Towards Bose-Einstein Condensation of Sodium - Kirchhoff-Institut ...
Towards Bose-Einstein Condensation of Sodium - Kirchhoff-Institut ...
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Faculty <strong>of</strong> Physics and Astronomy<br />
University <strong>of</strong> Heidelberg<br />
Diploma thesis<br />
in Physics<br />
submitted by<br />
Anton Piccardo-Selg<br />
born in Stockton, California, USA<br />
2008
Degenerate Quantum Gases:<br />
<strong>Towards</strong> <strong>Bose</strong>-<strong>Einstein</strong> <strong>Condensation</strong> <strong>of</strong> <strong>Sodium</strong><br />
This diploma thesis has been carried out by Anton Piccardo-Selg at the<br />
Kirchh<strong>of</strong>f <strong>Institut</strong> für Physik<br />
under the supervision <strong>of</strong><br />
Pr<strong>of</strong>. Dr. M. K. Oberthaler
Auf dem Weg zur <strong>Bose</strong>-<strong>Einstein</strong> Kondensation von<br />
Natrium<br />
Gegenstand der vorliegenden Diplomarbeit ist die Inbetriebnahme eines Experiments zur<br />
Erzeugung ultrakalter bosonischer und fermionischer Quantengase. Fermionisches 6 Li<br />
soll mit Hilfe von bosonischem 23 Na sympathetisch gekühlt werden. Es wird die magnetoopische<br />
Falle, die Magnetfalle und das evaporative Kühlen von 23 Na untersucht. Eine<br />
anfängliche Thermalisierungszeit von 500 ms wurde in der Magnetfalle gemessen. Mittels<br />
evaporativem Kühlen konnte das runaway Regime erreicht werden. Hierbei wurde die<br />
Phasenraumdichte um zwei Größenordnungen auf 10 −4 erhöht. Der Einsatz eines optischen<br />
Stöpsels wird die erforderlichen letzen vier Größenordnungen zum Erreichen eines<br />
<strong>Bose</strong>-<strong>Einstein</strong> Kondensats ermöglichen.<br />
<strong>Towards</strong> <strong>Bose</strong>-<strong>Einstein</strong> <strong>Condensation</strong> <strong>of</strong> <strong>Sodium</strong><br />
The topic <strong>of</strong> this diploma thesis, is the implementation <strong>of</strong> an experiment for ultracold<br />
bosonic and fermionic quantum gases. Fermionic 6 Li is going to be cooled sympathetically<br />
with the help <strong>of</strong> bosonic 23 Na. The experiment’s magneto-optical trap, magnetic<br />
trap and evaporative cooling are discussed. An initial thermalization time <strong>of</strong> 500 ms<br />
was measured in the magnetic trap. Using evaporative cooling, the runaway regime was<br />
observed. The phase-space density was raised by two orders <strong>of</strong> magnitude to 10 −4 . An<br />
optical plug will allow for the last four orders <strong>of</strong> magnitude needed to obtain a <strong>Bose</strong>-<br />
<strong>Einstein</strong> condensate.
I<br />
Contents<br />
1 Introduction 1<br />
1.1 Degenerate <strong>Bose</strong> gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2<br />
1.2 Degenerate Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3<br />
1.3 The NaLi experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />
2 The first cooling stages 7<br />
2.1 Light forces and atomic physics . . . . . . . . . . . . . . . . . . . . . . . 7<br />
2.1.1 The scattering force . . . . . . . . . . . . . . . . . . . . . . . . . . 7<br />
2.1.2 The dipole force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />
2.1.3 Atoms in static magnetic fields . . . . . . . . . . . . . . . . . . . 9<br />
2.1.4 Cold collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />
2.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />
2.2.1 Facts about 23 Na . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />
2.2.2 Laser setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />
2.2.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 15<br />
2.2.4 The Zeeman Slower . . . . . . . . . . . . . . . . . . . . . . . . . . 17<br />
2.3 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17<br />
2.4 The magneto-optical trap (MOT) . . . . . . . . . . . . . . . . . . . . . . 18<br />
2.4.1 MOT theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18<br />
2.4.2 MOT temperature and Doppler limit . . . . . . . . . . . . . . . . 20<br />
2.4.3 Density limitations and loss mechanisms in a MOT . . . . . . . . 21<br />
2.4.4 Dark spot MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24<br />
2.4.5 Loading <strong>of</strong> the MOT for different temperatures . . . . . . . . . . 24<br />
2.5 Sub-Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25<br />
2.5.1 σ + -σ − cooling basics . . . . . . . . . . . . . . . . . . . . . . . . . 26<br />
2.5.2 Sub-Doppler cooling in the experiment . . . . . . . . . . . . . . . 28<br />
3 The magnetic trap 31<br />
3.1 Types <strong>of</strong> magnetic traps . . . . . . . . . . . . . . . . . . . . . . . . . . . 31<br />
3.2 Setup <strong>of</strong> the magnetic trap . . . . . . . . . . . . . . . . . . . . . . . . . . 33<br />
3.2.1 The Feshbach coils . . . . . . . . . . . . . . . . . . . . . . . . . . 33<br />
3.2.2 The circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34<br />
3.2.3 The fast turn-<strong>of</strong>f . . . . . . . . . . . . . . . . . . . . . . . . . . . 37<br />
3.2.4 The field switch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38<br />
3.3 Trappable states and the first steps in the magnetic trap . . . . . . . . . 40
II<br />
Contents<br />
3.3.1 Trappable states <strong>of</strong> 23 Na . . . . . . . . . . . . . . . . . . . . . . . 40<br />
3.3.2 Mode matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40<br />
3.3.3 Adiabatic compression . . . . . . . . . . . . . . . . . . . . . . . . 42<br />
3.4 Loss mechanisms and heating . . . . . . . . . . . . . . . . . . . . . . . . 45<br />
3.4.1 One-body losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45<br />
3.4.2 Two-body losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51<br />
3.4.3 Three-body losses . . . . . . . . . . . . . . . . . . . . . . . . . . . 53<br />
3.4.4 Lifetime measurement . . . . . . . . . . . . . . . . . . . . . . . . 53<br />
3.5 Rethermalization measurement . . . . . . . . . . . . . . . . . . . . . . . 54<br />
3.6 The plugged quadrupole field . . . . . . . . . . . . . . . . . . . . . . . . 56<br />
3.6.1 Plug in the NaLi experiment . . . . . . . . . . . . . . . . . . . . . 56<br />
3.6.2 Compensation <strong>of</strong> astigmatism . . . . . . . . . . . . . . . . . . . . 59<br />
4 Evaporative cooling 61<br />
4.1 Principle <strong>of</strong> evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . 61<br />
4.2 Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . . . . 66<br />
4.2.1 The microwave setup . . . . . . . . . . . . . . . . . . . . . . . . . 66<br />
4.2.2 Evaporative cooling in the experiment . . . . . . . . . . . . . . . 67<br />
5 Conclusion and Outlook 71<br />
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71<br />
5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71<br />
Bibliography 78
III<br />
List <strong>of</strong> Figures<br />
1.1 Quantum statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2<br />
1.2 Principle <strong>of</strong> Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . 5<br />
2.1 Breit-Rabi diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />
2.2 Level scheme <strong>of</strong> <strong>Sodium</strong> . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />
2.3 Setup <strong>of</strong> the laser table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />
2.4 Setup <strong>of</strong> the experimental table . . . . . . . . . . . . . . . . . . . . . . . 16<br />
2.5 Working principle <strong>of</strong> the MOT . . . . . . . . . . . . . . . . . . . . . . . . 19<br />
2.6 Time-<strong>of</strong>-flight series <strong>of</strong> the MOT . . . . . . . . . . . . . . . . . . . . . . . 21<br />
2.7 Light-assisted collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23<br />
2.8 Loading rate <strong>of</strong> the <strong>Sodium</strong> MOT for different temperatures . . . . . . . 25<br />
2.9 Clebsch Gordan coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 26<br />
2.10 Principle <strong>of</strong> sub-Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . 27<br />
2.11 Measurement <strong>of</strong> sub-Doppler cooling . . . . . . . . . . . . . . . . . . . . 28<br />
3.1 Magnetic field <strong>of</strong> the Feshbach coils . . . . . . . . . . . . . . . . . . . . . 34<br />
3.2 Electric circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36<br />
3.3 Current regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37<br />
3.4 PSpice simulation <strong>of</strong> the current in the Feshbach coils . . . . . . . . . . . 38<br />
3.5 Pre-logic <strong>of</strong> the IGBT drivers . . . . . . . . . . . . . . . . . . . . . . . . 39<br />
3.6 Mode matching measurement . . . . . . . . . . . . . . . . . . . . . . . . 42<br />
3.7 Adiabatic compression <strong>of</strong> the magnetic trap . . . . . . . . . . . . . . . . 44<br />
3.8 Majorana losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47<br />
3.9 Lifetime due to Majorana losses . . . . . . . . . . . . . . . . . . . . . . . 48<br />
3.10 Lifetime measurements with Majorana losses . . . . . . . . . . . . . . . . 49<br />
3.11 Fourier spectrum <strong>of</strong> the current . . . . . . . . . . . . . . . . . . . . . . . 50<br />
3.12 Lifetime measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54<br />
3.13 Rethermalization <strong>of</strong> the magnetic trap . . . . . . . . . . . . . . . . . . . 55<br />
3.14 Combined magnetic and optical potential . . . . . . . . . . . . . . . . . . 58<br />
3.15 Plug shot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59<br />
4.1 Principle <strong>of</strong> evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . 62<br />
4.2 Patch antenna and the S11 parameter . . . . . . . . . . . . . . . . . . . . 67<br />
4.3 Phase-space density and atom number during evaporation . . . . . . . . 68<br />
4.4 Phase-space density vs. atom number . . . . . . . . . . . . . . . . . . . . 69
IV<br />
Fundamental Constants<br />
Quantity Symbol Value Unit<br />
Speed <strong>of</strong> light c 2.99792458 · 10 8 ms −1<br />
Boltzmann constant k B 1.3806503(24) · 10 −23 JK −1<br />
Planck constant h 6.62606876(52) · 10 −34 Js<br />
h/2π 1.054571596(82) · 10 −34 Js<br />
Bohr magneton µ B 9.27400949(80) · 10 −24 JT −1<br />
Specific data <strong>of</strong> <strong>Sodium</strong>-23<br />
Mass m 3.81754023(30) · 10 −26 kg<br />
Saturation intensity I 0 6.2600(51) mWcm −2<br />
D2-line width Γ 2π × 9.795(11) MHz<br />
Doppler temperature T D 235 µK<br />
Recoil temperature T Rec 2.3998 µK
1<br />
Chapter 1<br />
Introduction<br />
In the 5th century B.C. Democritus developed his theory <strong>of</strong> the “atomos” – the indivisible<br />
entity that builds up the world in limitless varieties <strong>of</strong> shapes and sizes. Other cultures<br />
had developed this theory even earlier. Modern occidental atomic theory began in the<br />
19th century with John Dalton. His ideas were developed further by Thomas, Rutherford<br />
and many others. With the development <strong>of</strong> quantum mechanics, it was finally possible<br />
to describe processes in atoms that were previously not intuitively accessible.<br />
Elementary and composed particles are categorized into bosons and fermions, having<br />
symmetric and antisymmetric wave functions, respectively. They are characterized by<br />
their integer and half-integer spin properties. This little difference has great effects on<br />
the behaviour <strong>of</strong> ensembles <strong>of</strong> such particles. It is <strong>of</strong> quantum mechanical nature and<br />
in the classical limit not apparent. For extreme situations as in ultracold gases, where<br />
the particles’ wavepackets start to overlap, this categorization is crucial. This is shown<br />
in figure (1.1). The phase-space density <strong>of</strong> the system determines whether a gas can be<br />
described in terms <strong>of</strong> classical physics or has to be treated quantum mechanically.<br />
If the phase-space density <strong>of</strong> the system is on the order <strong>of</strong> one, identical bosons<br />
undergo a phase transition called <strong>Bose</strong>-<strong>Einstein</strong>-<strong>Condensation</strong> (BEC) after Satyendra<br />
Nath <strong>Bose</strong> and Albert <strong>Einstein</strong>, who first discovered the underlying quantum statistics.<br />
The <strong>Bose</strong>-<strong>Einstein</strong> Condensate is described as a macroscopic occupation <strong>of</strong> the system’s<br />
ground state.<br />
After the publications <strong>of</strong> <strong>Bose</strong> [1] and <strong>Einstein</strong> [2], it would take about 70 years to<br />
realize such a <strong>Bose</strong>-<strong>Einstein</strong> condensate experimentally. In 1995 the group <strong>of</strong> Eric Cornell<br />
and Carl Wiemann at the JILA [3], the group <strong>of</strong> Wolfgang Ketterle at the MIT [4] and<br />
the group <strong>of</strong> Randy Hulet [5] at the RICE University accomplished this goal. The alkali<br />
metals 87 Rb, 23 Na and 7 Li were used back then.<br />
Identical fermions, on the other hand, do not show this behaviour. They are governed<br />
by the Pauli-principle that prohibits the occupation <strong>of</strong> the same quantum state for two<br />
identical fermions. Therefore the condensation process cannot happen. Rather, they<br />
start to fill up the different states, starting from the ground state. This leads to the socalled<br />
Fermi sea. The gas does not pass through a phase transition. A degenerate Fermi<br />
gas is obtained when the phase-space density is on the order <strong>of</strong> one 1 . The concept <strong>of</strong><br />
the Fermi sea is, for example, used in nuclear physics, the physics <strong>of</strong> metals and neutron<br />
stars.<br />
1 Degeneracy, in the case <strong>of</strong> fermions, means that the energy states are filled up with atoms from the<br />
ground state on.
2 1.1 Degenerate <strong>Bose</strong> gases<br />
Some years after the first production <strong>of</strong> <strong>Bose</strong>-<strong>Einstein</strong> Condensates, there was great<br />
effort to produce degenerate Fermi gases. First success was reported in 1999 [6] using<br />
40 K. Other groups followed using 6 Li [7].<br />
d)<br />
a) b)<br />
c)<br />
300K<br />
0K<br />
temperature<br />
Figure 1.1: a) Atoms at room temperature behave classically and can be described as “pointlike”<br />
particles. b) With decreasing temperature and increasing density the wave nature<br />
<strong>of</strong> the atoms cannot be disregarded. Collective effects come into play. c) In the case<br />
<strong>of</strong> fermions the Pauli-principle holds and they fill up the trap energy levels from the<br />
lowest state on. d) At ultra-low temperatures bosons occupy the lowest energy state <strong>of</strong><br />
the trap.<br />
1.1 Degenerate <strong>Bose</strong> gases<br />
In the framework <strong>of</strong> quantum statistical mechanics the distribution function for <strong>Bose</strong><br />
particles is given by<br />
1<br />
n(E ν ) =<br />
exp [(E ν − µ)/k B T ] − 1 . (1.1)<br />
µ is the chemical potential <strong>of</strong> the system. It describes the energy needed if a particle<br />
is added to the ensemble. E ν denotes the energy <strong>of</strong> the particle in state ν. It can be<br />
easily seen that the chemical potential must be less than the energy E 0 <strong>of</strong> the lowest<br />
state. Else unphysical, negative occupation numbers would arise. When µ approches E 0<br />
for small decreasing temperatures, the occupation <strong>of</strong> the ground state enhances. The<br />
critical temperature<br />
( ) 2/3 n h 2<br />
T c =<br />
(1.2)<br />
ζ(3/2) 2πmk B<br />
determines when condensation takes place. Here ζ(3/2) denotes the Riemann zeta function,<br />
n the density and m the mass. This equation holds for non-interacting, ideal gases.<br />
The description <strong>of</strong> a non-ideal <strong>Bose</strong> gas is immensely more difficult. Mean-field<br />
theories have been developed to take interactions into account. Commonly the model <strong>of</strong>
1.2 Degenerate Fermi gases 3<br />
Gross [8] and Pitaevskii [9] is used. It holds for T = 0, large atom numbers and weak<br />
interactions. The time evolution is given by the Gross-Pitaevskii equation (GPE)<br />
i d [<br />
]<br />
dt Ψ(⃗r, t) = − 2<br />
2m ∇2 + V (⃗r) + U 0 |Ψ(⃗r, t)| 2 Ψ(⃗r, t) (1.3)<br />
with U 0 = 4π 2 a 0 /m, where the s-wave scattering length a 0 gives the scale <strong>of</strong> the interaction.<br />
The last term is the contact potential and describes the short-ranged, weak<br />
interactions <strong>of</strong> the degenerate <strong>Bose</strong> gas. Ψ(r, t) is not a single particle wave function but<br />
rather the condensate wave function. The particle density is given by n(⃗r, t) = |Ψ(⃗r, t)| 2 .<br />
The stationary equation is<br />
]<br />
[− 2<br />
2m ∇2 + V (⃗r) + U 0 |Ψ(⃗r)| 2 Ψ(⃗r) = µΨ(⃗r) (1.4)<br />
where µ is the chemical potential. An interesting limit <strong>of</strong> the GPE is the Thomas-<br />
Fermi approximation. This limit is obtained when the kinetic energy <strong>of</strong> the particles is<br />
negligible compared to the interaction energy <strong>of</strong> the ensemble. The kinetic energy term<br />
in the GPE is canceled, resulting in<br />
n(⃗r) = 1 U 0<br />
[µ − V (⃗r)] (1.5)<br />
for µ > V (⃗r) and n(⃗r) = 0 otherwise. It can be seen that the density distribution <strong>of</strong><br />
the BEC is just given by the shape <strong>of</strong> the confining potential. In a physical perspective<br />
the Thomas-Fermi approximation means that the chemical potential is assumed to be<br />
uniform over the whole sample, such that adding a particle costs the same everywhere<br />
in the trap.<br />
1.2 Degenerate Fermi gases<br />
The distribution function for fermions is given by<br />
n(E ν ) =<br />
1<br />
exp [(E ν − µ)/k B T ] + 1 . (1.6)<br />
where E ν is defined as above. For T = 0 the distribution function is a Heaviside stepfunction<br />
that is unity for E ν < µ and vanishes otherwise. The energy at the edge is<br />
called Fermi energy E F and is equal to the chemical potential at zero temperature. With<br />
this energy the Fermi temperature T F = E F /k B can be defined which gives a measure<br />
<strong>of</strong> when a fermionic system can be called degenerate. Values for this temperature vary<br />
dramatically in different fields <strong>of</strong> physics. As described in many textbooks [10, 11], the<br />
Fermi energy <strong>of</strong> atoms in the homogenous case <strong>of</strong> a box is<br />
E F = (6π2 ) 2/3<br />
2<br />
2<br />
m n2/3 . (1.7)<br />
To obtain the density distribution in a trap, the Thomas-Fermi approximation is applied<br />
again. The cost for adding a particle is assumed to be the same all over the trap. Local
4 1.2 Degenerate Fermi gases<br />
Fermi energies E F (⃗r) can be defined. Setting the sum <strong>of</strong> the local Fermi energy and the<br />
trapping potential equal to the chemical potential [12],<br />
n(⃗r) = 1<br />
6π 2 [ 2m<br />
2 [µ − V (⃗r)] ] 3/2<br />
(1.8)<br />
is obtained. The density distribution is different from equation (1.5) and shows an<br />
enhanced pr<strong>of</strong>ile in the trap center.<br />
The production <strong>of</strong> a degenerate Fermi gas is much more complicated than the production<br />
<strong>of</strong> a <strong>Bose</strong>-<strong>Einstein</strong> Condensate, since collision processes in the ultracold regime<br />
are suppressed between fermionic same-state particles. Therefore several sophisticated<br />
methods have been developed to overcome this difficulty. Jin et al. used several spin<br />
components <strong>of</strong> 40 K atoms to reach a degeneracy <strong>of</strong> T/T F = 0.5 [6]. A more recent experiment<br />
<strong>of</strong> Ketterle et al. showed a degeneray <strong>of</strong> T/T F = 0.09 [13]. Here the cooling scheme<br />
made use <strong>of</strong> cold bosonic 23 Na atoms. The 23 Na atoms can be cooled efficiently with<br />
evaporative cooling which will be explained in chapter 4. 6 Li atoms are cooled indirectly<br />
via collisions with the cold 23 Na sample. The two-species gas reaches a temperature that<br />
is way below the initial temperature <strong>of</strong> the lithium sample. The 23 Na atoms are used as<br />
sort <strong>of</strong> refrigiator. This cooling method is known as sympathetic cooling.<br />
Feshbach resonances<br />
In recent years a main focus has been put on the studies <strong>of</strong> interactions between particles.<br />
The occurance <strong>of</strong> Feshbach resonances allows one to tune the s-wave scattering length<br />
and hence gives a method to control the interaction strength <strong>of</strong> fermion-, fermion-bosonor<br />
boson-pairs.<br />
A Feshbach resonance occurs when the kinetic energy <strong>of</strong> two scattering particles<br />
entering the open channel is close to the total energy <strong>of</strong> a bound interatomic state in<br />
the closed channel as depicted in figure (1.2). The channel is called closed if the energy<br />
<strong>of</strong> the particles is below the dissociation threshold. This resonance behaviour can be<br />
realized by tuning an external magnetic field. Different states have different magnetic<br />
moments, hence the field shifts these levels differently. The two channels can be tuned<br />
with respect to each other to have the same energy. The dependence <strong>of</strong> the scattering<br />
length on the external magnetic field is given by<br />
[<br />
a(B) = a 0 1 − ∆B ]<br />
B − B 0<br />
(1.9)<br />
[14] for small scattering energies, where ∆B is the width <strong>of</strong> the resonance, B 0 the field at<br />
which the resonance occurs and a 0 the non-resonant s-wave scattering length. Feshbach<br />
resonances have been found in many alkalis such as 23 Na [15], 6 Li [16] and in interspecies<br />
mixtures [17]. Some resonances are shown in table (1.1). There are many more resonances<br />
than the ones listed and research is being done to find new ones at higher fields.<br />
The most important one for 6 Li is the 834 G resonance as it is very broad and therefore<br />
tuning through it can be done easily.
1.3 The NaLi experiment 5<br />
energy<br />
d)<br />
c)<br />
b)<br />
a)<br />
scattering length<br />
repulsive<br />
interactions<br />
0<br />
attractive<br />
interactions<br />
position<br />
magnetic field B<br />
Figure 1.2: left: a) and b) are an open and a closed channel <strong>of</strong> the interaction potential,<br />
respectively. The particle arriving in the open channel experiences at c) a coupling<br />
between the two channels and can make a transition to the closed channel. This happens<br />
if the particle’s total energy is close to the energy <strong>of</strong> a bound state in the closed channel.<br />
right: The dependence <strong>of</strong> the s-wave scattering length on the external magnetic<br />
field in the vincity <strong>of</strong> a Feshbach resonance is shown. The scattering length diverges<br />
at the resonance. In the case depicted the scattering length changes sign when passing<br />
through the Feshbach resonance. Repulsive interactions are present if the sacttering<br />
length is positive, whereas for negative scattering lengths the interaction is attractive.<br />
Table 1.1: Feshbach resonances <strong>of</strong> 6 Li and 23 Na. The data is from [15, 16, 17].<br />
Species B 0 [G] ∆B[G] Species B 0 [G] ∆B[G]<br />
6 Li- 6 Li 159.14 0.40 6 Li- 23 Na 746.0 -<br />
6 Li- 6 Li 185.09 0.40 6 Li- 23 Na 759.6 -<br />
6 Li- 6 Li 214.28 0.40 6 Li- 23 Na 795.6 -<br />
6 Li- 6 Li 543.28 0.40 23 Na- 23 Na 907 -<br />
6 Li- 6 Li 822 . . . 834 23 Na- 23 Na 855 -<br />
1.3 The NaLi experiment<br />
The <strong>Sodium</strong>-Lithium experiment (NaLi) aims to produce ultracold Fermi gases. Large<br />
samples <strong>of</strong> fermionic 6 Li are intended to be cooled far into degeneracy. The attained<br />
degeneracies mentioned before certainly determine a goal for us. Models describing
6 1.3 The NaLi experiment<br />
interactions beyond the established mean-field theories need to be investigated. Locally,<br />
the theory department <strong>of</strong> the university <strong>of</strong> Heidelberg does research on renormalizationgroups<br />
in quantum field theory [18, 19].<br />
But not only pure Fermi gases are <strong>of</strong> interest in this experiment. Having bosons in<br />
the vacuum chamber makes the study <strong>of</strong> interspecies effects appealing. With the help <strong>of</strong><br />
interspecies Feshbach resonances it is possible to study boson-mediated interactions in a<br />
gas <strong>of</strong> dilute fermions. Fermions propagating through the bath <strong>of</strong> bosons are a subject <strong>of</strong><br />
interest here. Other fields <strong>of</strong> interest are gases in lower dimensional traps and possibly<br />
optical lattices. But these projects still lie far in the future.<br />
So far the first steps have been taken. The experimental chamber has been set up<br />
as well as the main laser system. Several more lasers are about to be installed in the<br />
coming months. The cooling steps have been implemented and our first milestone – a<br />
23 Na BEC – seems to be in reach.<br />
In this thesis I will report on the progress made since the experimental framework<br />
<strong>of</strong> NaLi has been set up. Several cooling steps were introduced during my stay. In<br />
particular I will describe the control <strong>of</strong> the electrical circuit, the change <strong>of</strong> configuration<br />
<strong>of</strong> the current and the fast turn-<strong>of</strong>f <strong>of</strong> the magnetic trap. The thesis is not structured<br />
as <strong>of</strong>ten done in theory and experiment, but rather follows chronologically the different<br />
steps <strong>of</strong> the experiment.
7<br />
Chapter 2<br />
The first cooling stages<br />
The first part <strong>of</strong> the atoms’ path to ultracold temperatures is governed by laser cooling.<br />
It seems counterintuitive to cool with light, since light is normally associated with heat.<br />
Still, by almost no other means than laser cooling in combination with evaporative<br />
cooling (see chapter 4) it has been possible to reach temperatures as low as the µK and<br />
nK regime. This makes the lab (on a small scale) one <strong>of</strong> the coldest places in the entire<br />
universe (to the best <strong>of</strong> our knowledge). 1<br />
2.1 Light forces and atomic physics<br />
The atoms <strong>of</strong> interest for most groups in the field <strong>of</strong> ultracold atoms are alkalis, foremost<br />
due to their hydrogen-like electronic structure. These atoms have a pseudo-closed twolevel<br />
system which makes the theoretical description <strong>of</strong> the processes in the experiment<br />
significantly simpler.<br />
The equations governing the temporal evolution <strong>of</strong> the density matrix ρ <strong>of</strong> the system<br />
are the optical Bloch equations that can be derived from the von-Neuman equation<br />
i dρ<br />
dt<br />
= [H, ρ]. (2.1)<br />
The effect <strong>of</strong> spontaneous emmision is introduced by hand [20]. H is the system’s Hamiltonian.<br />
The Hamiltonian H can be related with help <strong>of</strong> the Ehrenfest theorem to the<br />
force acting on the atom<br />
〈 〉<br />
⃗F = − ⃗∇H . (2.2)<br />
The expectation value in equation (2.2) is evaluated by taking the trace <strong>of</strong> ρ ⃗ ∇H. This<br />
results for the steady-state in two light forces. First, the scattering force which is <strong>of</strong><br />
dissipative nature and is used for cooling; second, the conservative dipole force used for<br />
trapping and manipulating atoms.<br />
2.1.1 The scattering force<br />
The scattering force is the underlying force <strong>of</strong> the magneto-optical trap (MOT) and<br />
the Zeeman slower. It results from the scattering properties <strong>of</strong> the atom with light.<br />
1 By means <strong>of</strong> adiabatic demagnetization and the dilution refrigiator comparable temperatures can<br />
be achieved.
8 2.1 Light forces and atomic physics<br />
The atoms can absorb light that is at or close to the transition frequency ω 0 . The frequency<br />
range around ω 0 , at which the light can have an effect on the atom is determined<br />
by the linewidth Γ, ω 0 itself as well as the frequency and intensity <strong>of</strong> the light. The<br />
absorbed photon does not only excite the electron but also transfers the momentum<br />
⃗ k = (ω 0 /c)⃗e prop to the atom where ⃗ k is the wave vector <strong>of</strong> the photon propagating in<br />
the direction <strong>of</strong> ⃗e prop . The excited atom decays to the ground state and emits a photon,<br />
causing again a change <strong>of</strong> momentum. The force depends on how <strong>of</strong>ten a photon scatters<br />
from the atom. The scattering force results therefore in<br />
The scattering rate is [20]<br />
F scatt = (momentum <strong>of</strong> a photon) × (scattering rate). (2.3)<br />
R scatt = Γ 2<br />
I/I 0<br />
1 + I/I 0 + 4∆ 2 /Γ 2 . (2.4)<br />
∆ = ω − ω 0 is the detuning <strong>of</strong> the laser light ω with respect to the transition frequency.<br />
In the case <strong>of</strong> moving atoms, the Doppler shift has to be taken into account as is done<br />
in subsection 2.4.1. I 0 = (hcπΓ)/(3λ 3 ) is the saturation intensity <strong>of</strong> the transition and<br />
is determined by the life time <strong>of</strong> the excited state. λ is the wavelength corresponding to<br />
the transition frequency ω 0 . At I 0 one-quarter <strong>of</strong> the atomic ensemble is in the excited<br />
state if there is no detuning <strong>of</strong> the light. At very high intensities the scattering rate<br />
does not increase further but causes undesired effects like saturation broadening. The<br />
Lorentzian shape <strong>of</strong> the scattering rate is just a consequence <strong>of</strong> the lineshape <strong>of</strong> the<br />
atomic transition. Finally, the scattering force can be written as<br />
⃗F = ⃗ k Γ 2<br />
I/I 0<br />
1 + I/I 0 + 4∆ 2 /Γ 2 (2.5)<br />
with a maximal force <strong>of</strong> | ⃗ F max | = | ⃗ k Γ 2 | when the light is in resonance and I/I 0 ≫ 1.<br />
2.1.2 The dipole force<br />
As mentioned before, the dipole force is not dissipative and hence cannot be used for<br />
cooling atoms. It does not rely on spontaneous emission, which causes the irreversibilty<br />
in the case <strong>of</strong> the scattering force. The force is given by [20]<br />
⃗F DIP = − ∆ 2<br />
Ω<br />
∆ 2 + Ω 2 /2 + Γ 2 /4 ⃗ ∇Ω. (2.6)<br />
In contrast to the scattering force the dipole force vanishes for resonant light. The<br />
on-resonance Rabi frequency Ω = −(eE 0 /) 〈e| ⃗r⃗e prop |g〉 depends on the electric field<br />
amplitude E 0 <strong>of</strong> the light. e is the electron charge. It is a measure for the electric<br />
dipole transition strength between the ground state |g〉 and the excited state |e〉. The<br />
dipole force is used in experiments with far-<strong>of</strong>f resonant traps. It is useful to deduce the<br />
approximated potential (Γ ≪ |∆|, Ω ≪ |∆|)<br />
U DIP = Γ 8<br />
Γ I<br />
(2.7)<br />
∆ I 0
2.1 Light forces and atomic physics 9<br />
for this far-<strong>of</strong>f resonance region. The scattering rate drops as R scatt ∝ I/∆ 2 , therefore<br />
the dipole force ceases to be slower than the scattering rate. The force depends on the<br />
sign <strong>of</strong> the detuning. For blue-detuned light (∆ > 0) the force is repulsive and the atom<br />
will be driven to minima <strong>of</strong> the light field, whereas in the case <strong>of</strong> red detuning (∆ < 0)<br />
the atom experiences an attractive force to regions where the light field maximum lies.<br />
This force is commonly used to create optical trapping potentials. In future experiments<br />
NaLi will make use <strong>of</strong> these optical dipole traps. Two different traps will be<br />
installed. A high power laser at 1064 nm and a dye laser, which is only several nm away<br />
from a transition in 6 Li, will be used. The first laser creates an equal potential for both<br />
species, whereas the latter can produce a potential that is significantly stronger for 6 Li<br />
than for 23 Na.<br />
2.1.3 Atoms in static magnetic fields<br />
If atoms are exposed to an external magnetic field, the two will couple. The linear part<br />
<strong>of</strong> the interaction-Hamiltonian is given by<br />
H JB = − µ B<br />
g J ⃗ J ⃗ B ext (2.8)<br />
The total angular momentum J ⃗ = L ⃗ + S ⃗ <strong>of</strong> the electron orbiting around the nucleus<br />
is coupled to the external magnetic field B ⃗ ext with µ B being Bohr’s magneton. It is<br />
composed <strong>of</strong> the angular momentum L ⃗ and the spin S ⃗ <strong>of</strong> the electron. The Landé<br />
g-factor is given by<br />
g J = 1 +<br />
J(J + 1) + S(S + 1) − L(L + 1)<br />
. (2.9)<br />
2J(J + 1)<br />
For most field strengths equation (2.8) is sufficient, but for high magnetic fields also<br />
the quadratic Zeeman effect becomes important. Another relevant part in the total<br />
Hamiltonian is the L ⃗ S-coupling ⃗ term resulting from the coupling between the electron’s<br />
spin and the magnetic field produced by the nucleus being a moving charge in the<br />
momentary rest-frame <strong>of</strong> the electron. The Zeeman effect exists for external magnetic<br />
fields if<br />
µ B g JJ ⃗ Bext ⃗ ≪ µ n g II ⃗ Bint ⃗ (2.10)<br />
is fulfilled. ⃗ Bint is the internal magnetic field at the nucleus caused by the orbiting<br />
electron. The nuclear spin ⃗ I produced by the nucleons couples to the internal magnetic<br />
field. Analogous to the electron, µ n is the nuclear magneton and g I the nuclear g-factor.<br />
If equation (2.10) is fulfilled, then the coupling between ⃗ I and ⃗ J is stronger than the<br />
coupling <strong>of</strong> each single one <strong>of</strong> them to the external magnetic field. The total atomic<br />
angular momentum can then be defined as<br />
⃗F = ⃗ I + ⃗ J. (2.11)<br />
The projection states m F <strong>of</strong> the total atomic angular momentum determine the Zeeman<br />
energy shift<br />
∆E Zeeman = µ B g F m F | ⃗ B ext |. (2.12)
10 2.1 Light forces and atomic physics<br />
in the external magnetic field. If the external field is very strong, not fulfilling equation<br />
(2.10), then ⃗ I and ⃗ J couple separately to the external magnetic field. The energy splitting<br />
in this Paschen-Back regime is<br />
with<br />
∆E Paschen-Back = µ B g J m J | ⃗ B ext | + Am I m J (2.13)<br />
A = − µ Iµ J | B ⃗ int |<br />
. (2.14)<br />
IJ<br />
| ⃗ B int | is the in avarage internal magnetic field.<br />
In the intermediate regime the two different forms <strong>of</strong> coupling need to be treated as<br />
equal. This is described by the Breit-Rabi formula which can be found in textbooks [21].<br />
The result for 23 Na and 6 Li is shown in figure (2.1). Some states bend with increasing<br />
magnetic field strength which is important to keep in mind when thinking about magnetic<br />
trapping.<br />
energy[a.u.]<br />
F=2<br />
a) b)<br />
m j<br />
=+1/2<br />
energy[a.u.]<br />
F=3/2<br />
m j<br />
=+1/2<br />
F=1<br />
m j<br />
=-1/2<br />
F=1/2<br />
m j<br />
=-1/2<br />
~300<br />
B[G]<br />
~27<br />
B[G]<br />
Figure 2.1: a) shows the Breit-Rabi diagram for sodium. The bending <strong>of</strong> the<br />
|F = 1, m F = −1〉 state occurs at about 300 G which is not relevant for our means. b)<br />
shows the Breit-Rabi diagram <strong>of</strong> lithium. Here the bending <strong>of</strong> |F = 1/2, m F = −1/2〉<br />
occurs at about 27 G, making this state not efficiently trappable.<br />
When setting up a magnetic trap it is essential to know in which regime the atoms<br />
are captured. In the magnetic trap a gradient <strong>of</strong> up to 660 Gcm −1 can be achieved. 2 If<br />
the cloud <strong>of</strong> 23 Na atoms is assumed to have a diameter <strong>of</strong> less than 3 mm, then the field<br />
in which they are captured is at about 100 G, which is safely in the Zeeman regime. 6 Li<br />
atoms, on the other hand, exhibit a change <strong>of</strong> regimes already at low magnetic fields.<br />
2.1.4 Cold collisions<br />
Cold and ultracold atomic collisions play a crucial role in reaching a BEC. Inelastic,<br />
exoergic collisions in a light field or in the magnetic trap can lead to trap loss. Elastic<br />
collisions are an essential part <strong>of</strong> evaporative cooling, the final cooling stage before<br />
achieving a BEC.<br />
2 The gradient corresponds to the strong axial direction <strong>of</strong> our quadrupole field. The transversal<br />
gradient is just half the value <strong>of</strong> the axial gradient.
2.1 Light forces and atomic physics 11<br />
Two atoms passing each other experience an interaction potential that effects their<br />
wave functions. The total wave function <strong>of</strong> these two atoms is given by the incoming<br />
plane wave and a scattered spherical wave, being modulated by the scattering amplitude<br />
f(E, θ) for a spherical symmetric potential. The scattering amplitude depends on the<br />
relative energy E and the angle <strong>of</strong> deflection θ with respect to the original relative motion.<br />
The total wave function becomes<br />
[<br />
Ψ(⃗r) ∝<br />
exp(i⃗r · ⃗k in ) + exp(ikr) f(E, θ)<br />
r<br />
]<br />
〈⃗r|i〉 〈⃗r|j〉 (2.15)<br />
with the internal states |i〉 and |j〉 <strong>of</strong> the scattering atoms. This describes elastic scattering<br />
where internal changes do not happen and the relative kinetic energy <strong>of</strong> the two<br />
atoms stays constant. A common method for treating scattering problems is the partial<br />
wave expansion. The incoming plane wave is expanded in terms <strong>of</strong> spherical waves. The<br />
resulting wave function exhibits for r → ∞ a plane wave behaviour with a phase shift<br />
induced by the interaction potential. An important result is the total cross-section σ(E)<br />
that can be linked to the phase shifts δ l induced by each partial wave<br />
σ(E) = 4π<br />
k 2<br />
∞<br />
∑<br />
l=0<br />
(2l + 1) sin 2 δ l . (2.16)<br />
Due to the spherical symmetry <strong>of</strong> the interaction potential, the angular part <strong>of</strong> the<br />
wave function is given by spherical harmonics. Therefore, the index <strong>of</strong> summation in<br />
equation (2.16) corresponds to a relative angular momentum <strong>of</strong> the two atoms. The<br />
entire information <strong>of</strong> the potential lies in the phase shifts. The description is general<br />
so far and the advantage for cold collisions arises from the fact that for these collisions<br />
the sum has a cut-<strong>of</strong>f. In the most extreme case only the l=0 component <strong>of</strong> the sum,<br />
the so-called s-wave, has to be considered. The cut-<strong>of</strong>f is due to the centrifugal barrier<br />
2 l(l + 1)/(2µr 2 ) that arises from the Hamiltonian <strong>of</strong> this problem. The relative mass µ<br />
<strong>of</strong> the two colliding atoms is in our case µ = m Na /2. For large relative angular momenta<br />
the centrifugal barrier rises and the classical turning point is shifted far away from the<br />
collision center. The potentials <strong>of</strong> relevance are <strong>of</strong>ten <strong>of</strong> the form C 6 /R 6 , created by the<br />
3S − 3S state. This potential vanishes rapidly and therefore higher angular momenta do<br />
not experience the potential. The lower the relative kinetic energy is, the fewer partial<br />
waves take part in the scattering process.<br />
For an entire treatment <strong>of</strong> the scattering process, the internal degrees <strong>of</strong> freedom<br />
need to be included as well. The interaction potential depends on the set <strong>of</strong> quantum<br />
numbers <strong>of</strong> the colliding atoms. Further, the symmetrization <strong>of</strong> the two bosons needs to<br />
be taken into account [22].<br />
The s-wave domain<br />
For low relative kinetic energies (k → 0) a scaling law for the phase shift with respect<br />
to the relative kinetic energy can be found [23]. This depends on the power n <strong>of</strong> the<br />
potential C n /R n . For partial waves which obey 2l < (n − 3),<br />
lim<br />
k→0 k2l+1 cot δ l = 1 (2.17)<br />
a l
12 2.2 Setup<br />
holds. a l is the scattering length <strong>of</strong> the l th partial wave. For the other partial waves the<br />
relation is replaced by<br />
lim<br />
k→0 kn−2 cot δ l = const. (2.18)<br />
These partial waves vanish rapidly for the potentials <strong>of</strong> interest. For 23 Na bosons with<br />
the same quantum number, as in the case in our magnetic trap (|F = 1, m F = −1〉),<br />
only even partial waves contribute to the scattering process 3 . Therefore it follows from<br />
equations (2.17) and (2.18) that only the s- wave is relevant for the captured 23 Na.<br />
The s-wave scattering length has been measured in [24]. Table (2.1) shows the s-wave<br />
scattering lengths <strong>of</strong> 23 Na for different internal states <strong>of</strong> the two colliding atoms. The<br />
scattering lengths are given in units <strong>of</strong> Bohr radii a Bohr .<br />
Table 2.1: Scattering lengths for 23 Na- 23 Na.<br />
m a F m b F Scattering length [a Bohr ]<br />
1(−1) 1(-1) 52.98<br />
1(-1) 0(0) 52.98<br />
1 -1 49.23<br />
0 0 51.12<br />
2(-2) 2(-2) 62.51<br />
For k → 0 and with equation (2.16) the elastic cross-section can be approximated by<br />
σ = 8πa 2 0. (2.19)<br />
There was a factor <strong>of</strong> two inserted here, as the treatment above does not account for<br />
the atoms’ boson nature. The s-wave scattering length determines the rethermalization<br />
behaviour <strong>of</strong> an ultracold gas. By these means, the scattering length in ultracold gases<br />
can be measured.<br />
2.2 Setup<br />
When an experiment on ultracold atoms is to be designed, many things need to be<br />
considered. The choice <strong>of</strong> atom determines the laser system. The cooling strategies have<br />
to be adapted to the particular species.<br />
2.2.1 Facts about 23 Na<br />
23 Na is an essential part <strong>of</strong> human life. An adult human being consists <strong>of</strong> approximately<br />
0.1% sodium, corresponding to 80 g (enough to run the experiment for almost a year).<br />
3 Odd spherical harmonics have negative parity and it is not possible to construct fully symmetric<br />
wave functions under particle exchange.
2.2 Setup 13<br />
But why is 23 Na interesting for our experiment? 23 Na is not the main focus <strong>of</strong> investigation<br />
in NaLi. 6 Li will be cooled to degeneracy with the sympathetic cooling scheme<br />
mentioned before. For this purpose we need large numbers <strong>of</strong> atoms with similar particle<br />
mass <strong>of</strong> the colliding species. This ensures an efficient energy transfer between the 23 Na<br />
sample and the 6 Li sample during sympathetic cooling. Since 6 Li is followed by 23 Na in<br />
the alkali group <strong>of</strong> the periodic table, the prerequisite <strong>of</strong> similar mass is fulfilled quite<br />
well. Further, it is possible to produce 23 Na BECs with atom numbers exceeding 10 7<br />
[25].<br />
The level structure <strong>of</strong> 23 Na consists <strong>of</strong> the D1 and D2 lines at 589.756 nm and<br />
589.158 nm, respectively, having almost the same linewidths <strong>of</strong> approximately 2π ×<br />
10 MHz. The wavelength is not easily accessible with common laser technologies and<br />
to date there are only dye lasers commercially available which can be tuned to these<br />
wavelengths. Unfortunately, dye lasers are very expensive and quite difficult to handle.<br />
They need to be adjusted on a daily basis, which is not the case with diode lasers used<br />
in lithium and rubidium experiments.<br />
F’=3 (g F’<br />
=2/3)<br />
~14.5 MHz<br />
3 2 P 15.944 MHz 58.326 MHz<br />
3/2<br />
~92 MHz<br />
34.244 MHz<br />
F’=2 (g F’<br />
=2/3)<br />
F’=1 (g F’<br />
=2/3)<br />
589.158 326 4 nm<br />
508.848 716 3 THz<br />
3 2 S 1/2<br />
0.664359 GHz<br />
15.809 MHz<br />
1.771626GHz<br />
~350 MHz<br />
a) b) c)<br />
F’=0 (g F’<br />
=2/3)<br />
F=2 (g F<br />
=1/2)<br />
F=1 (g F<br />
=-1/2)<br />
Figure 2.2: Level scheme <strong>of</strong> sodium: The laser frequencies depicted will be further explained<br />
in subsection 2.2.2. a) is the repumper light used in MOT and molasses stages. b) is<br />
the Zeeman slower light that is necessary for the very first cooling stage. c) is the MOT<br />
light.<br />
It is favorable to have a closed two-level system to operate a MOT. In the case <strong>of</strong><br />
23 Na the transition between F = 2 and F ′ = 3 is used. There was also success in the<br />
production <strong>of</strong> a MOT using the transition between F = 1 and F ′ = 0 [26, 27]. The MOT<br />
will be discussed in more detail in subsection 2.4.1. The level scheme <strong>of</strong> 23 Na and the<br />
transitions we use in the experiment are depicted in figure (2.2).
14 2.2 Setup<br />
2.2.2 Laser setup<br />
The light needed for our experiment is provided by two dye lasers using Rhodamin 6-G<br />
for 23 Na and DCM 4 for 6 Li. The dye lasers are pumped by an Yb:YAG 5 laser producing<br />
2 × 10 W. To date, only the light for 23 Na has been used. The dye laser’s output can<br />
range between 1 W and 650 mW depending on how good the laser is adjusted and how<br />
“fresh” the dye is. Details on the lasers can be found in [28]. The light is split into several<br />
paths and manipulated with acousto-optical modulators (AOMs) and an electro-optical<br />
modulator (EOM). Figure (2.3) shows a schematic diagram <strong>of</strong> the optical setup. There<br />
are several separate sections that are described briefly in the following.<br />
Fiber<br />
Coupler<br />
Plug<br />
Polarizing Mirror λ/2 λ/4 Lens Glass<br />
Beamsplitter<br />
plate<br />
Yb:YAG<br />
MOT<br />
Repumper<br />
Imaging<br />
Photodiode<br />
Repumper for<br />
dark spot MOT<br />
Slower<br />
Dye laser<br />
AOM<br />
1.72GHz<br />
AOM<br />
AOM<br />
AOM<br />
AOM<br />
spectroscopy<br />
cell<br />
to laser-lock system<br />
Figure 2.3: The fiber couplers correspond to the couplers in figure (2.4). The black dot on the<br />
side <strong>of</strong> the AOMs depicts the location <strong>of</strong> their transducers. We have the possibility to<br />
couple repumper light into the MOT fiber hence producing a bright MOT, or sending<br />
everything through the dark spot repumper fiber.<br />
• The laser system is locked onto the crossover peak between F ′ = 2 and F ′ = 3.<br />
We use a 23 Na heat-pipe which is heated to 160 ◦ C. The crossover peak is located<br />
29 MHz below the cycle transition. The spectroscopy signal is detected by a regular<br />
photodiode and sent to the laser-lock system. The spectroscopic method is based<br />
upon frequency modulation spectroscopy [29].<br />
• The MOT beams need to be red-detuned with respect to the cycle transition as<br />
will be shown later. For other atoms like rubidium the detuning <strong>of</strong> the MOT is<br />
several linewidths. In our case, this is not possible since the hyperfine splitting<br />
4 4-dicyanomethylene-2-methyl-6-p-dimethylaminostyryl-4H-pyran<br />
5 The active medium is an ytterbium(Yb)-doped yttrium aluminum garnet (YAG) crystal.
2.2 Setup 15<br />
between F ′ = 2 and F ′ = 3 is only 58 MHz and the linewidth is Γ/(2π) = 10 MHz.<br />
Hence the influence <strong>of</strong> the MOT beam on the F ′ = 2 state would be too strong.<br />
If the atom is in the excited F ′ = 3 state, it can only decay to the F = 2 state<br />
(considering only electric dipole transitions). The frequency detuning <strong>of</strong> the MOT<br />
beams with respect to the cycle transition is currently set at −14.5 MHz.<br />
• As the excited hyperfine states F ′ = 2 and F ′ = 3 are close to each other and the<br />
linewidth is large compared to the splitting, there can be non-resonant excitations<br />
from the F = 2 state to the F ′ = 2 state. Once in this state, there can be a decay<br />
to the F = 1 hyperfine ground state, producing a dark state. This is the reason a<br />
repumper beam is needed. It drives ground state atoms back to the F ′ = 2/F ′ = 1<br />
state 6 , where they can decay to the F = 2 ground state and take part in the<br />
cycle process again. As indicated in figure (2.3) there is the possibility to couple<br />
repumper light into the MOT fiber or into the dark spot MOT fibers when the<br />
experiment is run with a dark spot MOT (see subsection 2.4.4).<br />
• The initial slowing process is done with a Zeeman-Slower. For this apparatus we<br />
need a beam that is red-detuned by several hundert MHz with respect to the cycle<br />
transition. The light is matched to the design <strong>of</strong> the Zeeman slower. An EOM<br />
modulates repumper sidebands onto the Zeeman slower light. The EOM is located<br />
on the experimental table depicted in figure (2.4).<br />
• Light which is resonant on the cycle transition is used for imaging. When taking<br />
an image the repumper is turned on as well. The atoms are excited several times<br />
during one shot, hence a better signal can be achieved.<br />
• Finally, we have a plug beam for the magnetic trap. This beam has a wavelength <strong>of</strong><br />
515 nm and is therefore blue-detuned. The light is taken from the Yb:YAG pump<br />
laser. Due Majorana losses at low magnetic fields in magnetic traps this beam is<br />
essential when producing ultracold samples. The dipole force <strong>of</strong> the beam pushes<br />
the atoms out <strong>of</strong> the region <strong>of</strong> low magnetic fields.<br />
The laser light is prepared on a separate laser table and sent via single-mode fibers to<br />
the experimental table.<br />
2.2.3 Experimental setup<br />
The experiment begins with the atoms being in a steel oven which is heated up to 380 ◦ C<br />
for 23 Na and about 400 ◦ C for 6 Li. The situation is depicted in figure (2.4). The mixing<br />
nozzle in between the ovens needs to be heated sufficiently to avoid the deposit <strong>of</strong> 23 Na.<br />
It is constructed at an angle to enable condensed 23 Na to flow back into its reservoir.<br />
The sodium oven is filled with approximately 25 g <strong>of</strong> 23 Na which lasts less than half a<br />
year at these temperatures. At the end <strong>of</strong> this two-species oven, there is an oven nozzle<br />
which creates together with an aperture an atomic beam. In this first section, there is an<br />
ion-getter pump able to pump 55 ls −1 , creating a pressure <strong>of</strong> nearly 10 −8 mbar. Details<br />
on the ovens and the vacuum setup can be found in [30].<br />
6 Details on why the frequency <strong>of</strong> the repumper is set between F ′ = 1 and F ′ = 2 and how both<br />
states are used for repumping can be found in [28]
16 2.2 Setup<br />
<strong>Sodium</strong><br />
Oven<br />
Lithium<br />
Oven<br />
Valves<br />
Zeeman<br />
Slower<br />
coils<br />
TSP<br />
Glass window<br />
mixing<br />
nozzle<br />
IGP<br />
IGP<br />
coils for<br />
magnetic<br />
field<br />
EOM<br />
Imaging MOT Dark-Spot Slower Plug<br />
CCD-Camera<br />
Figure 2.4: Top view <strong>of</strong> the experimental table: The beams in the vertical direction are not<br />
sketched here. The different light sources are depicted by the fiber couplers. TSP<br />
denotes the titan-sublimation pump and IGP the ion-getter pumps.<br />
After this section, there is another aperture for beam collimation followed by two differential<br />
pumping stages that consist <strong>of</strong> tubes. They are located between two valves that<br />
can seal ultra-high vacuum to atmospheric pressure and are needed when the samples<br />
in the ovens are replaced. To improve the differential pumping, there is an ion-getter<br />
pump located between the two tubes.<br />
The ultra-high vacuum, starting after the second valve, has a pressure <strong>of</strong> less than<br />
10 −11 mbar. The first cooling stage, the Zeeman slower, has a length <strong>of</strong> approximately<br />
75 cm. It is a so-called spin-flip Zeeman slower. At the end <strong>of</strong> the Zeeman slower is<br />
the glass cell where the experiment takes place. The glass cell consists <strong>of</strong> quarz glass<br />
with a refractive index <strong>of</strong> n = 1.4585 at a wavelength <strong>of</strong> λ = 587.6 nm and a Brewster<br />
angle <strong>of</strong> θ B = 55.58 ◦ . The advantage <strong>of</strong> this cell is the good optical access in the table<br />
plane. Furthermore, no eddy currents can arise in a glass cell. This is a problem with<br />
octagon-chambers which are made <strong>of</strong> glass and metal.<br />
Finally, at the end <strong>of</strong> the vacuum chamber, there is a window where the laser light<br />
for the Zeeman slower enters. For the ultra-high vacuum there is a combination <strong>of</strong> an<br />
ion-getter pump and a titan-sublimation pump which is able to pump 200 ls −1 . The<br />
window needs to be heated up to 200 ◦ C to avoid deposit <strong>of</strong> lithium that could damage<br />
or blur it.<br />
Up to now there are ten beams on the table and the lithium part has not been set<br />
up yet. Two optical dipole traps are yet to come and possibly there will be an extra<br />
imaging direction. This would easily double the amount <strong>of</strong> beams on the table.
2.3 Imaging 17<br />
2.2.4 The Zeeman Slower<br />
The very first cooling stage <strong>of</strong> the experiment is the Zeeman slower. Via a magnetic field<br />
the Zeeman sublevels <strong>of</strong> the incoming atoms are always kept in resonance with the laser<br />
light <strong>of</strong> the Zeeman slower beam. This works for a wide range <strong>of</strong> velocities. The main<br />
goal is to compensate the Doppler shift <strong>of</strong> the moving atoms by the Zeeman shift due<br />
to the magnetic field. For a uniform deceleration and red-detuned light the necessary<br />
magnetic field is [20]<br />
B(z) = B 0<br />
(1 − z z 0<br />
) 1/2<br />
+ B Bias (2.20)<br />
where z 0 is the length <strong>of</strong> the path through the magnetic field. B 0 needs to be adapted<br />
to the maximal incoming velocity and B Bias can be used to build a Zeeman Slower in<br />
different configurations. We use the spin-flip configuration that has several advantages<br />
over other existing configurations as is explained in [31]. Our spin-flip slower has been<br />
designed to decelerate two species. The average incoming velocity <strong>of</strong> the 23 Na atoms<br />
is about 800 ms −1 and the slowed beam has an average velocity <strong>of</strong> 30 ms −1 , enough to<br />
be trapped in our MOT. A disadvantage <strong>of</strong> the spin-flip configuration is the zero-field<br />
crossing. This causes a flip <strong>of</strong> the quantization axis. We use an EOM at 1.771 GHz to<br />
pump the atoms again in the right state when they cross the magnetic field zero. The<br />
EOM produces first-order sidebands at about 30% <strong>of</strong> the carrier wave amplitude.<br />
2.3 Imaging<br />
To obtain knowledge about the captured and cooled atoms, images need to be taken<br />
or fluorescent light has to be collected. There are several standard ways to do this,<br />
e.g. through fluorescence, absorption and phase contrast imaging. Just recently a group<br />
imaged a BEC with the help <strong>of</strong> a scanning electron microscope [32]. In our case we use<br />
absorption imaging and some measurements are done by collecting the fluorescence light<br />
with a photodiode.<br />
Principally, absorption imaging is counting the photons that never made it to the<br />
camera. All that is needed is a weak collimated laser beam which is resonant for the F =<br />
2 to F ′ = 3 transition. A beam with circular polarization is used and the quantization<br />
axis is provided by turning on a bias magnetic field along the direction <strong>of</strong> our imaging<br />
system. The imaging pulses have lengths <strong>of</strong> about 30µ s at imaging intensities that are<br />
around 0.1 mWcm −2 . The beam is sent through an optical system onto a CCD camera.<br />
Each picture consists <strong>of</strong> three shots with a certain intensity information. The intensities<br />
<strong>of</strong> shots with atoms I atoms (y, z), without atoms I ref (y, z) and finally without the laser<br />
beam turned on I back (y, z) are collected. The background shot is subtracted from the<br />
actual shot and the reference shot to get rid <strong>of</strong> stray light that might permanently exist.<br />
The optical density <strong>of</strong> the sample is then<br />
( )<br />
Iatoms (y, z) − I back (y, z)<br />
OD(y, z) = − ln<br />
. (2.21)<br />
I ref (y, z) − I back (y, z)<br />
The change <strong>of</strong> the laser beam’s intensity through the sample<br />
dI = −hνR scatt ρ(x, y, z)dx (2.22)
18 2.4 The magneto-optical trap (MOT)<br />
depends on the scattering rate R scatt and the density ρ(x, y, z) <strong>of</strong> the sample. As the<br />
imaging intensities used are small, the optical density can be brought into direct relation<br />
with the column density <strong>of</strong> the sample. This is just the integrated density along the axis<br />
<strong>of</strong> the laser beam. For this limit the column density [33]<br />
[<br />
n(y, z) = − 2I ( ) ] 2<br />
0 2∆<br />
1 + × OD(y, z) (2.23)<br />
Γω Γ<br />
is directly proportional to the optical density. Equation (2.23) is a form <strong>of</strong> the Lambert-<br />
Beer law [20]. Taking into account the symmetries <strong>of</strong> the trap, it is straightforward<br />
to calculate the total atom number from this information. The equations need to be<br />
modified if there is a magnification through the lens system. The relations above hold<br />
for every pixel <strong>of</strong> the CCD camera chip. Therefore, these images provide information<br />
about the density, the size and the total atom number <strong>of</strong> the sample.<br />
Taking time-<strong>of</strong>-flight series [34] gives an insight on the temperature <strong>of</strong> the trapped<br />
sample. The atoms get released and are imaged at different times. The temporal evolution<br />
<strong>of</strong> the width<br />
√<br />
σ(t) = σ 0 + k BT<br />
m t2 (2.24)<br />
<strong>of</strong> the sample shows for large times a linear behaviour. σ 0 is the initial width at t = 0.<br />
2.4 The magneto-optical trap (MOT)<br />
The first success using a MOT was reported in 1987 by Pritchard et al. [35]. Back then<br />
10 7 atoms were trapped at a temperature <strong>of</strong> 600 µK with a peak density <strong>of</strong> approximately<br />
10 11 cm −3 . Since then several variations <strong>of</strong> this technique have been developed. For 23 Na<br />
the so-called dark spot MOT is <strong>of</strong>ten used [36].<br />
2.4.1 MOT theory<br />
Before the first MOTs were implemented, there were experiments with optical molasses<br />
[37]. In optical molasses the atoms move in red-detuned light fields produced by three<br />
perpendicular counterpropagating laser beam pairs. This configuration provides a viscous<br />
confinement and a cooling <strong>of</strong> the 23 Na atoms.<br />
For atoms moving along a beam axis with velocity v, the detuning in such an optical<br />
molasses is ∆ MOL = ω − ω 0 ± kv depending on the motion <strong>of</strong> the atom with respect<br />
to the propagation direction <strong>of</strong> the laser beam. The light is more resonant for atoms<br />
moving towards the photons and less for atoms that are moving away from the laser<br />
light. Therefore the atom scatters more light from the direction it is moving to. Since<br />
spontaneous emission is isotropic but absorption is directional, there is a net-force against<br />
the motion <strong>of</strong> the atom. In the case <strong>of</strong> optical molasses no magnetic fields are involved,<br />
therefore the polarization <strong>of</strong> the laser beams does not play a role 7 . This net force is<br />
to first order linear in the velocity v. The force does not have a spatial dependence,<br />
therefore trapping is not possible.<br />
7 This is not true in the case <strong>of</strong> sub-Doppler cooling. There the polarization plays a crucial role.
2.4 The magneto-optical trap (MOT) 19<br />
A spatial dependence <strong>of</strong> the trapping force can be introduced by applying a special<br />
magnetic field. Two coils in an anti-Helmholtz-like configuration provide a magnetic<br />
quadrupole field which has a linear field gradient in the region <strong>of</strong> interest. The field is<br />
given by B = B ′ z with the gradient B ′ . The detuning above needs to be modified by<br />
adding a Zeeman shift caused by the magnetic field<br />
with the additional Zeeman shift<br />
∆ MOT = ω − ω 0 ± kv + ∆ Zeeman (z) (2.25)<br />
∆ Zeeman (z) = µ B<br />
(g F’m F’ − g F m F )B ′ z (2.26)<br />
as in equation (2.12). For the MOT to work, the polarization is crucial. In our multilevel<br />
MOT, the cycle transition is from F = 2 to F ′ = 3 with 5 and 7 Zeeman sublevels,<br />
respectively. Schematically, the situation is depicted in figure (2.5) for the ground state<br />
sublevel |F = 2, m F = 2〉. The situation is analogous for the other ground state sublevels.<br />
Atoms at a positive position experience an increasing detuning for the σ + transition and<br />
a decreasing one for the σ − transition as they move further away from the trap center.<br />
Therefore the scattering rate will increase for the later transition and the σ − laser beam<br />
will push the atoms to the trap center. The situation is reversed for atoms on the<br />
negative position side. They are more resonant for the σ + beam and also get pushed<br />
into the center <strong>of</strong> the trap.<br />
m F’<br />
=3<br />
a)<br />
m F’<br />
=1<br />
b)<br />
energy[a.u.]<br />
ω<br />
σ + 0<br />
ω σ-<br />
m F<br />
=2<br />
0<br />
position[a.u.]<br />
Figure 2.5: Working principle <strong>of</strong> the MOT: The MOT process is explained in the text. ω 0 is<br />
the transition frequency at zero field and ω is the frequency <strong>of</strong> the red-detuned laser.<br />
The dashed line a) is a measure for the detuning <strong>of</strong> the σ − transition with respect to<br />
the laser frequency. For atoms on the positive position side this detuning is smaller<br />
than for atoms on the negative position side. b) is the same for the σ + transition.
20 2.4 The magneto-optical trap (MOT)<br />
The force in a MOT [20] in 1D<br />
F = −αv − αβ<br />
k z (2.27)<br />
consists <strong>of</strong> a term that depends linearly on the velocity and a term that depends linearly<br />
on the position <strong>of</strong> the atoms. The coeffiecients<br />
α = 4k 2 I I 0<br />
−2∆/Γ<br />
(1 + I/I 0 + (2∆/Γ) 2 ) 2 (2.28)<br />
and<br />
β = (g F’ m F’ − g F m F ) µ B<br />
· B′ (2.29)<br />
are dependent on the parameters <strong>of</strong> the MOT. This is for the case <strong>of</strong> perfect alignment<br />
and beam balancing. The atoms’ motion in the MOT corresponds to an overdamped<br />
oscillation in a harmonic potential. They are accumulated in the center <strong>of</strong> the trap and<br />
simultaneously cooled.<br />
The MOT is known to be a very robust trap that does not depend sensitively on the<br />
setup. The explanations above suggest that atoms could be arbitrarily cooled down and<br />
densely trapped. But there are limitations to both which makes the use <strong>of</strong> other cooling<br />
mechanisms necessary.<br />
2.4.2 MOT temperature and Doppler limit<br />
In MOT and optical molasses techniques, photons <strong>of</strong> a directed beam are redistributed in<br />
random directions. Spontaneous emission imposes a heating on the atoms. The process<br />
is <strong>of</strong> stochastic nature and can be described by a random walk process 8 . A steady state<br />
value between the heating process due to the random walk in momentum space and the<br />
cooling process explained above can be found [20]. The resulting temperature<br />
T D = Γ 1 + (2∆/Γ) 2<br />
4k B −2∆/Γ<br />
(2.30)<br />
is called Doppler limit and is a theoretical limit for this cooling process. A detuning <strong>of</strong><br />
∆ = −Γ/2 results in the minimal temperature, the Doppler temperature. For 23 Na this<br />
is 235 µK. 6 Li has a lower Doppler temperature <strong>of</strong> 144 µK due to a smaller linewidth <strong>of</strong><br />
6 Li compared to 23 Na.<br />
Figure (2.6) shows a time-<strong>of</strong>-flight series <strong>of</strong> the MOT atoms. The temperature is<br />
calculated by evaluating the temporal evolution <strong>of</strong> the cloud’s width as mentioned before.<br />
The fitted results give a temperature <strong>of</strong> 405 µK ± 17 µK. For the detuning <strong>of</strong><br />
∆ = −14.5 MHz the theoretical expected temperature is 388 µK. The two temperatures<br />
match within the error, hence the sample is cooled to its theoretical value. More careful<br />
alignment <strong>of</strong> the beams may cause sub-doppler cooling within the trap as well.<br />
The inital width 1.2 mm <strong>of</strong> the fit does not correspond to the observed values. This<br />
discrepancy results from the assumption that the initial density distribution is Gaussian<br />
due to the potential ascociated with equation (2.27) 9 . The density distribution <strong>of</strong> the<br />
8 This corresponds to a diffusion in the momentum space.<br />
9 The velocity dependent term is neglected.
2.4 The magneto-optical trap (MOT) 21<br />
trapped sample can only be assumed to be Gaussian for atom numbers smaller than<br />
4 × 10 4 [38]. Higher atom numbers will cause different density pr<strong>of</strong>iles in the MOT due<br />
to effects explained in the next subsection.<br />
4<br />
3.5<br />
T=(405±17)µK<br />
3<br />
σ[mm]<br />
2.5<br />
2<br />
1.5<br />
1<br />
0 2 4 6 8 10<br />
t[ms]<br />
Figure 2.6: Time-<strong>of</strong>-flight series <strong>of</strong> the MOT<br />
2.4.3 Density limitations and loss mechanisms in a MOT<br />
Just as the temperature <strong>of</strong> a MOT is limited, there are processes limiting the achievable<br />
density. The main reason for this is the reabsorption <strong>of</strong> photons emitted by neighbouring<br />
atoms. Further, light-assisted collisions can lead to trap loss.<br />
Radiation trapping<br />
Radiation trapping [39] produces a repulsive force between the atoms and limits the<br />
achievable atomic density in a MOT by regular means. To fully understand the underlying<br />
process, the attractive attenuation force [40] has to be taken into account.<br />
The attenuation force F ⃗ att arises from the absorption <strong>of</strong> the laser beam across an<br />
optically thick gas. The laser intensity diminishes when travelling across the sample.<br />
Therefore, the net-force <strong>of</strong> the two counterpropagating beams acting on the atom is<br />
always in favour <strong>of</strong> the beam which pushes the atom to the trap center. This attenuation<br />
gives rise to an overall compression <strong>of</strong> the MOT. The divergence <strong>of</strong> the resulting force<br />
can be formulated as<br />
⃗∇ · ⃗F att = −6σLI 2 n<br />
∞ (2.31)<br />
c
22 2.4 The magneto-optical trap (MOT)<br />
with c being the speed <strong>of</strong> light [39]. A higher density n will lead to a greater intensity<br />
difference <strong>of</strong> the counterpropagating beams at positions next to the center <strong>of</strong> the trap.<br />
I ∞ is the initial intensity <strong>of</strong> a single beam. The cross-section σ L for absorption <strong>of</strong> photons<br />
from the laser beam depends on the detuning and the intensity <strong>of</strong> the laser beams. The<br />
negative sign implies the compression <strong>of</strong> the cloud.<br />
In the case <strong>of</strong> low atomic densities, the radiation trapping force is weak. The emitted<br />
photons pass almost without absorption through the sample. If the atoms have a mean<br />
distance d from each other and are in an light field <strong>of</strong> intensity I then the force <strong>of</strong> the<br />
emitting atom on the reabsorbing atom is<br />
|F rad | = 1 c × σ R × σ LI<br />
4πd 2 (2.32)<br />
The last part describes the intensity <strong>of</strong> the emitted fluorescence light at the position <strong>of</strong><br />
the reabsorbing atom. The cross-section σ R for reabsorption can be different than the<br />
cross-section σ L for absorption <strong>of</strong> the laser beam. This is due to the shifted frequency <strong>of</strong><br />
the emitted photon. The divergence <strong>of</strong> the radiation trapping force<br />
⃗∇ · ⃗F rad = 6σ R σ L I ∞<br />
n<br />
c<br />
(2.33)<br />
has a similar form to the attenuation force. Only the cross-section for reabsorbtion <strong>of</strong><br />
the fluorescence light differs. The sign is positive, causing a repulsion <strong>of</strong> the atoms.<br />
For low temperatures the velocities are small and the first part <strong>of</strong> equation (2.27)<br />
can<br />
[<br />
be neglected. Only a trapping force <strong>of</strong> the form F ⃗ trap = −κ⃗r remains. Setting<br />
⃗∇ · ⃗Ftrap + F ⃗ ]<br />
att + F rad = 0 the steady state density distribution <strong>of</strong> the MOT can be<br />
easily found.<br />
With a laser detuning <strong>of</strong> ∆ 0 = −14.5 MHz and a single beam intensity <strong>of</strong> about<br />
I ∞ = 2.5 mWcm −2 , the cross-sections are σ L = 2.9 × 10 −13 m 2 and σ R = 4.4 × 10 − 13 m 2 .<br />
This leads to an achievable density in our MOT <strong>of</strong> about 3.6×10 11 cm 3 . This limit has not<br />
been observed in our group so far. The estimation is done with a homogenous intensity<br />
pr<strong>of</strong>ile. Further we do not have a perfect beam balancing and the counterpropagating<br />
beams are slightly misaligned, since this optimized the total number <strong>of</strong> atoms in the<br />
MOT. Other groups with a similar setup report on densities <strong>of</strong> about 1 × 10 −11 cm −3 in<br />
a dark spot MOT [25]. Even with this additional technique it was not possible for them<br />
to raise the density in the MOT.<br />
Loss mechanisms<br />
Aside from the principal limit due to the radiation trapping force, there are several loss<br />
processes involved that limit an infinite loading <strong>of</strong> a MOT.<br />
One-body losses: These losses are due to collisions with background gas and fast<br />
atoms from the atomic beam. Atoms with velocities faster than 689 ms −1 are not decelerated<br />
by the Zeeman slower [31]. They fly without hindrance to the experimental cell<br />
and collide with the MOT atoms. There is an atomic beam shutter installed to block<br />
the beam, but <strong>of</strong> course this stops the loading <strong>of</strong> the MOT as well. Another part arises<br />
from the limited value <strong>of</strong> the pressure in the experimental cell. The rate for one-body<br />
losses is independent <strong>of</strong> the MOT’s density as they are collisions between MOT and<br />
external atoms. They are characterized by a pure exponential decay <strong>of</strong> the number <strong>of</strong>
2.4 The magneto-optical trap (MOT) 23<br />
atoms in the MOT when the atomic shutter is closed. A remedy against these losses is<br />
a good vacuum, which is in our case about 1 × 10 −11 mbar. Background gas collisions in<br />
magnetic traps will be treated in subsection 3.4.1.<br />
energy<br />
energy<br />
ΔE=ћω 1<br />
-ћω 2<br />
a) b)<br />
3S 1/2<br />
+3P 3/2<br />
A<br />
ћω 2<br />
ћω 1<br />
ћω<br />
B<br />
3S 1/2<br />
+3P 3/2<br />
ΔE FS<br />
3S 1/2<br />
+3P 1/2<br />
3S 1/2<br />
+3S 1/2<br />
3S 1/2<br />
+3S 1/2<br />
R˜<br />
position[R]<br />
R<br />
R<br />
position[R]<br />
Figure 2.7: Light-assisted collisions: a) Radiative escape b) Fine-structure-changing collision<br />
Two-body losses: Two atoms inside the MOT take part in these collisions and therefore<br />
the rate depends on the density <strong>of</strong> the MOT. There are several mechanisms and<br />
just two so-called light-assisted collisions are presented briefly. The laser light field <strong>of</strong><br />
a MOT can induce inelastic, exoergic collisions between MOT atoms. The energy released<br />
is enough for atoms to escape the trap [41]. At the beginning <strong>of</strong> the processes<br />
two neighbouring MOT atoms are both in the ground state. They experience a ground<br />
state molecular potential corresponding to 3S 1/2 + 3S 1/2 with a position dependent energy<br />
curve E 3S1/2 +3S 1/2<br />
(R) depicted in figure (2.7). At position R a photon from the<br />
MOT laser beam is absorbed and the molecular potential changes to 3S 1/2 + 3P 3/2 with<br />
E 3S1/2 +3P 3/2<br />
(R). The two atoms are accelerated towards each other. During this process<br />
radiative escape [42] and fine-structure-changing collisions can occur as seen in figure<br />
(2.7).<br />
• Radiative escape: As the two atoms are accelerated towards each other, the excited<br />
atom can undergo spontaneous emission back to the ground state at position ˜R.<br />
The energy difference ∆E = (E 3S1/2 +3P 1/2<br />
(R)−E 3S1/2 +3S 1/2<br />
(R))−(E 3S1/2 +3P 1/2<br />
( ˜R)−<br />
E 3S1/2 +3S 1/2<br />
( ˜R)) <strong>of</strong> the two photons is equally distributed as kinetic energy between<br />
the atoms.<br />
• Fine structure change: In this case the excited atom does not undergo spontaneous<br />
decay to the ground state but turns around at point A in the molecular potential.<br />
At the point <strong>of</strong> the fine structure crossing it can undergo a change <strong>of</strong> levels marked<br />
as point B. When the atoms part again, the energy difference ∆E F S <strong>of</strong> the fine<br />
structure splitting is distributed between the two MOT atoms, causing a trap<br />
escape.
24 2.4 The magneto-optical trap (MOT)<br />
Besides these excited state losses there also exist ground state losses due to hyperfine<br />
changing collisions [22]. These collisions are more important in magnetic traps and in<br />
the process <strong>of</strong> evaporative cooling. Light-assisted collisions can be suppressed in a dark<br />
spot MOT. The losses in our MOT are thouroughly treated in [30].<br />
2.4.4 Dark spot MOT<br />
To achieve higher densities in MOTs, several techniques have been developed. For 87 Rb a<br />
compressed MOT is <strong>of</strong>ten used [43]. In the case <strong>of</strong> 23 Na the MIT group [36] introduced a<br />
technique <strong>of</strong> trapping atoms in a dark state – called the dark spot MOT. For this purpose<br />
they simply created a repumper hollow sphere and not a repumper volume. The atoms<br />
in the middle <strong>of</strong> the hollow sphere decay eventually to the lower ground state and are<br />
transparent for the MOT light since there is no repumper light present inside the hollow<br />
sphere. Therefore, radiation trapping and light assisted collisions are strongly suppressed<br />
in the middle <strong>of</strong> the MOT. If the atom is moving to the periphery <strong>of</strong> the MOT, it gets<br />
repumped and pushed back to the middle. The dark spot technique has also been used<br />
with other alkali metals, but <strong>of</strong>ten a depumping scheme needs to be applied, since the<br />
decay to the dark state is not strong enough [44]. For lithium this scheme cannot work<br />
since the hyperfine ground state splitting is too small to produce a true dark state.<br />
Experimentally the dark spot is produced by imaging a black dot on a glass plate<br />
onto the middle <strong>of</strong> the MOT. In our case we just project this black dot onto the MOT,<br />
creating a dark tube. Diffraction effects cause a rather grey dot, however. The intensity<br />
ratio between the bright ring and the dark inside is about 300. A weakly glowing MOT<br />
can be observed. Since we create dark tubes which would lead to a dark loss tunnel, we<br />
need to use two repumper beams that are crossed. This creates the desired repumper<br />
hollow sphere.<br />
The dark spot MOT we setup, seems to be not as efficient as we hoped for. Within<br />
regular measurements there is hardly a difference visible. The rethermalization measurement<br />
in subsection 3.5 revealed the same thermalization times for the regular and the<br />
dark spot MOT. If we had a dark spot MOT, the density would rise significantly. This<br />
higher density would cause a higher collision rate, hence the thermalzation time would<br />
decrease 10 . This implies that the dark spot is not working as well as it should. At some<br />
point it might be reasonable to go back to this step and improve the density in the dark<br />
spot MOT.<br />
2.4.5 Loading <strong>of</strong> the MOT for different temperatures<br />
The loading rate <strong>of</strong> the MOT depends strongly on the flux <strong>of</strong> atoms coming from the<br />
oven [30]. The flux is about 10 17 s −1 for a temperature <strong>of</strong> 381 ◦ C and 6 × 10 16 s −1 for<br />
a temperature <strong>of</strong> 364 ◦ C. This is the flux out <strong>of</strong> the oven nozzle. The velocity limit <strong>of</strong><br />
the Zeeman slower is fixed to a value <strong>of</strong> about 689 ms −1 for 23 Na atoms. The Maxwell-<br />
Boltzmann distribution states that only 69.02% and 70.80% <strong>of</strong> the atoms in the beam are<br />
decelerated. Of the decelerated atoms 60% are captured in the MOT [31]. The resulting<br />
fraction <strong>of</strong> captured atoms at 381 ◦ C and 364 ◦ C are 41.41% and 42.50%, respectively.<br />
10 The collision time τ = 1/(nvσ) is proportional to the thermalization time. n denotes the density, v<br />
the mean velocity and σ the elastic collision cross-section. A gain in density <strong>of</strong> one order <strong>of</strong> magnitude<br />
would cause a decrease <strong>of</strong> one order <strong>of</strong> magnitude <strong>of</strong> the thermalization time.
2.5 Sub-Doppler Cooling 25<br />
The ratio <strong>of</strong> fluxes at these temperatures can be estimated to 1.65. This is equal to the<br />
ratio <strong>of</strong> the loading rates at these temperatures.<br />
The loading rate <strong>of</strong> atoms in the MOT is given by the slope <strong>of</strong> the loading curve at<br />
t = 0. The loading curve <strong>of</strong> the MOT can be expressed as N(t) = N 0 (1 − exp(−t/τ)<br />
with N 0 being the final number <strong>of</strong> atoms and 1/τ the one-body loss rate. The ratio <strong>of</strong><br />
the loading rates in figure (2.8) has a value <strong>of</strong> 2.16 ± 0.06. The measured ratio does not<br />
correspond to the estimated ratio <strong>of</strong> the loading rates.<br />
16<br />
fluorescence signal[a.u.]<br />
14<br />
12<br />
10<br />
8<br />
6<br />
4<br />
2<br />
0<br />
T=381ºC<br />
T=364ºC<br />
-2<br />
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5<br />
time[s]<br />
Figure 2.8: Loading rate <strong>of</strong> the <strong>Sodium</strong> MOT for different temperatures<br />
The used vapor pressure model [45] is only a rough approximation, hence the estimation<br />
<strong>of</strong> the fluxes is only vague. Further the transversal motion <strong>of</strong> the atoms moving<br />
through the Zeeman slower has not been considered. This leads to a temperature dependent<br />
deviation <strong>of</strong> the calculated fluxes. Transversal motion is treated in detail in [31].<br />
As the temperatures were measured several hours apart from each other the photodiode<br />
might have been accidentally dislocated. Due to the changed effective area <strong>of</strong> the photodiode<br />
with respect to the incoming light a comparison <strong>of</strong> the two measurements would<br />
not be possible. A temporal changing laser power output, as it is the case sometimes<br />
with our dye laser, would cause a similar effect.<br />
2.5 Sub-Doppler Cooling<br />
The theoretical limit <strong>of</strong> regular laser cooling is given by the Doppler temperature. With<br />
our detuning this yields a temperature <strong>of</strong> about 388 µK. Still, it is possible to cool<br />
the atomic sample further down by means <strong>of</strong> sub-Doppler cooling. This technique was<br />
observed in some <strong>of</strong> the first MOTs. The measured temperature was below the expected<br />
Doppler temperatures. Many techniques have been developed, some being able to cool<br />
below the recoil limit given by the energy <strong>of</strong> a single photon. We tried to make use
26 2.5 Sub-Doppler Cooling<br />
<strong>of</strong> sub-Doppler cooling after the MOT was loaded. The magnetic field was turned <strong>of</strong>f<br />
and homogenous stray fields were cancelled out by making use <strong>of</strong> bias coils. MOT light,<br />
which was even further red-detuned, was shone in for about 5 ms. As we have circular<br />
polarized laser light, the cooling mechanism is the so-called σ + -σ − cooling [46].<br />
2.5.1 σ + -σ − cooling basics<br />
The σ + -σ − cooling mechanism is based on two counterpropagating, coherent waves with<br />
opposite circular polarization. The combination <strong>of</strong> the two circular polarized waves<br />
produces a linear polarized wave that rotates around the beam axis, forming a helix.<br />
The length <strong>of</strong> one turn is just given by the wavelength. Since the polarization always<br />
remains linear, the light shift <strong>of</strong> the states stays the same in space. Therefore there is<br />
no potential gradient involved like in other sub-Doppler cooling mechanisms.<br />
m F’=3<br />
:<br />
-3 -2 -1 0 1 2 3<br />
1/2 1/3 1/5 1/10 1/30<br />
-1/6<br />
-4/15<br />
-3/10<br />
-4/15<br />
-1/6<br />
1/30<br />
1/10 1/5 1/3<br />
1/2<br />
m F=2<br />
:<br />
-2<br />
-1 0 1 2<br />
Figure 2.9: Clebsch-Gordan coefficients for the cycle transition [45]<br />
An atom at rest experiences a local polarization <strong>of</strong> the field along the quantization axis<br />
labeled with y. The wave function is given as eigenfunctions <strong>of</strong> F y being the y-component<br />
<strong>of</strong> the total angular momentum. The only transition driven is the π-tranistion, and the<br />
steady state distribution is N 0 = 400/922, N ±1 = 225/922 and N ±2 = 36/922. 11 The<br />
atom’s probability to be in the m F = 0 state is enhanced compared to the other states.<br />
In the case <strong>of</strong> an atom moving along the z-axis with velocity v, the situation changes.<br />
In the rest frame <strong>of</strong> the atom the polarization is linear and rotates with an angular<br />
frequency <strong>of</strong> kv. A transformation into a rotating frame where the polarization is fixed<br />
causes an inertial field along the axis <strong>of</strong> propagation. The Hamiltonian is then perturbed<br />
by an additional term<br />
V pert = kvF z (2.34)<br />
which is proportional to the velocity <strong>of</strong> the atom 12 . This term gives rise to a coupling<br />
11 The populations were normalized to N 0 + N 1 + N −1 + N 2 + N −2 = 1.<br />
12 The change to a rotating frame can be achieved by a unitary transformation T (t) = exp(−ikvF z /).<br />
The system’s transformed states are Ψ ′ = T (t)Ψ. The transformed Schrödinger equation has the form<br />
i d dt T (t)Ψ(⃗r, t) = T (t)H(t)T † (t)T (t)Ψ(⃗r, t). Evaluating the left hand side <strong>of</strong> the equation yields the<br />
inertial term.
2.5 Sub-Doppler Cooling 27<br />
between the ground states <strong>of</strong> the atom in the rest frame, since the non-diagonal matrix<br />
elements <strong>of</strong> the perturbation do not vanish. The result is a velocity-dependent mixing <strong>of</strong><br />
the ground states for the moving atom. The steady state expectation value <strong>of</strong> the total<br />
angular momentum in the propagation direction is given by [46]<br />
〈F z 〉 ∝ kv<br />
∆ 0<br />
(2.35)<br />
where ∆ 0 is the light shift <strong>of</strong> the m Fy = 0 state. This means that in terms <strong>of</strong> the<br />
eigenbasis <strong>of</strong> F z there is an imbalance <strong>of</strong> the states. These populations shall be denoted<br />
by Π i (i ∈ (−2 . . . 2)). For a negative ground state light shift and a positive velocity the<br />
imbalance is in favor <strong>of</strong> atoms with negative angular projection numbers, meaning<br />
2∑<br />
Π −i ><br />
i=1<br />
2∑<br />
Π i . (2.36)<br />
i=1<br />
For a negative velocity the population imbalance is just reversed as depicted in figure<br />
(2.10).<br />
population[a.u.]<br />
v0<br />
m F=2<br />
: -2 -1 0 1 2 m F=2<br />
: -2 -1 0 1 2<br />
σ + σ -<br />
population[a.u.]<br />
population[a.u.]<br />
m F=2<br />
: -2 -1 0 1 2<br />
v=0 z<br />
Figure 2.10: σ + -σ − -cooling: The atoms at rest have a steady state population that is equally<br />
populated around m F = 0. Whereas in the moving case, there is a mixing <strong>of</strong> the states<br />
causing the populations depicted.<br />
This motion-induced population imbalance is the reason for the cooling in this scheme.<br />
For atoms having a positive velocity with the corresponding population probability, the<br />
Clebsch-Gordan coefficients (see figure (2.9)) favour the absorbtion <strong>of</strong> the σ − -light, which
28 2.5 Sub-Doppler Cooling<br />
they are moving towards (depending, <strong>of</strong> course, on the projection state). This causes a<br />
radiation pressure, hence damping the atoms. The force on the atoms is<br />
F ∝ k 2 〈R scatt〉<br />
∆ 0<br />
v (2.37)<br />
with the mean scattering rate 〈R scatt 〉 <strong>of</strong> the ground state atoms. For red-detuned light,<br />
the light shift is negative therefore decelerating the atoms. A full treatment <strong>of</strong> the<br />
problem can be found in [47] and [48].<br />
2.5.2 Sub-Doppler cooling in the experiment<br />
To use the σ + -σ − cooling scheme, bias fields, resulting from earth’s magnetic field and<br />
stray fields, need to be compensated, since they would create a fixed quantization axis.<br />
The cooling scheme, however, exists because <strong>of</strong> the rotation <strong>of</strong> the quantization axis.<br />
We installed rectangular bias coils for each spatial direction (x,y,z) with the possibilty <strong>of</strong><br />
sending 1 A with arbitrary polarity through the coils. The field created by the coils is on<br />
the order <strong>of</strong> 1 G. This is enough to compensate earth’s magnetic field. The currents were<br />
adjusted to have a symmetric, slow expansion <strong>of</strong> the cloud when the magnetic gradient<br />
field was turned <strong>of</strong>f. We never managed to obtain a very good symmetric expansion,<br />
probably due to imperfect beam alignment. A further reason could be parts close to the<br />
experimental chamber that are magnetizable, producing inhomogenous stray fields which<br />
cannot be compensated by simple bias coils 13 . We observed a small MOT produced by<br />
four magnetizable screws in the construction <strong>of</strong> our bias coils. They were replaced by<br />
teflon screws.<br />
2.5<br />
MOT: T=(133 ±12) μK<br />
sub-Doppler: T=( 97 ±17)μK<br />
2<br />
σ [mm]<br />
1.5<br />
1<br />
0 2 4 6 8 10<br />
time[ms]<br />
Figure 2.11: Time-<strong>of</strong> flight series <strong>of</strong> MOT and sub-Doppler cooled sample.<br />
13 It is though possible to compensate linear magnetic field gradients [49]
2.5 Sub-Doppler Cooling 29<br />
Figure (2.11) shows a time-<strong>of</strong>-flight series <strong>of</strong> a bright MOT compared to a time-<strong>of</strong>flight<br />
series <strong>of</strong> a sub-Doppler cooled atomic sample. The measured temperatures are<br />
T MOT = (133 ± 12) µK and T sub = (97 ± 17) µK, showing the disturbing fact that our<br />
sub-Doppler cooling works only very inefficiently. The ratio <strong>of</strong> temperatures was only<br />
0.72 ± 0.14. The limiting temperature for σ + -σ − cooling is on the same order as for lin⊥<br />
lin cooling [46]. With this method a temperature <strong>of</strong> (40 ± 20) µK was measured [50].<br />
This yields a temperatures ratio <strong>of</strong> 0.16 ± 0.08 with respect to the Doppler temperature.<br />
The absolute temperature <strong>of</strong> the MOT is in a range that suggests that its measurement is<br />
not working very well. The measurement was taken after installing a new magnification<br />
<strong>of</strong> our imaging system. It was changed from a 2 : 1 to a 1 : 1 imaging onto the CCD chip<br />
to have a better resolution when imaging the small dimensions <strong>of</strong> evaporated clouds.<br />
Compared to the area <strong>of</strong> the CCD chip, the MOT is a large object. During expansion<br />
atoms move outside <strong>of</strong> the field <strong>of</strong> view, obscuring the atom number and limiting the<br />
time an expanded cloud can be observed. Therefore, it seems not surprising that the<br />
absolute values deviate from the measured value <strong>of</strong> the MOT temperature in subssection<br />
2.4.2. As can be seen in figure (2.11) the initial widths <strong>of</strong> MOT and sub-Doppler cooling<br />
differ by about 0.5 mm. In the sub-Doppler cooling phase the atoms experience a force<br />
that damps their expansion, hence there cannot be a large deviation <strong>of</strong> the initial widths.<br />
This gives a hint that our inefficient sub-Doppler cooling scheme might come from bad<br />
beam balancing and misalignment. If the beams passed each other, they would tear the<br />
cloud apart in the most extreme case.<br />
For a dark spot MOT it is questionable if it makes sense to use a molasses sequence.<br />
Due to the higher densities in the dark-spot MOT it would not be possible to use regular<br />
sub-Doppler cooling. Other groups suggested to use a dark form <strong>of</strong> sub-Doppler cooling,<br />
where the cooling process happens on the shell <strong>of</strong> the hollow sphere [36]. Nowadays<br />
groups using sodium make use <strong>of</strong> the dark spot technique but not <strong>of</strong> sub-Doppler cooling<br />
[25, 51]. We tried to implement a sub-Doppler cooling phase with the dark spot setup<br />
but saw no positive effect. From this data and the resulting phase-space density in the<br />
magnetic trap we decided not to use sub-Doppler cooling.
30 2.5 Sub-Doppler Cooling
31<br />
Chapter 3<br />
The magnetic trap<br />
After having cooled the atoms in a laser light field to or below the Doppler temperature,<br />
they are transferred into a magnetic trap. In the first production <strong>of</strong> BECs a magnetic<br />
trap was used. In the meanwhile some groups use optical dipole traps instead [52].<br />
3.1 Types <strong>of</strong> magnetic traps<br />
The first magnetic trapping <strong>of</strong> neutral atoms was reported with a gas <strong>of</strong> 23 Na atoms in<br />
1985 [53]. The lifetime <strong>of</strong> the sample was limited to 0.87 s due to collisions with hot<br />
background gas. This was done with a quadrupole trap. Later, different trap configurations<br />
like the I<strong>of</strong>fe-Pritchard trap [54] or the TOP trap (time-orbiting potential trap)<br />
[55] were used to capture neutral atoms. The critical temperature T C when condensation<br />
occurs depends on the trapping geometry and the dimensionality [56].<br />
The quadrupole trap<br />
One <strong>of</strong> the simplest trap designs is the quadrupole trap, consisting <strong>of</strong> a pair <strong>of</strong> coils in<br />
an anti-Helmholtz-like configuration. It has the advantage that it can be easily built.<br />
Further, the simple design provides great optical access. The current flows anti-parallel<br />
through the two coaxial coils and produces a cylindrical symmetric field. For analytical<br />
calculations <strong>of</strong> the entire field, elliptical integrals are needed [57] but power series<br />
expansions can be found [58]. Up to third order this is<br />
[ ]<br />
B z (z, ρ) = b 1 z + b 3 z 3 − 3zρ2<br />
(3.1)<br />
2<br />
B ρ (z, ρ) =<br />
[<br />
ρ ρz<br />
2<br />
−b 1<br />
2 − b 3<br />
B φ (z, ρ) = 0<br />
2 + 3ρ2<br />
8<br />
for an infinitesimal thin wire in cylindrical coordinates. The coefficients are<br />
b 1 =<br />
]<br />
3µ 0 IdR 2<br />
(R 2 + d 2 ) 5/2 (3.2)<br />
b 3 = 5(4d2 − 3R 2 )<br />
6(A 2 + ρ 2 ) 2 b 1
32 3.1 Types <strong>of</strong> magnetic traps<br />
where R is the loop radius and 2A is the distance between the two coaxial coils that are<br />
supplied by a current I. For small distances from the trapping center the potential can<br />
be approximated in the cylindrical basis as a linear field<br />
⎛ ⎞<br />
⃗B = b 1<br />
2<br />
2z<br />
⎜<br />
⎝ ρ<br />
⎟<br />
⎠ (3.3)<br />
0<br />
where the field gradient is twice as large in the axial than in the transversal direction <strong>of</strong><br />
coil configuration. A disadvantage <strong>of</strong> this trap is the magnetic zero field at the bottom<br />
<strong>of</strong> the trap that can lead to Majorana losses due to non-adiabatic transitions. There are<br />
several ways to get rid <strong>of</strong> the zero field. Our way is to use an optical plug. This scheme<br />
was first demonstrated in [4].<br />
The I<strong>of</strong>fe-Pritchard Trap<br />
The I<strong>of</strong>fe-Pritchard trap produces a magnetic field that is strong enough to define a<br />
quantization axis everywhere in the trap, hence Majorana losses are not relevant. The<br />
trap consists <strong>of</strong> two different parts. The first part are the so-called I<strong>of</strong>fe-bars that create<br />
a linear quadrupole field. They are formed in a rectangular shape. The second part is<br />
to get rid <strong>of</strong> the zero magnetic field. So-called pinch coils are added and run with the<br />
same polarity with a distance from each other that is greater than the Helmholtz-mode.<br />
They produce a bias and a curvature field. The bias field is undesirable, as it shifts the<br />
bottom <strong>of</strong> the trap out <strong>of</strong> the geometric center. Therefore anti-bias coils are added to<br />
compensate this shift. The trapping field is given by [58]<br />
⎛ ⎞ ⎛ ⎞ ⎛<br />
⎞<br />
0 x<br />
−xz<br />
⃗B = B 0<br />
⎜<br />
⎝0<br />
⎟<br />
⎠ + B′ ⎜<br />
⎝−y<br />
⎟<br />
⎠ + B′′<br />
⎜<br />
2 ⎝ −zy<br />
⎟<br />
⎠ . (3.4)<br />
1 0 z 2 − 1 2 (x2 + y 2 )<br />
The I<strong>of</strong>fe-Pritchard trap can be approximated by an anisotropic harmonic field. This<br />
is very useful, as the harmonic oscillator is a textbook standard in quantum mechanics<br />
and therefore easy to describe.<br />
This trap is used in an experiment [59] aiming for similar goals as the NaLi experiment.<br />
They were able to prepare a BEC <strong>of</strong> 23 Na in the stretched |F = 2, m F = 2〉<br />
state. This state is favorable for sympathetic cooling <strong>of</strong> 6 Li in the |F = 3/2, m F = 3/2〉<br />
state. Spin-exchange collisions are forbidden for these states, since they are both in the<br />
stretched state. The total projection <strong>of</strong> the spin is conserved in this case. Furthermore<br />
this state can be trapped at high magnetic field values, whereas the lithium state <strong>of</strong><br />
|F = 1/2, m F = −1/2〉 can only be trapped in magnetic fields up to about 27 G (see<br />
figure (2.1)). To prepare the 23 Na BEC in the stretched state it is necessary to get rid <strong>of</strong><br />
the |F = 2, m F = 1〉 and |F = 2, m F = 0〉 (quadratic Zeeman-shift) states that are also<br />
captured in the magnetic trap. This was done by spectrally separating them with a<br />
homogeneous field <strong>of</strong> B 0 = 80 G using the bias field <strong>of</strong> the I<strong>of</strong>fe-Pritchard trap. The<br />
splitting allowed for selective removal <strong>of</strong> the undesired states.<br />
This was not considered when designing the magnetic trap <strong>of</strong> our experiment. We<br />
do not have an extra bias field that could produce 80 G, but in combination with optical<br />
traps, the purification <strong>of</strong> the upper-hyperfine state could be possible.
3.2 Setup <strong>of</strong> the magnetic trap 33<br />
The TOP trap<br />
Finally, it should be mentioned that the very first BEC was achieved using a TOP trap<br />
[55]. The essential part is a non-static magnetic bias field that is added transversally to<br />
a quadrupole field. A static addition <strong>of</strong> a bias field only causes a shift <strong>of</strong> the magnetic<br />
field zero. Eventually the atoms follow this shift and will spill out <strong>of</strong> the trap. To avoid<br />
this, the bias field rotates with a speed which the atoms can no longer follow. The<br />
atoms experience a time-averaged, orbiting potential with a minimum having a non-zero<br />
magnetic field. To first order the trapping potential is harmonic. Atoms travelling far<br />
out <strong>of</strong> the time-avaraged center experience the moving hole, which forms a circle and get<br />
lost. This circle is referred to as the “Circle <strong>of</strong> Death” and can be used for evaporative<br />
cooling.<br />
3.2 Setup <strong>of</strong> the magnetic trap<br />
The setup <strong>of</strong> our magnetic trap is quite simple. The field is produced by quadrupole<br />
coils that can be supplied with 440 A. The coils are not only used for the magnetic trap<br />
but provide also the quadrupole field for the MOT. In future experiments they will be<br />
used to produce homogeneous fields to make use <strong>of</strong> Feshbach resonances.<br />
3.2.1 The Feshbach coils<br />
The Feshbach coils were custom made by Oswald Elektromotoren GmbH. There are four<br />
coils, each with a double layer <strong>of</strong> windings, in the double pancake configuration. Four<br />
stacks <strong>of</strong> windings are above and below the glass cell. The copper wire has an edge length<br />
<strong>of</strong> 5 mm and a borehole <strong>of</strong> 1.5 mm radius to allow for water cooling. Each coil has 15<br />
windings with an inner radius <strong>of</strong> 18 mm and an outer radius <strong>of</strong> 65 mm. The inductances<br />
<strong>of</strong> the coils vary between 14 µH and 15 µH. We use a Riedel 25 l chiller to cool the coils.<br />
It can deliver an output pressure <strong>of</strong> 4 bar, ensuring a flow <strong>of</strong> 1.5 lmin −1 . The de-ionized<br />
water is stabilized to a temperature <strong>of</strong> 18 ◦ C with an accuracy <strong>of</strong> about 1 K. For any<br />
value <strong>of</strong> the current, no significant heating <strong>of</strong> the coils was measured. Figure (3.1) shows<br />
the magnetic field produced by the coils in a Helmholtz-like and in an anti-Helmholtz-like<br />
configuration using 5 A current. The field was measured in the axial and in the radial<br />
direction with respect to the symmetry axis.<br />
The measured gradients are (3.60±0.05) Gcm −1 for the radial and (6.86±0.13) Gcm −1<br />
for the axial direction. The expected values were 3.75 Gcm −1 and 7.5 Gcm −1 , respectively.<br />
The deviations result possibly from inaccurate measurements. Further, the calculations<br />
were made for an infinitesimal small wire whereas the extension <strong>of</strong> the wire is not negligible<br />
at these distances.<br />
For the Helmholtz-like mode, polynomial functions were fitted to estimate the region<br />
<strong>of</strong> linearity. The fit for the axial direction is a simple parabola. The fit parameters are<br />
depicted in figure (3.1). A region <strong>of</strong> linearity can be defined for which the magnetic field<br />
changes only by 1 <strong>of</strong> the center value. For the axial direction this yields 1.4 ± 0.1 mm.<br />
This value can be considered only an esimate since the measured points are not very<br />
dense in the region <strong>of</strong> interest. For the radial direction a polynomial fit to the fourth<br />
order was used, as depicted in figure (3.1). The resulting region <strong>of</strong> linearity is estimated<br />
to be 314 ± 140 µm. Since the fit coefficients have large errors, the value for the linearity
34 3.2 Setup <strong>of</strong> the magnetic trap<br />
is only a rough estimate. According to the measurement, the magnetic field at 440 A<br />
would be 1760 G.<br />
a)<br />
b)<br />
20<br />
10<br />
B[G]<br />
10<br />
0<br />
B[G]<br />
0<br />
-10<br />
-20<br />
-5 0 5<br />
position[cm]<br />
-20<br />
c)<br />
fit: y=ax<br />
a=6.86±0.13<br />
-10<br />
-5 0 5<br />
position[cm]<br />
5<br />
-19.35<br />
d)<br />
fit: y=ax<br />
a=3.60±0.05<br />
-22<br />
0<br />
B[G]<br />
B[G]<br />
-24<br />
-26<br />
-28<br />
fit: y=a+bx 2<br />
a=-20.24±0.40<br />
b=-0.93±0.14<br />
-3 0 3<br />
position[cm]<br />
B[G]<br />
-5<br />
-10<br />
-15<br />
-19.75<br />
-0.6 0 0.6<br />
position[cm]<br />
fit: y=a+bx 2 +cx 4<br />
a=-19.72±0.03<br />
b=-0.22±0.21<br />
c=0.80±0.62<br />
-20<br />
-8 -4 0 4 8<br />
position[cm]<br />
Figure 3.1: Magnetic field <strong>of</strong> the Feshbach coils: a) and b) is the magnetic field in axial and<br />
radial direction in the anti-Helmholtz-like mode. c) and d) show the axial and radial<br />
direction for the Helmholtz-like mode, respectively. The inlet in d) is a zoom from<br />
−0.8 cm to 0.8 cm<br />
3.2.2 The circuit<br />
During the work on this thesis, the electrical part <strong>of</strong> the experiment was installed. The<br />
electrical system can be categorized into two major parts. The first part is responsible<br />
for the current’s turn-<strong>of</strong>f and the change <strong>of</strong> the coil’s configuration. The second part is<br />
the control <strong>of</strong> the current by a feedback loop. The circuit is depicted in figure (3.2).<br />
The first part is composed <strong>of</strong> so-called insulated-gate bipolar transistors (IGBTs)<br />
that are a hybrid <strong>of</strong> regular bipolar transistors and MOSFETs. They have the capacity<br />
to allow high currents and fast switching. Power-MOSFETs can be used instead. They<br />
are highly sensitive to overvoltage across the emitter-collector (max. 1200 V) and gatecollector<br />
(max. 20 V) connections. In the experiment high voltages occur when the
3.2 Setup <strong>of</strong> the magnetic trap 35<br />
current is turned <strong>of</strong>f. Faraday’s law <strong>of</strong> induction<br />
U ind = − d Φ(t) (3.5)<br />
dt<br />
states that an induced voltage U ind occurs when the magnetic flux Φ changes. The minus<br />
sign means physically that the system wants to keep up the status quo. When the current<br />
is switched on, the rising current gets damped, whereas for a turn-<strong>of</strong>f <strong>of</strong> the current, the<br />
induced voltage tries to keep the current alive. The transient voltages can be as high as<br />
several thousand volts. Therefore the systems needs to be protected with varistors and<br />
transient-voltage-suppressor diodes (TVS). The used varistor across emitter-collector <strong>of</strong><br />
the IGBT clamps at 1056 V. The TVS diode is bipolar and shows clamping at 15 V.<br />
It is used to clamp the voltage across the gate-emitter connections. The coils must be<br />
protected as well, since they can be damaged when switching <strong>of</strong>f the currents. The<br />
varistors have clamping voltages <strong>of</strong> 475 V and are accompanied by ring-down resistors<br />
with R = 10 Ω. These resistors are used to damp the induced current. Commercially<br />
available drivers for the IGBTs were used and built into a circuit for remote control.<br />
The IGBT (MBI600U4-120) and the drivers (VLA) are both from FUJI Electric.<br />
The current flows first through a coil-IGBT stretch depicted in the upper part <strong>of</strong><br />
figure (3.2). After this stretch, the current passes the high-accuracy current transducer<br />
(LEM IT 600-S). This transducer is specified to follow the current with d I = 100 A/µs<br />
dt<br />
and has a linearity <strong>of</strong> < 1 ppm from 0 A up to 600 A. The transducer works with<br />
fluxgate technology and convertes the measured current down with a ratio <strong>of</strong> 1500 : 1.<br />
The maximal current <strong>of</strong> 440A corresponds therefore to an output <strong>of</strong> 293 mA. This signal<br />
is detected in the Sens-Box that mainly consists <strong>of</strong> a 10 Ω high-precision burdon resistor,<br />
resulting in a maximal output voltage <strong>of</strong> 2.93 V. Unfortunately, the signal cannot be<br />
raised by using a higher burdon resistor as the power specifications <strong>of</strong> the transducer do<br />
not allow higher values. Finally, the voltage is monitored on a groundfree oscilloscope<br />
and sent to the current regulator box.<br />
The next stage after the LEM sensor is the passbank, which is responsible for the<br />
current control. This is an array <strong>of</strong> NPN-Darlington transitors that are in a parallel<br />
connection in a common-collector follower configuration. The basis resistor values were<br />
adapted to the maximum output power <strong>of</strong> the current regulator box that sends its setting<br />
signal to the basis <strong>of</strong> the passbank. For the emitter resistor, paper clips were used<br />
as they have a small resistance and can dissipate a lot <strong>of</strong> power. Altogether there<br />
are 64 Darlington transistors in parallel. The transistors are water cooled. Although<br />
the maximum current for each transistor lies at 20 A, many are needed, as the power<br />
dissipation is quite bad and overheating can damage them. The worst situation for the<br />
passbank are intermediate currents. This can be understood as follows. The resistance <strong>of</strong><br />
each single coil is about 5 mΩ. Including the wiring the total resistance can be estimated<br />
to be between (30 − 40) mΩ. The voltage is constant at 15V. Therefore, the power<br />
dissipated in the passbank is given by<br />
P passbank = [ 15I − R coil I 2] W. (3.6)<br />
Maximizing the power in the passbank yields values between (1.4 − 1.9) kW at currents<br />
between (180 − 250) A. This is only an orientation, since the power dissipated in the<br />
IGBT has not been taken into account.
36 3.2 Setup <strong>of</strong> the magnetic trap<br />
R1<br />
L1<br />
U<br />
V1<br />
C1<br />
Agilent<br />
6690A<br />
C1<br />
Z1<br />
Z3<br />
U U<br />
V2<br />
V2<br />
R1<br />
L2<br />
U<br />
C1<br />
V1<br />
C1<br />
Z2<br />
Z4<br />
U<br />
U<br />
V2<br />
V2<br />
earthfree!<br />
...<br />
LEM-Sensor<br />
Q1<br />
R3<br />
Q1<br />
R3<br />
Sens-Box<br />
Current-Regulator<br />
SENS<br />
DRIVE<br />
R5<br />
R5<br />
CNTRL<br />
MON<br />
CPU<br />
Figure 3.2: Electric circuit: R i and L i correspond to the passive parts in the circuit. The<br />
IGBTs and their drivers are denoted by Z i and C i , respectively. For overvoltage protection,<br />
most components are protected with varistors V i . The Darlington transistors<br />
Q1 control the current.<br />
The current regulator<br />
The current regulator was built by the electronic division. A schematic diagram is<br />
depicted in figure (3.3). The signal from the sensor box is sent to the SENS input and is<br />
amplified by an instrumentation amplifier. The gain resistor R G is selected to project the<br />
output span <strong>of</strong> (0−2.93) V from the sensor box to (0−10) V, hence the sensitivity <strong>of</strong> the<br />
control voltage is raised. The preamplified signal can be observed at the MON output<br />
where the signal is fed through an isolation amplifier. The amplification ratio is 1 : 1<br />
and it is used only by means <strong>of</strong> galvanic isolation. The signal is sent to an integrator
+<br />
3.2 Setup <strong>of</strong> the magnetic trap 37<br />
circuit, where the capacitor was chosen such that the current showed a fast response<br />
without any oscillations. The CNTRL input provides the setpoint. Finally, the output<br />
stage drives the integrator output. Isolation amplifiers always deliver an <strong>of</strong>fset voltage.<br />
Therefore the control voltage needs to be set slightly negative to actually achieve zero<br />
current.<br />
15V<br />
current<br />
regulator<br />
voltage<br />
in<br />
CNTRL<br />
-<br />
+<br />
Integrator<br />
Output Stage<br />
passbank<br />
drive<br />
DRIVE<br />
current<br />
monitor<br />
out<br />
MON<br />
Isolation Amplifier<br />
R G<br />
sensor<br />
input<br />
SENS<br />
Isolation Amplifier<br />
-<br />
Instrumentation Amplifier<br />
Figure 3.3: Current regulator<br />
The amplifier shows a response time <strong>of</strong> about 7 ms when a current <strong>of</strong> 440 A is switched<br />
on. To reduce oscillations for such extreme events we use tailored ramp forms for the<br />
CNTRL input.<br />
3.2.3 The fast turn-<strong>of</strong>f<br />
Switching <strong>of</strong>f the magnetic field is essential for our measurements. The high field shifts<br />
the Zeeman levels, hence it is impossible to image all the atoms at once in a gradient field.<br />
Therefore, the magnetic field is supposed to decay as fast a possible. As equation (3.5)<br />
shows, the system wants to keep the current alive when it is turned <strong>of</strong>f, creating high<br />
voltages that may damage the surrounding equipment through sparkovers. A turn <strong>of</strong>f <strong>of</strong><br />
440 A in 10 µs would create an induced voltage <strong>of</strong> more than 2500 V. IGBTs normally<br />
switch in a few ns time, but the induced voltage is not that easy calculable and needs to<br />
be simulated by finite element methods. The estimated voltages above are too high for<br />
the epoxy <strong>of</strong> the coils which is rated to withstand voltages <strong>of</strong> approximately 1500 V. The<br />
insulation <strong>of</strong> the coils could be damaged. Therefore Epcos varistors <strong>of</strong> type S20K175<br />
with clamping voltages <strong>of</strong> about 450 V are used over each coil pair.<br />
A direct measurement <strong>of</strong> the current through the coils was not possible since the<br />
clamp-on ammeters were not specified for this current. Therefore the voltage across the<br />
coils was measured. The voltage does not relate directly to the current as can be seen
38 3.2 Setup <strong>of</strong> the magnetic trap<br />
in the simulation in figure (3.4). The voltage stays at a level <strong>of</strong> about 375 V for 25 µs.<br />
This shows that the varistor clamps at a voltage that is below the specified level. The<br />
decay time defined as the time the voltage drops from 90% to 10% <strong>of</strong> the maximum value<br />
is 90 µs. This is mainly due to the long voltage tail. This tail still is 15 V at 200 µs.<br />
Eventually, the tail extincts at times between (400−450) µs. A finite element simulation<br />
using PSpice 1 shows that the expected voltage drops in less than 50 µs to zero voltage.<br />
The simulated current through the coils drops linearly in about the same time.<br />
To date it is not quite clear why the measured voltage exhibits this tail. Removing<br />
this tail is crucial to be able to image at very short times. The simulations showed that<br />
the use <strong>of</strong> higher calmping voltages is the only effective way to reduce the turn-<strong>of</strong>f time.<br />
For too high ring-down resistors the current starts to oscillate and if the resistors are<br />
selected to low the turn-<strong>of</strong>f time increases drastically. The voltage drop accros the whole<br />
system is about 750 V. If a secure shielding around the coils’ ends was installed, the<br />
current varistors could be replaced by varistors with higher clamping voltages, allowing<br />
shorter turn-<strong>of</strong>f times. Nevertheless, this would not solve the problem <strong>of</strong> the voltage tail.<br />
I[A] /U[V]<br />
500<br />
450<br />
400<br />
350<br />
300<br />
250<br />
200<br />
150<br />
100<br />
50<br />
measured voltage<br />
simulated voltage<br />
simulated current<br />
90%<br />
10%<br />
0<br />
0 ~30 100 ~120<br />
200<br />
time[µs]<br />
Figure 3.4: PSpice simulation <strong>of</strong> the current in the Feshbach coils: The voltage across the<br />
coils was measured during the turn-<strong>of</strong>f <strong>of</strong> the current. A PSpice simulation shows the<br />
expected transient voltage and the decaying current.<br />
3.2.4 The field switch<br />
As was mentioned previously, the magnetic field is needed in two different configurations:<br />
the gradient field and the homogeneous field. Having only one pair <strong>of</strong> coils an H-bridge<br />
with four IGBTs is used as depcited in figure (3.2). Two diagonal IGBTs form a pair that<br />
is always switched in the same manner. The two different pairs, however, are switched<br />
in an opposing mode. If one pair is switched on, the other has to be switched <strong>of</strong>f. This<br />
1 by Cadence
3.2 Setup <strong>of</strong> the magnetic trap 39<br />
causes a change <strong>of</strong> the current’s polarity through the second coil. The method is also<br />
used in our home-built laser beam shutters [60].<br />
The driver unit has two inputs: one to select the polarity and the other to turn <strong>of</strong>f<br />
all four IGBTs. The pre-logic <strong>of</strong> the IGBT drivers is depicted in figure (3.5). It can<br />
be catigorized into two main parts. First, the switching part and, second, a frequency<br />
limitation part. The pre-logic elements consist <strong>of</strong> integrated circuits <strong>of</strong> the 74LSXXX<br />
series.<br />
a) b)<br />
Mon<strong>of</strong>lop<br />
ON/OFF<br />
A<br />
Field switch<br />
B<br />
Mon<strong>of</strong>lop<br />
to IGBT drivers<br />
Figure 3.5: a) The first part is for changing the current’s polarity through the second coil<br />
and for the turn-<strong>of</strong>f event. b) The second part is the frequency limitation part.<br />
The first part is constructed only with an inverter and an AND-gate. In the truth<br />
table (3.1) the first two rows represent one polarity mode and the other two rows the<br />
opposing polarity mode. A true signal at the ON/OFF input always sets the outputs to<br />
false. In either polarity mode, one pair is true and the other false, provided the On/OFF<br />
input is false.<br />
Table 3.1: Truth table <strong>of</strong> the field switch<br />
Field switch ON/OFF A B<br />
1 1 0 0<br />
1 0 1 0<br />
0 1 0 0<br />
0 0 0 1<br />
The frequency limitation part was integrated since an IGBT can only be switched<br />
up to a certain limiting frequency. In case <strong>of</strong> longer operation times, a high frequency<br />
can lead constantly to a high current density and hence damage the IGBT. In our case,<br />
the limiting frequency is specified to be about 40 kHz. We do not intend to switch that<br />
fast, but as our computer control system has a sampling rate <strong>of</strong> 1 MHz, there is a chance<br />
that the IGBTs might become accidentally damaged. A combination <strong>of</strong> a mon<strong>of</strong>lop and<br />
an OR-gate produces pulses that have a length <strong>of</strong> at least 100 µs. For frequencies higher
40 3.3 Trappable states and the first steps in the magnetic trap<br />
than 12 kHz the output stays constant, therefore protecting the IGBT. Finally, the signal<br />
is sent to an inverter again. This is needed to send the proper signal to the IGBT drivers.<br />
They charge the gate emitter capacity in less than 1 µs.<br />
3.3 Trappable states and the first steps in the magnetic<br />
trap<br />
Loading into a magnetic trap is actually a straightforward task. Even so, there were<br />
some difficulties when we first implemented it. Parameters for procedures like mode<br />
matching and adiabatic compression had to be selected in order to improve the loading<br />
and compression <strong>of</strong> the magnetic trap.<br />
3.3.1 Trappable states <strong>of</strong> 23 Na<br />
As was seen in subsection 2.1.3, the atoms are in the Zeeman regime and therefore<br />
equation (2.12) holds. The magnetic moment <strong>of</strong> the atom precesses around the local<br />
magnetic field B(⃗r) with the Larmor frequency<br />
ω L = m F g F µ B<br />
B<br />
. (3.7)<br />
If the precession is rapid compared to the change <strong>of</strong> the magnetic field, it can follow<br />
adiabatically and the interaction potential is<br />
V (⃗r) = m F g F µ B B(⃗r) (3.8)<br />
Earnshaw’s theorem [50] prohibits the existance <strong>of</strong> a static magnetic feld maximum in<br />
free space, therefore only atoms in low-field seeking states can be magnetically trapped.<br />
Low-field seeking atoms fulfill g F m F > 0.<br />
For 23 Na the possible candidates are |F = 1, m F = −1〉 with g F = −1/2 for the F = 1<br />
ground state and for the F = 2 ground state |F = 2, m F = 2〉, |F = 2, m F = 1〉 and<br />
very weakly |F = 2, m F = 0〉 with g F = +1/2. The first 23 Na BEC was realized in the<br />
F = 1 ground state. Just a couple <strong>of</strong> years ago, a BEC in the F = 2 ground state was<br />
accomplished [61]. In the latter, effects between the different Zeeman sublevels lead to<br />
an extra loss channel.<br />
3.3.2 Mode matching<br />
After the first cooling stages with lasers, the atom cloud needs to be transferred into a<br />
magnetic trap. This transfer is easily a source <strong>of</strong> heating and loss <strong>of</strong> phase-space density<br />
if it is not mode matched. It is important that the atoms experience a similar potential<br />
in the magnetic trap to the one they experienced in the MOT. The density is Gaussian<br />
for MOTs with atom numbers below 4 × 10 4 . Therefore perfect mode matching with a<br />
harmonic trap should be possible. A too steep gradient causes heating whereas a too<br />
shallow trap can cause a loss in density. In the case <strong>of</strong> large MOTs the density has<br />
different pr<strong>of</strong>iles as seen in [38]. In addition, the density distribution depends on the<br />
alignment <strong>of</strong> the beams. Imperfectly aligned beams can push in a direction <strong>of</strong> preference
3.3 Trappable states and the first steps in the magnetic trap 41<br />
and deform the MOT. Since all projection states m F are equally present in the MOT,<br />
the maximal transfer <strong>of</strong> MOT atoms into the magnetic trap can be 33% if atoms are<br />
captured in the F = 1 state. Depending on the condition <strong>of</strong> the experiment, there are<br />
up to 2 × 10 8 atoms in the magnetic trap.<br />
The condition for mode-matching can be expressed as<br />
P SD MOT = P SD MAG (3.9)<br />
where the phase-space density is P SD ∝ T −3/2 N/V . For a transfer that does not change<br />
the temperature, the volume <strong>of</strong> the magentic trap would have to be three times smaller<br />
than the volume <strong>of</strong> the MOT. However, normally the volumes are set equal since the<br />
assumption <strong>of</strong> the same temperature is mostly not fulfilled.<br />
In our quadrupole trap, mode matching is done by selecting a field gradient such<br />
that the trapping volume corresponds more or less to the MOT volume. The density <strong>of</strong><br />
atoms in the classical case is given by<br />
( ) 3/2 [<br />
2πmkB T<br />
n(⃗r) =<br />
z exp − U(⃗r) ]<br />
h<br />
} 2<br />
k<br />
{{ }<br />
B T<br />
n 0<br />
(3.10)<br />
with z the fugacity and n 0 the peak density. The quadrupole potential is U(⃗r) =<br />
m F g F µ B B ′√ x 2 + y 2 + 4z 2 . An effective trapping volume can be defined as V eff = N/n eff .<br />
The effective density n eff = ∫ n 2 (⃗r)dV/ ∫ n(⃗r)dV = n 0 /8 accounts for the spatial dependence<br />
<strong>of</strong> n and gives a mean value <strong>of</strong> the density. Evaluating this with the anisotropic<br />
quadrupole potential leads to<br />
( ) 3 kB T<br />
V eff = 32π<br />
. (3.11)<br />
m F g F µ B B ′<br />
A higher temperature means that more atoms are at higher field gradients – the occupied<br />
volume enlargens. An increase in the field gradient is an effective compression which<br />
makes the trapping volume smaller. The relation is V eff ∝ (T/ω) 3/2 for the harmonic<br />
trap, with ω being the trapping frequency 2 .<br />
The Gaussian density distribution for the MOT atoms can be written as<br />
]<br />
n(⃗r) = n 0 exp<br />
[− x2 + y 2 + z 2<br />
(3.12)<br />
2σ 2<br />
where σ can be related via equation (2.27) with the temperature and the MOT parameters.<br />
The effective volume <strong>of</strong> the MOT can be equated to<br />
V MOT = (4π) 3/2 σ 3 (3.13)<br />
with σ ∝ T 1/2 . Setting equations (3.13) and (3.11) equal, the initial gradient <strong>of</strong> the<br />
magnetic trap can be easily estimated. For MOTs with a diameter <strong>of</strong> 6 mm, the strong<br />
gradient would need to be set to 66 Gcm −1 . This seems to be a too small gradient,<br />
2 This shows the advantage <strong>of</strong> the quadrupole trap. It has a tighter confinement than the other traps<br />
which is preferable for evaporative cooling
42 3.3 Trappable states and the first steps in the magnetic trap<br />
especially when considering the gradients <strong>of</strong> 130 Gcm −1 used in [4]. The result is not<br />
surprising as the Gaussian approximation fails for large atom number MOTs.<br />
The phase-space density and the atom number in the magnetic trap for different<br />
field gradients were compared in figure (3.6). The phase-space density rises linearly with<br />
decreasing magnetic field gradient. At a certain point the phase-space density should fall<br />
as the density decreases, but these regions were not measured. After the loading process<br />
the magnetic trap was ramped to the same value for all initial field gradients. The<br />
atomnumber in figure (3.6) seems to have a maximum at about 140 Gcm −1 . The change<br />
is only 20% whereas the phase-space density has a change <strong>of</strong> almost 50%. Currently the<br />
experiment uses an initial magnetic field gradient <strong>of</strong> 79 Gcm −1 which is only 13 Gcm −1<br />
<strong>of</strong>f <strong>of</strong> the estimated value.<br />
phase-space<br />
1.6 x density<br />
atom 10-6<br />
6.0 x number<br />
linear fit:<br />
107 1.5<br />
(16.8-2.6xB’ 5.8<br />
init<br />
)x10 -7<br />
1.4<br />
5.6<br />
1.3<br />
5.4<br />
1.2<br />
5.2<br />
1.1<br />
5.0<br />
4.8<br />
1.0<br />
4.6<br />
0.9<br />
4.4<br />
0.8<br />
4.2<br />
0.7<br />
4.0<br />
50 100 150 200 50 100 150 200<br />
initial gradient [Gcm -1 ]<br />
initial gradient [Gcm -1 ]<br />
Figure 3.6: Mode matching measurement: The phase-space density shown in a) increases<br />
when the inital gradient is lowered. The atom number depicted in b) appears to have<br />
a maximum at a gradient <strong>of</strong> 140 Gcm −1 .<br />
Although there is a dependence <strong>of</strong> the phase-space density on the initial gradient, it<br />
is not very strong. The initial position <strong>of</strong> the MOT center and the magnetic trap center<br />
appeared to be far more critical. With bias coils we were able to adjust the position <strong>of</strong><br />
the MOT to achieve better mode matching.<br />
3.3.3 Adiabatic compression<br />
Immediately after turning on the magnetic trap, we want to compress the cloud to<br />
increase the density. This results in a higher elastic collision rate which is necessary for<br />
the evaporative cooling stage to be efficient. It is <strong>of</strong> course not possible to increase the<br />
field gradient instantaneously, as this would give the atoms a kick and cause a loss in<br />
phase-space density. Since phase-space density is the figure <strong>of</strong> merit in our experiment,<br />
we do not want to lose any <strong>of</strong> it at any time. In an adiabatic compression where the<br />
atom remains in its trapped state, the phase-space density stays constant. To compress<br />
adiabatically, the trapping length L assocciated with the trapping volume by L ∝ V 1/3<br />
has to change more slowly than the mean velocity <strong>of</strong> the trapped sample [33]. In this
3.3 Trappable states and the first steps in the magnetic trap 43<br />
case, the atoms do not experience the change <strong>of</strong> the potential and can adapt to the new<br />
compressed trap. This adiabatic criterion can be expressed as<br />
dL<br />
∣ dt ∣ ≪ v. (3.14)<br />
or equivalently |dL/dt| = ɛv with the adiabaticity parameter ɛ ≪ 1. The trapping<br />
volume V eff ∝ T 3 /B ′3 from equation (3.11) can be expressed as an effective trapping<br />
length<br />
L ∝ T/B ′ . (3.15)<br />
This length has to change slowly in order to be adiabatic. If the process is adiabatic,<br />
then the volume and the temperature <strong>of</strong> an ideal gas fulfill the adiabatic relation [62]<br />
V T 1/(κ−1) = const. (3.16)<br />
κ is the adiabatic exponent and in the case <strong>of</strong> an ideal gas it is 5/3. Therefore the<br />
trapping length obays<br />
L ∝ T −1/2 . (3.17)<br />
With equations (3.17) and (3.15) the adiabatic length scale behaviour can be worked into<br />
our physical situation. The following quantities having a subscript denote the initial<br />
values <strong>of</strong> these quantities. The quantities without subscript are timedependent, e.g.<br />
L = L(t). The mean velocity <strong>of</strong> the atoms scales as v ∝ T 1/2 . Using this scaling and<br />
equations (3.17) and (3.15), the following relations<br />
( ) B<br />
′ −1/3<br />
L = L 0 (3.18)<br />
B ′ 0<br />
( ) B<br />
′ 1/3<br />
v = v 0 (3.19)<br />
can be obtained. This relates the thermodynamical properties <strong>of</strong> the gas in the trap to<br />
the field gradient B ′ during an adiabatic compression. The field gradient can be chosen<br />
at will. Equations (3.17) and (3.15) need to be inserted into the adiabatic criterion<br />
(3.14). This yields<br />
B ′2/3 ′−5/3 dB′<br />
0 B = 3v 0<br />
ɛ. (3.20)<br />
dt L 0<br />
Solving this equation is straightforward and gives the final time-dependence <strong>of</strong> the adiabtaic<br />
compression<br />
B ′ (t)<br />
B ′ 0<br />
=<br />
B ′ 0<br />
1<br />
(1 − (2v 0 /L 0 )ɛt) 3/2 (3.21)<br />
At first, the change <strong>of</strong> compression is slow and accelerates until infinite compression can<br />
be reached in a finite time. By compressing the atoms, they get heated (correspondingly<br />
the density rises such that the phase-space density stays equal), causing a rise in the mean<br />
velocity. Therefore the field gradient can be ramped up faster, causing an even higher<br />
increase in temperature, and so on. The process is self-energizing, hence explaining this<br />
paradox time behaviour.<br />
In our computer control system we have functions to simulate the compression <strong>of</strong><br />
the magnetic trap. An exponential ramp <strong>of</strong> the form A exp(t/τ) + A0 was chosen
44 3.3 Trappable states and the first steps in the magnetic trap<br />
which is ramped up in 300 ms. Figure (3.7) depicts the adiabtic curve for our trapping<br />
parameters and the exponential ramp with time constants τ = 7 ms, τ = 70 ms<br />
and τ = 400 ms. The corresponding adiabaticity parameter <strong>of</strong> the adiabatic curve is<br />
ɛ = 0.01, fulfilling criterion (3.14). This means that when this ramp up process is<br />
driven from the initial value <strong>of</strong> the trap to the fully compressed value corresponding to<br />
B ′ (t = 300 ms)/B 0 ′ = 660 Gcm −1 /79 Gcm −1 = 8.33, the compression can be assumed<br />
adiabatic. The time constant τ = 400 ms used for a long time in the experiment shows<br />
that it fulfills the adiabtic criterion better at the end <strong>of</strong> the ramp than at the beginning.<br />
This means that the slope <strong>of</strong> the exponential gradient evolution is larger than the slope<br />
<strong>of</strong> the adiabatic gradient evolution at the end <strong>of</strong> the ramp up process, and reversed at<br />
the beginning. Very small time constants, however, fulfill the adiabatic criterion better<br />
at the beginning <strong>of</strong> the ramp up process than towards the end.<br />
8<br />
7<br />
6<br />
Exponential ramp with τ =10 ms<br />
Exponential ramp with τ =70 ms<br />
Exponential ramp with τ =400ms<br />
Adiabatic curve with ε =0.011<br />
5<br />
B(t)/B 0<br />
4<br />
3<br />
2<br />
1<br />
0<br />
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35<br />
t[s]<br />
Figure 3.7: Adiabatic curve and experimental ramps<br />
The compression parameter τ was scanned from 7 ms to 3 s. There was no significant<br />
difference in the phase-space denstiy or atom number visible. For τ > 400 ms the form<br />
<strong>of</strong> the curve during the 300 ms ramp time does not change anymore. Small τ are limited<br />
by the speed <strong>of</strong> the current regulator, hence we could not measure below τ = 7 ms.<br />
To get an idea <strong>of</strong> why there is no difference visible, the adiabaticity <strong>of</strong> the different<br />
ramps has to be considered. Using equation (3.15), the change <strong>of</strong> length is<br />
dL<br />
dt ∝ 1 dT<br />
} B {{ ′ dt}<br />
A<br />
− T B ′2 dB ′<br />
dt<br />
} {{ }<br />
B<br />
. (3.22)<br />
Term A causes an expansion <strong>of</strong> the volume <strong>of</strong> the cloud whereas term B lets the volume
3.4 Loss mechanisms and heating 45<br />
shrink. For a simple estimation, term A is neglected. Further, it is assumed that the<br />
temperature is the same for the different ramps. Physically this is not true, but the<br />
change in temperature after compression is normally less than 20% whereas the field<br />
gradient changes by more than 800%. Neglecting term A means that the volume shrinks<br />
faster than it would normally do, hence the estimation <strong>of</strong> the adiabaticity gives an upper<br />
boundary.<br />
The slope <strong>of</strong> the exponential ramps and the adiabatic curve is calculated and compared<br />
at the point t = ˜t <strong>of</strong> the greatest deviation between the slopes <strong>of</strong> the two curves.<br />
Dividing the change <strong>of</strong> lengths <strong>of</strong> the exponential ramp by that <strong>of</strong> the adiabatic curve<br />
yields with equation (3.14) the ratio <strong>of</strong> the adiabaticity parameters<br />
dL exp /dt<br />
dL adiab /dt =<br />
dB exp(˜t) ′ /B ′ dt exp(˜t) 2<br />
= ɛ exp<br />
(3.23)<br />
dB adiab ′ (˜t)<br />
/B ′ dt adiab (˜t) 2 ɛ adiab<br />
where the adiabatic curve and the exponential ramps are denoted by subscripts. For a<br />
time constant <strong>of</strong> 10 ms the ratio is 5.5, resulting in a maximal adiabaticity parameter <strong>of</strong><br />
ɛ exp = 0.055. This small value explains why there was no change <strong>of</strong> phase-space density<br />
visible when τ was varied. All the other measured time constants have an even smaller<br />
adiabaticity parameter.<br />
3.4 Loss mechanisms and heating<br />
On the way to a <strong>Bose</strong>-<strong>Einstein</strong> condensate there are several processes that work against<br />
us. Loss mechanisms impose a time-scale in which the cooling process and the experiment<br />
has to take place. They can be characterized more or less by the number <strong>of</strong> particles<br />
inside the trap that are involved in each loss mechanism. Heating, on the other hand,<br />
does not lead directly to loss, but can hinder condensation.<br />
3.4.1 One-body losses<br />
One-body losses are the main loss feature for trapped atoms in dilute gases. Only one<br />
atom in the trap is involed in contrast to two- or three-body losses which result from<br />
interactions <strong>of</strong> the atoms within the sample. There are several causes for one-body losses.<br />
The most prominent is the collision with background gas, but losses due to stray light<br />
can be significant as well. The one-body loss rate Γ 1body is density independent. This is<br />
modeled by<br />
dN<br />
dt = −Γ 1bodyN (3.24)<br />
resulting in an exponential decay <strong>of</strong> the atom number. Majorana losses can be treated<br />
as one-body losses since they do not result from the interaction between two particles in<br />
the trap.<br />
Background gas collisions<br />
Heating and trap loss can be caused by collisions <strong>of</strong> the trapped sample with background<br />
gas. The background gas is at room temperature, hence having a mean velocity that is<br />
a thousand times larger than that <strong>of</strong> the trapped sample. The energy transfer is large
46 3.4 Loss mechanisms and heating<br />
enough to expel atoms from the trap. Using the loss rate R = nvσ Na-Na [63] a rule <strong>of</strong><br />
thumb for the lifetime<br />
6.2 × 10−9<br />
τ = s (3.25)<br />
p[mbar]<br />
at a pressure p can be obtained. The lifetime for a pressure <strong>of</strong> 10 −11 mbar is around<br />
10 minutes. The gauge in the experimental section shows a pressure <strong>of</strong> this magnitude.<br />
Still, the lifetime in our experiment is more than an order <strong>of</strong> magnitude away from this<br />
value. The measurement <strong>of</strong> the pressure takes place inside the last pumping section.<br />
The distance between the glass cell and gauge is relatively long. Therefore the walls<br />
have a pumping effect and create a difference between the measured pressure and the<br />
pressure in the glass cell. Even so, this explanation does not account for this big <strong>of</strong><br />
a difference. The atomic beam is blocked sufficiently by the mechanical beam shutter.<br />
It was checked if there is a difference in lifetime when the mechanical beam shutter is<br />
opened or closed. The difference was less than a factor <strong>of</strong> two. To check if the shutter<br />
was not closing properly, the lifetime <strong>of</strong> the atoms was measured with a closed valve<br />
at the differential pumping stage (see figure (2.4)). This measurement showed that the<br />
atomic beam shutter works just fine.<br />
Majorana losses<br />
As previously mentioned, the magnetic field zero causes Majorana losses [64] in the<br />
quadrupole trap. In a spatially changing magnetic field the orientation <strong>of</strong> the magnetic<br />
moment needs to be preserved with respect to the local magnetic field while the particle<br />
is moving. The magnetic moment has to reorientate in an adiabatic manner or else a<br />
non-adiabatic transition to untrapped states is possible. If the change in the magnetic<br />
field is too rapid the precessing magnetic moment cannot follow the field. The criterion<br />
for adiabaticity can be formulated as [65]<br />
ω L >><br />
∣ B<br />
d<br />
B dt<br />
∣ . (3.26)<br />
For small magnetic fields the Larmor frequency, given by equation (3.7), tends to zero<br />
and the criterion is not fulfilled anymore. Therefore the non-adiabatic region is called<br />
the “hole” <strong>of</strong> the magnetic trap. For an estimation <strong>of</strong> this region’s effect on the sample,<br />
the trajectories <strong>of</strong> the atoms have to be considered. For harmonic traps, the motion<br />
is characterized by the trapping frequencies ω Trapi (i = 1, 2, 3) <strong>of</strong> the potential. For the<br />
linear potential, trapping frequencies that correspond to circular motion around the trap<br />
center are introduced [66]. A simple classical estimation [55] uses a particle <strong>of</strong> mass m<br />
being in the z = 0 plane, passing the trap center with velocity v at a distance ρ as<br />
shown in figure (3.8). The atom moves on a circular orbit with ω = v/ρ resulting from<br />
the Lorentz and the centripetal force. In the rest frame <strong>of</strong> the atom the direction <strong>of</strong><br />
the magnetic field is changing in time with v/ρ = | d B/B|, hence the Larmor frequency<br />
dt<br />
must exceed the trapping frequencies to avoid losses. An approximated radius<br />
[ ] 1/2<br />
v<br />
ρ 0 ≈<br />
. (3.27)<br />
g F m F µ B B ′
3.4 Loss mechanisms and heating 47<br />
results if equation (3.7) is set equal to v/ρ. For the Doppler temperature <strong>of</strong> 23 Na at the<br />
maximal transversal gradient <strong>of</strong> B ′ = 330 Gcm −1 , the radius is ρ 0 = 2.27 µm and for the<br />
lowest gradient used in the magnetic trap the radius becomes ρ 0 = 6.52 µm.<br />
a) b)<br />
c) d)<br />
Figure 3.8: Majorana losses: The dashed lines depict iso-magnetic fields and the solid line<br />
the limit <strong>of</strong> the non-adiabatic region. The arrows depict the direction <strong>of</strong> the field<br />
gradient. a) An atom is orbiting outside <strong>of</strong> the non-adiabatic region. b) The atom<br />
is still orientated with respect to the local field and remains trapped. c) An atom is<br />
orbiting inside the non-adiabatic region. The change <strong>of</strong> magnetic field is too rapid for<br />
the magnetic moment to follow. d) Since the atom cannot follow the orientation <strong>of</strong> the<br />
field the spin is flipped with respect to the local field orientation. The atom is then in<br />
an untrapped state and can escape.<br />
The loss rate can be estimated if the elipticity <strong>of</strong> the loss region is neglected. The<br />
flux <strong>of</strong> atoms through the Majorana loss region is given by<br />
F Majorana = N V eff<br />
A Majorana v (3.28)<br />
with N being the total atom number, A Majorana = πρ 2 0 the cross-section <strong>of</strong> the Majorana<br />
loss volume and v the velocity. The effective volume in equation (3.11) is related to an<br />
effective length <strong>of</strong> L 0 = (3/(4π)V eff ) 1/3 . Relating the velocity and the effective length via<br />
the virial theorem (〈U〉 /2 = 〈E kin 〉) yields v = √ g F m F µ B B ′ L 0 /m. Combining all this,<br />
the Majorana loss rate can be written as<br />
Γ Majorana = 24 1/3<br />
m 32<br />
[ ]<br />
gF m F µ B B ′ 2<br />
. (3.29)<br />
k B T
48 3.4 Loss mechanisms and heating<br />
The dependence <strong>of</strong> the lifetime on the temperature τ ∝ T 2 can be easily understood. A<br />
decreasing temperature causes the density (∝ T 3 ) to rise. This means that the probability<br />
<strong>of</strong> being in the trap center increases, therefore reducing the lifetime. The radius <strong>of</strong><br />
the hole (∝ T 1/2 ) shrinks, since slower atoms see a slower change <strong>of</strong> the field, resulting<br />
in an increasing effect on the lifetime. Finally, the velocity (∝ T 1/2 ) reduces, therefore<br />
the atoms pass less <strong>of</strong>ten through the Majorana loss region. Figure (3.9) depicts the<br />
dependence <strong>of</strong> the lifetime on the field gradient and the temperature.<br />
τ[s]<br />
10<br />
9<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
B’[Gcm -1 ]<br />
300<br />
250<br />
200<br />
150<br />
100<br />
50<br />
10 20 30 40 50 60<br />
T[µK]<br />
Figure 3.9: The lifetime τ depends quadratically on the temperature and the gradient (measured<br />
in the radial direction). The dark blue regions correspond to gradient values that<br />
are not used during the magnetic trap. The temperature dependence is plotted only<br />
up to about 55 µK.<br />
The resulting lifetime at the Doppler temperature with the maximal field gradient<br />
is 183 s. Below a temperature <strong>of</strong> 20 µK, the lifetime reduces to less than a second with<br />
this model. For a cold gas, Majorana losses can be neglected, whereas in the regime <strong>of</strong><br />
ultracold temperatures, they represent a severe loss channel.<br />
The lifetime <strong>of</strong> the trap is (21.10±0.60) s at a temperature <strong>of</strong> (402±5) µK as depicted<br />
in figure (3.10). Majorana losses are negligible at this temperature. Making use <strong>of</strong><br />
evaporative cooling, explained in chapter 4, the atoms were cooled and the lifetime and<br />
the temperature was measured. Two different sweeps were used, resulting in a lifetime<br />
<strong>of</strong> (7.15 ± 0.55) s at a temperature <strong>of</strong> (99 ± 2) µK and a lifetime <strong>of</strong> (4.72 ± 0.22) s at a<br />
temperature <strong>of</strong> (63 ± 7) µK. After each sweep, the microwave radiation used for this<br />
cooling sheme, was turned <strong>of</strong>f. Obviously, there is an effect <strong>of</strong> the hole. To obtain<br />
the Majorana loss rate, the measured rates need to be corrected by the background<br />
loss rate Γ Majorana = Γ measured − Γ background . The corresponding lifetimes are τ Majorana =<br />
(6.08 ± 0.37) s and τ Majorana = (10.81 ± 1.27) s for the two measurements. The theory<br />
states though that the lifetimes should be 11.46 s and 31.20 s at these temperatures.<br />
The simple treatment above accounts only for the region in the trap with a transition<br />
probability to an untrapped state on the order <strong>of</strong> one. Other low-field regions in the trap<br />
produce losses as well, therefore reducing the lifetime. It is improbable, though, that this
3.4 Loss mechanisms and heating 49<br />
correction could account for the difference between theory and experiment. Finally, it is<br />
interesting that the measurements suggest rather a linear than a quadratic dependence<br />
<strong>of</strong> the Majorana lifetime on the temperature 3 . This is, <strong>of</strong> course, vague as there are only<br />
two data points.<br />
10 0<br />
normalized<br />
atom number<br />
τ =(4.720.22)s<br />
T=(637)μK<br />
τ =(7.150.55)s<br />
T=(992)μK<br />
τ =(21.10.6)s<br />
T=(4025)μK<br />
10 -1 0 5 10 15 20 25 30 35 40<br />
t[s]<br />
Figure 3.10: Lifetime measurements with Majorana losses: The atom numbers were normalized<br />
to the maximum value for comparison reasons. Cooling the atoms reveals the<br />
influence <strong>of</strong> the hole. The lifetimes are shorter than estimated with the simple model.<br />
Parametric heating<br />
Heating is not directly a source <strong>of</strong> losses but can prevent the production <strong>of</strong> a BEC. A<br />
source <strong>of</strong> heating are fluctuations in the current and therefore also in the magnetic field.<br />
Depending on the frequency <strong>of</strong> this technical noise and the trap geometry the noise can<br />
lead to a continuous excitation <strong>of</strong> the atoms from a lower to a higher trap state. This<br />
process is referred to as parametric heating, similar to the parametric oscillator that is<br />
driven by a changing oscillation frequency. For the case <strong>of</strong> harmonic traps, the heating<br />
rate was calculated in [67]. The following discussion follows this treatment. Our case<br />
is not that easily accessible since coordinates are coupled and the wavefunctions are<br />
not known analytically. Still, a 1D model can give some insight. The following is not<br />
intended to give a stringent introduction and only wants to show the effect <strong>of</strong> current<br />
fluctuations on the sample. The Hamiltonian for a parametric linear potential is given<br />
as<br />
H = p2<br />
2m + µ Fg F µ B B<br />
} {{ }<br />
′ (1 + ɛ(t))|x| (3.30)<br />
A<br />
with the relative current fluctuation ɛ(t) = (I(t)−I 0 )/I 0 . I 0 is the DC part <strong>of</strong> the current<br />
and I(t) is the entire time-varying current including AC parts. The fluctuation can be<br />
treated as a perturbation. The averaged transition rate for a transition from |n〉 to |m〉<br />
3 10.81 s<br />
6.08 s<br />
99 µK<br />
(99 µK)2<br />
≈ 1.8 and<br />
63 µK<br />
≈ 1.6, whereas<br />
(63 µK)<br />
≈ 2.5<br />
2
50 3.4 Loss mechanisms and heating<br />
is given by<br />
( ) 2 A π<br />
R n→m =<br />
4 | 〈m| |x| |n〉 |2 × 2 ∫ ∞<br />
dτ cos ω mn < ɛ(t)ɛ(t + τ) ><br />
π 0<br />
} {{ }<br />
S(ω mn)<br />
(3.31)<br />
with S(ω) being the power spectral density <strong>of</strong> the perturbation. In the case <strong>of</strong> our modulus<br />
potential, the matrix element is not known. For a harmonic potential, the matrix<br />
elements 〈m| x 2 |n〉 can be calculated. Due to parity considerations, the transitions exist<br />
only from states with quantum number n to states with a quantum number n ± 2i<br />
(i = 1, 2, 3 . . . ). As the wave function overlap <strong>of</strong> close-by states is greatest, only transitions<br />
with n → n ± 2 are relevant. This case is also relevant for the NaLi experiment.<br />
If the quadrupole trap is combined with the optical plug the resulting potential can be<br />
considered harmonic for small temperatures (see subsection (3.6.1)). The heating rate<br />
in the harmonic case is<br />
Γ Heating = π 2 f 2 S(2f), (3.32)<br />
giving rise to an exponential growth <strong>of</strong> the heating in time. f denotes the trapping<br />
frequency <strong>of</strong> the atoms in the pockets <strong>of</strong> the plugged quadrupole trap. They are calculated<br />
with equation (3.39). The factor 2 in the power spectral density accounts for the<br />
n → n ± 2 condition in the harmonic oscillator case.<br />
ΔI/I 0<br />
10 -3<br />
10 -4<br />
10 -5<br />
10 -6 10 2 10 3 10 4<br />
f[Hz]<br />
Figure 3.11: Fourier spectrum <strong>of</strong> the current<br />
With the equipment available, only the spectrum <strong>of</strong> the current was measured. Still, if<br />
there are spikes in the spectrum plot they are also present in the power spectral density,<br />
hence the spectrum is a good measure whether there are obvious sources <strong>of</strong> heating<br />
present. Figure (3.11) shows the relative current fluctuations in the frequency domain.<br />
The spectrum was measured from 30 Hz to 90 kHz. The measured values were normalized<br />
with respect to the DC current, therefore giving the relative current fluctuation<br />
∆I/I 0 . After 10 kHz there is clearly a peak structure visible. The largest resonance is
3.4 Loss mechanisms and heating 51<br />
located at 40 kHz with an amplitude <strong>of</strong> 2.5×10 −4 . Between 100 Hz and 10 kHz the amplitude<br />
decays steeply from 2×10 −4 to 10 −6 . The trapping frequencies in the harmonic plug<br />
potential are calculated with equation (3.39). They are 576 Hz, 993 Hz and 1.743 kHz<br />
for the different directions in the plugged trap. There are no resonances visible at twice<br />
or four times the value <strong>of</strong> the calculated trapping frequencies. In this region the current<br />
fluctuations are strongly suppressed. Nevertheless, the spikes at higher frequency can<br />
affect the precission <strong>of</strong> to what value the magnetic field is set to. Most spikes are way<br />
below a value <strong>of</strong> 10 −4 . Still, the source <strong>of</strong> this noise should be found in order to get rid<br />
<strong>of</strong> it.<br />
Stray light<br />
Stray light from the laser table can be a severe loss source. To prevent effects from stray<br />
light, we installed a light-pro<strong>of</strong> curtain made <strong>of</strong> pond foil. It is necessary to seal the<br />
experimental table, as even small non-detectable intensities reflected from somewhere in<br />
the lab can have grave effects. From equation (2.4) the influence <strong>of</strong> this stray light can<br />
be deduced. The most “dangerous” light is resonant light. For F = 1 state atoms this<br />
would be the repumper light. Whereas for F = 2 the imaging light as well as the MOT<br />
light, due to the small detuning, would cause the greatest loss. For resonant light the<br />
lifetime is<br />
τ = 2 ( )<br />
I0<br />
Γ I + 1 . (3.33)<br />
For an intensity <strong>of</strong> 200 pWcm −2 the resulting lifetime is already less than one second.<br />
This can hardly be detected, hence we have to rely on thorough light sealing.<br />
3.4.2 Two-body losses<br />
Apart from one-body losses due to stray light and background gas collisions, two-body<br />
losses are an important loss feature for a gas at high densities. We only trap atoms in<br />
one hyperfine ground state. When transferring the atoms from the MOT to the magnetic<br />
trap, the repumper light is turned <strong>of</strong>f several milliseconds before the MOT light. This<br />
ensures that all atoms are depumped in the lower hyperfine ground state F = 1. This can<br />
be verified if the sample is imaged without using the repumper. In case <strong>of</strong> a mixture <strong>of</strong><br />
states there should be a signal <strong>of</strong> the F = 2 state. Similarly, the upper hyperfine ground<br />
state can be prepared if the repumper is turned <strong>of</strong>f after the MOT beam. The different<br />
trappable states were discussed in subsection 3.3.1. As there are several trappable states<br />
in F = 2 this gives rise to extra loss channels.<br />
Since the interaction occurs between two atoms <strong>of</strong> the sample, two-body losses are<br />
density dependent. This can be formulated analogous to equation (3.24)<br />
dN<br />
dt = −Γ 1bodyN − β<br />
∫<br />
n 2 dV (3.34)<br />
with the two-body loss coefficient β. The integral can be replaced by making use <strong>of</strong> the<br />
effective volume V eff and the effective density n eff introduced in subsection 3.3.2. The<br />
solution to the quadratic differential equation is [68]<br />
N(t) = N 0<br />
Γ 1body exp(Γ 1body t)<br />
Γ 1body + N 0 ˜β(1 − exp(−Γ1body t))<br />
(3.35)
52 3.4 Loss mechanisms and heating<br />
with the initial atom number N 0 and ˜β = β/V eff . In contrast to pure one-body losses<br />
the number <strong>of</strong> atoms will show an over-exponential decay, provided the density is high<br />
enough. The two-body loss rate is composed <strong>of</strong> several loss mechanisms explained below.<br />
Spin exchange interaction<br />
During a spin changing collision, the atoms can change their hyperfine ground state<br />
while maintaining the total projection <strong>of</strong> the spin. For two atoms denoted by |1〉 and<br />
|2〉, the relative angular momentum l, the total spin angular momentum F = F 1 + F 2<br />
and its projection M F = m F1 + m F2 is conserved. If all the atoms are in the upper hyperfine<br />
state, a transition to the lower hyperfine state cannot occur. The only other spin<br />
exchange process is the collision <strong>of</strong> two atoms in |F = 2, m F = 1〉 with the exit channel<br />
corresponding to the (almost) untrapped |F = 2, m F = 0〉 state and the |F = 2, m F = 2〉<br />
state [69]. Spin-polarized samples in the stretched |F = 2, m F = 2〉 state are not affected<br />
by the spin exchange.<br />
Spin dipolar interaction<br />
Apart from spin exchange collisions that conserve the angular momentum and its projection<br />
onto the quantization axis, there are dipolar relaxations that conserve the total<br />
angular momentum but not its projection. Therefore the transition to a weaker bound<br />
state or to an unbound state can occur. The spin flip causes a change <strong>of</strong> the angular momentum<br />
l CMS <strong>of</strong> the center <strong>of</strong> mass system [70]. The internal spin gets transferred to the<br />
angular momentum <strong>of</strong> the center <strong>of</strong> mass system. In subsection 2.1.4 it was shown that<br />
for ultra-cold collisions only s-wave scattering dominates. Howerever, since the dipolar<br />
interaction potential<br />
U dip-dip = µ 0 (g F µ B ) 2 (<br />
⃗F1 · ⃗F 2<br />
)<br />
( ) ( )<br />
− 3 ⃗F1 · ⃗e ⃗F2 r · ⃗e r<br />
(3.36)<br />
4πr 3<br />
is a 1/r 3 potential, higher partial waves are allowed as can be seen from equations (2.17)<br />
and (2.18). Here |r| ⃗e r is the relative interatomic distance and F ⃗ i are the total angular<br />
momentum operators <strong>of</strong> the two colliding atoms. This equation describes the interaction<br />
<strong>of</strong> the dipole moment <strong>of</strong> one atom with the magnetic field created by the magnetic dipole<br />
moment <strong>of</strong> the other atom. In [71] it was shown that in a first-order Born-approximation<br />
<strong>of</strong> the inelastic cross-section only changes with ∆m F = ±1 <strong>of</strong> the single atom play a role.<br />
For the two-atom sytem the maximum change is ∆M F = 2. Such a spin flip releases<br />
an energy amount <strong>of</strong> ∆E = ∆m F µ B g F B ′ x. If the exit channel is a d-wave, the relative<br />
energy <strong>of</strong> the two colliding atoms needs to be large enough to surpass the centrifugal<br />
barrier.<br />
Atoms in the stretched state <strong>of</strong> F = 2 can be transferred into the |F = 2, m F = 1〉<br />
state with additional energy. From there other processes may happen. The atom can<br />
now interact again via spin exchange as explained in the previous subsection. Through<br />
plain elastic collisions it can redistribute its energy, hence heating up the sample. Highenergetic<br />
atoms have a higher probability <strong>of</strong> being far away from the trap bottom than<br />
low-energetic atoms. If evaporative cooling is employed this leads to a loss <strong>of</strong> atoms, as<br />
atoms in the margins <strong>of</strong> the trap are removed (see chapter 4). Atoms prepared in the
3.4 Loss mechanisms and heating 53<br />
|F = 1, m F = −1〉 state are directly transferred to an unbound state. The atom in the<br />
unbound state can only transfer energy on its way out <strong>of</strong> the trap.<br />
The loss rate <strong>of</strong> sodium in the stretched state is estimated to be smaller than Γ dip =<br />
4 × 10 −14 cm 3 s −1 [72].<br />
3.4.3 Three-body losses<br />
This loss mechanism gets important just before reaching the BEC. Three trapped atoms<br />
are involved in the process, as the name suggests. Three-body loss is a molecule formation<br />
process in which the extra atom is needed to carry away the released binding energy.<br />
The single atom is heated and eventually leaves the trap. The molecule has a different<br />
magnetic moment than the atom and depending on the configuration will leave the trap<br />
as well. The loss rate is a function <strong>of</strong> density squared<br />
Γ 3body = γn 2 . (3.37)<br />
since three particles are involved. The rate coefficient γ was calculated [73] and measured<br />
[74] for 23 Na and is typically on the order <strong>of</strong> 10 −28 cm 6 s −1 . For the initial densities <strong>of</strong><br />
(10 10 − 10 11 ) cm −3 this loss rate is negligible compared to one-body and two-body loss<br />
rates. But for typical BEC densities <strong>of</strong> the order <strong>of</strong> 10 14 cm −3 , the lifetime is dominated<br />
by three-body losses. Therefore the density is slightly reduced during the process <strong>of</strong><br />
evaporative cooling and in the final trap. This is just done by reducing the magnetic<br />
field gradient, therefore decompressing the trap.<br />
3.4.4 Lifetime measurement<br />
The lifetime <strong>of</strong> the different hyperfine ground states was measured. To directly compare<br />
the results between the two hyperfine ground states, the difference <strong>of</strong> the trapping<br />
strength has to be taken into account. The gradients <strong>of</strong> the F = 2 measurement were<br />
set to half the value <strong>of</strong> the F = 1 measurement. The lifetime <strong>of</strong> the F = 2 state at full<br />
gradient was measured to observe possible effects at increased compression.<br />
The results <strong>of</strong> the measurement are depicted in figure (3.12). The lifetime <strong>of</strong> the F = 1<br />
state and the corresponding F = 2 state show an exponential decay curve, suggesting<br />
that the samples were only affected by one-body losses. The lifetimes <strong>of</strong> (5.8 ± 0.8) s for<br />
the F = 1 state and (5.5±0.5) s for the F = 2 state are extremly short and are due to bad<br />
vacuum. There were severe problems with leaks in our experiment when the lifetimes<br />
were measured. The problems have been fixed and now the lifetime is more than 20 s.<br />
The measurement <strong>of</strong> the F = 2 state at full gradient shows a different behaviour. The<br />
curve suggests two-body losses, since there appears to be an over-exponential decay at the<br />
beginning. A direct fit with equation (3.35) did not converge properly, hence the problem<br />
had to be treated indirectly by parting the data. For long times the density decreases<br />
and the decay behaviour is dominated by one-body losses. For this reason an exponential<br />
decay curve was fitted to the data values at times larger than 1 s, resulting in Γ 1body =<br />
(0.161 ± 0.032) s −1 . This value was set constant in the fit function given by equation<br />
(3.35). The evaluation yields ˜β = (1.652 ± 0.798) × 10 9 s −1 . The effective volume for a<br />
typical magnetic trap temperature <strong>of</strong> 400 µK (as seen in figure (3.10)) is 2.549×10 −4 cm 3 .<br />
This yields a two-body loss rate coefficient <strong>of</strong> (6.48 ± 3.13) × 10 −14 cm 3 s −1 .
54 3.5 Rethermalization measurement<br />
a) b)<br />
10 8<br />
atomnumber<br />
10 8<br />
0 2 4 6 8 10 0 2 4 6 8 10<br />
t[s]<br />
t[s]<br />
c)<br />
10 8<br />
one-body fit<br />
two-body fit<br />
10 7<br />
atomnumber<br />
τ = (5.8±0.8)s<br />
atomnumber<br />
τ = (5.5±0.5)s<br />
0 2 4 6 8 10<br />
t[s]<br />
Figure 3.12: Lifetime measurement: a) shows the decay curve <strong>of</strong> an atomic sample prepared<br />
in the F = 1 state. The empty circles correspond to data that has not been used in<br />
the fit process. It has the same lifetime as the sample prepared in the F = 2 state<br />
with a corresponding gradient, seen in b). In c) the F = 2 sample experiences twice<br />
the compression than the sample in b).<br />
The value corresponds to the theoretical value for dipolar relaxation Γ dip = 4 ×<br />
10 − 14 cm 3 s −1 <strong>of</strong> the stretched state. Figure (3.12) shows that the fit function only<br />
corresponds to the data due to the large errors <strong>of</strong> the measurement. The prepared<br />
sample consisted not only <strong>of</strong> the stretched state, since we do not have the possibility to<br />
spin-polarize the sample. Therefore spin-exchange collisions were probably involved as<br />
well.<br />
3.5 Rethermalization measurement<br />
When ramping up the magnetic field gradient too quickly, it was noticed that in time-<strong>of</strong>flight<br />
measurements the evolution <strong>of</strong> the width <strong>of</strong> the gas behaves differently for the x-<br />
and y-direction. This means that the different directions had different “temperatures”.
3.5 Rethermalization measurement 55<br />
5<br />
5<br />
σ [mm]<br />
4.5<br />
4<br />
3.5<br />
3<br />
2.5<br />
σ x<br />
:<br />
σ y<br />
:<br />
σ [mm]<br />
4.5<br />
4<br />
3.5<br />
3<br />
2.5<br />
σ x<br />
:<br />
σ y<br />
:<br />
2<br />
2<br />
1.5<br />
1.5<br />
1<br />
1<br />
0.5<br />
0 2 4 6 8 10<br />
5<br />
0.5<br />
0 2 4 6 8 10<br />
t[ms]<br />
t[ms]<br />
a) b)<br />
5<br />
4.5<br />
4<br />
3.5<br />
σ x<br />
:<br />
σ y<br />
:<br />
4.5<br />
4<br />
3.5<br />
σ x<br />
:<br />
σ y<br />
:<br />
σ[mm]<br />
3<br />
2.5<br />
σ[mm]<br />
3<br />
2.5<br />
2<br />
2<br />
1.5<br />
1.5<br />
1<br />
1<br />
0.5<br />
0 2 4 6 8 10<br />
4.5<br />
t[ms]<br />
c)<br />
0.5<br />
0 2 4 6 8 10<br />
550<br />
t[ms]<br />
d)<br />
σ[mm]<br />
4<br />
3.5<br />
3<br />
2.5<br />
σ x<br />
:<br />
σ y<br />
:<br />
T[µK]<br />
500<br />
450<br />
400<br />
T x<br />
2<br />
350<br />
1.5<br />
1<br />
300<br />
T y<br />
0.5<br />
0 2 4 6 8 10<br />
t[ms]<br />
e)<br />
250<br />
0 0.1 0.2 0.3 0.4 0.5<br />
t[ms]<br />
f)<br />
Figure 3.13: Rethermalization <strong>of</strong> the magnetic trap: a)-e) are measurements with hold times<br />
<strong>of</strong> 30 ms, 100 ms, 200 ms, 300 ms and 500 ms, respectively. The evolution <strong>of</strong> the temperature<br />
is depicted in f).<br />
During the ramp process the trapped sample experienced a kick and was not in<br />
thermal equilibrium, hence the difference between the evolution <strong>of</strong> the widths σ x and<br />
σ y . By increasing the hold time in the magnetic trap, the sample was given the chance
56 3.6 The plugged quadrupole field<br />
to rethermalize again. The evolution <strong>of</strong> the widths is compared in figure (3.13). This<br />
measurement was done using the dark spot MOT.<br />
At 500 ms, the difference in temperature <strong>of</strong> the two directions is less than 10 µK. It<br />
is reasonable to estimate the thermalization time to be close to 500 ms. In [75] it was<br />
shown that each particle needs about 4-5 collisions to reach thermal equilibrium. The<br />
ratio <strong>of</strong> the thermalization time and the collision rate was estimated to be 2.7 [76]. This<br />
leads in our case to a collision rate <strong>of</strong> 5.4 s −1 .<br />
The expected value at the densities measured can be calculated with<br />
R scatt = nvσ (3.38)<br />
where n is the density, v the mean velocity and σ from equation (2.19). The density<br />
can be extracted from the images. The mean density is estimated only roughly to be<br />
(1.25 − 7.5) × 10 9 cm −1 . The peak density was divided by a factor <strong>of</strong> 8 to obtain the<br />
mean density in the trap. For a temperature <strong>of</strong> 390 µK the mean velocity is 0.6 ms −1 .<br />
The scattering length a 0 = 2.8 nm is nowadays well known. This results in a collision<br />
rate <strong>of</strong> (0.15 − 0.90) s −1 . The value is about one order <strong>of</strong> magnitude <strong>of</strong>f <strong>of</strong> the measured<br />
thermalization rate.<br />
With the measured values for the collision rate, the temperature and the density, the<br />
scattering rate yields (5.9 − 14.4) nm. This is on the same order as the acctual scattering<br />
length. First estimations in the 1990’s gave a value <strong>of</strong> (1.9 − 8.1) nm [77].<br />
The temperature and the thermalization time are measured with an accuracy that<br />
would not correspond to such deviations. An underestimated density is very likly the<br />
reason for the deviation <strong>of</strong> the collision rate. The density <strong>of</strong> the sample is extracted from<br />
the time-<strong>of</strong>-flight series just like the temperature. During the time-<strong>of</strong>-flight a reduction<br />
<strong>of</strong> the atom number was observed. Since the cloud was quite large compared to the CCD<br />
chip <strong>of</strong> the camera, more and more atoms moved out <strong>of</strong> the field <strong>of</strong> view. Not having the<br />
total atom number, obscurrs the fit and leads to an underestimation <strong>of</strong> the density. The<br />
measurement suggests that we had a mean density on the order <strong>of</strong> (5 × 10 10 − 10 11 ) cm −3<br />
in the trap.<br />
The same measurement was done with a regular MOT setup. The thermalization<br />
time was similar to the one measured for the dark spot MOT. This implies that there is<br />
no significant increase in density in the dark spot MOT.<br />
3.6 The plugged quadrupole field<br />
Majorana losses can be avoided by using a bias magnetic field as in the case <strong>of</strong> the I<strong>of</strong>fe-<br />
Pritchard trap or an effective bias field like in the case <strong>of</strong> the TOP trap. In our case, the<br />
optical plug method is used. This method does not allow the atoms to get in the region<br />
<strong>of</strong> low magnetic fields. It was applied in the first sodium BEC [4]. The advantage <strong>of</strong> this<br />
configuration is the tighter confinement <strong>of</strong> this trap compared to others.<br />
3.6.1 Plug in the NaLi experiment<br />
A blue-detuned, intense laser beam is needed to produce a repulsive potential. The<br />
potential <strong>of</strong> the beam follows equation (2.7). It depends on the detuning and the intensity<br />
<strong>of</strong> the laser beam. We use about 350 mW <strong>of</strong> 515 nm light produced by a frequency
3.6 The plugged quadrupole field 57<br />
doubled Yb:YAG laser. The light is sent via a single-mode fiber to the experimental<br />
table and is focused to a beam waist <strong>of</strong> about w 0 = 10 µm with a corresponding Rayleigh<br />
length <strong>of</strong> z R = 607 µm. This is realized with a Melles Griot 120 mm achromatic lens. The<br />
laser beam enters the glass cell under Brewster’s angle. This imposes an astigmatism<br />
on the beam. The solution to this is presented in subsection 3.6.2. For conveniance, the<br />
plug direction shall be the y-axis in this section; previously this had been the direction<br />
<strong>of</strong> the atomic beam.<br />
The combined potential <strong>of</strong> the magnetic field and the laser beam is depicted in figure<br />
(3.14). The plot shows the potential in the x-z-plane. The potential minimum is located<br />
x min = 16 µm from the trap center. It has a height <strong>of</strong> 19.5 µK if the the maximal field<br />
gradient is used. The height <strong>of</strong> the potential at the Majorana radius with respect to<br />
the minima <strong>of</strong> the potential is 128.2 µK. An ultracold gas will not be able to penetrate<br />
the region where Majorana losses are relevant. As depicted in figure (3.14) the gas<br />
will eventually be located in two potential pockets, provided it is adequately cold. The<br />
barrier between the two minima has a height <strong>of</strong> about 17 µK relative to the potential<br />
minima, i.e the pockets. The plug height is approximately 250 µK. In the pockets <strong>of</strong> the<br />
minima the potential can be approximated by an anisotropic harmonic oscillator with<br />
trapping frequencies [4]<br />
ω x =<br />
√<br />
(4 x2 min<br />
w0<br />
2<br />
− 1)ω y<br />
√<br />
gF m F µ B B<br />
ω y =<br />
′<br />
2mx min<br />
ω z = √ 3ω y . (3.39)<br />
In our case, the oscillation frequencies at the maximal field gradient are f x = 1.743 kHz,<br />
f y = 993 Hz and f z = 574 Hz. These frequencies seem quite high, but right before<br />
degeneracy the field gradient will be decreased to avoid three-body collisions. The lower<br />
gradient will reduce the frequencies. The small beam waist is needed since the power in<br />
the plug beam is limited by the single-mode fiber in use. A high power fiber would allow<br />
for more power in the plug, hence larger beam waists could be possible. This would be<br />
advantageous when adjusting the plug, since it is difficult fo find a small hole with a<br />
small plug beam.<br />
The plug is adjusted by evaporatively cooling the atoms, explained in chapter 4. If<br />
the temperature drops, the atom cloud shrinks and atoms pour out through the hole.<br />
The loss rate <strong>of</strong> the uncooled sample needs to be compared to the loss rate <strong>of</strong> the cooled<br />
sample to check if the hole has a measureable effect on the atoms. Such a measurement<br />
is shown in figure (3.10).<br />
The outcoupler <strong>of</strong> the plug-fiber is mounted on a piezo stage and can be controlled<br />
via a high-voltage piezo controller. The effective scanning length <strong>of</strong> the plug beam at the<br />
location <strong>of</strong> the atoms is about (30 − 40) µm. First though, the beam has to be adjusted<br />
manually with micrometer screws.
58 3.6 The plugged quadrupole field<br />
z[µm]<br />
U[µK]<br />
25<br />
20<br />
15<br />
10<br />
5<br />
0<br />
−5<br />
−10<br />
−15<br />
−20<br />
−25<br />
250<br />
200<br />
150<br />
100<br />
50<br />
a)<br />
−20 −10 0<br />
x[µm]<br />
10 20 30<br />
b)<br />
U[µK]<br />
70<br />
60<br />
50<br />
40<br />
30<br />
20<br />
U[µK]<br />
250<br />
200<br />
150<br />
100<br />
50<br />
0<br />
z[µm]<br />
−50<br />
−40 −20 0 20 40<br />
x[µm] x[µm]<br />
50<br />
Figure 3.14: Combined magnetic and optical potential: a) shows a contour plot <strong>of</strong> the plugged<br />
quadrupole trap. The region in the middle is cut <strong>of</strong>f. There are two seperate pockets<br />
clearly visible. The pocket structure will determine the BEC shape as given by<br />
equation (1.5). In b) the potential shows a height <strong>of</strong> about 250 µK in the middle.<br />
The adjustment procedure is done by using resonant light instead <strong>of</strong> the plug beam<br />
and “shooting away” atoms in the MOT. Once the plug is adjusted, the field gradient<br />
is raised, hence compressing the MOT and repeating the same procedure again. After<br />
several iterations, the procedure is done in the magnetic trap. Once the beam is adjusted<br />
to the scale <strong>of</strong> the insitu trap size, the piezos are used. Figure (3.15) shows an insitu
3.6 The plugged quadrupole field 59<br />
image <strong>of</strong> atoms in the magnetic trap, shot away by resonant imaging light. The vertical<br />
position can be adjusted very well, whereas the other directions cannot be positioned<br />
very precisely. After this procedure the resonant light gets switched <strong>of</strong>f and with the<br />
help <strong>of</strong> the piezo stage a position grid <strong>of</strong> the blue-detuned plug beam is scanned. At each<br />
grid point the sample is cooled and then imaged. The resulting images are compared to<br />
check if there is an increase in atom number visible. An increase in atom number would<br />
mean that Majorana losses are suppressed. In this case, the plug is at the position <strong>of</strong> or<br />
at least close to the hole.<br />
130<br />
260<br />
390<br />
z[µm]<br />
540<br />
670<br />
800<br />
930<br />
130 260 390 540 670 800 930 1060 1190 13201450<br />
y[µm]<br />
Figure 3.15: Plug shot: The picture shows that we were using too high intesities and/or had<br />
not positioned the focus <strong>of</strong> the laser beam onto the cloud. The width <strong>of</strong> the cut is<br />
52 µm, whereas the beam waist is only 10 µm.<br />
Up to now, we have not found any increase in atom number. There are several reasons<br />
that the mentioned method could not have worked as easily as it sounded. Firstly, the<br />
fibers for the plug beam and the resonant light were two separate fibers. They were<br />
manually placed into the fiber outcoupler. It appeared that the foci <strong>of</strong> the two beams<br />
were at different positions. The fiber heads were located differently inside the fiber<br />
outcoupler. Therefore the grid scans searched through the wrong region. This was<br />
solved by coupling both wavelenghts in the same fiber. Secondly, there is a displacement<br />
<strong>of</strong> the resonant beam with respect to the plug beam, when they pass through the lens<br />
and the glass cell. The displacement is about 70 µm, almost twice as large as our scan<br />
width. Considering this displacement it should be possible to plug the hole.<br />
3.6.2 Compensation <strong>of</strong> astigmatism<br />
Since we want the light <strong>of</strong> the plug beam to be reflected as little as possible from the<br />
surface <strong>of</strong> the glass cell, it needs to enter under Brewster’s angle, which is approximately
60 3.6 The plugged quadrupole field<br />
55 ◦ . The refractive index <strong>of</strong> the cell wall is n = 1.462 at 515 nm. Using the right polarization<br />
yields a maximum <strong>of</strong> transmission into the glass cell. This advantage imposes an<br />
astigmatism on the beam. Fortuntaly, this astigmatism can be compensated by methods<br />
<strong>of</strong>ten used in folded laser resonators, as is explained in the following.<br />
Light travelling under Brewster’s angle through the cell wall experiences an altered<br />
width d, given by [78]<br />
d sagittal = d√ n 2 + 1<br />
n 2 (3.40)<br />
d tangential = d√ n 2 + 1<br />
n 4 . (3.41)<br />
In the sagittal plane, the plane perpendicular to the optical table, the beam sees a<br />
larger width than in the tangential plane, the plane <strong>of</strong> the optical table. Since the beam<br />
experiences different widths, a displacement <strong>of</strong> the foci <strong>of</strong> the beam in the sagittal and<br />
the tangential plane will occur.<br />
A second optical device, imposing again an astigmatism, can be used to compensate<br />
the astigmatism produced by the glass cell. If the achromatic lens is tilted by an angle<br />
θ with respect to the propagation direction <strong>of</strong> the beam, the focus f <strong>of</strong> the lens changes<br />
to [79]<br />
f sagittal =<br />
f<br />
cos θ<br />
(3.42)<br />
f tangential = f cos θ. (3.43)<br />
Tilting the achromatic lens creates an astigmatism. These two effects can now be compensated<br />
by chosing the right angle θ. The compensation angle is given by<br />
sin θ tan θ = Nd<br />
2f<br />
(3.44)<br />
where N = (n2 −1) √ n 2 +1<br />
. Our glass cell is 4 mm thick and the focal length <strong>of</strong> the achromatic<br />
lens is 120 mm. Solving equation (3.44) yields an angle <strong>of</strong> θ = 7.1 ◦ n 4<br />
.
61<br />
Chapter 4<br />
Evaporative cooling<br />
The last cooling step before obtaining a BEC is evaporative cooling. This was first<br />
suggested in 1986 for cooling hydrogen [80]. The ideas were adopted later for alkali<br />
metals [81]. A comprehensive model <strong>of</strong> evaporative cooling was developed by the MIT<br />
group [82]. A very rigorous and complex treatment is done in [83].<br />
4.1 Principle <strong>of</strong> evaporative cooling<br />
In the evaporative cooling scheme, energetic atoms are selectively removed from the<br />
atomic trap. After the removal <strong>of</strong> the energetic atoms, the gas will thermalize again<br />
with a lower temperature than before. If this is done succesively, the process can lead to<br />
ultra-cold temperatures and eventually to a BEC. Evaporative cooling is a lossy method,<br />
hence large numbers <strong>of</strong> atoms are needed to begin with. The process can be applied to<br />
magnetic [82] as well as to optical traps [52].<br />
For the classical case, the population <strong>of</strong> energy states follows the Maxwell-Boltzmann<br />
distribution. Depending on the type <strong>of</strong> trap, there are several methods to remove<br />
the most energetic atoms, thus cutting <strong>of</strong>f the tail <strong>of</strong> the Maxwell-Boltzmann distribution.<br />
After some time this non-equilibrium state will thermalize, resulting in a temperature<br />
T final that is smaller than the inital temperature T init at the cost <strong>of</strong> losing<br />
N Loss = N init − N final atoms. Normally the process is continuous, hence the atoms only<br />
have the possibility to thermalize incompletely.<br />
In optical traps, the trap depth is lowered, hence energetic atoms are no longer<br />
trapped. The trap depth can be controlled easily by reducing the power <strong>of</strong> the laser<br />
beam. For magnetic traps, several schemes <strong>of</strong> evaporative cooling have been developed.<br />
High-energetic atoms have a higher probability <strong>of</strong> being at high magnetic fields than<br />
low-energetic atoms. In the TOP trap presented in subsection 3.1, the radius <strong>of</strong> the<br />
“circle <strong>of</strong> death” is diminished. Only high-energetic atoms can reach this circle and get<br />
lost. To efficiently cool down the sample, the circle is contracted towards the bottom <strong>of</strong><br />
the trap.<br />
In our case, the removal <strong>of</strong> atoms is done by shining in microwave radiation, causing<br />
a transition from the trapped state to the untrapped states. High-energetic atoms at<br />
high magnetic fields have a larger Zeeman shift than low-energetic atoms at the bottom<br />
<strong>of</strong> the trap. The frequency <strong>of</strong> the radiation is chosen such that it is only in resonance for<br />
atoms at high magnetic fields. The atoms are removed and the frequency is adapted to<br />
further cool the sample. In our experiment we scan the frequency through approximately
62 4.1 Principle <strong>of</strong> evaporative cooling<br />
150 MHz below the center frequency <strong>of</strong> 1771 MHz, this being the zero field energy splitting<br />
<strong>of</strong> the F = 1 and F = 2 hyperfine ground states. This frequency range cools the<br />
|F = 1, m F = −1〉 state. If we were to cool the |F = 2, m F = 2 > state, the scan would<br />
have to start above the center frequency.<br />
The physical situation is depicted in figure (4.1). Only the transition to one other<br />
trapped state has been taken into account. The whole situation, however, involves more<br />
states. In the dressed-state picture the trapped and untrapped levels approach each<br />
other and produce avoided level crossings.<br />
energy<br />
|F=2;m F=2<br />
=-2><br />
energy<br />
|F=1;m F=1<br />
=-1><br />
a)<br />
position<br />
b)<br />
position<br />
Figure 4.1: a) The standard way <strong>of</strong> looking at evaporative cooling is by a simple transition<br />
from a bound state to an unbound state. The frequency is adapted so that only energetic<br />
atoms at high magnetic fields experience the radiation. b) Equivalently, the atom-light<br />
system is treated in the dressed-state picture. There is a level shift leading to avoided<br />
crossings. For small Rabi frequencies the splitting is small, hence the atom has a nonvanishing<br />
probability to undergo a Landau-Zener transition to the upper dressed-state.<br />
There it can stay in the trapped state. If the Rabi frequencies are high, though, the<br />
Landau-Zener transition probability vanishes. In this case, the energetic atom follows<br />
adiabatically the dressed-state potential. This eventually leads the atom to a potential<br />
that is no longer trapped. To avoid Landau-Zener transitions, high power rf- and<br />
microwave-amplifiers are used.<br />
After the removal <strong>of</strong> atoms with energies higher than E, a temperature T arises after<br />
a certain thermalization time. The relation between the cut-<strong>of</strong>f energy and the resulting<br />
temperature is called truncation parameter η = E/(k B T ) [82]. An important quantity<br />
is the efficiency parameter<br />
γ = d ln T<br />
d ln N =<br />
˙ T /T<br />
Ṅ/N . (4.1)
4.1 Principle <strong>of</strong> evaporative cooling 63<br />
It relates the relative change in temperature to the relative change in the number <strong>of</strong> atoms<br />
during evaporation [83]. Large γ correspond to an efficient cooling process. The relative<br />
change <strong>of</strong> other quantities scales with Ṅ/N. For the quadrupole trap, the volume scales<br />
as V ∝ T 3 as stated in equation (3.11). The proportionality factor shall be denoted by<br />
α. This can be used to relate the relative change <strong>of</strong> atom number to the relative change<br />
<strong>of</strong> quantities like particle density n, phase-space density P SD and so on. For the particle<br />
density this is<br />
ṅ<br />
n = d ln n (4.2)<br />
dt<br />
= d ln N<br />
αT 3<br />
= (1 − 3γ) Ṅ<br />
N .<br />
It is straightforward to extend this scaling behaviour to other relevant quantities as shown<br />
in table (4.1) [82]. It should be mentioned that the scaling depends on the geometry <strong>of</strong><br />
the trap.<br />
Table 4.1: Scaling factors<br />
Quantity<br />
Exponent<br />
Temperature T<br />
γ<br />
Volume V<br />
3γ<br />
Density n<br />
1 − 3γ<br />
Phase-space density PSD 1 − 9/2γ<br />
Elastic collision rate R 1 − 5/2γ<br />
The important quantity in our experiment is the phase-space density. All our cooling<br />
methods aim at increasing the phase-space density, hence the efficiency parameter γ is not<br />
<strong>of</strong> direct interest. Since we compress and decompress the magnetic trap adiabatically,<br />
the temperature changes, but the phase-space density stays constant. Therefore an<br />
increasing γ can have several origins and should not be confused with an increase in<br />
phase-space density. A better measure is the phase-space efficiency parameter<br />
P SD/P<br />
˜γ = −<br />
˙ SD<br />
. (4.3)<br />
Ṅ/N<br />
With evaporative cooling, it is possible to decrease the atom’s temperature while<br />
simultaneously increasing the collision rate in the sample. This process is called the<br />
runaway regime. Since the temperature decreases, the effective volume (∝ T 3 ) shrinks.<br />
Depending on how efficiently the atoms are removed, this can give rise to a net gain in<br />
collision rate. This is treated in the next subsection.
64 4.1 Principle <strong>of</strong> evaporative cooling<br />
Evaporation speed and runaway evaporation<br />
An important parameter <strong>of</strong> the evaporative cooling scheme is the cooling time. In<br />
the ideal, loss-free situation, a very large cooling time can be chosen. If the Maxwell-<br />
Boltzman distribution is truncated far into its tail, there is an exponentially vanishing<br />
probability that an atom will have an energy higher than the cut-<strong>of</strong>f energy. This atom<br />
would be removed and would carry away almost the entire energy <strong>of</strong> the system. The time<br />
scale on which this process happens increases exponentially. In the non-ideal situation<br />
with losses, they impose a timescale on the cooling process. Using forced evaporation,<br />
in which the Maxwell-Boltzmann distribution is cut-<strong>of</strong>f by a temporally changing truncation,<br />
it is possible to cool the sample down to degeneracy in less than the loss-induced<br />
lifetime.<br />
The temporal evolution <strong>of</strong> evaporation and the role <strong>of</strong> losses are treated in the kinetic<br />
approach to evaporative cooling and can be estimated by the principle <strong>of</strong> detailed balance<br />
as is done in [83]. The following treatment is a combination <strong>of</strong> the Amsterdam model [84]<br />
and the MIT model [85]. The treatment starts with the Amsterdam model to motivate<br />
the origin <strong>of</strong> the atom number’s temporal evolution during evaporation. Scaling laws<br />
and basic insights are gained from the MIT model.<br />
The change <strong>of</strong> the distribution function f(ɛ) in time is governed by the Boltzman<br />
transport equation [11] and can be written as<br />
ρ(ɛ 4 ) f(ɛ ˙ 4 ) = mσ ∫<br />
dɛ<br />
π 2 3 1 dɛ 2 dɛ 3 δ [ɛ 1 + ɛ 2 − ɛ 3 − ɛ 4 ] × (4.4)<br />
ρ(min [ɛ 1 , ɛ 2 , ɛ 3 , ɛ 4 ]) · (f(ɛ 1 )f(ɛ 2 ) − f(ɛ<br />
} {{ } 3 )f(ɛ 4 )).<br />
} {{ }<br />
A<br />
B<br />
Here ρ(ɛ) is the density <strong>of</strong> states and ɛ i the energies <strong>of</strong> the coliding atoms. The deltafunction<br />
accounts for energy conservation in the collision process. The physically interesting<br />
parts are terms A and B. Term A corresponds to collisions <strong>of</strong> two atoms with<br />
energies ɛ 1 and ɛ 2 . This results in the production <strong>of</strong> two atoms with energies ɛ 3 and ɛ 4 .<br />
The process has a positive sign since a particle with ɛ 4 is produced. Term B describes<br />
the collision <strong>of</strong> atoms with ɛ 3 and ɛ 4 . Atoms with ɛ 1 and ɛ 2 are produced. The negative<br />
sign accounts for the loss <strong>of</strong> an atom with energy ɛ 4 . Up to now this treatment is general.<br />
Evaporative cooling truncates the distribution function<br />
f(ɛ) trunc = f(ɛ)θ(E − ɛ) (4.5)<br />
with the heaviside function θ(E − ɛ). E is the energy at which the distribution function<br />
is truncated. In the following discussion, the energies are defined as ɛ 4 > E > ɛ 2 , ɛ 1 ,<br />
hence ɛ 3 has to be the lowest energy due to energy conservation. Every atom with ɛ 4 is<br />
lost immediately from the trap. The rate <strong>of</strong> evaporation is<br />
Ṅ ev = −<br />
∫ ∞<br />
E<br />
dɛ 4 ρ(ɛ 4 ) ˙ f(ɛ 4 ). (4.6)<br />
Equations (4.4) and (4.5) have to be inserted into equation (4.6). But since all atoms<br />
with energies greater than E are removed from the distribution, term B in equation (4.4)<br />
does not contribute to the evaporation rate. This is trivial, since atoms that are lost<br />
from the trap cannot be removed again. Equation (4.6) combined with equations (4.4)
4.1 Principle <strong>of</strong> evaporative cooling 65<br />
and (4.5) states the following: the rate <strong>of</strong> producing evaporated atoms is the number <strong>of</strong><br />
atoms with energy greater than E (produced in collisions via term A) divided by their<br />
collision time τ el . In [83] this is calculated in full detail.<br />
The rate <strong>of</strong> evaporating atoms can be expressed in terms <strong>of</strong> the evaporation time τ ev<br />
Ṅ ev = − N τ ev<br />
. (4.7)<br />
The Amsterdam model connects the evaporation time τ ev with the elastic collision time<br />
τ el and the truncation parameter η. For large truncation parameters, this can be written<br />
as<br />
√<br />
2 exp η<br />
τ ev = τ el . (4.8)<br />
η<br />
Up to now, losses have been left aside. But one-body losses can be easily incorporated<br />
into the quantity <strong>of</strong> interest. Firstly, the quantity is calculated in the loss free picture<br />
with the help <strong>of</strong> table (4.1). Secondly, a loss rate 1/τ loss is subtracted from the ideal<br />
calculation [82]. For the elastic collision rate this yields<br />
Ṙ<br />
R = Ṅ [1 − 5 ]<br />
}{{} N 2 γ − 1<br />
(4.9)<br />
τ loss<br />
−τ −1<br />
el<br />
It should be mentioned that there is a further loss channel. The atoms cannot only<br />
be expelled from the trap by one-body losses and evaporation, but they can also be<br />
spilled from the trap. These atoms do not get lost by means <strong>of</strong> elastic collisions as in<br />
evaporation but simply have a higher energy than the energy E to which the potential<br />
is truncated. These atoms do not take part in the process <strong>of</strong> thermalization. Spilling is<br />
treated thoroughly in the Amsterdam model.<br />
To obtain a sustained evaporation it is necessary to get into the runaway regime<br />
where the collision rate increases with time. This is the case if the left side <strong>of</strong> equation<br />
(4.9) is positive. In terms <strong>of</strong> the efficiency parameter, this can be stated as<br />
γ > 2 (<br />
1 + τ )<br />
ev<br />
. (4.10)<br />
5 τ loss<br />
As was mentioned before, the efficiency parameter is not <strong>of</strong> interest, since the temperature<br />
can change while the phase-space density remains the same. In the same manner<br />
as equation (4.9) was derived, the phase-space efficiency parameter can be expressed<br />
P SD/P<br />
˜γ = − ˙ SD<br />
Ṅ/N<br />
= − τ ev<br />
−1<br />
(<br />
1 −<br />
9<br />
2 γ) + τ −1<br />
loss<br />
τ −1<br />
ev<br />
+ τ −1<br />
loss<br />
= 9 2 γ (<br />
1 + τ ev<br />
τ loss<br />
) −1<br />
− 1.<br />
(4.11)<br />
This equation relates the two parameters γ and ˜γ. The combination <strong>of</strong> this relation<br />
and the requirement for runaway evaporation given by equation (4.10) yields the simple<br />
condition<br />
˜γ > 4 (4.12)<br />
5
66 4.2 Experimental realization<br />
for our quadrupole trap. This condition states that if four orders <strong>of</strong> magnitude in phasespace<br />
density are gained, maximally five orders <strong>of</strong> magnitude in atom number may be<br />
lost to have runaway evaporation. This condition is different for other traps, since the<br />
scaling factors in table (4.1) change. The same calculation for the harmonic trap yields<br />
˜γ > 2. The tighter confinement <strong>of</strong> the quadrupole trap compared to harmonic traps<br />
benefits the conditon <strong>of</strong> runaway evaporation.<br />
Often, evaporative cooling is mentioned in one word with the ratio <strong>of</strong> “good” to<br />
“bad” collisions R gb = τ loss /τ el . The entire treatment above can be done with this ratio<br />
by making use <strong>of</strong> the connection between τ ev and τ el in equation (4.8). Elastic collisions<br />
are reffered to as “good” whereas “bad” collisions are simply trap-loss collisions. The<br />
ratio <strong>of</strong> “good” to “bad” collisions changes during the time <strong>of</strong> evaporation, and if the<br />
initial ratio is sufficiently large, the ratio will show an accelerated growth. If the initial<br />
ratio is too small, though, the evaporation process ceases. For quadrupole traps, the<br />
ratio <strong>of</strong> good to bad collisions needs to be on the order <strong>of</strong> 100 − 200 to have runaway<br />
evaporation [85].<br />
Since the hole is not yet plugged, this ratio will decrease during evaporation due to<br />
Majorana losses. Considering only one-body losses, the results <strong>of</strong> subsection 3.5 and<br />
3.4.1 can be used to estimate an initial ratio. The results were τ el = (1/5.4) s = 0.19 s<br />
and τ loss = 21.1 s. The initial ratio <strong>of</strong> “good” to “bad” collisions is then about R gb =<br />
21.1/0.19 ≈ 111. This value suggests that runaway evaporation in our experiment is<br />
possible. The loss rate increases, though, once two- and three-body losses come into<br />
play.<br />
4.2 Experimental realization<br />
The driven transition for evaporative cooling in our experiment is from |F = 1, m F = −1〉<br />
to |F = 2, m F = −2〉. The detuning <strong>of</strong> these states is in the microwave range. The<br />
first experiment with 23 Na BECs used rf-induced transitions within the lowest hyperfine<br />
ground state. The scan was swept from 30 MHz to a final value <strong>of</strong> 1 MHz in about 7 s<br />
[4]. This is certainly the easier way to implement evaporative cooling, since the design<br />
<strong>of</strong> microwave antennas is more complicated, and trial and error methods do not really<br />
work well. Rf-cooling is not used because there is also 6 Li in the experimental chamber.<br />
The upper hyperfine ground state <strong>of</strong> 6 Li will be trapped since the trappable state <strong>of</strong> the<br />
lower hyperfine ground state exhibits the bending structure shown in figure (2.1). These<br />
hyperfine states exhibit a splitting similar to the 23 Na atoms and therefore are resonant<br />
for rf-radiation. This leads to transitions to untrapped states. Therefore a frequency<br />
range needs to be used for which the 6 Li atoms are transparent.<br />
6 Li does not have<br />
any energy splitting in the microwave range at the magnetic fields used. This makes<br />
microwave transitions the candidates for our evaporative cooling scheme.<br />
4.2.1 The microwave setup<br />
The microwave antenna was designed and built by Lizzy Brama. The design is similar<br />
to patch antennas used in cellular phones. Unfortunately they emmit radiation several<br />
MHz above our desired frequency range and are not broadband. Therefore they did not<br />
suit our needs. The antenna has an E-shape and uses teflon as dielectric material as can
4.2 Experimental realization 67<br />
be seen in figure (4.2). When the antenna was built, the design focused on evaporating<br />
atoms in the upper hyperfine state <strong>of</strong> 23 Na. It is impedance-matched for the frequency<br />
range above 1771 MHz. This is the sweep region if the upper hyperfine ground state is<br />
cooled. Currently the lower hyperfine ground state is cooled which needs frequencies<br />
below 1771 MHz.<br />
The antenna’s signal is produced by a high accuracy Rhode&Schwarz signal generator<br />
having a frequency range from 300 kHz − 2000 MHz. This signal is amplified by a 10 W<br />
broadband amplifier. Before reaching the antenna, the signal passes a MECA Electronics<br />
isolator to prevent destruction <strong>of</strong> the amplifier by reflections. Figure (4.2) shows the S11<br />
parameter <strong>of</strong> the antenna in use. The S11 two-port parameter measures the reflection <strong>of</strong><br />
the microwave sent to the patch antenna. For the frequency range (1751−1866) MHz the<br />
reflection is less than −7 dB. This means that 20% <strong>of</strong> the power gets reflected and cannot<br />
be used for evaporative cooling. From 1650 MHz to 1720 MHz the reflection is greater<br />
than −3 dB, hence more than 50% <strong>of</strong> the power gets reflected. This is compensated by<br />
the fact that we use a high-power amplifier. With the corresponding S11 parameters<br />
from figure (4.2), (4 − 8) W <strong>of</strong> the microwave power can be brought to the atoms. The<br />
antenna is set less than a wavelength <strong>of</strong> the microwave radiation away from the atoms,<br />
hence the sample is in the near field region <strong>of</strong> the microwave field.<br />
0<br />
a) b)<br />
-2<br />
-4<br />
S11 [dB]<br />
-6<br />
-8<br />
-10<br />
-12<br />
-14<br />
-16<br />
1.60<br />
1.70 1.80 1.90 2.00<br />
f [GHz]<br />
Figure 4.2: Patch antenna and the S11 parameter: a) illustrates the S11 parameter <strong>of</strong> the<br />
antenna shown in b). Originally designed for frequencies above 1771 MHz, the antenna<br />
is fed with frequencies below 1771 MHz. The silvery marks on the antenna in b) were<br />
used to fine-tune its parameters.<br />
4.2.2 Evaporative cooling in the experiment<br />
The temporal evolution <strong>of</strong> the phase-space density and the atom number <strong>of</strong> an evaporatively<br />
cooled sample is depicted in figure (4.3). The frequency <strong>of</strong> the microwave radiation<br />
was ramped up within 7.6 s from 1625 MHz to 1745 MHz. The atom number decreases
68 4.2 Experimental realization<br />
in the first 5 s with a time constant τ = (3.22 ± 0.21) s. After this period, the decrease<br />
in atom number is steeper, probably due to Majorana losses. The evaporation time can<br />
be estimated from this data for the first 5 s <strong>of</strong> evaporation. The measured time constant<br />
needs to be corrected by the loss time τ loss = 21.1 ± 0.6 s that was extracted from figure<br />
(3.10). τ −1 = τev<br />
−1 + τ −1<br />
loss<br />
describes the different components <strong>of</strong> the measured loss rate.<br />
This results in τ ev = 3.8 ± 0.3 s. It is, however, an estimation that accounts for neither<br />
spilling nor two- and three-body losses. The evolution <strong>of</strong> the phase-space density is<br />
smoother than the evolution <strong>of</strong> the atom number. The time constant <strong>of</strong> the phase-space<br />
density decay for the time between 1 s and 5 s is τ = 1.61 ± 0.34. Until 1 s the increase<br />
in phase-space density is smaller.<br />
N<br />
a) b)<br />
PSD<br />
10 -4<br />
10 7<br />
10 -5<br />
10 -6<br />
10 6 N exp(-t/τ)<br />
τ =(3.22±0.21)s<br />
0 2 4 6 8<br />
t[s]<br />
t[s]<br />
10 -7<br />
PSD exp(t/τ)<br />
τ =(1.61±0.34)s<br />
0 2 4 6 8<br />
Figure 4.3: Phase-space density and atom number during evaporation: The atom number in<br />
a) shows a steep decay after about 5 s <strong>of</strong> evporation. The phase-space density in b)<br />
rises constantly.<br />
Figure (4.4) shows the phase-space density vs. the number <strong>of</strong> atoms at different times<br />
<strong>of</strong> the evaporation process. It is the same measurement as depicted in figure (4.3). The<br />
phase-space density rises drastically with decreasing number <strong>of</strong> atoms. In the double<br />
logarithmic plot, the data points can be fitted with the fit function P SD = αN −β<br />
with the fit parameters α and β. During the first part <strong>of</strong> the sweep the slope is β =<br />
−2.10±0.26. This means that two orders <strong>of</strong> magnitude in phase-space density are gained<br />
while losing one order <strong>of</strong> magnitude in atom number. The fit parameter β is nothing<br />
else than the phase-space density efficiency parameter ˜γ defined by equation (4.3). The<br />
phase-space density rises to about 10 −4 , which is still more than 4 orders <strong>of</strong> magnitude<br />
away from condensation. The increase in phase-space density flattens until a constant<br />
level is reached. The flattening is due to Majorana losses. As the sample gets cooler and<br />
cooler Majorana losses become more important. The loss rate due to Majorana losses<br />
needs to be added to the loss rate caused by background gas collisions. Finally, further<br />
cooling <strong>of</strong> the atoms leads to an entire loss <strong>of</strong> the sample. At this point the optical plug<br />
needs to be used. This is essential to gain the last four orders <strong>of</strong> magnitude necessary<br />
for reaching a BEC.<br />
Condition (4.12) is fulfilled by a factor <strong>of</strong> 2.6 during the first couple <strong>of</strong> seconds. Even
4.2 Experimental realization 69<br />
the more stringent requirement for the harmonic potential would probably be fulfilled.<br />
If the slope <strong>of</strong> β = 2.1 can be maintained during evaporation, we should be able to<br />
produce a BEC with an atom number on the order <strong>of</strong> 10 4 . The MIT group produced a<br />
23 Na condensate with 10 7 atoms to sympathetically cool down the 6 Li atoms [86]. This<br />
suggests that once the BEC is achieved, we will have to go back a few steps and increase<br />
the atom number in the MOT.<br />
10 -4<br />
PSDN -β<br />
β=2.10±0.26<br />
10 -6<br />
phase-space density PSD<br />
10 -5<br />
10 6 atom number N<br />
10 7<br />
Figure 4.4: Phase-space density vs. atom number: The phase-space density rises untill the<br />
point where Majorana losses dominate the losses.
70 4.2 Experimental realization
71<br />
Chapter 5<br />
Conclusion and Outlook<br />
5.1 Conclusion<br />
Having all necessary components installed and being able to bring the sample into the<br />
runaway regime, a sodium BEC seems to be in close reach. Several weeks after my<br />
arrival at the NaLi experiment, we produced the first sodium MOT in Germany. In the<br />
meantime the magnetic trap as well as evaporative cooling have been implemented.<br />
The installed sub-Doppler cooling scheme for the regular MOT showed moderate<br />
but promising results that give a starting-point for future optimization. In order to<br />
increase the density, a dark spot MOT was installed. A comparison <strong>of</strong> the thermalization<br />
behaviour <strong>of</strong> atoms in the magnetic trap revealed that a sample loaded from a dark spot<br />
MOT showed the same thermalization times as observed for an initial cloud produced<br />
by a regular MOT. Therefore it can be concluded that the density gain in the dark<br />
spot MOT configuration is marginal. The dark spot MOT needs to be reconsidered and<br />
eventually remodeled.<br />
The magnetic trap was analyzed and showed a lifetime <strong>of</strong> more than 20 s. The<br />
different trap steps are examined and compared with theoretical models. A comparison<br />
<strong>of</strong> different hyperfine ground states in the trap revealed that for the F = 2 state twobody<br />
losses are apparent. The setup <strong>of</strong> the magnetic trap is described in detail. The<br />
source <strong>of</strong> the current noise, that appears above 10 kHz, needs to be pinned down.<br />
By means <strong>of</strong> evaporative cooling the cloud was cooled to a temperature <strong>of</strong> 63 µK,<br />
being the lowest temperature so far in the experiment. The phase-space efficiency parameter,<br />
describing the gain <strong>of</strong> phase-space density vs. the loss <strong>of</strong> atoms during evaporation,<br />
was measured to ˜γ = 2.1 ± 0.26. This is significantly higher than the theoretical<br />
minimum requirement for the runaway regime, being ˜γ = 4/5. With the achievement<br />
<strong>of</strong> the runaway regime, one <strong>of</strong> the last requirements for reaching degeneracy has been<br />
fulfilled; however, Majorana losses hindered a further cooling <strong>of</strong> the atoms. Therefore<br />
an optical plug beam needs to be installed. If the hole <strong>of</strong> the magnetic trap is finally<br />
plugged, then, the first condensation <strong>of</strong> sodium atoms in Germany can take place.<br />
5.2 Outlook<br />
Once a BEC is achieved the next steps seem to be clear. Enhancing the number <strong>of</strong> atoms<br />
in the BEC will eventually require to revisit all earlier cooling stages, especially the dark
72 5.2 Outlook<br />
spot MOT, since at this cooling stage optimization would be most effective. Alongside<br />
the lithium laser system is installed and will produce a lithium MOT soon. The two<br />
species will be confined in the same trap, allowing for sympathetic cooling <strong>of</strong> the lithium<br />
atoms.<br />
The first planned experiments at NaLi concern interspecies effects between fermions<br />
and bosons. A Nd:YAG laser will create an optical dipole trap for the captured sodium<br />
and lithium atoms. A second dipole trap, created by a dye laser, is set to a frequency such<br />
that it produces a potential which is significantly stronger for lithium than for sodium.<br />
The lithium atoms inside the tight trap propagate through a bath <strong>of</strong> bosonic sodium<br />
atoms. Inter-species Feshbach resonances will allow the tuning <strong>of</strong> the interaction strength<br />
between lithium and sodium atoms. This eventually effects the oscillation frequency <strong>of</strong><br />
the lithium sample in the tight dipole trap. In a way, the changing interaction strength<br />
produces a changing effective mass <strong>of</strong> the lithium atoms inside the trap. This effect is<br />
known in condensed matter physics as superheavy fermions.
73<br />
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81<br />
Danksagung<br />
Zu guter Letzt möchte ich mich bei den Leuten bedanken, die mich während der Zeit<br />
meines Studiums und meiner Diplomarbeit begleitet und unterstützt haben. Mein besonderer<br />
Dank gilt hierbei:<br />
• Pr<strong>of</strong>. Markus K. Oberthaler für die Aufnahme in seine Arbeitsgruppe und die<br />
Möglichkeit am neuen NaLi Experiment viele Erfahrungen zu sammeln. Seine<br />
Ideen haben das Experiment immer weitergebracht und in so mancher Sackgasse<br />
wusste er, was zu tun war. Für die erfahrene Unterstützung, sein Vertrauen in<br />
mich und dieses wertvolle Jahr bin ich sehr dankbar.<br />
• Pr<strong>of</strong>. Annemarie Pucci für die Begutachtung meiner Diplomarbeit.<br />
• meinem NaLi-Team:<br />
– den ehemaligen Diplomanden der ersten Generation – Marc Repp, Stefan Weis<br />
und Jan Krieger – die mit vollem Einsatz das Experiment aufgebaut haben.<br />
– der zweiten Generation – Valentin Volchkov, Raphael Scelle und Bernhard<br />
Huber – die mit gleichem Eifer NaLi vorantreiben. Raphael danke ich ausserdem<br />
für die Hilfe bei Messungen.<br />
– der neuen Doktorandin Fabienne Haupert, die einen frischen Wind in den<br />
Männerhaufen bringt.<br />
– und zu guter letzt Jens Appmeier, dem alten Platzhirsch am NaLi Experiment.<br />
Er kennt jede Schraube am Experiment und war sich nie zu schade mir die<br />
Dinge, wenn nötig, auch ein zweites Mal zu erklären.<br />
– Lizzy Brama für die raffiniert gebaute Mikrowellenantenne.<br />
• Pr<strong>of</strong>. Peter Krüger, der nicht nur im Labor, sondern vor allem auch ausserhalb<br />
des Labors immer gute Ratschläge wusste und bei kleineren Katastrophen stets<br />
die Ruhe bewahrt hat. Danke und alles Gute für deine Zeit in Nottingham!<br />
• allen Mitgliedern der Arbeitsgruppe für die stets gute Atmosphäre, die Hilfe bei<br />
technischen Problemen und die Grill-, Poker- und ”<br />
Zwitscherstub“-abende.<br />
• meinem Elektronikguru und Yogalehrer Sri Maha Sri“Jürgen Schölles. Wenn<br />
”<br />
440 A mal wieder nicht so wollten, wie ich wollte, wusste er mich stets zu beruhigen.<br />
• der mechanischen Werkstatt (insbesondere Herrn Spiegel) und der gesamten Elektronikabteilung.
82 Bibliography<br />
• allen, die sich die Zeit genommen haben, diese Arbeit zu korrigieren.<br />
• meinen Freunden und meiner gesamten Familie, die dafür Verständnis hatten, dass<br />
ich das letze Jahr ein bißchen weniger Zeit für sie hatte als sonst.<br />
• vor allem meinen Eltern, Bonnie Piccardo-Selg 1 und Erwin Selg, die stets für mich<br />
da waren und mich immer unterstützt haben, gleichgültig welche Entscheidungen<br />
ich in meinem Leben getr<strong>of</strong>fen habe. DANKE!<br />
1 Illegitimi non carborundum :-)
Erklärung:<br />
Ich versichere, dass ich diese Arbeit selbstständig verfasst und keine anderen als die<br />
angegebenen Quellen und Hilfsmittel benutzt habe.<br />
Heidelberg, den<br />
Unterschrift