and higher atom numbers in the initial MOT
and optical trap. The optical trap was changed, using a laser that with a larger
detuning, which means fewer scattered photons and less heating for the atoms
in the trap. The vacuum system was changed so that the pressure could be
further reduced. These changes are described here. The original set-up as de-
scribed in [Buc01, Dum03a] will be described very briefly only where needed.
The changes to the 87 Rb isotope and for higher atom numbers in the MOT
can also be found in [Len04], the diploma thesis of A. Lengwenus, who assisted
the author and took over the experiment after the author’s first stay, and in
[M¨05, Geh05], which contain information about the improved optical trapping
and evaporation.
While the standard approach to reach Bose-Einstein condensation is to trap
the atoms in the minimum of a magnetic potential, such an approach does not
allow one to trap all possible atomic states. Magnetic fields cannot be designed
to have a local maximum, so states that have a negative magnetic quantum
number mF , so called strong field seeking states, cannot be trapped. Precision
experiments for metrology with atoms in the magnetically neutral states, mF =
0, are not possible with magnetic traps. This also means that optical traps
allow one to collect atoms in different magnetic states at the same time, and
to create a BEC consisting of several components. A further advantage of
optical dipole traps over magnetic traps is that they can be used to study
the influence of homogeneous magnetic fields, which are needed to implement
Feshbach resonances [Chi05b]. For applications in atom interferometry based
on microstructured potentials, a technical advantage of optical dipole traps is
the use of micro-optical arrays that are positioned outside the vacuum chamber
154

Chapter 6: The All-Optical BEC Experiment: Apparatus
and have their foci imaged into the chamber. This allows a quick change of the
experimental set-up without the need to break the vacuum [Dum02a, Dum02b].
The first all-optical BEC was realised in 2001 [Bar01]. It used crossed
beams of a CO2 laser to produce a condensate of 87 Rb atoms. The same iso-
tope was used in an experiment that reached BEC in a single beam of a CO2
laser [Cen03]. A trap where the rubidium atoms were pre-cooled in an opti-
cal lattice and then condensed in a compressible trap has been demonstrated,
using a Nd:YAG laser [Kin05]. A BEC of Cs was created within a so-called
“dimple” trap [Web03]. Here the focussed 1064 nm light from a YAG laser
was added to the crossed beams of a CO2 laser. A two dimensional surface
trap using blue detuned light was also successful in condensing Cs [Ryc04].
These experiments demonstrate the importance of optical dipole traps, as Cs
atoms in the magnetically trappable state could not be condensed due to high
losses caused by a high rate of inelastic two-body collisions [Kok98]. A further
element to be trapped optically and then cooled down to BEC was ytterbium.
The trap here was created by crossed beams of frequency doubled light (523
nm) from a YAG laser [Tak03]. All of these experiments are technically very
demanding: they either work with light of a very long wavelength, needing spe-
cial optics and windows, with frequency-doubled light, or with an intermediate
cooling stage in an optical lattice.
Our goal is to set up an apparatus that allows the creation of an all-optical
BEC by a much simpler means, using standard optics and no intermediate
stages.
This chapter begins with the vacuum system which provides the environ-
ment to cool and trap atoms in an optical dipole trap. This is followed by a
description of the laser systems used to load atoms into and to trap and cool
the atoms in a magneto-optical trap. The next section covers the laser system
155

6.2. The Vacuum System
used for trapping by the dipole force of the light. The chapter ends with a
short description of the atom detection.
6.2 The Vacuum System
The vacuum system has been described in detail in [Buc01]. The changed set-
up is depicted in Fig. 6.1. To achieve a low background pressure in the space
experiment
chamber
Ion pump
Ti: sublimation
pump & cold
finger (LN )
2
valve &
diff. pumping
stage
Turbo
pump
oven
chamber
Figure 6.1: Schematic view of the vacuum system. The atomic beam starts
from the oven on the right and passes through the differential pumping stage
into the experiment chamber. The sublimation pump and cold finger are ad-
ditions to the set-up described in [Buc01].
of the atom traps, the system consists of two chambers. One chamber holds
the source of the Rubidium atoms. This oven chamber is continually pumped
by a turbo pump [E 22], leading to a typical pressure of 2.5 × 10 −7 Torr. The
oven heats the Rubidium atoms which create an atomic beam. Between the
oven and the experiment chamber a differential pumping stage with an inner
156
oven

Chapter 6: The All-Optical BEC Experiment: Apparatus
diameter of 3 mm and a length of 15 cm is placed and the atomic beam is
directed through it. The experiment chamber is connected to an ion pump
[E 23] and to a Titanium sublimation pump [E 24]. Close to the sublimation
pump a cold finger for liquid nitrogen has been installed. These last two devices
were not part of the original set-up of [Buc01] and allow a further temporary
reduction in the pressure if needed. The design of two chambers connected by
a differential pumping stage allows the ion pump to maintain a pressure of less
than 1 × 10 −10 Torr in the experiment chamber.
As previous experiments on this set-up only used light close to 780 nm,
four windows were anti-reflection coated for the new used wavelength of 1030
nm.
6.3 The Diode Laser Systems for Magneto-
Optical Trapping
As both the experiments at Swinburne and Hannover work with the same
isotope, the laser systems employed in the experiments only differ in minor
details, mostly in the techniques to detune and stabilise the frequency of the
emitted light. The work of [M¨05, Len04] contain details about the laser sys-
tems. Each system is based on diode laser technology with a wavelength of 780
nm and labelled according to their use in the experiment. Unless noted, their
roles are equivalent to the systems described in chapter 4.2. Additionally there
is a set of two lasers which are used to decelerate the atomic beam so that the
velocity of the atoms is slow enough to be captured by the MOT [Ert85]. In
turn, they are labelled “Chirp lasers”.
157

6.3. The Diode Laser Systems for Magneto-Optical Trapping
6.3.1 The main laser
As in the experiment described in chapter 4 the main light source for this
experiment is a commercial tapered amplifier, type TA100 [E 25]. It shows
similar spectral characteristics, but is different in the specifics of the stabilisa-
tion and detuning schemes. The set-up up to the fibre coupling is depicted in
Fig. 6.2. The frequency of this laser is stabilised using saturated absorption
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PBS
ν=227 MHz
AOM
shutter
AOM ν=233 MHz
λ/4
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ν=121 MHz
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Figure 6.2: The main laser system: tapered amplifier, spectroscopy of the
master laser and path of the amplified light.
spectroscopy with a frequency modulated pump beam in conjunction with a
commercial lock-in amplifier [E 26] and a home-built PID regulator. The fre-
quency is modulated by an AOM [E 27]; the small modulation amplitude is
added to a fixed frequency of ν1 = 121 MHz. As only the light of the saturation
158
λ/2
PBS

Chapter 6: The All-Optical BEC Experiment: Apparatus
beam is detuned, but not the detected light, the frequencies of the two beams
now differ by 2 · ν1. The spectroscopy does not address atoms of velocity class
v = 0, but rather those that have a Doppler detuning of ν1. Overall, the out-
put light of the master laser is detuned by a frequency −ν1 with regard to the
lock point of the spectroscopy. This lock point is the Lamb dip of the 87 Rb D2
line which is created by the crossover resonance |F = 2〉 → |F ′ = 1, 2〉; here
a peak-lock technique is used. As this line is less energetic than the cooling
transition |F = 2〉 → |F ′ = 3〉 by 346 MHz, the overall detuning of the laser
light frequency to the cooling transition is 467 MHz.
This light is used to seed the tapered amplifier crystal of the TA100, which
is current and temperature controlled [E 28]. The amplified light is split at
a polarising beam splitter: one arm is used for the absorption imaging, the
other as the cooling light of the MOT. Each arm passes an AOM twice for fast
switching and detuning to the required frequencies, and a mechanical shutter
[E 10] to fully block the light from the experiment before the light of each arm
is coupled into a fibre to be transferred to the experimental chamber. The
light for cooling and trapping the atoms is detuned by the AOM [E 29], so
that the frequency of the light is 13 MHz below the resonance. This leads to
a final power of ≈ 50 mW available at the position of the MOT. The other
arm double-passes an AOM [E 30], so that this light is resonant with the
|F = 2〉 → |F ′ = 3〉 transition with a maximum power of slightly less than 200
µW available for imaging.
6.3.2 The repumping laser
To close the |F = 2〉 → |F ′ = 3〉 cooling cycle a home-built diode laser system
is used. This system is described fully in [Buc01]. It consists of a grating
stabilised diode laser in Littrow arrangement, which is temperature and current
159

6.3. The Diode Laser Systems for Magneto-Optical Trapping
stabilised by home-built electronics. The frequency is controlled by saturated
absorption spectroscopy similar to that described in the section above, locking
directly onto the |F = 1〉 → |F ′ = 2〉 transition. Only the saturation beam
double-passes an AOM which is driven at ν = 72 MHz, leading to an overall
detuning of the same magnitude. This light can be used to injection-lock a
second laser diode for higher power. One result of the experiments was that
for the loading of the dipole trap a minimal amount of this repumping light
is beneficial (see chapter 7). For some experiments a grey filter was used to
reduce this intensity; for others the light of the master laser has been used
directly without any amplification. A second AOM, also running at ν = 72
MHz, is used to tune the frequency of the light back to resonance and to
quickly switch the light. The light of the zeroth order is fed into a scanning
etalon to monitor the quality of the injection lock. A mechanical shutter in
the path can be used to completely block the repumping light.
6.3.3 The chirp lasers
The atoms which are finally trapped in the MOT originate from an oven at
several hundred Kelvin. The resulting beam of hot thermal atoms has to be
decelerated so that the slowed atoms can be captured by the MOT forces.
In this set-up this is done by chirp cooling the beam [Ert85]. This needs a
cooling and a repumping laser. The light sources here are home-built, grating
stabilised diode lasers, whose frequencies are stabilised by saturated absorption
spectroscopy on the required wavelengths. The cooling laser consists of a
master-slave system, where the light of the frequency stabilised master laser is
used to injection-lock a single laser diode. The repumping light comes from a
single stabilised diode laser. The chirp which detunes the lasers to keep them
in resonance with the slowed atoms is created by a ramp voltage which is
applied to the PZT of both the cooling master laser and the repumping laser.
160

Chapter 6: The All-Optical BEC Experiment: Apparatus
This chirp signal is then repeatedly applied to the lasers for a quasi-continuous
loading of the MOT.
6.4 The Dipole Trap
The aim of this experiment was to reach degeneracy of the bosonic atomic
ensemble by trapping in a purely optical trap without the help of any magnetic
fields. This requires light of sufficient power and detuning from the transition
to create a trap deep enough to trap many atoms and so far-detuned that
the heating rate of the atoms due to scattering of photons from the beam is
sufficiently small. While a Ti:Sa laser allows the creation of deep traps, the
small detuning leads to a high scattering rate and allows temperatures just
below 1 µK. CO2 lasers are available with high powers and wavelengths of
10.6 µm, and have been used for evaporation and creation of BECs [Bar01,
Cen03]. The long wavelength requires the use of special materials for the
optical elements, such as windows made of ZnSe. We use a Yb:YAG disk-laser
[E 31] with an output wavelength of 1030 nm as the light source. Its emission
wavelength is sufficiently far away for low scattering rates, the output power
is high enough for a sufficiently deep trap, and standard optical components
can be used. This laser was also chosen because its output is single mode and
single frequency. Inside its folded cavity, an etalon and a Lyot-filter are used as
frequency selective elements. These features are beneficial for the experiments
as the trap created by the light has a high stability which is an important factor
in reducing heating effects. The laser has a maximum output power of 50 W
(without frequency selective elements) and 25 W (in single mode operation).
To trap atoms, the focus of the laser beam has to be imaged into the atomic
cloud. The set-up of the optics to achieve this is shown in Fig. 6.3. The
emitted light first passes a combination of a half-wave plate and a polarising
161

6.5. Detection of Atoms
beam splitter for a rough intensity control. A telescope then reduces the beam
diameter by a factor of two. An AOM is used for fine control of the light
intensity, so that the power in the beam and thus the trap depth can be
ramped down in a controlled way. A lens of f = 200 mm focusses the beam.
Before this focus is imaged in the experiment chamber, the beam is split into
two by a polarising beam splitter. The lenses of f = 250 and f = 300 mm
each create an image of the focus. The two beams are aligned so that they
cross in their foci in the atomic cloud. The perpendicular polarisation of the
beams reduces interference effects in the crossed trap. The light in one of the
arms is then used to control and monitor the laser. This light is split and sent
to a photodiode and to a scanning etalon. The photodiode feeds its signal to a
PI-regulator, which in turn uses the AOM to stabilise the intensity, increasing
the long term stability of the intensity. The signal of the scanning etalon shows
whether the laser is emitting a single mode or several modes of light.
6.5 Detection of Atoms
At the top of the chamber, a CCD camera [E 32] is mounted. This camera
can be used for detecting the atoms either by their fluorescence or by their
absorption. In this experiment fluorescence detection was used only in the
early stages and for the MOT. All images of atoms trapped in the dipole trap
were taken using absorption imaging. The camera used in this experiment
has a pixel size of 9 × 9 µm 2 . The magnification of the imaging optics has
been determined to 1.06 ± 0.1, so that each pixel gathers light from an area of
(8.5 ± 0.8 µm) 2 .
162

PD
Chapter 6: The All-Optical BEC Experiment: Apparatus
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Figure 6.3: Set-up of the dipole trap. The focus of a f = 200 mm lens is
imaged into the experiment chamber by a f = 250 mm and a f = 300 mm
lens. The atomic beam enters the chamber from the right.
163

6.5. Detection of Atoms
164

Chapter 7
The All-Optical Bose-Einstein
Condensate Experiment:
Results
7.1 Overview
In this chapter the results from the all-optical BEC experiment located at
the Institut für Quantenoptik in Hannover, Germany, are presented. As the
author worked on the experiment as a non-resident researcher, these results also
appear in colleagues’ theses [M¨05, Geh05, Len04]. Apart from two exceptions
only original work for which the author actively contributed to the experiment
is included in this thesis. The exceptions are marked as citations in the text
and figures.
The typical experiment starts with the trapping of atoms in a magneto-
optical trap. This trap is loaded from a chirp cooled atomic beam, and it is
possible to load up to 10 9 87 Rb atoms at a temperature of as low as 35 µK. The
optical trap is created by two crossed beams of 8 W of power at a wavelength
165

7.2. The Magneto-Optical Trap
of 1030 nm and is loaded directly from the MOT, with both traps active during
the loading. After 30 ms loading time, we find a maximum of 1.5·10 5 atoms in
the dipole trap. Evaporation takes place during three linear ramps, in which
the trap depth is decreased by two orders of magnitude. Typical values before
the evaporation are 6 · 10 4 atoms at a phase space density of about 2 · 10 −4 .
After the evaporation ramp, we are left with ≈ 10 3 atoms and a phase space
density of about 0.2.
In the following sections, the results achieved so far will be described, start-
ing with the characterisation of the magneto-optical trap. This is then followed
by the description of the optical trap. Here the first section deals with the
loading and the different steps to optimise the loading process. The chapter
ends with the results from the evaporative cooling implemented so far. Un-
fortunately quantum degeneracy was not reached at the time this thesis was
written.
7.2 The Magneto-Optical Trap
The magneto-optical trap in the Hannover experiment is a six-beam MOT.
Three beams are separated before the experiment chamber and each is retrore-
flected onto itself after passing the chamber and a quarter-wave plate. The
atom source in this experiment is a chirp cooled atom-beam, where the actual
Rb-oven is located in a second chamber and the beam is guided through a
differential pumping stage. The quadrupole field is created by anti-Helmholtz
configured coils inside the vacuum chamber. The results presented here were
achieved during the author’s first stay in Hannover. The data was taken to-
gether with A. Lengwenus and our results also are part of his thesis [Len04],
where they are discussed in greater detail.
A part of the fluorescence from the captured atoms is captured by a photo-
166

Chapter 7: The All-Optical BEC Experiment: Results
diode, allowing an online measurement of the number of atoms. This was
used to measure the loading and loss characteristics of this trap. The loading
process is shown in Figure 7.1, (a). Use of equation (5.1) allows us to calculate
number of atoms
x 108
8
7
6
5
4
3
2
1
0
0 5 10 15 20 25 30 35 40 45
time / s
number of atoms
x 108
8
7
6
5
4
3
2
1
0
0 5 10 15 20 25 30 35 40 45
time / s
Figure 7.1: Loading and decay curve of the Hannover MOT. (a) Loading:
fitted to the data is equation (5.1), with the loading rate RL = 6.4 · 10 7 atoms
s −1 , and the loss γ = 0.08 s −1 ; (b) Decay: equation (5.2) has been fitted to
the data, with a loss rate of γ = 0.06 s −1 .
the loading rate RL = 6.4 · 10 7 atoms per second from the fit of the equation
to the data. The loss rate from the trap is fitted to γ = 0.08 s −1 . The highest
number of atoms that were trapped at this stage was more than 1.3·10 9 atoms.
The loading process can be turned off by shutting off the chirp lasers, so
the atoms in the beams are too fast to be captured. The atom number in the
trap then begins to decay according to equation (5.2). This is shown in Figure
7.1, (b). The loss coefficient here has been determined to be γ = 0.06 s −1 .
The lifetime of the MOT is the inverse of the loss parameter, τ = 14.3 s for
the mean of both values of γ. Compared to the MOT in Melbourne, this set-up
has a faster loading rate but a shorter lifetime. The first can be explained by
the more efficient loading from an atomic beam. The shorter lifetime probably
has its origin in the fact that these measurements were taken with a detuning
167

7.3. Loading the Dipole Trap
from the atomic resonance of about 6 MHz only. The atoms in the trap thus
are not cooled as much. The detuning was then changed to a higher value of
about 15 MHz when it came to load the atoms into the dipole trap.
Before the loading into the optical trapping potential, the atoms are cooled
by a phase of optical molasses. The fluorescence of the atoms was used in time
of flight measurements to determine the temperature of the atomic cloud.
Using equation (5.6), the temperature reached after this stage was measured
as T = 34 µK.
7.3 Loading the Dipole Trap
In this section the loading of atoms from the MOT into the optical dipole
potential is described. This work was done during the author’s second and
third stay in Hannover. The results have thus also been published in the
theses of the author’s colleagues at that time [Geh05, M¨05] in greater detail
and among their individually taken results.
To achieve quantum degeneracy by evaporative cooling in optical traps,
a high initial number of trapped atoms is even more important than it is
for the experiments with magnetic traps. This is because the trap widens
during the evaporation ramp, and thus the time for rethermalisation increases
with shallower traps. Indeed, the mean time between two elastic collisions
scales with the inverse of the product of the trap frequencies of each direction
[Geh05]. This need for a high initial atom number led to the optimisation of
several parameters. These results are presented here, where all data was taken
by absorption measurements.
For an isotropic confinement of the atoms, the trapping potential was cre-
ated by two crossed, red detuned laser beams. Unless noted, a power of 8W
per beam of wavelength 1030 nm was focussed down to a size of 60 µm. This
168

Chapter 7: The All-Optical BEC Experiment: Results
corresponds to a trap depth of about U = kB ·300 µK or less than 1/10th of the
depth of the magnetic trap in section 5. This trap was loaded directly from the
magneto-optical trap. This is shown in Figure 7.2, where the trap formed by a
single beam is loaded from the MOT. The absorption light’s frequency here is
Figure 7.2: Absorption image of a dipole trap and the MOT during the loading
stage for a single beam optical trap. The detuning of the absorption light is 4
MHz.
shifted by 4 MHz from the atomic resonance. The atoms in the MOT and the
dipole trapped atoms show their peak absorption at different laser frequencies.
This difference of 6 MHz is due to the ac-Stark shift that the energy levels of
the atoms in the optical trap experience which is absent for the atoms in the
MOT.
The lifetime in the crossed trap was measured for both hyperfine states of
the ground state, F = 1 and F = 2. The rate equation for the atom number
N takes into account constant losses α and losses that are linearly density
dependent, like two-body-collisions, β:
dN
dt
= −αN − βN 2
(7.1)
Solving this equation for N yields an exponential function in α for an initial
atom number of N0 (see equation (5.9)). This formula was fitted to the data
169

7.3. Loading the Dipole Trap
number of atoms
x 104
7
6
5
4
3
2
1
0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
time / ms
Figure 7.3: Lifetime measurements of the dipole traps for both ground states
F = 1 and F = 2.
of the lifetime measurements, and the results are shown in Figure 7.3. For
both ground states, the constant loss parameter α was negligible. The density
dependent loss was measured to be βF =1 = 4.6·10 −6 s −1 for the lower hyperfine
state and βF =2 = 3.8 · 10 −5 s −1 for the upper state. The parameter for the
state with higher energy is thus about one order of magnitude larger and these
atoms have a reduced lifetime in the trap. The reason for this is that the atoms
in the F = 1 state cannot undergo hyperfine-state changing collisions, while
the atoms in F = 2 can. These hyperfine-state changing collisions and photo
association of atoms to molecules are the main loss mechanisms in the dipole
trap. Scaled with the initial atom number, the density dependent losses of the
F = 1 state are comparable with the losses in the magnetic trap of section 5.3,
while N0βF =2 ≈ 2.5 s −1 is about one order of magnitude larger.
During the loading of the trap, which is under the influence of the near-
resonant light of the MOT lasers, light assisted collisions dominate the losses.
Here we again find collisions that change the hyperfine state of the atom, but
now losses due to radiative escape are by far the most dominating [Kup00]. In
this case, one of the two colliding atoms has to be in the upper ground state,
170
F = 1
F = 2

Chapter 7: The All-Optical BEC Experiment: Results
F = 2, as the detuning of the light for the MOT is too large to couple from
the lower state F = 1 to a molecular state during a collision. Thus, the rate
equation here will have a loss term β ∝ NF =2 · (NF =1 + NF =2). As it is the
role of the repumping laser of the MOT to close the cooling cycle by driving
the transition F = 1 → F ′ = 2 → F = 2 one can already see that during the
loading of atoms into the dipole trap a reduced power in the repumping light
is advisable as it minimizes β by minimizing NF =2. The detailed calculation
can be found in [Kup00].
The dynamics of the loading can be deduced from the loss equation by
adding a term to equation (7.1) to cover the transfer of atoms from the MOT.
dN
dt = −αN − βN 2 + R0 · e −γt
(7.2)
Here R0 is the initial loading rate from the maximum sized MOT. The MOT
size will decrease though as the settings of the cooling and repumping lasers for
an optimum transfer from the MOT to the dipole trap are different from the
settings for a standard MOT. One has to keep in mind that the parameters
α and β here are under the influence of the near-resonant light, and hence
β = 3.55 · 10 −4 s −1 is about two orders of magnitude larger than without the
light [M¨05]. The parameter γ = 10 s −1
can be determined by measuring the
lifetime of the MOT with the settings used for the transfer.
The number of atoms in the dipole trap was measured as a function of
loading time without the help of the author [M¨05]. The results are shown in
Figure 7.4. The loading parameter R0 is given by the initial gradient of the
loading curve and was measured to be R0 = 9.2·10 6 s −1 . The maximum number
of atoms here was Nmax = 1.4 · 10 5 atoms. This is largely consistent with
an approximated equilibrium value for N from equation (7.2). The density
independent losses α are negligible, and if one assumes the MOT to be constant
during the loading (γ ≡ 0) then we can approximate the maximum number of
171

7.3. Loading the Dipole Trap
number of atoms
x 104
15
10
5
0
0 100 200 300 400 500 600 700 800 900 1000
time / s
Figure 7.4: Loading curve of the dipole trap from the MOT: number of opti-
cally trapped atoms as a function of time. The markers are results from the
experiment, the solid line is the numerical fit of equation (7.2), taken from
[M¨05].
atoms Nmax,approx from equation (7.2) to
Nmax,approx = � R0/β ′ = 1.6 · 10 5
(7.3)
Also, from the graph, the optimum loading time can be deduced to be about
30 ms for our experimental setting.
The loading and loss parameters depend on the settings of the MOT lasers.
A higher detuning of the cooling laser leads to a cooler atomic cloud to load
from, increasing the number of atoms in the dipole trap. On the other hand,
if the detuning is chosen to be too high, then the atom number in the MOT
itself decreases and thus the loading characteristics of the optical trap decrease.
Figure 7.5 (a) shows the number of optically trapped atoms in a trap depth of
about 300 µK (a crossed trap with a waist of w0 = 45 µm and 3.5 W per beam)
as a function of the detuning of the MOT light. The effect of the detuning was
examined without the help of the author [Geh05].
The optimum detuning can be determined to be −50 MHz from the atomic
172

Chapter 7: The All-Optical BEC Experiment: Results
resonance. Increasing the light’s power in the arms of the trap will cause the
distribution to flatten up to the point where no influence of the detuning on the
atom number can be seen anymore for high power [Geh05]. The reason why
the optimum detuning here differs by about a factor of three from experiments
that use CO2 lasers to create the trap [Cen03] (the detuning there is about
−150 MHz) is that the energy levels are affected differently by the radiation’s
ac-Stark shift. In the case of the quasi-static trap with a radiation wavelength
of 10 µm all energy eigenvalues are shifted to lower energies. In the case of the
Rb D2 line the differential Stark shift there can detune the transition by about
60 MHz towards the red [Cen04]. In our case the upper energy level is shifted
further up while the lower state is even lowered. Here the transition energy
is increased, and the transition is blue detuned. If we take into account the
nominal detuning of the light towards the unperturbed atomic energy levels
and the shift of the energy levels, then the optimum detuning for CO2 traps
is about −90 MHz while it is between −70 and −80 MHz in our case. The
difference then is comparable.
The intensity of the MOT’s repumping light will directly affect the loading
by pumping atoms into a hyperfine state that is trapped with a shorter lifetime.
The number of atoms in the trap will thus increase for a lower intensity of this
laser light, although it has to be kept sufficiently high to keep the magneto-
optical trap working. Figure 7.5 (b) shows the number of atoms in the dipole
trap for different powers of repumping light. Here the frequency of the light
is fixed to the F = 1 → F ′ = 2 transition frequency of the atoms. One
can clearly see an optimum power at about 50 µW. This corresponds to an
intensity of about 30 µW/cm 2 and a factor of 10 less than the power that is
usually applied to the MOT.
173

7.4. Evaporative Cooling in the Dipole Trap
number of atoms
3.5
3
2.5
2
1.5
1
0.5
x 104
4
0
−80 −70 −60 −50 −40 −30 −20
detuning / MHz
x 104
6.5
6
5.5
5
4.5
4
10 0
3.5
10 1
power / µW
Figure 7.5: Number of atoms in the optical trap: (a) as a function of the
detuning of the MOT cooling laser, taken from [Geh05], (b) as a function of
the power in the MOT’s repumping light on a logarithmic scale.
7.4 Evaporative Cooling in the Dipole Trap
Evaporative cooling is possible in an optical trap, although it works differently
from the case of a magnetic trap. In the magnetic case, the trap remains
unchanged and the high energy atoms are pumped into untrapped mF -states
by RF-radiation. In optical traps, one has the possibility to reduce the trap
depth directly by reducing the intensity of the trapping light. As the beam
waist is constant regardless of the intensity in the beam, this results in a
change of the trapping gradient and curvature. This shallower trap increases
the rethermalisation times needed in the evaporation process. This fact has
to be taken into account when the form of the evaporation ramp is chosen.
Figure 7.6 shows how the the intensity in the trapping beams is decreased over
a ramp of four seconds in four steps. The first step is a 100 ms long phase
of free evaporation at the maximum power of 17 W trapping light (or a trap
depth of 390 µK). This is followed by three linear ramps to a final value of 110
mW or a trap depth of 2 µK in the lowest case. For laser powers of 70 mW or
less, it is expected that the optical potential is insufficient to overcome gravity
174
number of atoms
10 2
10 3

trapping light power / W
18
16
14
12
10
Chapter 7: The All-Optical BEC Experiment: Results
8
6
4
2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
time / s
Figure 7.6: The power in the trapping beams as a function of time: the evap-
oration ramp.
and trap atoms. The ramps were optimized by varying the ramp length for a
chosen final value of that ramp. The figure of merit was the atom number of
the atomic cloud. This number first increased with a longer ramp, as such a
ramp reduced the spilling [Geh05]. After an optimum ramping time, the atom
number decreased again due to the finite lifetime of the trap.
In our case, the best results were achieved with a focus of about 40 µm.
In this case, 6 · 10 4 atoms are loaded from the MOT into the optical trap.
This results in an initial phase space density of 3 · 2 · 10 −4 ; the factor of three
stems from the distribution over all three Zeeman sublevels. Assuming these
sublevels are populated evenly, the starting phase space density is reduced to
2 · 10 −4 .
The final laser power has been varied from 210 mW to 110 mW. At the
end of the ramp, absorption images have been taken to determine the atom
number and the temperature of the atomic cloud. To increase the signal to
noise ratio the images of three identical experiment runs have been summed
up before further evaluation. These sum images are shown in Figure 7.7.
The temperatures were determined using the time of flight technique. The
175

7.4. Evaporative Cooling in the Dipole Trap
213 mW 183 mW 170 mW
153 mW 143 mW 113 mW
Figure 7.7: Absorption images of the atomic cloud after the evaporation ramp
for different final values of the trapping power. Each image shown is the sum
over three absorption images after 10ms TOF.
results are depicted in Figure 7.8. Although a rather sudden decrease in the
temperature can be seen for trapping powers less than 170 mW, this is not
a signature of the phase transition to a BEC as it is not accompanied by a
sufficient increase in the atomic density: the density for the 210 mW trap is 6 ·
10 12 cm −3 and increases to 8.5·10 12 cm −3 for the 150 mW trap. The phase space
density in all images is of the order of 0.2 (assuming equal distribution over all
mF sublevels). Compared to the initial value of the PSD before evaporation
(2 · 10 −4 ) this is an increase by a factor of 1000, which is twice what would
be expected by the scaling laws for optical traps [O’H01]. Unfortunately, this
176

temperature / nK
700
600
500
400
300
200
100
Chapter 7: The All-Optical BEC Experiment: Results
0
100 120 140 160 180 200 220
final power / mW
Figure 7.8: Temperatures from the TOF measurements of the images in Fig.
7.7
factor can be explained by experimental errors, as this factor of two in phase
space density is equivalent to an error of 25% in the temperature measurement.
As we look at only a few thousand atoms in the clouds and as is obvious from
Fig. 7.8, this error is of the order of the experimental error. To this we can
add the error in the density measurement, as the focal size also can only be
measured with a 30% accuracy. Still, the evaporation works and only one
order of magnitude in phase space density has to be gained to reach quantum
degeneracy. This should be achievable by improving the parameters during
the loading and an optimised focal and thus trap size. Employing a so-called
“dark” MOT allows one to increase the density of atoms before loading them
into the dipole trap [Ket93, Tow96]. Changing the experimental set-up to
create such a MOT was considered but postponed until other enhancements
are implemented.
177

7.4. Evaporative Cooling in the Dipole Trap
178

Chapter 8
Summary and Outlook
8.1 Summary and Discussion
In this thesis, two atom optics experiments and their results have been pre-
sented. The far goals of these experiments are to set up atom interferometers
in microstructured potentials. Near goals were to achieve quantum degeneracy
of the trapped bosonic atoms. Unfortunately, the phase transition to BEC was
reached in only one of the experiments. Both experiments have in common
that they work with the same atomic species of 87 Rb, an isotope which has
become a standard in atom optics. Yet, they use different interactions to trap
the atoms.
The first experiment is the permanent magnetic chip experiment at Swin-
burne University of Technology. Here the trapping potential was created by
magnetic fields. In this experiment a new hybrid atom chip was used, in which
current-carrying structures and a magneto-optical film were combined. This
film was produced in the group, and consists of multiple layers, where layers
of the ferrimagnetic alloy TbGdFeCo alternate with layers of Chromium. It
allows the trapping of atoms in a micron sized potential and their cooling down
179

8.1. Summary and Discussion
to Bose-Einstein condensation. Not only is this chip new, the whole experi-
mental apparatus was first set up as part of this thesis. In chapter 4 the whole
apparatus, starting with the laser systems, including the vacuum system, and
ending with the above-mentioned chip, is described in detail.
The results of this experiment, from the initial collection of atoms in a
MOT up to quantum degeneracy of the atoms, can be found in chapter 5. In
the MOT, we trap 5·10 8 atoms at about 90 µK initially and cool to 40 µK after
compression to match the atomic cloud with the magnetic trapping potential.
After optically pumping all atoms into a magnetically trappable substate, we
collect 8 · 10 7 atoms at 50 µK in an initial Z-wire magnetic trap. This trap
is compressed, and then the atoms are cooled by forced evaporation using RF
radiation. This leads us to a BEC with up to 10 5 atoms. We were also able
to trap atoms without the current-carrying wires and with only the magneto-
optical film. Chapter 4 ends with a presentation of the early results of these
traps. The work on this experiment, together with the theoretical work on
the double well atom interferometer, finally led to a double well system in the
potential of the magnetic film using a BEC capable of measuring asymmetries
between the wells [Hal07b].
In the second experiment the trap’s potential was provided by spatially
intensity-dependent light, a dipole trap. As part of this thesis the existing
experimental set-up in Hannover was improved (see chapter 6). The number
of trapped atoms was improved for each step in the experiment to 10 9 atoms
during the MOT phase and 1.5 · 10 5 atoms in the dipole trap. A much better
transfer seems unrealistic, as the dipole trap’s radius is of the order of 10
µm and thus several orders of magnitude smaller than the MOT from where
the atoms are loaded. Cooling by evaporation of atoms from the trap was
demonstrated. With this the temperature of the atomic cloud could be reduced
from about 30 µK in the MOT to below 200 nK. Unfortunately, the atom
180

Chapter 8: Summary and Outlook
number could not be sufficiently increased to demonstrate beyond doubt that
Bose-Einstein condensation in this set-up was achieved, although an increase
in the phase space density by a factor of 1000 was shown. With this set-up,
only a further factor of 10 in the phase space density is required to reach
quantum degeneracy. The complete results of the dipole trap experiment can
be found in chapter 7.
In both experiments presented here atoms are trapped in micron-sized po-
tentials. With both kinds of trapping the splitting and merging of traps have
already been demonstrated [Cas00, H¨01c, Dum02b]. A beamsplitter and a
recombiner are the main components of optical interferometry; so with the
splitting and merging of the traps the two main requirements for double well
atom interferometers are available. Some theoretical models for such an inter-
ferometer exist, but these in general work with an overly optimistic assumption.
In chapter 3, this point is addressed. It is shown that for a single atom in a
double well, a two mode approximation and the Bloch equations are helpful to
understand the physics of a realistic double well interferometer. The process of
phase accumulation has been identified as Larmor precession. The two mode
Bloch model itself is well known. The results from this model are thus not
surprising, though they are counter-intuitive if one has worked with the old
and too optimistic models. The model presented here can easily be adapted to
further problems, for example, in some quantum computation schemes or in
the field of solid state physics. In addition, an easy to implement and robust
way to read out a double-well atom interferometer is presented.
8.2 Outlook and Future
The use of both magnetic and optical double well potentials for interferometry
with Bose-Einstein condensates has already be shown in principle [Shi04, Shi05,
181

8.2. Outlook and Future
Jo07]. Here one of the demonstrations using the magnetic field of an atom
chip [Jo07] examined increasing losses of the interference signal as a function
of the merging time, as has been treated theoretically for single atoms in this
thesis in part 3.3.2. The experimental data was compared with the results
of the model and showed remarkable agreement. This publication shows how
important a deeper understanding of the underlying physics is, as the authors
only speculate about the reasons for the reducing signal. To understand the
physics of a single atom in a double well is an important and crucial step to
understand the behaviour of a BEC in such a system. The other magnetic
trap experiment suffered from a non-repeatable phase relation between the
BECs in the two wells [Shi05]. As the double well of optical dipole traps has
comparable parameters [Shi04], the main difference seems to be the splitting
time, which is a factor of 40 higher in the magnetic trap. The failure of being
able to reproduce a phase relation between the wells can be attributed to the
localisation of the atoms. This number squeezing leads to a phase diffusion.
This magnetic version was implemented on an atom chip with two wires. Just
as the edge of a permanent magnet is equivalent to a single current-carrying
wire, a system of two current-carrying wires can be replaced by the two edges of
a slab of permanent magnetic material. The use of two slabs, equivalent to four
wires, allows a field with a guiding potential between the slabs without the need
for any external bias fields. One possible future application of our magneto-
optical film is thus in structures like single and double slabs. These can then be
used as waveguides. A double well created by a permanent magnetic film has
been demonstrated [Hal07b]. Although not an atom interferometric process,
the experiment allows the measurement of asymmetries between the wells. An
accuracy of δg
g = 10−4 for gravity fields has been inferred, and a read out
process that was based on adiabatic splitting similar to the one presented in
section 3.3.3 has been implemented. A more complicated structure of crossed
182

Chapter 8: Summary and Outlook
slabs has been proposed to create a 2D array of magnetic traps [Gha06]. This
has possible applications mainly in the field of quantum information.
Permanent magnets have many advantages over current-carrying wires.
The most important is the absence of thermal noise that is always found in the
current of a wire and leads to fluctuations in the magnetic field. These fluctu-
ations can cause decoherence and are unwanted for atom interferometry. Also,
it is possible to construct a vacuum chamber that does not need any electrical
connections. The permanent magnet can act as a wire, and the desorption of
atoms from the chamber surface by light can act as the atom source [And01].
This could create a device that should be robust enough to be taken into the
field. An atom interferometer using these techniques could well be taken to
measure gravitational gradients outside the laboratory.
All-optical Bose-Einstein condensates have already been created [Bar01,
Web03, Cen03, Kin05]. The most important up to date double well interfer-
ometer with a BEC works with a BEC that is trapped optically, although here
the BEC was created in a magnetic trap [Shi04]. When optical traps are used
to manipulate the atoms for interferometry or quantum computation, the cre-
ation of a BEC in situ in these traps would be a considerable simplification
of the experiment. Once a BEC is trapped optically, micro-optical devices
offer a plethora of possibilities. The BEC could be loaded into waveguides
[Dum02a, Kre04], as has been demonstrated already with cold atoms, to cre-
ate a spatial atom interferometer. Loading the BEC into an array of microtraps
leads to even more options. Each of these traps could be split and merged, so
that an array of atom interferometers could be created. A possible scheme has
been proposed in [Dud03]. Here a single measurement could include several
tens of interferometers with one experimental run, and thus reduce the number
of measurements needed to reduce statistical errors. Another possible applica-
tion of a BEC loaded into an array of traps is in quantum information. Using a
183

8.2. Outlook and Future
BEC can be beneficial when the loading of the array can be achieved in a way
similar to the transition between a BEC and a Mott insulator in an optical
lattice [Gre02a]. This would lead to a fixed and well defined number of atoms
in each trap and most importantly there would be the same number of atoms
in each trap. Proposals exist for using an array of optical traps for quantum
computation, including the localisation of a single atom in a double well as the
qubit [Mom03]. Also, many proposals for atoms in optical lattices suffer from
the drawback that the traps are not addressable. An array of microtraps can
overcome this problem.
184

Appendix A
Reprints of selected
Publications by the Candidate
• A. I. Sidorov, R. J. McLean, F. Scharnberg, D. S. Gough, T. J. Davis, B.
J. Sexton, G. I. Opat and P. Hannaford. Permanent-magnet microstructures
for atom optics. Act. Phys. Pol. B 33, 2137-2155 (2002).
• J. Y. Wang, S. Whitlock, F. Scharnberg, D. S. Gough, A. I. Sidorov,
R. J. McLean and P. Hannaford. Perpendicularly magnetized, grooved
GdTbFeCo microstructures for atom optics. J. Phys. D: Appl. Phys. 38,
4015-4020 (2005). Included as appendix A.3.
• B. V. Hall, S. Whitlock, F. Scharnberg, P. Hannaford and A.I. Sidorov.
Bose-Einstein condensates on a permament magnetic film atom chip. In:
Laser Spectroscopy; Proceedings of ICOLS 2005, E. A. Hinds, A. Ferguson
and E. Riis (Editors), page 275-282 (World Scientific, Singapore,
2005).
• B. V. Hall, S. Whitlock, F. Scharnberg, P. Hannaford and A.I. Sidorov.
A permanent magnetic film atom chip for Bose-Einstein condensation.
J. Phys. B: At. Mol. Opt. Phys. 39, 27-36 (2006). Included as appendix
A.2.
• A. I. Sidorov, B. J. Dalton, S. Whitlock and F. Scharnberg. The asymmetric
double-well potential for single-atom interferometry. Phys. Rev.
A 74, 023612 (1-9) (2006). Included as appendix A.1.
185

A.1. Asymmetric double-well potential for single-atom interferometry
A.1 Asymmetric double-well potential for singleatom
interferometry
SIDOROV et al. PHYSICAL REVIEW A 74, 023612 2006
PHYSICAL REVIEW A 74, 023612 2006
harmon ic well app ears with asepa ration bet wee n minima of
2 2 . For zero and for large x the symmet ric pot ential
app roximate s tha t for aqua ntu m harm onic oscillato r with
freque ncy 0 and mass m. Key results in the pape r would
still app ly if oth er suitable forms for the symmet ric pot ential
are used .
We den ote the eigenvectors ofH ˆ as i and the ir ene rgy
eigenvalues asEi, where i= 0,1,2,..., and Ei+1 Ei. The cor-
Sec. III tha t provide s an adeq uat e descripti on of the dynam -
ics of the splitting, holding, and recombi nation process es in
the pres ence of an asymmet ric component , and is the n used
Sec. IV in describing the interferometric process. The validity
of the two-mo de appr oach is explored in Sec. V
throug h the comparison of predictions of the Bloch vecto r
mode l with the results of direct nume rical simulations of the
time-depend ent multimod e Schrödinger equation .Adiscussion
of results follows in Sec. VI and include saschem e to
mea sure the popul ation of the excited state . Theoretical
details are dealt with in the Append ix.
Asymmetric double-well potential for single-atom interferometry
A. I. Sidorov, * B. J. Dalton, S. M. Whitlock, and F. Scharnbe rg †
ARC Centre of Excellence for Quantu m-Atom Optics
and Centre for Atom Optics and Ultrafast Spect roscopy, Swinburn e University of Technology , Melbou rne, Victoria 3122, Australia
Received 20 March 2006; published 18 August 2006
II. THEORETICALFRAME
In gene ral, the evolut ion of asingle ato m in an interferomete
r must be describe d using athree-dimensio nal quan tum
treatment . Howeve r, for asystem of cylindrical symmetryas
is prese nt in typical ato m chip experime ntsit is possible to
ignore excitations associate d with the two tightl y confined
dime nsions, as long as the dynamic s through out the proces s
is restricted to the dimension of weak confineme nt longitudinal
splitting . In this system it is pos sible to redu ce the
quantu m trea tmen t to tha t of aone-dime nsiona l problem .
We consider the one- dime nsiona l evolution of asingleato
m system due to atime-depend ent Hamiltonian Hˆ t that
can be written as the sum of asymmetric HamiltonianH ˆ
0 t
and an asymmetric potentialV ˆ as xˆ
Hˆ t = Hˆ 0 t + Vas ˆ xˆ
= pˆ 2
2 + V0 ˆ xˆ,t + Vˆ as xˆ , 1
ˆ
xˆ
V0 xˆ,t = 1 + t − 2 2 1/2
, 2
2
where aspecific form for the symmetric potentialV ˆ
0 is chosen
26 . The Hamiltonian and oth er physical quantit ies have
bee n writte n in dime nsionles s quantu m harmo nic oscillator
units associated with atomi c massm and angu lar frequency
0. With the original quan tities denote d by primes we have
xˆ = xˆ
, pˆ =
a0 a respo ndin g qua ntities for the symmet ric compo nen t of the
Hamiltonian, Hˆ 0,
will be den ote d Si and ESi. Both sets of
eigenvecto rs are orthonor mal, and all ene rgies and eigenvectors
are time depe nde nt.Hˆ 0 is symmet ric and the ground
state S0 is symmet ric and deno ted asS , while the first
excited stat e S1 is ant isymmet ric and den ote d asAS . Their
ene rgies are den ote dE S and EAS. The one -dimensional nat ure
of the system allows real eigenfunctions i x , Si x to be
chosen. In this case the geo met ric pha se27 is zero.
We can illustrate the gen eral beh avior of the lowest few
ene rgy eigenvalues E0,E1,E2,E3, ... , of the tota l Hamilton
ian H
0
pˆ ,
t = 0t , a0 = . 3
m 0
The dimensio nless Hamiltonians, potential s, and ene rgies are
obtaine d by dividing the original quantitie s by 0. Vas ˆ will
be taken as alinear function ofxˆ.
The symmetric potenti al depend s on a time-de pen den t
splitting paramete r , whose chan ge from zero to alarge
value and back to zero aga in convenientl y describ es the splitting,
holding, and recom binati on processe s with periods Ts, Th, and Tr, respec tively. For zero the symmetric pot ential
involves asingle quar tic well. When it is large adoub le
ˆ as the splitting parame ter is increased from zero to
a large value and back. At the begi nning and the end of the
process whe n 0 the ene rgy eigenvalues are well sep arated
. Here the effect of asymmetryV ˆ
as is small and the
eigenvalues and eigenvecto rs resemb le tho se for the symmet -
ric Hamiltonian. When the splitting paramet er increase s and
the trap ping pot ential change s to adou ble well, pairs of eigen
values E0 and E1, E2 and E3, etc. beco me very close. At
this stag e the qua ntu m system is very sensit ive to the presence
of Vˆ as which breaks the symmet ry, allows tran sitions
bet wee n 0 and 1 to occur, and causes the eigenvecto rs
0 and 1 as well as 2 and 3 to be localized in the
individual wells in this far split regime.
Initially, the ato m is prepare d in the lowest ene rgy eigenstate
0 of the single well. Transitions to excited state s are
supp ressed if the time scale for splitting and recombinatio n is
much longer tha n the inverse freque ncy gap bet wee n the relevant
state s. The ene rgy gap bet wee nE 0 and E2 is always
larger tha n the gap bet wee nE 0 and E1, and by choo sing
app ropriate time scales we can isolate the two lowest ene rgy
eigensta tes 0 and 1 from highe r excited state s 2 ,
3 , etc. , but still allow for tran sitions bet wee n the two
lowest ene rgy eigensta tes to occur. As aconsequenc e the
dyna mics of the DWAI can be treate d unde r the two-mode
app roximation, in which only the two lowest ene rgy eigenstate
s of the tota l HamiltonianHˆ and the symmet ric Hamilton
ian Hˆ 0 nee d to be considered . In this case the first excited
stat e probab ility a mea surable qua ntitycan vary from zero
to one .Aproposal for mea suring the excited stat e popu lation
is out lined in Sec. VI.
Using the two-mod e app roximation expression s for the
lowest two ene rgy eigenvalues E0,E1 and eigenvecto rs
0 , 1 for the Hamiltonian Hˆ will be obt ained .Astan -
dard mat rix mech anics app roach will be used , but instea d of
using the symmet ric pot ential ene rgy eigenvecto rs S , AS
as basis vectors, we use the left, rightL-R basis vectors L ,
R , which are defined by
023612-2
We conside r the evolutio n of asingle-ato m wave function in atime-depen den t dou ble-well interferomete r in
the presence of aspatially asymmet ric pot ential. We examine acase where asingle trapp ing pot ent ial is split
into an asymmet ric dou ble well and the n recomb ined again. The interferomete r involves ameas urem ent of the
first excited stat e pop ulation as asensitive mea sure of the asymm etric pot ent ial. Based on atwo-mode approximation
aBloch vecto r mode l provides asimple and satisfactory des cription of the dynami cal evolution.
We discuss the roles of adiabaticity and asymmet ry in the doubl e-well interferomete r. The Bloch mode l allows
us to accoun t for the effects of asymmet ry on the excited stat e pop ulation throug hou t the interferometric
process and to choos e the app ropriate splitting, holding, and recombin ation periods in order to maximize the
outpu t signal. We also compare the out come s of the Bloch vector mode l with the results of num erical
simulation s of the multistat e time-dep enden t Schrödinger equatio n.
DOI: 10.1103/PhysRevA.74.023612 PACS num ber s : 39.20. q, 03.75.Dg, 03.75.Be
trap ping pot ential. An ato m is initially prepared in the
grou nd stat e of asingle symmet ric trap ping pot ential, which
is the n split into asymmet ric doub le well. Aspatially asymmet
ric pot ent ial is the n app lied and anona diabatic evolution
leads to tran sitions bet wee n grou nd and excited state s. The
asymmetry is the n switched off and the doubl e well is recomb
ined into the original pot ential. The popul ation of the
excited stat e mea sures the effect of the asymmetric pot ential.
The DWAI can be considered as a qua ntu m-stat e Mach-
Zehn der interferomet er where the evolution of the qua ntu m
stat e via the two sepa rate d wells is analo gou s to the propagat
ion of an opt ical field via two pat hways.
However, thes e treatm ent s ignore the effect of asymmet ry
during the splitting and merging stag es. In reality asymmetry
is always present and could be the result of imperfect horizontal
splitting introducing agravity-based asymmetry, extern
al spatially variable magn etic and electric fields or different
left and right trap freque ncies. We show tha t the presence
of small asymmet ries has dram atic consequ ences on the interferom
etric process. We have produce dasimple mode l in
term s of aBloch vector evolution tha t ena bles us to consider
a splitting-ho lding-merging sequ ence involving a doub le
well and takes into accoun t the presence of asymmetry at all
stag es. The two key paramet ers are the ene rgy gap bet wee n
the lowest two state s of the symmet ric comp one nt of the
trap ping pot ential and an asymmetry parame ter, which is related
to mat rix elem ent s of the asymmet ric compo nen t of the
pot ential. Nona diabat ic evolution only occurs during the
splitting and recombining stag es whe n the torq ue vector
change s much more rapidly compa red to the Larmor precession
of the Bloch vector. It is impo rtan t tha t the torq ue vector
rema ins constan t during the holding stag e, and tha t this period
is long compa red to the splitting and recombin ation
times. In this case the final excited stat e popu lation is asinusoidal
function of the holding time, with aperiod det ermined
via the asymmetry paramet er.
In this pap er we consider the dyna mics of asingle ato m in
an asymmetric DWAI, with the basic theo retical framew ork
being out lined in Sec. II. Using atwo-mod e app roximation
we develop aBloch vector mode l for time-depen den t DWAI
I. INTRODUCTION
The evolution of aquantu m system in adou ble-well potenti
al has bee n the sub ject of num erou s theore tical studi es.
These include trea tment s of Joseph son-like oscillations1,2 ,
dynami c splitting 3 , and interference 4 of Bose-Einstein
condensate s BECs . The interpreta tion of thes e effects is
base d on the app roach5 , in which interference patte rns are
seen to evolve as aresult of successive boso n mea sureme nts
which do not ident ify the originating cond ensate . The product
ion of cold atom s and BEC in microtraps on ato m chips
6–8 and in micro-opti cal system s 9 has stimulate dagrea t
interes t towa rds novel implementa tions of ato m interferomete
rs 10–15 tha t are base d on the use of dou ble-well potentials.
Double-well ato m interferome ters DWAI of bot h the
single-atom and the BEC varieties are well suited to implemen
tat ion on an ato m chip. Here microfabricate d structures
allow us aprecise control on asubmi cron scale over the
splitting and merging processes . The key processe s of splitting
16–18 and merging 19 of cold ato mic clouds and
even interfe rence of aBEC after splitting in adoubl e well
20–22 have bee n already demo nstrat ed. Althoug h the
impleme ntati on of a DWAI using a BEC can lead to a
N-fold enhance men t in precision mea sureme nts23 , pha se
diffusion 24 associate d with mea n field effects is of concern
20 . Double-well interfe rometry with asingle ato m can
allow us alonge r mea surem ent time and in this regar d has a
potentia l advanta ge. An on-chip single-atom interferome ter
can be integra ted with the source of ato ms in a ground
state —the Bose-Einstein condens ate —and be used for
sensitive mea surem ent s of gravitationa l fields. DWAI may
also be app lied to mea sure collisiona l phas e shifts induc ed
by the ato m-ato m interaction, which is useful for qua ntum
compu tatio n processe s25 .
Two proposed scheme s of asingle-atom DWAI involve
time-depend ent transverse10 and axial 11 splittings of a
186
*Email addres s: asidorov@swin.edu.au
†
Also at IQO, University of Hannov er.
1050-2947/2006/742 /023612 9 023612-1
©2006 The American Physical Society

SIDOROV et al. PHYSICAL REVIEW A 74, 023612 2006
ASYMMETRIC DOUBLE-WELL POTENTIAL FOR ? PHYSICAL REVIEW A 74, 023612 2006
Chapter A: Reprints of selected Publications by the Candidate
t = C L L + CR R . 12
1
E0 = 2 E 1
S + EAS − 2 ,
20
1
0.8
Our interferomet er will be described in term s of the Bloch
vector and its dynamics determ ined via Bloch equat ions. We
now introduce Pauli spin ope rato rs and the Bloch vector.
Time-depen dent Pauli spin ope rato rs ˆ
a a=x,y,z are defined
in the Schrödinger picture
ˆ
x = R L + L R ,
0.7
1
E1 = 2 E 1
S + EAS + 2 , 8
15
0.5
0.6
as
where the quantit y gives the ene rgy gap for the total
Hamiltonian Hˆ and is defined by
V
0.5
,
)
0
x
R L − L R ,
ˆ
1
y =
i
= 0 2 2
+ Vas = E1 − E0. 9
In term s of the laser-driven two-level ato m analogy, would
be ana logou s to the gene ralized Rabi frequency.
The ortho norma l ene rgy eigenve ctors for the tota l Hamilton
ian Hˆ are given by
(
10
0
ψ
0.4
V
,
0.3
5
-0.5
0.2
(a)
0.1
ˆ
z = R R − L L . 13
From its mat rix represen tat ion in theL-R basis 5 , the
dimensionless Hamiltonian ope rato rHˆ in the Schrödinger
picture can be expressed in term s of the Pauli spin ope rato rs
as
0
-6 -4 -2 0 2 4 6
-1
0 2 4 6 8 10 12
splitting parameter
0
R ,
1 − V
2
L +
1 + V
2
0 =
x
20
1
R , 10
1 + V
2
L −
1 − V
2
1 =
FIG. 1. Energy difference between ground and first excited
states solid line for Vˆ as=
0.02xˆ as afunction of the splitting paramete
r . Dotted line—energy difference 0 for symmetric Hamiltonian,
dashe d line—asymmetry quantityV as.
15
0.5
Hˆ = 1
2 01ˆ + ˆ
x x + ˆ
y y + ˆ
z z , 14
where the effect of asymmet ry is now represente d by the
variable
)
whe re
x
(
S + AS ,
V
10
0
ψ
L = 1
2
0 = E S + EAS ,
V = V as . 11
x = − 0, y = 0, z = Vas. 15
It is conve nient to introduce aso-called torque vector, defined
as = x, y, z .
The Bloch vector is defined to have comp one nts which
are the expe ctation values of the Pauli spin ope ratoˆrsa in
the qua ntu m state . These compon ent s will be deno ted as
a. Hence in the Schrödinger picture
a = t a
ˆ t t a = x,y,z . 16
The Bloch vector is defined as = x, y, z . The Bloch
comp one nts are bilinear functions of the amp litudes CL and
CR. The evolution of the DWAI system is now described by a
set of real variables x, y, z and each of thes e variables has
a certain physical mea ning. The compon ent z is amea sure
of the imbalance of the atom ic popu lation of the localized
state s L , R . The compon ent x is amea sure of the atom ic
pop ulation imbalance betwee n the delocalized stateS s ,
AS , as can be seen if the qua ntu m stat e is expan ded in the
symmet ric basis. For x= + 1 all the popul ation is in the symmet
ric state S , for x= −1 it is all in the antisymme tric state
AS . It is thu samea sure of the excitation of the first excited
stat e in the unsplit trap regime.
Equa tions for the compon ent s of the Bloch vector can be
obtained from Heisenberg equati ons for the Pauli spin ope rators.
The derivation must take into accoun t the present situation
where the Pauli spin ope rato rs are explicitly time depen
dent since the basis vectorsL , R chang e with time.
This differs from the stan dard situation of time indepe ndent
basis vectors 28,29 . However, the add itional term in the
Heisenberg equati ons can be shown to cont ribute zero to the
Bloch equ ations due to the two eigenfunctions in the sym-
On substituting for L , R the eigenve ctors for the total
Hamiltonian can be related to thos e for the symmetric
Hamiltonian Hˆ 0.
At the beginn ing and end of the interferom -
ete r process we find tha t the asymmet ry parame ter Vas is
small comp ared to the symmetric ene rgy gap0. For V 1
the eigenve ctors 0 , 1 become similar to S and AS ,
respe ctively. For Vas 0 V 1 , the eigenve ctors 0 ,
1 become equa l to L , R , respe ctively, the localized
eigenve ctors for the separa te wells.
The behavior of the quantit ies , Vas, and 0 as the splitting
paramet er is chan ged is shown in Fig.1 for the case
where the asymmetric potenti alV as x varies linearly with
the coordinatex, spe cifically withˆ Vas = 0.02xˆ. The symmetric
ene rgy gap 0 become s close to zero for 4 and the n the
actua l ene rgy gap is appr oximately given byV as. The energy
eigenfun ctions 0 x and 1 x for different splitting
paramete rs are depicted in Fig.2, again with Vˆ as=
0.02xˆ.
The beha vior of the potenti alV x =V0 x +Vas x is also
shown . For small Fig. 2 a the potentia l is asingle well
and the eige nfunctions are app roximately symmetric and antisymmet
ric. For larger Fig. 2 c the potent ial is adouble
well, which still appea rs to be symmetric. However, even
with asmall asymmetry in the potenti al the eigenfun ctions
are no longer symme tric and antisymme tric, but instea d are
each localized in separa te wells. This sensitivity of the
eigenfun ctions to asmall asymmetry is critical to the perfo rman
ce of the presen t interfe romete r.
5
-0.5
187
(b)
0
-6 -4 -2 0 2 4 6
-1
x
20
1
R = 1
S − AS . 4
2
The stat es L , R are orthonorm al and for large correspon
d to an atom localized in the left or right well, respectively.
However, even for a single well theL-R basis is still
applicable. The matrix for the HamiltonianHˆ in the L-R
basis is given by
15
0.5
, 5
− Vas − 0
− 0 + Vas
1
+
2
1 0
0 1
Hˆ L−R = 1
2 ES + EAS )
x
(
10
0
ψ
where the order of the columns and rows isL,R and we
define the convenient real quantities
V
5
-0.5
0 = E AS − ES, 6
(c)
Vas = R Vas ˆ R − Lˆ Vas L
0
-1
-5 0 5
x
FIG. 2. Stat ionary eigenfunctions of the ground statedasheddotted
line and the first excited state dotted line for different
values of the splitting paramete r 1 a , 2.5 b , and 5 c . The
potentialV x is shown as the solid line.
II. BLOCH VECTOR MODEL
We can expres sagen eral time-depen dent normalized stat e
vector as aquantu m sup erposition of the state sL and
R
Hamiltonian transition frequency 0 is analogous to the
Rabi frequency, while the quantityV as is analogous to
the detuning.
The energy eigenvalues for the total HamiltonianH ˆ are
obtained from the dete rminental equation as the eigenvalues
of the matrix Hˆ L−R , and are given by
=− S Vˆ as AS + AS Vas ˆ S . 7
The derivation of the Hamiltonian matrix uses the
symmetry prop erties of S , AS , and the reality of the
eigenfunctions. The total energy for the symmetric Hamiltonian
is given by ES+EAS, and the energy gap is given
by 0. The quantity Vas describes the asymmetry of the
system, and would be zero if the Hamiltonian was symmetric.
The second equation relatesV as to off-diagon al elements
of the asymmetric contribution to the Hamiltonian, indicating
its role in causing transitions between the eigenstate s
S , AS of the symmetric Hamiltonian. The Hamiltonian
matrix 5 is analogous to tha t for a two-level atom interacting
with amon ochromatic light field28 . The symmetric
023612-4
023612-3

A.1. Asymmetric double-well potential for single-atom interferometry
SIDOROV et al. PHYSICAL REVIEW A 74, 023612 2006
1
0.8
0.6
0.4
0.2
0
- 0.2
- 0.4
- 0.6
(a)
- 0.8
-1
0 10 20 30 40 50 60
time
1
0.8
ASYMMETRIC DOUBLE-WELL POTENTIAL FOR ? PHYSICAL REVIEW A 74, 023612 2006
σ x , σ y , σ z
0.6
0.4
(b)
0.2
β / 12.
5,
V,
P
1
0 10 20 30 40 50 60
time
0
FIG. 4. a Time evolution of the Bloch vector componen ts x
solid line , y dashed line, and z dotted line for Ts=Th=Tr = 20 and max= 12.5; b time evolution of the first excited state
popu lation P1 dotted line, the asymmetry paramete rV dashed
line , and the splitting paramete r / m ax solid line .
the n chang ed linearly to zero during the recombination
period. The dynamical beha vior of the Bloch vector
comp onent s is shown in Fig.4 a along with the time
depend ence of the asymmetry paramete rV=V as/ , the
aligned with the x axis. For small values of the splitting
paramete r the abso lute value of the torqu e vector is mainly
dete rmined by the symmetric ene rgy gap0 Fig. 1 and its
direction remains along the −x direction. The pos ition of the
Bloch vector remains mostly along the +x direction Fig.
3 b during early stag es of the splitting process. When the
splitting paramete r is increased furthe r the decreasing ene rgy
gap 0 becomes comparable with and later much smaller
tha n the asymmetry quantityV as. As a result the torque vector
rotate s in ax-z plane until it is aligned along thez direction
0,0,V as . It is impor tan t to make this chan ge
nonadia batically so tha t the Bloch vector does not follow the
torqu e rotation. If the Bloch vector were to follow the
change s of the torqu e vector adiabatically the atom will always
stay in the ground stat e and no interference would be
observed.
During the holding stag e the torque vector is constan t and
the Bloch vector precesses around the torqu e vector with a
constan t ang ular velocityV as, and hen ce both thex and y
compo nent s oscillate with aperiod 2/V as Fig. 3 c . In an
ideal doub le-well interferomete r the splitting and recomb ination
stage s are short and the value of thex compo nen t does
not chang e much during the se stage s, so thatx T which
defines the final excited stat e popu lationis basically given
by its value at the end of the holding period. The simple
beha vior during the holding period indicate s tha t the excited
stat e popu lation would have aperiod 2/V as considered as a
function of holding time. A similar description in terms of
the evolution of a Bloch vector also app lies to the scheme
described in Ref. 11 , though the dynamical beha vior of the
Bloch vector is different .
The beha vior of the interferomete r may also be described
in terms of time-dependen t state sL , R , which during the
holding period represent atoms localized in the left and right
wells. The interferomete r process involves the transition
S 0 ? AS T , which involves two pathw ays S 0
? L T/ 2 ? AS T and S 0 ? R T/ 2 ? AS T , involving
two poss ible localized intermediate stat es associated
with the left or right wells. The overall transition amp litude
is the sum of amplitudes for the two pathw ays, and depend -
z
(a)
metric basis bein g real and having opposit e symmetrysee
Appendi x . The Bloch equ ation s are given by dt d/dt
Ω
d t x = − Vas y,
d t y = Vas x + 0 z,
d t z = − 0 y, 17
y
σ
and can be solved nume rically using the Runge -Kutt a
algorithm . In vecto r nota tion the Bloch equ ation s are
x
d t = . 18
This form of the Bloch equation s is adirect conseq uence
of the equiva lence of the two-mod e dou ble-well interferom -
ete r to aspin 1
2 system . The Bloch vecto r precesse s at the
Larmo r frequenc y arou nd the torq ue vecto r, which in detail
is
z
(b)
= − 0,0,Vas . 19
If ther e is no asymmetr y, thex compo nen t of the Bloch
vecto r rema ins unch ange d, while its compo nen t in the y-z
plane just rotate s abou t thex axis Fig. 3 .
Ω
188
y
σ
x
splitting paramete r and the excited stat e popu lationP 1
Fig. 4 b . The paramete rs used areV ˆ
as=
0.02xˆ, max= 12.5,
and Ts= 20, Th= 20, Tr= 20 in dimensionless harmonic
oscillator units. Here we observe comp lex oscillatory
beha vior for the x and y comp onent s of the Bloch vector
which occurs during the splitting and merging stag es. During
the holding stag e the y exhibit simp le periodic variations
ing on the relative phase betwee n the se amplitudes eithe r
with frequencyV as= 0.2. At the same time thez comp onent
constructive or destructive interference may occur. For maxi- develops a small neg ative value during splitting and
mum contrast it is desirable tha t the magnitudes of the two increases the abso lute value even furthe r during merging.
partial amp litudes be equal, so tha t during the holding period The x comp onen t reaches a neg ative value of −0.9 at the
the popu lations of the left and right well stat es should be end of the process. This corresponds to an excited state
abou t the same. After optimal splitting thez compo nen t of popu lation of 0.95 and represent s a case of constructive
the Bloch vector z should be kept close to zero during the interference.
holding period, however a phase difference between the lo- By monitoring the beha vior ofP 1 during the interferometcalized
state s accumu lates. Only at the end of the recomb iric process we can see whe n nonad iabatic evolution occurs.
nation stag e this phase is translated into the popu lation of the The pop ulationP1 change s from 0to 0.47 Fig. 4 b at the
excited state .
beginning of the splitting process and does not reach the
optimal value
V. RESULTS OF NUMERI CALSIMULATIONS
We studied the evolution of a Bloch vector and the
popu lation of the excited stat e by solving Eqs.17 numerically.
The splitting paramete r is chan ged linearly from zero
up to a maximum max during the splitting period. It is then
held constan t at max during the holding period, and
1
2 as a result of the nonzeroz comp onen t of the
Bloch vector. The variable P1 does not chang e during the
adiabatic precession of the Bloch vector around the torque
vector during the rest of the splitting, holding, and the begin -
ning of recomb ining stag es. It again exhibits drastic change s
in a short period during the remerging whe n the torque vector
rotate s rapidly and the Larmor frequency is relatively
small.
z Ω
(c)
IV. MODELOFA SINGLE-ATOM
DOUBLE-WELATOM INTERFEROMETER
In the single-ato m interferome ter unde r considerat ion the
ato m is always locate d in atrappin g pot ential, which changes
from a single well to adoubl e well—which, in gene ral, is
slightly asymmetric—and back again to the original single
well. The interferom ete r is used to mea sure the effects of this
asymmetr y, the cause of which may be of mea surable interest
e.g., as in agravity gradio mete r. The ato m is initially in the
groun d state 0 0 of the original unsplit pote ntial, and as
Vas is the n small compared to 0, 0 0 is the n appr oximatel
y the same asS 0 . The popul ation of the excited stat e
at the end of the recombinati on process is the mea surable
interferome ter out put . The probabil ityP 1 of findin g the atom
in the uppe r ene rgy eigenstat e at any time is given by
y
σ
x
2 P1 = 1 , 20
and this will rema in zero unless an asymmetr y is present
toget her with suitabl y short splitting and recom bining stages
for the interferom ete r process—so tha t transition s occur
bet ween 0 and 1 due to the presenc e ofVas ˆ .
We find that
· . 21
1
+
2
1
P1 =
2 1 + 2 1
zV − x 1 − V =
2
FIG. 3. Evolution of the Bloch vector and the torque vector
at differen t mome nts of the splitting stag e:a at the beg inning
when 0 Vas and − 0,0,0 , b when 0=Vas, and c ,
when Vas 0 and 0,0,Vas .
recombining stag es. At the start of the process the Bloch and
torq ue vectors are antiparallelFig. 3 a and app roximately
At the beg inning and end of the interferomete r proce ss
V 1 and henc e the probabil ityP 1 only dep end s on thex
compo nen t of the Bloch vecto r. The probabilityP 1 T thus
depend s on how this componen t has change d from its initial
value of 1. We can, there fore, describ e the dynamical behav -
ior of the single-ato m interferome ter in term s of the evolution
of the Bloch vector during the splitting, holding, and
023612-6
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SIDOROV et al. PHYSICAL REVIEW A 74, 023612 2006
ASYMMETRIC DOUBLE-WELL POTENTIAL FOR ? PHYSICAL REVIEW A 74, 023612 2006
Chapter A: Reprints of selected Publications by the Candidate
The equat ion of motion for the Bloch vector compone nts
can be derived using the Heisen be rg picture via
ˆ H
a 0 , A1
d
dt
tion betwee n out come s of two mod els. For large values of
the asymmetr y frequencyV as it is impossible to adiabat ically
isolate two lower state s and the transition s to highe r stat es
have to be taken into account.
0.9
1
(a)
1
(a)
d
a = 0
dt
0.8
0.7
0.5
whe re the Heisen be rg equati on of mot ion for the Pauli spin
operat or in dimensionless units is
VI. DISCUSION AND CONCLUSIONS
(c)
0.6
0.5
F
population P1
d
ˆ H
a = − i ˆ H
a , Hˆ H + ˆ
a
dt
t
H
. A2
The derivation involves the use of the commu tation rules
for the Pauli spin operat ors. For the first term , we have after
substi tuting forHˆ from Eq. 14
b a
ˆ H
, ˆ
b
0 a
ˆ ,1ˆ +
b=x,y,z
− i
2
− i a
ˆ H , Hˆ H =
= ˆ H a. A3
We have app lied aBloch vecto r mode l to des cribe the
quantum -stat e interference of asingle-ato m wave function in
a time-variable asymmetric double -well pote ntial. The probability
of findin g the ato m in the first excited stat e is closely
associate d with the mag nitude of the spatially dependen t externa
l pote ntial and could be used as amea sure of the app lied
asymmetr y. Transition s betwe en groun d and excited state s
occur during the splitting and recom binati on stages . Larmo r
precess ion of the Bloch vecto r during the holdin g stag e is
induc ed by the asymmet ry, will effect an interfe rome tric
phase , and dete rmine the final value of the excited state
population. The evolut ion of the Bloch vecto r during the
splitting and merging stag es is also importan t bec ause it will
(b)
0.4
0.3
0 10 20 30 40
Th
0
(d)
0.2
0.1
0
10 3
T s
10 2
10 1
10 0
(b)
1
FIG. 6. Depen den ce of the filling factorF on the duration of the
splitting stag eTs for max= 12.5 and various values of asymmet ry
Vˆ as=
0.01xˆ a , 0.02xˆ b , 0.05xˆ c , and 0.1xˆ d .
0.5
population P1
Hence the cont ribution from the first term in Eq.A2 is
given by
ˆ H
a 0 1 = a. A4
0 d
dt
For the contribution from the second term in Eq.A2 , we
may first write
are time independ ent
a a
KAB A B , where the KAB
ˆ
a=
A,B=L,R
coefficients tha t can be read from Eqs.13 , and then
H
.
A B + A
a
KAB = A,B=L,R
t B
t
ˆ
a
H
t
A5
Using Eq. 12 for the stat e vector and reverting to the
Schrödinger picture we have
0 t a
ˆ H a
0 2 = t KAB t A B + A
A,B=L,R
S
t B t
affect the mea surable probabilityP 1. We have also shown
tha t special requ irement s app ly to the durat ion of the splitting
and merging stage s in orde r to avoid excitation of high er
mode s for short times and partial adiabatic following if the
splitting is too long. Both thes e effects lead to adecre ase of
the mea sured signal. Interest ingly enoug h the y do not affect
the contras t of the interfe rence fringes if the first excited state
is not initially populate d.
Adiaba tic evolut ion of the Bloch vecto r can offer anew
way to mea sure the first excited stat e population after the
doubl e-well interferome tric process. We have already men -
tione d tha t in the far-split regime the excited stat e wave function
doe s not overlap with the groun d stat e wave function
and will predomina ntly occupy the high er ene rgy wellFig.
2 c . If at the end of the nonadia batic splitting, phase evolution,
and non adiabatic recom binati on process we also add
an addit ional stag e of adiabatic splitting in aknown asymmetrical
potenti al, the n the wavefunctions of the two state s
will be spatially separated. For recording the out putP 1 we
now simply mea sure the population of the high er ene rgy
well. To shorte n the adiabatic evolution time we nee d to
app ly the highest available asymmetryFig. 6 .
signal and aredu ced maximum popu lation of the first excited
state .
For splitting and merging timesTs=Tr= 20 Fig. 5 b
bot h mode ls show excellent agreemen t indicating asimple
sinusoidal variation of the first excited stat e popu lation with
holding time. This simple beh avior is also obs erved for long
splitting time Fig. 5 c , but with significantly redu ced amplitud
e of the oscillations. The redu ced fringing is att ributed
to the onset of adiab atic following of the Bloch vector during
splitting and recombinatio n which is shown by the presence
of anon zero z comp one nt Fig. 4 a . We not ed tha t our
num erical solution of the Bloch equa tion is robust with regard
to the variations of the signal but is fragile rega rding the
phas e. The error was accumulat ed during the splitting stag e
as aresult ofVas 0 in amerged trap and will scale with the
splitting time.
In the asymmetric doub le-well pot ent ial the ground state
eigenfunction will predom inantly occupy the lower well, and
the excited stat e eigenfunction will be localized in the upp er
well Fig. 2 c . In the slow splitting regime the onset of the
adiab atic evolution will lead to the unbalan ced distribution
of the ato mic wave function bet wee n the wells, which in turn
leads to aredu ction in the mea sured signal. In app lication to
interferomet ry it can be seen as intrinsic which-way information
whe n the ato m predom inantly follows one pat h after the
0 10 20 30 T 40
h
0
1
(c)
0.5
population P1
189
0 10 20 30 T 40
h
0
FIG. 5. Dependenc e of the first excited stat e popu lationP 1 at the
end of the interferometri c process on the duratio n of the holdin g
stage Th for various dura tions of the splitting and recombining
stage sTs=Tr= 5 a , 20 b , and 200 c . Results of the Bloch vector
mod el are represente d by solid lines and out come s of full num erical
simulation s are presented by circles.
a *
= KAB CDCB
D t A
A,B,D=L,R
*
+ CACD t B D . A6
To evaluate this result, a consideration of the four qua ntities
A t B , where A,B=L,R is requ ired, not ing also
tha t t B A = A t B * . These four qua ntities can be
expressed in term s of related mat rix elem ent s in the symmet -
ric basis Si t Sj , where i, j=0,1 . Note S0 S and
S1 AS .
Using the normalization and reality property, we first
show tha t the diagona l term sSi t Si are zero. For the
off-diagona l term s Si t Sj , the se are zero becau se Si
and t Sj have opp osite symmet ry. From the se considerations
we find tha t all mat rix elemen tsA t B , where
splitting. In gen eral,
ACKNOWLEDGMENTS
= a 0 + b 1 + i , 22
It is tempti ng to limit the evolution of the Bloch vecto r
and the relevan t phas e accum ulation during the splitting
and merging stage s by making thes e stage s shorte r. However,
this can lead to excitation s of highe r excited stat es. We
have compared out come s of the Bloch vecto r mode l with
the results of the numeri cal solution of a multist ate
Schrödinge r equatio n using theXMDS code 30 . The
behavi or of the excited stat e popula tionP 1 T at the end
We than k T.D. Kieu for fruitful discussions, and T.
Vaug han and P. Drummon d for the introdu ction to the
XMDS code . This work has bee n supp orte d by the ARC
Centr e of Excellence for Quant um-Atom Optics.
whe re i is alinear combination of all oth er excited state s.
We define afilling factor
F = 2ab, 23
APPENDIX: DERIVATION OF BLOCH EQUATIONS
The state vectors at timet and at time 0 are related via the
unitary evolution operat orU t ˆ as t =U t ˆ 0 . Operator
s in the Heisenbe rg and Schrödinge r pictures are related
viaÛ as ˆ H † = Û ˆ S Û . The expectatio n values of
operator s in the two pictures are related as ˆ
which will des cribe the balan ce of ground and excited state s
popu lations. The dep end ence of the filling factor on splitting
time is shown in Fig.6 for asplitting of = 12.5 and different
asymmetries. The results of the Bloch mod eldot ted line
and the multistate num erical simulationssolid line show
goo d agreemen t for splitting timesT s 20. For shorter splitting
stag es the two-mod e app roximation fails and excitations
into higher mod es take place. In the case of the high asym-
= t ˆ S t = 0 ˆ H 0 .
of the interfe romet er proces s as afunction of the holding
period Th is shown in Fig. 5 for the parameters Vas ˆ = 0.02xˆ,
max= 12.5, Ts=Tr= 5 Fig. 5 a , Ts=Tr= 20 Fig. 5 b ,
and Ts=Tr= 200 Fig. 5 c . In all cases the sinusoidal beh avior
of the excited stat e pop ulation as a function of the
holding period can be seen. Situations rangin g from
compl ete destructi ve interfe rence to perfect constru ctive
interference are bot h present . For short splitting timesFig.
5 a we observe d a discrepanc y bet wee n the two mod els.
Multistate nume rical simulations indicate the presenc e of
popul ate d high er ene rgy state s which the Bloch vecto r mod el
ignores. The full num erical calculations show an irregular
high freque ncy modulat ion of the fundame nta l frequenc y met ry Vˆ as=
0.1xˆ Fig. 6 d we obs erve asignificant devia-
023612-7
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A.1. Asymmetric double-well potential for single-atom interferometry
ASYMMETRIC DOUBLE-WELL POTENTIAL FOR ? PHYSICAL REVIEW A 74, 023612 2006
Combining bot h contributions we find tha t
d
a = a. A8
dt
Thus the Bloch equ ations can be expressed in vector form as
in Eq. 18 and in det ail as in Eqs. 17 .
A,B=L,R , are zero. Hence the secon d contributi on to the
equatio n of motio n for the Bloch vector componen t is zero
ˆ H
a 0 2 = 0. A7
0 d
dt
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Ertmer, G. Birkl, and A. Sanp era, Phys. Rev. Lett. 92, 163201
2004 .
16 D. Cassett ari, B. Hessmo, R. Folman, T. Maier, and J.
Schmiedmayer , Phys. Rev. Lett. 85, 5483 2000 .
190
023612-9

Chapter A: Reprints of selected Publications by the Candidate
A.2 A permanent magnetic film atom chip for
Bose-Einstein condensation
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: A TOMIC, MOLECULAR AND OPTICAL PHYSICS
28 BVHall et al
J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 27–36 doi:10.1088/0953- 4075/39/ 1/004
z 0B s
a
i
b
A permanent magnetic film atom chip for
Bose–Einstein condensation
z
y
x
BVHall, S Whitlock, FScharnberg 1 ,PHannaford and A Sidorov
ARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast
Spectroscopy, Swinburne University of Technology, Hawtho rn, Victoria 3122, Australia
Figure 1. A simple mode l des cribes the magn etic field of asemi-infinite, perpendic ularly
magneti zed thin film in comb ination with a uniform bias mag netic field.
E-mail: brhall@groupwise.swin.edu .au
material s offer the possibility of ultra-stable mag netic pote ntials due to their intrinsically low
magneti c field noise. Moreover, permanen t magn etic films are thin and relatively high in
resistanc e comp ared to current -carrying wires, properties which strongly supp ress the rmal
magneti c field noise [12]. Permanen t mag net ic mate rials with in-plane mag netization have
recentl y bee n used to demo nstra te trapp ing of cold atom13] s [ and to prod uce BEC on a
magneti c videot ape 14, [ 15]. Here we empl oy a novel mag netic mat erial with perpen dicular
anisotro py, develope d spec ifically for appl ications with ultraco ld atoms . Perpen dicularly
magnetize d mat erials allow arbitrary 2D patter ns to be writte n in the plane of the film and
provide magnet ic field configurati ons analogou s to thos e produced by planar microfabricated
wires [16, 17].
In this pap er, we repor t the realization ofapermanen t mag netic film /machined conductor
87 ato m chip which has bee n used to produ ce aRb
BEC. In section 2, we present a simple
mode l for a thin film of perpen dicularly magnetize d mat erial which results in straigh tforward
equat ions for the mag netic field nea r the edg e of the film. We the n desc ribe the principle
of trapp ing ultracold atom s in the potentia l formed by the permanen t mag netic film (the
film trap ). TbGdFeCo material s are the n introd uced in section3 with adesc ription of the
deposit ion process. We mea sured the bulk prop erties of the mag netic film using bot h a
superco nductin g quan tum interference device (SQUID) and a mag netic force microscope
(MFM). Section 4 describe s the constructio n of the ultra-hig h vacuu m (UHV) compa tible
ato m chip. This include sacurrent-carr ying struct ure to provide time-dep enden t control of
surface-ba sed pot entials and is used to form a conduc tor-base d mag netic trap (the wire trap).
The app aratus and experime nta l procedure s used for making a BEC indep enden tly with
the film trap or the wire trap are describ ed in section5. In section 6, we appl y the BEC
as a novel ultraco ld ato m mag net omet er by meas uring the spatia l decay of the mag netic
field from the film. High precision trap frequen cy meas uremen ts in conjunct ion with radio
frequen cy outp ut coupl ing also allow the direct det erminat ion of the associated mag netic field
gradient . In conclusion, we specul ate on future directions for our permanen t mag netic film
ato m chip.
Received 19 July 2005, in final form 25 July 2005
Published 5 Decem be r 2005
Online at stacks.iop.org/JPhysB/39/ 27
Abstract
We present a hybrid atom chip which combine s a permanen t mag netic film with
a micromachined current-carrying structure used to realize a Bose–Einstein
conden sate (BEC). A novel TbGdFeCo material with large perpendic ular
magne tization has bee n tailored to allow small scale, stable mag netic pot entials
for ultracold atoms. We are able to produce 87Rb BECinamag netic trap based
on either the permanen t mag netic film or the current-carrying structure. Using
the conde nsate as a mag netic field probe we perform cold atom mag netometry
to profile both the field magni tude and gradient as a function ofdistance from the
magne tic film surface. Finally, we discuss future directions for our permanen t
magne tic film atom chip.
191
2. The simple model of a permanent magnetic film
1. Introduction
A recent technological advance in the area of quan tum degen erate gases has bee n the
developmen t of the ‘atom chip’. These devices exploit tightly confining , mag netic pot entials,
created by low power current-carrying wires to simplify the production of Bose–Einstein
conden sates (BEC) [1, 2]. In addition, they provide the freedom to realize intricate mag netic
pot entials with features of the size comparable to the atomic de Broglie waveleng th. Atom
chips have bee n used to realize atomic waveguides and transpo rt devices for BEC[ 3, 4]. These
tools allow controllable manipulation of ultracold neu tral atoms, with pot ential app lications
in quan tum information processing 5, [ 6] and atom interferometry 7, [ 8].
For current-carrying wire-based atom chips, technical limitations are imposed by current
noise and spatial fluctuations in the current den sity leadin g to increased heating rates and
fragmen tation of cold clouds 9, [ 10]. In addition, nea r-field therma l noise in condu ctors
is respo nsible for afundamen tal atom loss mechanism 11]. [ Atom chips incorporating
perman ent magne tic materials are expected to overcome many of these difficulties. These
Consider a semi-infinite rectangula r magne t with the mag netization M and thickness h
(figure 1). The magne t lies in thexy plane with one edg e aligned along they axis. The
magne t is uniformly magnet ized in the z+ direction and the distance from the film is large
with respec t to the film thickness(z h). The mag netic field and the field gradient directly
1 Present addr ess: Institut für Quantenopt ik, Universität Hann over, 30167 Hann over, Germany.
0953-4075/06/010027 +10$30.00 © 2006 IOP Publishing Ltd Printed in the UK 27

A.2. A permanent magnetic film atom chip for Bose-Einstein condensation
30 BVHall et al
A perm ane nt magnet ic film ato m chip for Bose–Einstein condens ation 29
above the edg e can be written as
0.3
0.2
hM
. (1)
z2 and Bfilm =− µ0
2?
hM
z
Bfilm = µ0
2?
( T)
0.1
M
0
−0.1
magnetization, μ
0
−0.2
−0.3
−1.2 −0.8 −0.4 0 0.4 0.8 1.2
applied field, μ H (T)
0
Figure 2. Ahysteresis loop derived from SQUID mag neto met ry of a multilayer Tb 6Gd10Fe80Co4
magneti c film. The film mag netizat ion isµ0M S ? µ0M R = 0.28 T and the coercivity is
µ0H C = 0.32 T.
These expression s are analogo us to thos e derived using Biot–Savart’s law for the mag netic
field above an infinitely long and thin current -carrying wire. The similarity betw een a
permanen t mag netic film and a current -carrying wire can be explained using asimple mod el.
An unmag net ized film is comprise d of many small mag netic doma ins of rando m orientat ion.
The mag netic field prod uced by each dom ain is equivalent to tha t from an imaginary surface
current flowing along the doma in border s, perp endi cular to the mag netization vector 18]. [
For a uniformly mag net ized film with perpend icular anisot ropy all dom ains are aligned in the
same direction (out of plane) and within the bulk the mag netic fields of neighb ouring dom ains
cancel. A net effective current exists abo ut the perimet er of the film with amag nitude given
by the prod uct of the magneti zation and the film thickness (Ieff = hM).
The appl ication of auniform bias field(B bias)in the −x direction produces a radially
symmetric two-dimensional quadrupole mag netic field above the film edg e at the heigh z0, t
where the magnit udes ofBbias andBfilm are equal. To realizeathree-dimensional(3D)mag netic
trap for weak-field seeking atom s a non -uniform axial fieldBy is provided by two parallel
current s located beneat h and perpendic ular to the waveguide. Additionally, By supp resses
spin-flip losses by prevent ing the tota l mag net ic field at the trap botto m from going to zero.
This results in a3D harmoni c film trap at adistanc ez0 from the surface with the radial
frequency given by
goo d magneti c prop erties and increase the mag netic field strengt h nea r the surface we have
implem ente d a multilayer depo sition which produc es high quality TbGdFeCo mag netic films
with atot al thickness app roach ing 1µm. A glass slide subs trate was cleaned in an ultraso nic
bat h using a nitric acid solution the n carefully rinsed before being mounte d in the dep osition
chambe r. The base pressure was less tha n 5× 10− 8 Torr prior to introducing the argon buffer
gas (? 4 mTorr). The sub strate was the n hea ted to 100 ? C and a bon ding layer of chromium
(120 nm) was spu ttere d on the surface. This was followed by the depo sition of six bi-layers
of TbGdFeCo (150 nm) and Cr (120 nm) films.
The mag netic proper ties of the multilayer TbGdFeCo/Cr film were characterized by a
SQUID magnetomete r (figure2). The hysteres is loop indicate s a remanen t mag netization
of 0.28 T for a tot al magne t thickness of 900 nm hM ( = 0.20± 0.01 A). The comp lete
magnetizat ion of the film can be achieved by applying a field ? of0.8
T, while the film
magnetizat ion is robust in the presence of externa l fields below ? 0.1 T. The surface features
of the films have also bee n examined byahigh-resolution atom ic force microscope ope rating
in the magnet ic force mod e (figure3). An unmag netized sampl e shows micron-sized features
consisten t with domai n stripes , while an example of a uniformly mag net ized sample exhibits
excellent magnet ic homo geneit y.
192
, (2)
µBgFm F
mBy
hM
z0 2
2?f radial = µ0
2?
whereµB is the Bohr mag neton ,gF is the Landé factor,m F is the mag netic quant um number
and m is the atomi c mass. The ability to prod uce high quality, thick mag netic films with large
magneti zation is necessary to prod uce tightly confining mag net ic traps for ultracold ato ms.
3. Tb6Gd10Fe 80Co4 magneto-o ptical films and their properties
4. The atom chip design
This device represen ts the first ato m chip base d on a perpen dicularly mag net ized permanen t
magneti c film for trap ping ultracold ato ms. It has bee n designe d for the production and
manipulation of a BEC nea r the surface of the mag netic mat erial. Althoug h thes e films are
well suited for making tight and stable trapp ing pote ntials up to a few 100 µm from the
surface, the small volume of the film trap is not suitab le for efficient loading directly from a
magneto -optica l trap (MOT). To circumvent this difficulty a current -carrying wire structure
locate d beneat h the magneti c film provides an add itiona l trapp ing field. The comb ination of
bot h the magneti c film and the wire structur e represen ts the hybrid ato m chip des ign shown
schematicall y in figure4.
The des ire for large capa city information storage devices has enco urage d an extens ive
investmen t towards develop ing novel mag netic comp ositions. These are primarily opti mized
to achieve small scale, recordabl e pat ternin g of mag netic media . While it is poss ible to
benefi t from this experience , app lications with cold ato ms have several add itional yet very
specific requiremen ts. Firstly, a high Curie tem perature(TC)will prevent dem agnet ization
during the bake-out procedure, a necessary step in achieving UHV conditions. Secondly,
a high coercivity(H C)will prevent the loss of magne tization whe n app lying large external
magneti c fields. Finally, the remanen t magneti zation(M R)and the saturation mag netization
(M S) should be large and nearly equivalent, an indication of goo d mag netic homo gen eity.
These conditi ons are satisfied by Tb6Gd10Fe80Co4 mag neto -opti cal films which have a high
Curie temperat ure (TC ? 300 ? C), perpendic ular anisotropy and a square hysteresis loop.
TbGdFeCo films were prod uced using athin film depo sition system (Kurt J Lesker
CMS-18) equipped with magnetro n sputterin g and elect ron bea m evapo ration sources 19]. [
A compo site target with a nominal atomi c comp osition of 6Gd10Fe80Co4 Tb and ahigh purity
chromium target are used in the prod uction of the mag netic films. A systema tic study of the
influence of process parame ters over the prop erties of the film indicate d tha t det eriorat ion of
the mag netic anisotro py occurs for the film thickness of above 250 nm. In order to mainta in

32 BVHall et al
A perm ane nt magnet ic film ato m chip for Bose–Einstein condens ation 31
Chapter A: Reprints of selected Publications by the Candidate
The second layer ofthe hybrid chip is a wire structure which was produced using the micromachine
d silver foiltechniqu e develope d by Valet al[20]. A500 µm thick silver foil(99.99%
purity) was fixed with epo xy (Epot ek H77) to a 2 mm thick Shapal -M machineab le ceramic
base-plate . A comp ute r cont rolled Quick Circuit 5000 PCB mill was used to cut 500µm
wide insulating grooves in the foil. Each wire has a width of 1 mm which is broadene d to
6 mm far from the trapp ing region to facilitate goo d elect rical connections . After cutting, the
insulatin g channe ls were filled with add itional epo xy to increase the structural integrity and
therma l cond uctivity. The wire structur e including elect rical connection s hasatot al resistan ce
of 4.6 m . A continuou s current of 30 A can be app lied with an associated tem peratur e rise
of less tha n 40 ? C and negligible increase in vacuum pressure.
In convent ionalato m chips U- or Z-shap e wires are used for creating qua drupole and Ioffe–
Pritchard(IP)magneti c field geom etries to realize mirror MOTs and mag netic microtraps 21]. [
In the prese nt ato m chip, high current s are used to formatight trap relatively far from the wire,
thereb y avoiding unwante d collisions with the surface of the slide. Conse que ntly, the use of
broa d conduc tors proh ibit the use of sep arate U- and Z-shape wires. This is circumvent ed
with a plana r H-shap e structure , designe d to allow bot h U- and Z-shap e current path s with
goo d spatia l overlap of the associated traps . Axial confinemen t for the film trap is provided
by additiona l parallel cond uctor s separate d by 9.5 mm and located eithe r side of the H-shape
structure . The top surface of the machine d silver foil was later polished flat to supp ort the
glass slides.
During assembl y, the polished edg e of the TbGdFeCo film coat ed slide is aligned to the
midd le of the H-shape structure and set with epo xy. The second gold-coa ted slide is epo xied
adjacen t to the magnet ic film slide to comp lete the reflective chip surface. Two rubidium
dispe nsers are mounte d on two ceramic blocks (Macor) which are recesse d below the chip
surface. The two glass slides, mach ined silver foil and ceramic base -plate are the n fixed
to acopp er hea t sink. The comp leted chip is clamped to a19 mm diamete r solid copper
feed throug h (Cerama seal, 800 A rating) and mou nte d in the vacuu m cham be r. Electrical
conn ection s are mad e using a1.6 mm diamete r bare coppe r wire and BeCu barrel connectors
in conjunction with a 12 pin power feed throug h (Ceramaseal , 55 Arating ). A cold cathod e
gaug e indicate d a pressur e below× 110−
11 Torr after baking at 140 ? Cfor 4 days, highlight ing
the UHVcomp atibility of all mat erials.
−5
(a) (b)
−4
−3
−2
−1
0
1
position ( μ m)
2
3
4
5
Figure 3. The mag netic force microscope (MFM) image of a Tb6Gd10Fe80Co4 mag net ic film
surface. (a) The unm agne tized sampl e shows the dom ain structu re with micron-sized features.
(b) The uniformly mag netize d sampl e is free of any visible mag net ic structur e.
193
5. Bose–Einstein condensation on a permanent magnetic film
The reflective surface of the ato m chip is used to form amirror MOT and accomm oda tes
30 mm diamet er laser beam s provided byahigh-p ower diod e laser (Topti ca DLX110) locked
to the D2 (F = 2 ? 3)cooling transition of87Rb. The trappin g light is det une d 18 MHz
below resona nce and has an inten sity of 4 mW cm − 2 in each bea m. Arepum ping laser locked
to the D2 (F = 1 ? 2)trans ition is comb ined with the trapp ing light with an intens ity of
0.5 mW cm− 2 pe r beam . Two wate r-cooled coils mount ed outs ide the vacuu m cham ber
provide a quadrup ole magneti c field with gradient 0.1 Tm − 1 centered 4.6 mm below the chip
surface. To load the mirror MOT a current of 6.5Ais pulsed for 9.5 s throug h one resistively
heate d Rb dispe nser, allowing the collection of 2× 108 atoms . The atom s are held for a further
15 s while the UHVpressur e recovers, read y for trans fer to the chip-base d pot entia ls.
Transfer begins by simultaneousl y ramping a current thro ugh the U-shap e circuit
(IU = 0 ? 8 A), increasing the uniform fieldBbias and turning off the external qua drupole
magneti c field over 50 ms. This moves the atom s withou t loss, into a U-wire MOT locate d at
1.6 mm from the surface and increase s the radial gradient to 0.4 Tm − 1 . While this compression
increases the spat ial overlap with theIP potential , it also hea ts the cloud. To counte ract this,
Figure 4. Schematic view of the hybrid ato m chip. Inset: TbGdFeCo/Cr multilayer film and Au
overlayer. From the top down, glass slide coat ed with magn etic film, machined silver foil H-wire
and end wires, Shapa l-M base -plate and Cu hea t sink. Missing from the schema tic is the second
glass slide and two rubidium dispe nsers.
The top layer of the ato m chip consists of two adjacen t 300 µm thick glass slides which
are sturdy eno ugh to prevent warping . The long edg es of the glass slides were polished with
aluminium oxide grit prior to depo sition to remove visible chips. Amultilayer TbGdFeCo /Cr
film was depo sited on one slide using the proced ure out lined in section 3. Both slides were
the n coat ed withagold overlayer (170 nm) and toge the r formalarge reflective surface × (40
46 mm2 ). This allows the collection of a large num be r of atom s into a mirror MOT within a
single-cham be r UHVsystem . The glass slide coat ed with the TbGdFeCo/Cr multilayer film
was the n mag netized inauniform field of 1 T pend ing assembl y.

A.2. A permanent magnetic film atom chip for Bose-Einstein condensation
34 BVHall et al
A perm ane nt magnet ic film ato m chip for Bose–Einstein condens ation 33
1.5
ty
nsi
1
e
d
0.5
optical
0
−0.6 −0.3 0 0.3 0.6
−0.6 − 0.3 0 0.3 0.6
position (mm)
−0.6 −0.3 0 0.3 0.6
Figure 5. Typical absor ptio n images and opti cal den sity profiles of a ballistically expand ed atom
cloud. Each image is asingle realization of the experiment where evapora tion is performe d in
the perma nen t magnet ic film pot ential. After truncating the evaporat ion ramp , atom s are held for
150 ms and ballistically expande d for 30 ms befo re imaging. (a) RF final = 804 kHz—the rmal
cloud, (b) RFfinal = 788 kHz—part ially condens ed cloud, and (c) RFfinal = 760 kHz—an almost
pure condens ate.
cold atom s have bee n empl oyed to chara cterize the mag netic field produced by the film inside
the vacuum cham be r. This allowsadirect comp arison with the simple mod el desc ribed earlier.
A magnet ically trapp ed cloud of cold ato ms or a BEC beh aves as an ultra-sensitive probe to
the local mag net ic field. A measur e of the trap position as a functionBbias of det ermines
Bfilm(z), while an indepen den t measur e of the trap frequen cy is used to det ermine Bfilm (z).
Once the BEC is confin ed by the film trap it is poss ible to profile the mag netic field
depen den ce nea r the surface. The pot ent ial minimum is located at the poin t whe re the uniform
magneti c field is equal in mag nitud e to and cancels the field from the film (B bias = − Bfilm).
The uniform magneti c field can be increased (decreased ) to move the trap minimum closer to
(furthe r from) the film surface. The BECfollows the pot ent ial minimum and the measur ement
of the cloud posit ion with respec t to the film surface det ermines Bfilm(z). The stren gth ofBbias
is calibrated within the vacuu m cham be r using untr app ed atom s (far from the film) and a short
RF pulse resona nt with the Zeema n splitting. The pixel size in the imaging plane is calibrated
against the gravitationa l acceleration of freely falling atom s and agree s with the calibration
given by imaging areference rule external to the app aratus. Unfortunatel y thoug h, the glass
substrat e coate d with mag net ic mat erial has recessed appr oximat elyµm 50 beh ind the second
blank glass slide as aconse quenc e of unevenly cured epo xy. The exact position of the film
surface (in relation to the image) is theref ore unknown and present s an uncert ainty Bfilm(z). in
For this reaso nasecon d techni que has bee n app lied to provide more information abo ut the
magneti c field from the film.
Harmonic oscillation s with the small amplit ude and frequenc ies up to 10 kHz can be
measure d accura tely over many period s with a BECdue to low dam ping rate s and small spatial
extent . In this case, trap frequen cies are measure d by exciting radial centre of the mass motion
within the film trap and have bee n measure d to bet ter tha n1Hz ? 0.1% ( accuracy). These
excitation s were obser ved by rapidly increasing the uniform mag netic field by app roximately
5% befo re returnin g to the original position within 2 to5ms. The cloud position was measur ed
after 10 ms of free expansion and dat a have bee n taken over five periods of oscillation.
the radial gradient is reduc ed rapidly to 0.11 T m − 1 with the trap light off to minimize any
force on the atoms . Polarization gradien t cooling is applie d for 2 ms with 56 MHz red-detu ned
trap light to reduc e the tem peratur e from 140 µKto 40 µK. Both the MOT light and IU are
the n turned off leaving the cold ato ms inauniform mag netic field.
Next a 200µs opt ical pum ping pulse is appl ied to maximize the num ber of atom s in the
|F = 2,m F = +2 weak-field seeking stat e read y for mag netic trapp ing. A current (IZ)of
21.5 Ais switched on throug h the Z-shap e circuit whileBbias is increase d to 1.3 mTto form an
IP wire trap at the same posit ion. Atot al of 4× 107 atom s are held withabackground -limited
lifetime greate r tha n 60 s. Adiabatic compres sion of this trap is performed by ramping IZ up
to 31 A andBbias up to 4.0 mT over 100 ms. Furth er comp ression results in loss of ato ms to
the surface. The compres sed mag netic trap is 560µm from the film surface where the radial
and axial trap frequencies are 2?× 530 Hz and 2?× 18 Hz, resp ectively. The elastic collision
rate in this trap (?el ? 50 s− 1 ) is high enoug h to begin evapo rative cooling.
Forced evapo rative cooling to the BEC trans ition begins in the wire trap and is the n
tran sferred to the film trap during a single logarithmic radio frequen cy (RF) ramp . The first
8.85 s of this ramp is apreliminary cooling stag e in the wire trap down to atem peratur e of
? 5 µK. As the cloud is cooled the trap is compress ed furthe r to improve the evapo ration
efficiency by loweringIZ to 25 A, moving the trap to 350µm from the surface and increasing
the radial trap frequen cy to? 2? × 660 Hz. The RF amp litude is the n reduc ed to zero for
150 ms while the ato ms are trans ferred closer to the chip surface and finally to the film trap.
In this trapIZ is zero and axial confin eme nt on the mag netic film edg e is provided by the two
end wires, each with a current of 6 A. The trap botto m is tune d using an add itiona l mag netic
field para llel to the film edg e to minimize any discon tinuity in the RF evapo ration trajectory.
The radial and axial trap frequencie s are 2? × 700 Hz and 2? × 8 Hz, resp ectively. The
RF amplit ude is the n increase d again and evapo ration continues for 1sto the BEC pha se
trans ition.
Before imaging, the mag netic film trap is adiabatically moved 0.17 mm from the surface
to avoid excessive field grad ients from the film. The cloud is the n release d by switching
off Bbias and the ato ms fall unde r gravity with minor acceleration from the perman ent field
gradient . Resonan t opt ical absorptio n is used to image the atom s with a100 µs ? + light
pulse parallel to the gold surface and tun ed to the2 D(F
= 2 ? 3) transition. A CCD
camer a records the absorptio n image of the cloud using an achroma tic dou blet telescope with
a resolution of 5µm/pixel. Using the above proced ure a new conden sate of× 1105
atoms
is creat ed every 50 s. Figure 5 shows absorption image s and opt ical dens ity profiles after
30 ms of ballistic expans ion. The forced RF evapo ration is truncat ed at 804 kHz, 788 kHz
and 760 kHz revealing a therma l cloud, part ially conden sed cloud and nearly pure conden sate ,
respe ctively.
It is also possible to form a cond ensat e trapp ed solely by the wire trap. Here asingle,
uninterrupt ed, 10 s RF ramp results in a BECwith ato m num ber compa rable to tha t realized in
the film trap . This provides a unique possibility for stud ying the prop erties of aBEC in both
permanen t mag net ic and current -carrying trap ping environment s. In add ition, the formation
of a BEC inde penden t of the top layer will allow new mag netic structures or mat erials to be
replaced with ease . The wire trap can also be used to transp ort a BEC to regions on the chip
where the mag net ic field topolog y may be different from thos e nea r the substrat e edg e.
194
6. Magnetic field characterization
The mag netic prop erties of the TbGdFeCo film were mea sured prior to mountin g on the ato m
chip using a combination of SQUID and magneti c force microscopy. In situ techniques using

A perm ane nt magnet ic film ato m chip for Bose–Einstein condens ation 35
0.8
(a)
−3
T )
Chapter A: Reprints of selected Publications by the Candidate
0.6
0
( 1
0.4
th
g
n
0.2
stre
36 BVHall et al
ld
fie
0
Acknowledgments
(b)
12
We would like to than k J Wang and D Gough for carrying out the mag net ic film dep osition.
This projec t is supp orte d by the ARC Centre of Excellence for Quantum –Atom Optics and a
Swinburne University Strategic Initiative fund .
8
4
g r a d i e n t ( T T/
/ m )
References
f i e l d
0
0 50 100 150 200 250
distance from surface ( μm)
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[2] Ott H, Fort ágh J, Schlotterbe ck G, Grossmann A and Zimmerman n C 2001Phys. Rev. Lett. 23 230401
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Kr üge r P 2005Nature Phys. 1 57–62
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Figure 6. Measureme nts of the mag netic field streng th (a) and field gradient (b) as a function of
distanc e from the surface. The dat a (open circles) agree well with pred ictions (dotte d line) of the
simple mode l (see equat ion (1)). Experiment al errors are mostly det ermined by image resolut ion
and a small unce rtainty in the pixel size calibration.
195
In add ition, the trap bot tom was mea sured using RF out coupling with an accuracy bet ter than
10 mG(? 1%). The measure men t of trap frequen cy in comb ination with the trap bott(By) om
unambigu ously determi nes the local mag netic field gradient (see equa tion 2). This comb ined
with the trap posit ion meas uremen ts have bee n used to provide the mag netic field and the
magneti c field gradien t as afunction of heigh t above the surface (figure 6). These dat a are
consisten t withaprediction base d on the simple mode l whe re the film thickness -mag netization
produc t is given by the prior SQUID measure men t hM ( = 0.20 A).
7. Discussion and conclusion
We have dem onstrate d a hybrid ato m chip tha t exploits perpen dicularly mag netized film or
current -carrying wires for the prod uction ofaBEC. We have develop ed a multilayer mag netic
film structur e (TbGdFeCo/Cr) tha t provides large mag netization and thickness, important
for realizing tight and flexible mag netic microtrap s. We have used the BEC as asens itive
prob e to directly mea sure the local magneti c field and gradient associated with the mag netic
film. These measure ment s justify the use of the simple mod el for perpen dicularly mag net ized
magneti c microstructu res.
At prese nt we are extendin g the techniqu e of cold ato m mag neto metr y to the measur ement
of the spatia l dependenc e of the mag netic field along the film edg e. Spatially dep endent
magneti c field variations have bee n obser ved above microfabricate d wire-base d ato m chips
and have bee n att ribute d to spatia l deviations along the wire edg 10, e 22]. [ Similar phe nom ena
obser ved in permane nt mag netic structures may be caused by sub strat e roug hness , dep osition
irregularity or ultimate ly doma in reversal. Future stud ies are aimed at the interac tion betw een a
BEC and mag netic thin films. Acompa rison of the dec ohe rence rate s of conden sate s confined
in eithe r the film or wire-base d microtraps may reveal intriguing pos sibilities for coherent
manipulation of cold ato ms in microstructu red perman ent mag netic pot entia ls.

A.3. Perpendicularly magnetized, grooved GdTbFeCo microstructures for
atom optics
A.3 Perpendicularly magnetized, grooved GdTbFeCo
microstructures for atom optics
JYWang et al
2. Principles of magnetic atom optics with periodic
structures
A one-dime nsional periodic array of mag net s of alterna ting
polarity or aone- dime nsiona l periodi cally grooved magn etic
structur e produce samagneti c field pat tern tha t is well suited
to ato m optic s [2]. For such an array in the xz plane with
periodicity in thex direction , the mag nitude of the mag net ic
field depen ds on heig hty above the surface as given by 3, [ 4]:
|B(x,y)|= B0e −ky [(1− e −kb )
+ 1
3 (1− e− 3kb )e − 2ky cos 2kx +..], (1)
wherek− 1 = a/2? is the deca y leng th,a is the perio d of the
array,bis the thickness ofthe mag net s andB0 isacharacteristic
magneti c field tha t is define d by the magneti zation M 0 of
the mat erial. For an array of magn ets of alternat ing polarity
B0 = 8M 0 (Gaussian units) and for agrooved structure
B0 = 4M 0. The factors (1− e−nkb ) accoun t for the finite
thickness of the magneti c material . For heig htsy a/4?
above the surface equation 1) ( reduce s to
|B(x,y)|= B0(1− e −kb )e −ky it may be turne d into an ato mic mat ter wave diffraction
grating by app lying asmall bias magn etic field normal to the
microstruct ure surface to produce aspatial diffraction grating
[2,5] or by app lying an oscillating orthogona l mag netic field to
createatempora l diffraction grating for ato ms 6]. [ In addition,
it is possible to gen erate magn etic microtraps and waveguides
for low magn etic field-seeking ato ms by app lying app ropriate
dc bias fields to produce aseries of mag net ic field minima
above the array surface [7]. A mag netic tube for tran sporting
ato ms may be formed from acylindrically shape d periodic
magn etic structu re producin garadially varying magn etic field
tha t guides ato ms along the axis of the cylinder 8, [ 9].
The first ato mic mirror to retro -reflect cold ato ms was
based on aud iotape ont o which asinusoidal mag netic pat tern
of period 9.5µm had been recorded 10]. [ Subsequen tly, sine
waves with periods ofaround 15µm were recorded ont o floppy
disks [11, 12] and videota pe [13]. These magn etic recording
medi a magn etize in-plane, which limits the smallest period
pat tern tha t can be recorded and makes the recording ofpat terns
of arbitrary shape difficult.
Magn etic mirrors have also been constructe d based on
, (2) periodic arrays of perma nen t NdFeB [14] and SmCo [15]
magn ets of alternating polarity. Althoug h such magn ets
so tha t the magn itude of the magneti c field decay s
can produce large fields, they cann ot be used to produce
exponen tially with heig hty. The x andy component s of the
structu res with micron-scale periodicities. In our magn etic
magneti c field both vary sinusoidally in thex direction , with
ato m opt ics programme, we have previously att emp ted to
a pha se differenc e of?/2, so tha t they combine to produc e
construct periodic grooved microstructure s of ferroma gne tic
flat magn etic equipote ntials. When slowly moving atom s in
nickel, cobalt and alnico [16,17]. These microstructu res were
posit ive or low field-seeking mag netic state s mgF ( > 0, whe re
mad e by electron beam lithog raphy followed by sputt ering
m is the mag netic quant um num ber of the stat e and gF the
and electrop lating processes tha t resulted in the ent ire grooved
Land e g-factor) app roach the surface of such an array, they
structu re being mad e of the ferromag netic mat erial. In such
are repelle d by the increasing mag netic field strengt h and the
a structure , the se magn etic med ia also magn etize in-plane
array behaves as an atomi c mirror (figure1). The origin of
and have astrong preference for mag netizing parallel to the
the repuls ive force is the mag netic dipole interaction tha t has
direct ion of the grooves. However, for mag netic ato m opt ics,
potentialU int(x,y,z) =− µ·B(x,y,z) produci ng the gradien t
in order to produce the app ropriate magn etic field distribution
force Fgrad = ? (µ·B)= − mgFµB?B(x,y,z) , where µB is
above the grooved surface ofmateria ltha t magn etizes in-plane,
the Bohr magn eton.
it is nece ssary to magn etize the structu res at right ang les to the
Aperiodic magneti c array in the form ofamicrostruct ure
groove direct ion. The structu res failed to produce satisfactory
is also the basis of othe r ato m opti cs devices. For examp le,
results because of the difficulty in mag net izing the micronscale
protrusions between the grooves in this way. Magn etic
force microscope (MFM) images revealed dom ain structu re in
the protrusions tha t indicated incomp lete magn etization, and a
magn etic field tha t fell off withadeca y constant characteristic
of the mag netic dom ain size rather tha n the periodicity of the
structu re [17].
Mate rials with a perpen dicularmagn etic anisotro py do
produce the required magn etic field distribution whe n mag -
netized along the easy axis of mag netization. Microstructure
s compr ising Co0.8Cr0.2 films on anon -magne tic grooved
substrate and magn etized perpen dicular to the array surface
have been successfully used as mag netic mirrors 18,19]. [ For
ato m opt ics appl ications, however, the mag net ic properties of
Co0.8Cr0.2 films are inferior to tho se of GdTbFeCo magn eto -
opt ical films. In part icular, the shape of the hyste resis loop
indicate s tha t the rema nen t mag netization is only about one
quarte r of the satura tion magn etization and tha t the mag netic
dom ains are not comp letely orient ed, giving rise to magn etic
inhomo gen eities.
Magn eto -optical thin films, such as ferrimagnetic
Figure 1. Reflection of ato ms by aperiodic ally grooved structure (Gd,Tb)FeCo and (Dy,Tb)FeCo are widely used in mag-
coate d with magn etic film with perpen dicular magn etic anisot ropy. netic recording and device app lications due to the ir high
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS D: A PPLIED PHYSICS
J. Phys. D: Appl. Phys. 38 (2005) 4015–4020 doi:10.1088/00 22-3727/38/ 22/003
Perpendicularly magnetized, grooved
GdTbFeCo microstructures for atom
optics
JYWang,SWhitlock, FScharnberg ,DSGough ,AISidorov,
RJMcLean andPHannaford
Centr e for Atom Optics and Ultrafast Spectro scopy and ARC Cent re of Excellence for
Quantum- Atom Optics, Swinburne University of Technology, Hawthorn, 3122, Australia
Received 15 March 2005, in final form 26 Augu st 2005
Published 7Novemb er 2005
Online at stacks.iop.org/JPhysD/38/4015
Abstract
Periodically grooved, micron-scale structure s incorporating perpend icularly
magnetiz ed Gd10Tb 6Fe80Co4 magneto -optical films have bee n fabricated
and characterized . Such structure s produceamagn etic field having flat
equi pote ntials and whose magnitu de decays expone ntially with distanc e
above the surface, making them att ractive for manipulating ultracold ato ms
in ato m optics. The GdTbFeCo films have bee n deposited on aCr
under layer on asilicon (100) wafer and on agrooved silicon microstructure
using DC magnetro n sputt ering. The films are found to have excellent
magneti c properties for mag net ic ato m optics app lications, including high
remanen t magneti zation, high coercivity and excellent homog eneity.
196
(Some figure s in this article are in colour only in the electro nic version)
1. Introduction
practicalato m optics-based devices willincorporate permanent
magn ets.
Atom optics involves the manipulation of atom s, particularly Acommon requireme nt in magn etic ato m optical devices
cold atom s, in an ana logo us wayto the waylight is manipulate dis
that the mag netic struct ure be periodic and have feature s
in optics. High qua lity eleme nts are nee ded in atom on the scale of amicron. This is nece ssary to produc e a
optics for many purposes includin g the reflection, diffraction, ‘hard’ magn etic mirror where the magn etic field gradien t is
beam splitting, trapp ing, storag e and guiding of slowly movinglarge
eno ugh so that the ato m interacts with it over ashort
atom s and atomi c mat ter waves. The force commo nly used todistance
, and to use the periodic struct ure as the basis for a
manipulate the atom s derives from the interaction betw een the diffraction grating for ato ms, requiring the period ofthe grating
induced electric dipole mome nt of the ato m and the electricto
be comparable to the de Broglie wavelengt h of the ato m
field gradien t associated with a laser light field (see, for matte r waves for reasonab le diffraction ang les and inten sities.
example [1]), but there are many advant age s in exploiting theDifficulties
in micromachining and magn etizing materia ls on
force that results from the interaction betwee n the magnet ic this scale can be avoide d by the use ofmag net ic films deposited
dipole mom ent of the ato m and the magn etic field gradie nt on non- magn etic microstruct ures. In this paper we discuss
nea r, e.g. a current carrying wire or apermanen t mag net icthe
app lication of perpend icularly mag netized GdTbFeCo
structure [2]. These advanta ges include eliminating the nee d magn eto -optical films widely used in the recording indu stry
for alaser to generat e the inhomogene ous light field, and theto
the fabrication of periodic microstruct ures for magn etic
fact that the atom s can remain in the groun d stat e so that the ato m optics. In the following section we out line the principles
coheren ce-destroying process of spontaneo us emission doe sof
mag netic ato m optics with periodic struct ures and the use
not occur (see, for example [3]). Furthermore, perman ent of grooved microstruct ures for producing asuitable magn etic
magnet s have advant age s over the use of current carryingfield
gradien t, while in section3 we describe the production
wires to produc e the magneti c field gradien t that includeand
characterization of GdTbFeCo films and GdTbFeCo
eliminatin g the problems of heating, curren t instabilities andfilm-ba
sed microstruct ures with magn etic properties that are
short and open circuits. It is likely, therefore , that many attrac tive for ato m optics.
4016
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JYWang et al
Perpen dicularly magn etized, grooved GdTbFeCo microstructure s
(a)
Chapter A: Reprints of selected Publications by the Candidate
(a)
(b)
(b)
Figure 3. MFM micrograph s showing the domain structur e of a
Figure 2. AFM micrograph s of the surface morpho logy of 150 nm thick Gd10Tb6Fe80Co 4 film prepar ed on a140 nm thick Cr
(a) a 150 nm thick Gd10Tb6Fe80Co 4 film prepared on a140 nm thick underlayer on an Si wafer (a) unma gne tized and b) ( mag netized.
Cr underlayer on an Si wafer and (b) a 140 nm thick single-layer Cr
film deposite d on an Si wafer.
AFM and MFM imaging. Figure 5 shows typical AFM
top ograph y and MFM phas e images of the 3µm period
SQUID magn etome ter mea surement s of the magn eticgrooved
microstructu re obtained by scanning a) ( the surface
propertie s of the Gd10Tb6Fe80Co4 films with Cr underlayers for the AFM image and (b) 100 nm above the top surface of
deposite d on an Si wafer were also carried out , up to the magn etized grooved microstructu re for the MFM image.
a maximum applied field of 10 kOe. Figure 4 shows The AFM scan shows uniform periodic signals and indicates
a hysteresis loop mea sured at room tem perat ure in the directiontha
t the side walls are reasonably perpen dicular with some
perpe ndicular to the film. It has ashap e close to rectan gularroun
ding due to instrumen tal effects of the AFM. A statistical
and indicates the film has an intrinsic coercivity of about analysis indicates asurface roug hnes s of about 2 nm (rms).
− 3
2.7 kOe and a rema nen t magn etization of about 265 emu cm The MFM images indicate excellent magn etic homog eneity,
(3.3 kG). The GdTbFeCo films deposite d directly on Si were with no evidence of the domain structu re of figure 3(a). The
found to have inferior magneti c prope rties to thos e depo sitedvariation
of they (vertical) compo nen t of the mag net ic field (to
on the Cr underlayer, particularly in term s of coercivity which the MFM is sensitive) with distancex is appro ximately
(< 1.5 kOe).
sinusoidal (figure5(b)), even ataheight of only 100 nm (about
The surface topolog y and magn etic characteristics of the 0.2× a/2?) above the top surface. The observed sinusoidal
magneti zed grooved microstruct ures with periodicities of 1.5depen
dence at distances very close to the surface is att ribute d
and 3µm coate d with a150 nm thick GdTbFeCo film on to some roun ding of the top edges of the groove walls during
a 140 nm thick Cr underlayer were also investigated with the fabrication process, which significantly decrease s the
perpe ndicular magn etic anisotro py, high saturation magne -tempe
ratur e of around 300 ˚C [26]. The effect of process
tization and large coercivity (see, for example 20]). [ We paramet ers, such as argon gas pres sure, subst rate temp erat ure,
have fabricated periodically grooved microstruct ures based DC power and dep osition time, on mag netic properties of
on Gd10Tb6Fe80Co4 films with perpendi cular anisotro py and the films was investigated , and the dep osition conditions for
investigate d the ir properti es. Optimizing the magn etic prop erprepa
ring the bes t GdTbFeCo films within the capa bilities
ties of thes e films is crucial to successful device developmen t of the dep osition system were established using astatistical
and this deman ds careful prepar ation of the magn eto-opt ical design of experiment meth odo logy. The opt imal dep osition
films and charac terization of the ir prop erties. This pap er paramet ers were found to be an argon pressure of 4 mTorr,
repor ts investigation into thes e issues, and the results obtai nedsubst
rate tempe ratur e of 100 ˚C , dc discharge power of 150 W
indicate tha t the structure s are highly suitabl e for ato m opticsand
a dep osition time of 50 min. After opt imizing the
applica tions.
process paramet ers, a 140 nm thick chromium underl ayer was
Anothe r mag neto -opt ical materia l with perpe ndiculardep
osited ont o an Si (100) wafer and Si grating structure s
anisotr opy tha t is a promising candid ate for app lications(periodicitiesa=
1.5 and 3µm and groove dep th of 0.5µm)
in ato m optic s is CoPt. CoPt films usually compri se atadep osition rate of10 nm min
multiple alterna ting layers of Co and Pt. Such astruct ure
was recently mag netized with aview to produc ing atom ic
microtrap s above its surface by ruling pattern s of opposite
magneti c polarity in the film usingamag net o-optica lrecording
techniq ue [21]. We have used asimilar magneto -opt ical
recordin g techniq ue with thicker TbFeCo magn eto-opti cal
films [17, 18, 22] but the qua lity of the recorded pat tern s was
limited by dema gnetizat ion of the film adjacen t to the writing
laser during the recordin g process . Both CoPt and GdTbFeCo
films appea r to have excellent magn etic characteristic s for
ato m opti cs, altho ugh one poss ible limitation of CoPt may
be the relatively small thicknesse s (of the orde r of 50 nm) tha t
appea r to be necess ary if the perpe ndicular anisotro py is to be
maintained.
Many ato m opti cs experimen ts now involve miniatu rizing
and integrating ato m optica l elem ent s on the surface of an
‘atom chip’(see, for example [23]) and magn eto-opt ical films
will be useful in this tech nolog y as well. The use of permane nt
magneti c mat erials in ato m chips offers potent ial advant age s
in overcoming the instab ility of the mag netic potentia l due to
curren t fluctuat ions, as well as the problem s men tioned earlier
of heati ng from current s and short and ope n circuits. In our
group , cold Rb atom s have recently bee n successfully trapped
in amag net ic trap generat ed by amagn eto-opt ical film above
the surface of an ato m chip 24]. [ In this case, no grooved
structur e is involved, but the magneto -opt ical film is similar to
tha t used for the magn etic mirror microstructur e discussed in
this pape r.
3. Film preparation and characterization
The GdTbFeCo films were prepare d using a thin film
depositi on system (Kurt J Lesker CMS-18) capabl e of
magnetr on sputt ering and electro n bea m evapor ation.
A comp osite target with anominal atomi c composi tion of
Gd10Tb6Fe80Co4 and a chromium target were magnetr on
spu ttered in the system . The magn etic prop erties ofGdTbFeCo
film vary considerabl y with composi tion (see, for example
[25]). Briefly, the Curie temp eratur e increases with the Co/Fe
ratio, and the mag net ization and coercivity vary not only
with the Tb/Gd ratio but also with the amou nt of Co; the
degree of perpen dicular anisotr opy depe nds on the Tb/Gd
ratio and the rare earth to transition met al ratio. Film
prepar ation parameter s also influence the mag netic prop erties.
The composi tion selecte d was expected to produc e films
with high rema nen t magn etization and coercivity and aCurie
− 1 , followed bya150 nm thick
GdTbFeCo film at adep osition rate of 3 nm min − 1 . The base
pres sure of the chambe r was less tha n 5× 10− 8 Torr prior to
introdu cing the argon gas and the target to subs trat e distance
was 0.2 m. Finally, a 20 nm thickY2O3 film was dep osited ont o
the GdTbFeCo film as aprotect ive layer using electron beam
evapora tion. The distanc e bet wee n the evapo ration source and
the subst rate was 0.7 m.
Analysis of the film compo sition by indu ctively coupled
plasma spe ctroscopy gives an ato mic compo sition of
Gd9.6Tb6Fe80Co4.4, which is close to the nomina l compo sition
of the target . It is well known tha t having aCr underlayer
can pos itively influen ce the surface morpho logy and improve
the mag netic properties of magn etic films. To give some
insight into the value of prepa ring the magn eto -opt ical film
on a Cr und erlayer, a single-layer 140 nm thick Cr film
and asingle-layer 150 nm thick GdTbFeCo film were also
prepa red. All thre e films were prepare d on Si wafers rath er
tha n grooved structu res to facilitate the characterization. The
surface feature s of the films were examined immediately after
the sampl es were removed from the chambe r by an ato mic
force microscope (AFM) ope rating in high resolution, semicontact
mode .
Figure 2 shows AFM micrographs of the surface
morpho logy of the GdTbFeCo film on aCr underlaye r and
the single-layer Cr film. The two films exhibit similar surface
morpho logies. Both are den se and the ir surfaces are smooth .
The grain shape on bot h surfaces is found to be round , with
an average grain size of appro ximately 40 nm. The single
Cr layer possesses aslight ly larger grain size and is alittle
roug her tha n the GdTbFeCo film on aCr under layer. By
contrast, GdTbFeCo films dep osited direct ly ont o Si wafers
have roug her surfaces and larger grain sizes (50 nm). The
smaller grain size of the GdTbFeCo films whe n dep osited on a
Cr under layer may result from the enh anced surface roug hness
ofthe underlaye r,suggesting tha t it aids the fabrication ofden se
GdTbFeCo films with smaller grain size and smooth er surface.
Figure 3 shows MFMmicrographs of the domai n structu re
of the same film as in figure2 mad e by scanning ataheig ht of
100 nm above the film surface. In the unma gne tized state a), ( a
labyrinth of doma in pat terns with smooth surface contour s can
be clearly obser ved. These are typical of GdTbFeCo magn eto -
opt ical thin films with large perpen dicular anisotro py. In the
magn etized state b), ( the re is no domai n structure visible,
indicating tha t the film has excellent magn etic homog enei ty
down to the lower limit of resolution of the MFM (about
100 nm).
4017
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A.3. Perpendicularly magnetized, grooved GdTbFeCo microstructures for
atom optics
[3] Hind s E A and Hughes IG 1999 J. Phys. D:Appl. Phys. 32
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Preprint physics/0502073
[8] Myatt C J, Newbury N R, Guist RW, Luitzenhiser S and
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JYWang et al
in thezdirection. If an add itional field of 1 gauss is app lied in
thex direction to sup press spin-flips, the n the trap would have
a radial trapp ing frequency of 5.4 kHz for ground stateF = 2,
m = 2 87Rb atoms.
Perpen dicularly magn etized, grooved GdTbFeCo microstructure s
2
c)
1.5
u/
c
4. Conclusions
m
( e
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[25] Kryde r M H 1993 Annu. Rev. Mater. Sci 23 411
[26] Challene r W A 1997 Private communication
We have fabricate d magn etic microstructures tha t show
considerable promise as atom optical devices by depositing
Gd10Tb 6Fe80Co4 magn eto-optical film with perpendicular
magn etic anisotropy on grooved silicon microstructures with
periodicities of 1.5 and 3µm. When the magnet o-optical
mate rial was deposited onto a Cr underlayer the magne tic
characteristics of the films were found to be significant ly
improved, along with their surface topolo gy and density.
A single layer of the magn eto-optical film deposited on aCr
underlayer was measured at room temperature in the direction
perpendicular to the film to have an intrinsic coercivity
− 3
of 2.7 kOe and a remanen t magn etization of 265 emu cm
(3.3 kG). The periodically grooved microstructures coate d
with GdTbFeCo films exhibit reasonably perpendicular sidewalls
and uniform periodic modu lation in AFM and MFM
scans. Optimum magne tic characteristics were found for
multilayer structures of GdTbFeCo alternating with Cr. The
amplitude of the componen t of magneti c field in the direction
perpendicular to the grooved microstructure surface was found
to decrease exponen tially with heig ht above the surface with
a decay constan t consisten t with the the oretical value given
by the period of the microstructure. Such perpendicularly
magn etized grooved microstructures coate d with GdTbFeCo
films should be well suited for atom optical app lications.
1
l y ( relative)
on
n B
0.5
netizati
g
a
M
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Height above surface ( µm)
Figure 6. Plot of lnBy versus heigh t above the surface for the
a= 1.5µm perpendic ularly mag netize d grooved microstructu re.
The slope gives adec ay lengt h of 0.24µm.
Figure 4. Hysteresis loop measur ed by SQUID mag net ome try at
room temp erat ure in the perpendicula r directio n for a150 nm thick
Gd10Tb6Fe80Co4 film deposite d ona140 nm thick Cr underlaye r on
an Si wafer.
Acknowledgments
This work is funded by a Systemic Infrastructure Initiative
(SII) grant from the Department of Education, Science and
Training (DEST), the ARCCent re of Excellence for Quant um-
Atom Optics and a Swinburne University Strategic Initiative
grant . We tha nk Dr T Hicks and Mr D Robinson for their
help with the SQUIDana lysis at Monash University, Australia,
and W A Challene r and J Sexton of Imation Corp., USA for
supp lying the early GdTbFeCo samples.
References
[1] Metcalf H J and van der Strate n P 1999Laser Cooling and
Trapping (Heidelberg: Springe r)
[2] Opat G I, Wark S J and Cimmino A 1992 Appl. Phys. B
54 396
supe rior magn etic characteristics to films tha t had asingle
450 nm thick layer of GdTbFeCo. Withou t the Cr midlayers
present the remane nt mag netization and coercivity were
found to deter iorate at thicknesses larger tha n abo ut 200 nm,
probab ly due to the magn etic anisotro py direct ion bein g less
well defined ; but by using the multilayer structure the overall
thickness of the film could be built up beyond the thickness of
1µm while still preserving goo d magn etic properties.
A strong indicator of how the grooved microstructures
work as mag netic mirrors is the dep end ence of the magn etic
field on heig ht above the surface of the structu re. The
dep end ence on heighty above the 1.5µm period grooved
microstruct ure of they-compo nen t of the magn etic field was
test ed by making aseries of MFM scans with the magn etic
tip at different height s ranging from 100 to 1500 nm above the
top of the microstructure . From the MFM dat a it is possible
to plot the dep end ence ofBy on heighty (figure 6). The
values ofBy in figure 6, which are relative values dete rmined
from the amp litude of the MFM phas e signal, indicate tha t the
amp litude ofBy decre ases expon ent ially with heig ht above
the surface. The mag nitud e of the magn etic field is expect ed
to deca y with the same dep end ence. The slope yields a
deca y lengt h of (0.24 ± 0.02)µm, which is in agreemen t
with a/2? = 0.24µm given by the ory [2], with most of the
unce rtainty arising from the calibration of the vertical position
of the MFM tip.
The MFMmea sureme nts do not give absolute values ofthe
magn etic field above the surface, but we can use the value ofthe
magn etization from the SQUIDmea sureme nts to estimate how
Figure 5. Micrograph s ofaperpe ndicularly mag netize d grooved we expec t the ato m opt ical elemen ts based on the multilayer
microstruct ure fabricate d with a150 nm thick Gd 10Tb6Fe80Co4 film GdTbFeCo film to perform. From equation (2), the magn itude
ona140 nm thick Cr under layer onasilicon gratin g structure with a of the magn etic field at the surface of the structu re, using
perio d of 3µm. (a) AFM scan and (b) MFM scan. In ( b) the
a= 1.5µm,b= 0.45µm andM = 265 emu cm
grooves are represente d by the light regions and the inset shows a
cross-sectio n of the signal along the indicate d horizonta l line.
cont ribution of higher orde r spatial harm onics in the mag netic
potentia l [19].
The films depo sited ont o the grating struct ure for cold
ato m experiments were fabricated with amultilayer structu re
comprisin g thre e 140 nm thick layers of Cr alternat ing with
thre e 150 nm thick layers of Gd10Tb6Fe80Co4. This gave
− 3 , is abo ut
900 gau ss. Given tha t a87Rb ato m in theF = 2, m = 2
ground state dropp ed fromaheig ht of 10 mm will be reflected
by afield of 16 gau ss, the reflection should occur abo ut one
micron above the surface, well above any surface effects and
effects from the higher order term s in equat ion 1). (
If the film is used on the surface of an ato m chip, the n,
for example, a two-dime nsiona l quadrup ole magn etic trap is
produce d at aheight of 25µm above an edg e of the film in
the x direct ion by appl ying abias magn etic field of 10 gau ss
4019
198
4020

Appendix B
Determining the Temperature
from One Image
In the case in which the trap can be assumed to be harmonic, we have a relation
between the temperature T of the atoms and the trap frequency νT = ωT
2π from
equating the energies:
1
2 kBT = 1
2 m(ωt · q) 2
⇒ T = m
(2π · νT · σ) 2
kB
(B.1)
where the spatial variable q can be substituted by the measurable spread of
the atom cloud in the trap σ. This can be determined directly by an in-situ
absorption image of the atoms or can be calculated from the intercept of the
linear regression of the TOF images which also yield the temperature.
We already know that the harmonic approximation will not hold, especially
at the beginning of the evaporation process. We thus can not use equation
(B.1) to calculate the temperature from the size of the atomic cloud. On the
other hand, we have one set of TOF measurements of the temperature and
of the size of the atomic cloud in the trap for different RF frequencies for a
trap of a nominal frequency of ν1 = 217 Hz. If we plot the temperature of
the atomic cloud for the axial and the radial direction against the respective
spatial widths, as in Fig. B.1, we can use a commercial fitting software 1 to
interpolate between the points. As a fitting function, we chose the function
with the least number of free parameters that describes the behaviour to the
best degree in both directions. It showed that this was a function of the type
1 TableCurve 2D v5.01
T = (a + b ·
199
ln σi
σ2 )
i
−1
(B.2)

temperature in µK
300
250
200
150
100
50
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
atomic spread σ in mm
Figure B.1: To determine the temperature of an atomic cloud from a single
image, the temperature was plotted as a function of atomic cloud size for axial
(solid line, circles) and radial (dotted line, crosses) direction. Marked are the
data points. The fitted functions are explained in the text.
The fitted functions are also shown in the plot.
These functions can now be used to approximate the temperature for the
unsuccessful try. As we used a different trap with ν2 = 100 Hz, we can not
simply use the values for the atom cloud’s spread of those experiments in
the formula. Instead, the cloud size σi has to be corrected for the changing
steepness and curvature. This in a first approximation can be done if we
substitute the spread by ˜σi = ν2
ν1 · σi. Here we assume that the trap frequencies
scale the same in both radial and axial direction. The approximation was
chosen so that it will agree with the behaviour in a harmonic trap, where the
temperature T ∝ (ν · σ) 2 is proportional to the square of the product of the
frequency and the spread. The different number of atoms in the trap for the
different tries was ignored here, as the maximum height of a Gaussian should
not influence the standard deviation or width of that curve.
200

Appendix C
The Atom: 87 Rb
The isotope used in both experiments that are described in this thesis is 87 Rb.
Although the nucleons and electrons are fermionic, having an odd number of
nucleons and electrons adds to an overall even number. Their half-integral
spins couple to an integer spin, and the overall atom is bosonic. Together with
the collision properties and the easily obtained light sources to manipulate
the atoms, this is one of the reasons why 87 Rb has found a widespread use:
this isotope is probably the most common element to be found in atom optics
and Bose-Einstein condensation experiments. Although 87 Rb is unstable and
radioactive, the lifetime is more than 10 10 years and it can be considered as
stable for our purposes.
Rubidium, element 37, is an alkali-metal. The electronic configuration in
the ground state is [Kr]5s; the ground state is thus 5 2 S1/2. The strongest
transition is the D line which is fine split into the transitions to the two states
5 2 P1/2 and 5 2 P3/2, with wavelengths of 794 nm (D1 line) and 780 nm (D2
line). We here consider the D2 line only (see Figure C.1). The states show
a hyperfine splitting, with F = 1, 2 for the ground and F ′ = 0, 1, 2, 3 for the
excited state. These levels are themselves degenerate (2F + 1) times. When
examining absorption spectra, we find two Doppler broadened signals from the
transitions F = 1 → F ′ and F = 2 → F ′ with the different F ′ -states washed
out as their spectral difference is more than a factor of 10 smaller than the
separation of the ground states. Doppler free spectroscopy allows us to resolve
these transitions and their crossover peaks. Extensive information about these
transitions can be found in [Ste03].
201

Figure C.1: The energy levels of the 87 Rb D2 line, taken from [Ste03].
202

Appendix D
Technical Details of the Coils
for the Permanent Magnetic
Chip Experiment
The offset coils
Coils side length separation windings slope
(mm) (mm) (G/A)
Bx 240 335 10 0.235
2nd set 20 0.47
By 210 115 9 0.681
2nd set 20 1.515
Bz 300 55 7 0.49
2nd set 20 1.404
Table D.1: Dimensions and characteristics of the compensating offset coils.
The quadrupole coils
Coils centre separation windings width depth gradient
diameter
(mm) (mm) (mm) (mm) (G/(Acm))
Quad 160 280 400 16 35 ≈ 1
Table D.2: Dimensions and characteristics of the quadrupole coils.
203

The bias field coils
Coils inner diameter separation windings width depth slope
(mm) (mm) (mm) (mm) (G/A)
Bias 400 400 66 24 27.5 2.82
Table D.3: Dimensions and characteristics of the bias field coils for trapping
with a single wire.
trigger / V; current / A
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
(a)
0 5 10 15 20 25 30 35
time / ms
trigger / V; current / A
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
(b)
0 0.5 1 1.5 2 2.5 3 3.5
time / ms
Figure D.1: Switching times of the coils: (a) switch on, (b) switch off. Dotted/thin
line: trigger pulse; solid/thick line: current through the coils.
204

Appendix E
List of Equipment
[E 1]: tapered amplifier: TA 100, electronics: SC 100, DC 100, DCC
100, DTC 100; TOptica
[E 2]: acousto-optical modulator: 1205C-2; Isomet
[E 3]: PID Regulator: PID 100 ; TOptica
[E 4]: acousto-optical modulator: 1206C; Isomet
[E 5]: polarisation preserving fibre: PMJ-A3A-3AF-800-5/125-3-5-1;
Oz Optics
[E 6]: coupler and collimation: HPUC-2,A3A-780-P-11AS-11 and HPUCO-
23AF-800-P-6.2AS; Oz Optics
[E 7]: diode laser: DL 100, electronics: SC 100, DCC 100, DTC 100;
TUI Optics (now: TOptica)
[E 8]: lock-in regulator: LIR 100; TUI Optics (now TOptica)
[E 9]: current and temperature stabilisation: LDC500 and TED200;
Thorlabs
[E 10]: shutter and driver: LS612 and VMM-D1; UniBlitz
[E 11]: fast photodiode: Si Nanosecond Photodetector (1621); Newfocus
[E 12]: CCD camera: Micromax 1024CCD; Princeton Instruments
[E 13]: cold cathode gauge: TPG 300; Pfeiffer Vacuum
[E 14]: turbo pump: TMU 065 DN 63 CF-F, 1P (50 l/s); Balzers Pfeiffer
205

[E 15]: diaphragm pump: MD4T (3.3 m 3 /h); Vacuubrand, distributor:
Balzers Pfeiffer
[E 16]: ion pump: Varian Triode (75 l/s); Varian
[E 17]: Ti: sublimation pump and control unit: TSP 2140412 and 224-
0550; Physical Electronics
[E 18]: PCB mill: Quick Circuit 5000
[E 19]: RF synthesizer: SRS 200, Stanford
[E 20]: thin film deposition system: CMS-18; Kurt J. Lasker
[E 21]: digital and analogue I/O board: PCI-6733, BNC 2110; National
Instruments
[E 22]: turbo pump: TMU 260 (210 l/s); Pfeiffer Vacuum
[E 23]: ion pump: Varian Noble Diode; Varian
[E 24]: Ti: sublimation pump: SS-400/275; Thermionics Laboratory
Inc.
[E 25]: tapered amplifier: TA 100; TOptica
[E 26]: lock-in amplifier: Type 401A; Brookdeal
[E 27]: acousto-optical modulator: 3110-120; Crystal Technology
[E 28]: current, temperature controller: DCC 100, DTC 100; TOptica
[E 29]: acousto-optical modulator: 3200-124; Crystal Technology
[E 30]: acousto-optical modulator: 3200-121; Crystal Technology
[E 31]: disk laser: VersaDisk; ELS Elektronik Laser System GmbH
[E 32]: CCD camera: chip type: KAF0400; Photometrics Sensys
[E 33]: high power laser diode: DLX 100; TOptica
206

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