Atom  KIAS
Scalable quantum networks
with atoms, photons, and spins
Kyung Soo Choi
Research scientist
Korea Institute of Science and Technology
IQIM associate
California Institute of Technology
2012 Open KIAS Winter School on Quantum Information Science
Funding sponsors: NSF/North. Grumm./DoD/AFOSR MURI/IQIM/KIST Institutional Program
Lecture Outline
Lecture #1: Modelindependent verification
of nonclassicality and entanglement
Lecture #2: Experimental realizations of
quantum memory and quantum interfaces
Lecture #3: Scalable quantum networks
with atoms, photons, and spins
Quantum interface: light matter
Quantum memory
Storing a quantum state
without measuring it,
and reading on demand
write read
Quantum interface
A coherent and reversible transfer
between matter and light
Rudimentary interaction
Hamiltonian between matter & light
Freely propagating input EM fields
Useful quantum resource to be transported
Stationary QI
Physically stored in matter systems
Freely propagating output EM fields
Available for subsequent quantum operation
Quantum interface: light matter
Mapping quantum states of light into
quantum superposition of elements of the storing medium
ightMatter Interfaces : How ?
Photonic Qubit
Strategy: Mapping light quantum
tion into quantum superposition of
ents of the storing medium
write
QM
read
Complementary variables of light
(photon statistics and offdiagonal coherence)
stored in the quantum memory
Quantifying the performance of a quantum interface
•!
•!
•!
•!
Storage time
Storage & retrieval efficiency
Fidelity (conditional or not)
Bandwidth, wavelength, multimode capability
C. Simon et al. arXiv:1003.1107 (2010)
Motivation – atomlight interaction in the quantum regime
•!
Single atoms and photons are very simple systems
•! A hydrogen atom
2p
1s
•!
Harnessing atomlight interaction in the quantum regime allows us to examine
nontrivial quantum behaviors such as Schrodinger cat states and entanglement
for open quantum systems
•! Strong matterlight interaction at the single quanta level !
Quantum node
generate, process, store
quantum information locally
Quantum channel
transport / distribute
quantum entanglement over
the entire network
Quantum interface
map quantum resources into
and out of photonic channels
Distributed quantum computing
Scalable quantum
communication
Quantum resource sharing
Quantum simulation
Lecture #3
A major obstacle – scattering problem
•!
One major obstacle: single atoms and photons interact very weakly
Spinboson model:
•!
Scattering probability for a single photon from a single atom
Atom
OUT
IN
Radiation into 4!solid angle
Photonic
channel
INPUT
SINGLE ATOM
OUTPUT
A major obstacle – scattering problem
•!
One major obstacle: single atoms and photons interact very weakly
Spinboson model:
•!
Scattering probability for a single photon from a single atom
Atom
OUT
IN
Solution: develop techniques for enhancing the interaction rate g 0
Photonic
channel
INPUT
SINGLE ATOM
OUTPUT
Figure 1  Overview of the exper
Strong matterlight interaction
ct the optical cavities that we use in the lab from highfinesse mirrors
cavity •! Transport is shown in and Figure communication: 1.3. In orderPhotonics
meet the strong coupling
ant to maximize g, the scalar product of the atomic dipole Atom and the
•! Coherent storage and processing:
thin the cavity:
two or three level system (Atom)
OUT
c
d
porbital
Control logic D h
IN
sorbital
Write
Quantum no
a
r
g = ~µ · ~E ~!a
= µ , (1.4)
2✏ 0 V m
cavity Cavity modeQED volume, – Atomcavity is proportional molecule to the cavity length Atomic and the ensemble
strongly coupled single atom
strongly coupled single collective
ode waist. Thus, we should minimize the mode volume by building
and one photon
excitation and one photon
IM
nd using mirrors with a small radius of curvature. However, write
the fullimum
(FWHM) linewidth of a cavity is given by the ratio of its free
b
c
d
Experimental realizations of
quantum memory and quantum interfaces
Part 1 –
Cavity Quantum Electrodynamics
References:
• “Optical Coherence and Quantum Optics” by L. Mandel and E. Wolf
• “Quantum Optics” by M.O. Scully and M.S. Zubairy
• “AtomPhoton Interactions” & “Photons and Atoms”
by C. CohenTannoudji, J. DupontRoc, G. Grynberg
• “Exploring the quantum: Atoms, Cavities, and Photons” by S. Haroche and J.M. Raimond
Review articles:
• B. Lounis and M. Orrit. Single photon sources. Rep. Prog. Phys. 68, 1129 (2005)
• H.J. Kimble, The quantum internet, Nature 453, 1023 (2008)
• A.I. Lvovsky, B.C. Sanders & W. Tittel, Optical quantum memory.
Nature Photon. 3, 706 (2009).
• K. Hammerer, A.S. Sorensen, E.S. Polzik, Quantum interface between light and matter
Rev. Mod. Phys. 82, 1041 (2010)
• N. Sangouard, C. Simon, H. de Riedmatten and N. Gisin,
Quantum repeaters based on atomic ensembles and linear optics,
Rev. Mod. Phys. 83, 33 (2011)
Cavity QED in the optical domain
•!
Transport and communication: Photonics
Atom
•!
Coherent storage and processing:
two or three level system (Atom)
OUT
IN
Cavity QED – Atomcavity molecule
Quantum regime of
strongly coupled single atom and one photon
P in
P out
INPUT
OUTPUT
The electric JaynesCummings field The within atomcavity
interaction the system:
Hamiltonian cavity: describes the coupling of a twolevel
JaynesCummings coherent coupling model: coherent coupling
atom to a single cavity mode [3]:
cavity field
The atomcavity system:
coherent coupling
field raising operator
Atomcavity molecule operator
where â † â are coupling photonstrength
creation and annihilation operators, ˆ† and ˆ are atomic
cavity field
L
! A
raising where andV lowering m , theoperators, cavityand mode g is ! =
the volume, (spatially dependent) is proportional
g , 2 + ecoupling ,1 strength. to t
2
g, 2 ; e,1
2 2g0
e
g , 2 ! e,1
Here we have made the rotating ! ! wave L = approximation, as the cavity field is nearresonant
Figure with 1.1: Figure
! 1 ,0
A 1
g, 1 ; e,0
2g g , + e
0
g , 1 ! e,0
the Coupling g1.1: atomic
Coupling rates transition. for rates a model for
! L
When
a model cavity we
cavity QED include system. QED
terms
system. Atom for
Atom and excitations cavity and cavity couple in
coup
the
coherently coherently to one another to one another at
field raising operator 0
rate at g. rate
atomic
There g. are There
lowering
twoare g,0incoherent two incoherent mechanisms: mechanisms:
g,0
the cavity the cavit
atom short field andcavities decays cavity field at decays modes rate and at apple, asrate and well using apple, the as and atom for the mirrors adecays atom classical decays spontaneously probe with spontaneously field aatat small rate frequency at rate . radius . ! of c
operator
p , then we
E. T. Jaynes and F. W. Cummings, cavity Proc. IEEE 51, 89 (1963).
atom +
have the JaynesCummings Hamiltonian, field written here in the reference frame of the
product product basis where basisg, where ni and g, ni e, and n e, 1i nare 1i nexcitation are nexcitation states states with an with atom an atom in thein th
probe:
ground ground (excited) (excited) state and JaynesCummings state n (n and n 1) (nphotons 1) Hamiltonian
photons
Ĥ JC = ~ aˆ†ˆ + ~ c â † â + ~g(â †ˆ in theincavity. cavity. The interaction The interaction term ter
couples couples each pair each of nexcitation pair of nexcitation states, states, leading leading to
+âˆ†)+✏â
eigenstates to eigenstates and
+ ✏ ⇤ eigenvalues â and † , eigenvalues (1.2)
…
coupling strength
…
E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
r
g = ~µ · ~E ~!a
= µ ,
2✏ 0 V m
Ĥ int = ~g(â †ˆ +âˆ†),
(1.1)
atomic lowering
square of the mode waist. Thus, we should minimize th
width halfmaximum (FWHM) linewidth of a cavity is g
Diagonalization
…
where ! a and ! c are the atom and cavity frequencies, a = ! a ! p , c = ! c ! p ,
± n i =( ± ± p 4g 2 n + )g, ni +2g p n i =( ± p 4g 2 n + 2 )g, ni +2g p ne, n ne, 1i, n 1i,
and ✏ is the probe field drive strength.
E ±n = E ~
In the absence of a probe 2 (2n! c ± p ±n = ~ 2 (2n! c 4g± p 2 n + 4g 2 ), n + 2 ),
(! p = 0, ✏ = 0), we can diagonalize this Hamiltonian
…
JaynesCummings Hamiltonian
model:
into a master
steadystate
equation ˙⇢ = L⇢ for the
spectra
density matrix
system, where L is the Liouvillian superoperator [6]:
System: atomcavity molecule
+
Environment: transverse continuum modes, cavity output field
Liouvillian superoperator:
L = i[H JC , ⇢]+apple(2â⇢â † â † â⇢ ⇢â † â)+ (2ˆ⇢ˆ†
ˆ†ˆ⇢
⇢ˆ†ˆ).
Steadystate spectra
For a restricted basis set, the master equation can be solved numerically to fi
Bloch equations eigenvalues
No atom
P out
2 g
can be solved analytically [7].
Single atom
(splitting)
the linewidth of each peak is approximately apple+ 2
0
! = " c  " a
steady state density matrix and expectation values of various operators. In th
driving limit, in which the system is restricted to n = {0, 1}, the master eq
Figure 1.2 depicts the weak driving solution for the steadystate intracavity
number, proportional to the cavity transmission, as a function of probe freque
(! a = ! c ) for the parameters in our current cavity QED experiment. No
the frequencies of the two peaks correspond to the eigenvalues E ±1 /~ = ±g
. When g apple, , the twostructure
— known as the vacuumRabi splitting — is wellresolved. Our exper
Cesium hyperfine transition
D2 line, " = 852 nm
6S 1/2 , F=4 ! 6P 3/2 , F’=5
tical cavity with one
Observation of normal mode splitting
g operator atomic lowering
operator
Vacuum Rabi splitting
T. E. Northup 1 & H. J. Kimble 1
…
…
g , 2 + e,1
g, 2 ; e,1
2 2g0
g
n the flux limited by the rate ð2Þ
yz
ð0Þ due to these effects are g , 2small ! efor ,1 our parameters, as discussed
in the ofSupplementary spontaneous Information. decay g. In contrast,
ich cavitymediated schemes offer Withthe thesepossibility capabilities, g we of
, 1now photon
+ ereport ,0
emission measurements of g ð2Þ
yz ðtÞ for
y into a collimated spatial g, 1 mode ; the e,0light with transmitted high efficiency 2gby 0
a cavity and containing at a ratea single set bytrapped atom. We
g , 1 ! e,0
nd the cavity decay rate k, which tune can the probe much 1 y p to larger ðq p 2than q 0 Þ=2p g. ¼Achieving
234 MHz; near 2g 0 , and
acquire photoelectric counting statistics of the field 1
the photon blockade for a single atom in a cavity requires us to operate in
z t by way of
two avalanche photodiodes (D
of the regime of strong coupling, g,0 for which the g,0frequency 1 , D 2 ), as illustrated in Fig. 1c. From
the record of these counts, we are able toscale determine g 0 g ð2Þ
yz
ðtÞ by using
ht associated with reversible theevolution procedures of discussed the atom–cavity in ref. 22. Data system are acquired for each
er
exceeds the dissipative rates trapped (g, k) atom (ref. by 14). cycling through probing, testing, and cooling
ings, Proc. IEEE 51, 89 (1963).
Here we report observations intervals (of durations photon Dt blockade probe ¼ 500ms; in the Dt test light ¼ 100 ms and Dt cool ¼
1:4ms; respectively) using a procedure similar to that of ref. 21. The
Photon transmitted blockade by an optical in an cavity optical containing cavity one with atomone strongly
test beam is polarized along ^z and resonant trapped with the atom cavity. A
coupled to the cavity field. repumping For coherent beam transverse excitation to the atcavity the cavity axis and resonant with
Birnbaum et al. Nature 436, 87 (2005)
6S 1=2 ; F ¼ 3 ! 6P 3=2 ; F 0 ¼ 4 0
also illuminates the atom during the
probe and test intervals. This beam prevents accumulation of
er
cal
m–
rst
by
b
reur
ver
me
en
nic
cal
sed
be
a
er
IT)
the
of
has
gle
NATUREVol 4367 July 2005
Singlephoton nonlinearity with anharmonic JC ladder
population in the F ¼ 3 ground state caused by the probe offresonantly
exciting the F ¼ 4 ! F 0 ¼ 4 0 transition. All probing/cooling
cycles end after an interval Dt tot ¼ 0:3s; at which point a new
“Observation of normalmode splitting
for an atom in an optical cavity”
Thompson, Rempe, Kimble
Phys. Rev. Lett. 68, 1132 (1992)
loading cycle is initiated. We select for the presence of an atom by
requiring that T zz ðq p . q C1 Þ & 0:35 for the test beam. We use only
those data records associated with probing intervals after which the
presence of an atom was detected and for which the presence of an
atom was detected in all preceding intervals. If there is no atom and
the probe is tuned to be resonant with the cavity (q p ¼ q C1 ), then the
photon number in mode l y due to 1 y p is 0.21 and the polarizing beam
splitter at the output of the cavity (PBS in Fig. 1c) suppresses
detection of this light by a factor of ,94.
Figure 3 presents an example of g ð2Þ
yz
ðtÞ determined from the
recorded timeresolved coincidences at (D 1 , D 2 ). In Fig. 3a, the
manifestly nonclassical character of the transmitted field is clearly
observed with a large reduction in g ð2Þ
yz ð0Þ below unity, gð2Þ yz ð0Þ¼
ð0:13 ^ 0:11Þ , 1; corresponding to the subpoissonian character of
the transmitted field, and with g ð2Þ
yz ð0Þ , gð2Þ yz ðtÞ as a manifestation of
photon antibunching. We find that g (2) (t) rises to unity at a time
t . 45 ns; which is consistent with a simple estimate of t 2 ¼
2=ðg þ kÞ¼48 ns based upon the lifetime for the state j1;2l.
Although for small jtj our observations of g ð2Þ
yz ðtÞ are in reasonable
agreement with the predictions from our theoretical model, there are
significant deviations on longer timescales. Modulation that is not
present in the model is evident in Fig. 3b, which arises from the
centreofmass motion of the trapped atom. In support of this
assertion, Fig. 3c displays the Fourier transform ~gðf Þ of g ð2Þ
yz ðtÞ;
which exhibits a narrow peak at frequency f 0 . 535kHz just below
the independently determined frequency n 0 . 544 kHz for harmonic
motion of a trapped atom about an antinode of the FORT in the axial
direction x. This modulation is analogous to that observed in ref. 23
Unambiguous signature of
strong coupling regime
in the optical domain
LETTERS
for g (2) (t) for the light from a single ion, which arose from micromotion
of the ion in the radiofrequency trap.
Here, UðrÞ¼U 0 sin 2 ð2px=l C2 Þexpð22r 2 =w 2 C 2
Þ is the FORT
potential, which gives rise to an anharmonic ladder of vibrational
states with energies {E m }. Here m ¼ 0 to m max ¼ 99 correspond to
the pffiffiffiffiffiffiffiffiffiffiffiffiffi
bound states in the axial dimension for radial coordinate r ;
y 2 þ z
Cavity 2 ¼ 0: The anharmonicity leads to the observed offset
f 0 , n 0 due to the
QED
distribution
with
of
single
energies
trapped
for axial motion
atoms
in the
FORT well. Phys. Indeed, Rev. the frequency Lett. n93, min ¼ðE 233603 mmax 2 E mmax21Þ=h (2004) at the
top of the well is approximately half that at the bottom of the well,
n 0 ¼ðE 1 2 E 0 Þ=h: By comparing the measured distribution of frequencies
exhibited by ~gðf Þ with the calculated axial frequencies {n m },
ich we will feed into the master equation is a slight generalization of (2.11):
⇣
0 = ~ Aˆz + ~ F â † â + ~g(~r) â †ˆ
2 ⇣ 2
4 sin g(~r)tp 3
â † â
4 sin g(~r)t p ⌘ 3
â † â
â â pâ†â
† 5 ˆ (
pâ†â
† 5 ˆ (2.21)
Cavity QED:
i
dynamics
i
⌘
+âˆ+ +(" ⇤ â + "â † ). (2.37)
Consider the timeevolution
Next, we formally define an arbitrary initial state 23for n quanta of excitation in the system:
Next, we formally define an arbitrary initial state for n quanta of excitation in the sy
n this expression now corresponds to an15
external source driving the cavity
In Figure 2.3b we return to Equation (2.55) and treat the case of a variable probe
Initial  (t condition = 0)i =(↵  e, (t n= 0)i 1i + =(↵ g, e, ni) n , 1i + g, ni) , (2.22) (
frequency, P , as well as a variable atomcavity detuning AC = F A =(! F ! C )
Figure 1.2: Steadystate transmission of the atomcavity s
and
but
apply
a fixed
(2.21):
atomic resonance frequency. The resulting signal  a socalled “avoided crossing”
Figure 1.2: Steadystate and apply transmission (2.21): detuning
Timedomain of theRabi atomcavity the weak
oscillation system driving as limit. a function Parameters of probeare g =2
detuning  demonstrates in the weakhow driving the between vacuum limit. a single Rabi Parameters MHz, splitting atom and =2⇡ evolves are a single ⇥g =2⇡ 2.6 and photon MHz, develops ⇥ 33.9! asymmetry a MHz, = ! c apple . =2⇡ The as the ⇥cavity atomcavity
=2⇡detuning ⇥ 2.6  changes. MHz, (t)i = ! U b a 0 This (t) =
3.8 transm
MHz,

 asymmetry !
(t)i c . maximum
= The
=
0)i
U b cavity empty
0 (t)
will
 (t
be
=
important transmission cavity transmission;
0)i
as we discuss normalizedempty real atoms toand thecavity tra
the
maximum empty cavity transmission;
= ⇥ probe
↵ cos g(~r)t p = ⇥ empty detuning cavityis n i sin g(~r)t
↵ cos g(~r)t p transmission p plotted
n ⇤ for comparison. as
e, n 1i
n i sin g(~r)t p a
n ⇤ function of
di↵erential atomcavity detunings introduced via their complicated multilevel structures.
probe detuning is plotted for comparison.
e, n 1i
Finally, we will consider
+ ⇥ the
i↵ sin g(~r)t {g,
coupled
+ ⇥ p apple,
di↵erential
n + }. cos Mathematically, g(~r)t
i↵ sin g(~r)t p equations p n ⇤ for ↵(t)
g, ni . we
n + cos g(~r)t p canand n ⇤ treat
(t) ((2.49)
(2.23) dissipation b
g, ni . (
{g, apple, and }. (2.50)) Mathematically, in absence weofcan external probe (" = 0), but under the initial conditions
Cummings treat dissipation Hamiltonian by incorporating into a master theequation Jaynes ˙⇢ = L⇢ f
Cummings Not (↵(0), surprisingly, Hamiltonian
(0)) = we (0, find 1).
into the We The afrequencydomain want dynamics master
to look of strong the normalmode coupling temporal regime evolution splitting, which of thewas system calculated where the
Not surprisingly, we findsystem, equation
the frequencydomain where ˙⇢ = LL⇢ normalmode splitting, which was calcu
above, population corresponds is initially to a timedomain placed in g 0
aRabi superposition ( oscillation , apple,Tof of 1 is for the the Liouvillian density matrix superoperator of the [6]:
atomcavity population ). between eigenstates states (i.e., e, nentirely 1i in
system, where Labove, is thecorresponds Liouvilliantosuperoperator a timedomain Rabi [6]: oscillation of population between states e, n
and the g, atomic ni withexcited characteristic state). Solving frequency the di↵erential ⌦ E n, /~ equations =2 p ng(~r). for zero atomcavity
and g, ni with the characteristic frequency ⌦ = E n, /~ =2 p detuning,
From the master equation
˙ˆ⇢ = Lˆ⇢ with L = i[H JC , ⇢]+apple(2â⇢â † â ng(~r).
† â⇢ (2.38) ⇢â † â)+ (2ˆ⇢ˆ
we find:
L = i[H JC , ⇢]+apple(2â⇢â † â † â⇢ ⇢â † â)+ (2ˆ⇢ˆ†
ˆ†ˆ⇢ ⇢ˆ†ˆ). (1.3)
erent state of strength " and frequency ! P . In order to obtain interesting,
ar exponential and irreversible decay at characteristic rate apple. From our discus
ts from this equation in the presence of dissipation, it is instructive to add
nesCummings model, and because the coherent dynamics of the atomfield in
ich deposits additional energy into the system over time. This Hamiltonian
e the phenomena in which we are interested, it is important that we require
ame rotating with ! P , such that A ⌘ (! A ! P ) and F ⌘ (! F ! P ).
point
ent that
out
g
that 0 should
the master
dominate
equation
all dissipative
can be rewritten
rates, i.e.,
using the socalled
rmalism as:
n addition to the two previously discussed rates, ( , apple), at which information e
For a restricted basis set, the master equation can be so
(~r)
n(t
ibly into the 0 )=↵(t
environment, 0 )↵ ⇤ (t 0 (apple+ )t0
) = e
we have g2 h p
Liouvillian superoperator. A superoperator g 2 (~r) also 1/4(apple is included defined ) 2 sin2 g
T with 2 (~r) 1/4(apple )
, the respect the mean 2 to t 0i its (2.58)
lifetime
For a restricted basis set, the mastersteady equation state can density be solved matrix numerically and expectation to find values the of va
rd
the
quantum
field (i.e.,
operators, The rate of coherent
the average
i.e., interaction must exceed all relevant dissipative rates
length of time for which g(~r) > 0). In the limit des
steady state density matrix and expectation driving limit, valuesinofwhich various the operators. system isInrestricted the weakto n = {
Cavity QED: dynamics
FabryPerot cavity
P in
P out
INPUT
OUTPUT
Dominance of coherent,
reversible evolution
over irreversible
dissipative processes
E = electric field per photon
d = atomic transition dipole moment
n 0 ~ 10 3  10 5 photons
n 0 ~ 10 2  10 7 atoms
Overview of cavity QED with localized atoms
•!
QUANTUM INFORMATION SCIENCE
•! Quantum measurement
Quantum logic, computation, communication
•! Quantumclassical interface
Cavity QED with cold (neutral) atoms
•! H. J. Kimble, Caltech
•! G. Rempe, MPQ Garching
•! T. Kuga, University of Tokyo
•! M. Chapman, Georgia Tech
•! L. Orozco, U Maryland
•! D. Meschede, University of Bonn
•! S. Shahriar, Northwestern University
•! H. Mabuchi, Caltech
•! D. StamperKurn, UC Berkeley
•! !
Cavity QED with trapped ions
•! R. Blatt, University of Innsbruck
•! (H. Walther, MPQ Garching)
•! W. Lange, University of Sussex
•! C. Monroe & M. Chapman, U Michigan  GIT
•! !
Physical realization of cavity QED
Caltech quantum optics group
(Jeff Kimble)
1 cesium atom trapped in a cavity
Ye, Vernooy, Kimble, Phys. Rev. Lett. 83, 4987 (1999)
McKeever et al. Phys. Rev. Lett. 90, 133602 (2003)
50
Finesse ~ 500,000
Epoxy M 2 M 1
Shear PZT
Atomic localization
Trap lifetime ~ 3 s
Axial #z ~ 33 nm ! 8 nm
“Temperature” < 100 mK
Aluminum
•! Nonlinear interactions between individual atoms and single photons
Figure 3.2: Photograph (left) and schematic diagram (right) of the physics cavity assembly.
The physical separation between the mirrors, L = 42.2 µm, is barely visible in the
photograph.
•! Trapping that decouples internal and external degrees of freedom
• D 2
line of atomic Cs at 852 nm
expression for the size, d, of the aperture between the mirror surfaces:
• coherent coupling g / 2! = 34 MHz
[6S 1/2
,F = 4,m F
= 4 ! 6P 3/2
,F = 5,m F
= 5]
d = L L min , (3.1)
L min =2⇢
p
4⇢ 2 D 2 . (3.2)
• decay rates: atom ! "
/ 2" = 2.6 MHz
cavity # / 2" = 4 MHz
Here, L is the length of the cavity, D is the OD of the HR coated surface, and L min is the
Critical photon number
Critical atom number
Quantumstate transfer from one cavity to another
Stationary qubits + flying qubits
•! Quantum networks
•! Distributed quantum computation
•! Scalable quantum communication
Mirror reflectivity
R A
(t)
time
Mirror reflectivity
R () B
t
Timereversal condition
t d
time
is the time reversed version of
offset by the propagation delay
Cirac, van Enk, Zoller, Kimble, Mabuchi, Physica Scripta T76, 223 (1998)
Quantumstate transfer from one cavity to another
Reversible state transfer between matter and light
A. D. Boozer, A. Boca, R. Miller, T. E. Northup & H. J. Kimble
Phys. Rev. Lett. 98, 193601 (2007)
Adiabatic transfer of dark eigenstate
with
*Note that the excited state is absent
if the evolution is adiabatic
Initial:
final:
Classical fields – STIRAP
Oreg, Hioe, Eberly, Phys. Rev. A 29, 690 (1984)
Kuklinski, Gaubatz, Hioe, Bergmann, Phys. Rev. A 40, 6741 (1989)
*Quantum fields – vacuum induced STIRAP
Parkins, Marte, Zoller, Kimble, Phys. Rev. Lett. 71, 3095 (1993)
Singlephoton generation on demand
Deterministic generation of single photons
from one atom trapped in a cavity
McKeever, Boca, Boozer, Miller, Buck, Kuzmich, Kimble
Science 303, 1992 (2004)
HBT measurement
t cycle
Let the ratio of coherent coupling rate to the rate of
irreversible dissipation be
Cooperativity parameter:
i. Decay into other modes ii. Decay into the cavity field
~20 fold suppression of twophoton
Component relative to independent events
Optimal retrieval
efficiency
Deterministic generation
consistent with
for the intracavity field
Mapping coherent states to and from a single atom
coherent state
Reversible state transfer between matter and light
A. D. Boozer, A. Boca, R. Miller, T. E. Northup & H. J. Kimble
Phys. Rev. Lett. 98, 193601 (2007)
! () t 1,2
Coherence verification –
" Interference of $(t) and
field phasereferenced to %(t)
Examination of 1 st order coherence
! () t
i 1 (t)
APD 1
! () t
i 2 (t)
APD 2
Reference field
phase stable with !( t)
d: November 21, 2008)
created by driving a vacuumstimulated Raman adiabatic
Deterministic photonphoton passage (vSTIRAP) via aentanglement
polarized laser pulse address
an objective lens with a numerical aperture of 0.43 and a Zeeman state and the polarization of the emitted photon is
measured ingle rubidium resolution atom trapped of 1:3 within m. While a high this nesse technique optical alone
can entangled determine photons. theThe number entanglement of atoms is mediated with over by 90% the certainty,
l inequality we violation can further of S confirm =2: 5, as that well as exactly full quantumexceeding
F = 90%. The combination of cavityQED and
one atom is ing the Starkshifted F ¼ 2 $ F 0 ¼ 1 transition and the
trapped by measuring a perfect photon
tocol inherently deterministic  an essential
PhotonPhoton antibunching signal
step for the
Entanglement cavity frequency with a Single resonant Trapped with theAtom
F ¼ 1 $ F 0 ¼ 1 transition
[Fig. 2(d)] Mucke, [16]. Moehring, With a trapped Rempe atom coupled to the
in eenthe statistics nodes of a of distributed the emitted quantum photon network. Weber, stream Specht, [8]. Muller, The Bochmann,
combination of these two techniques allows us toPhys. discern Rev. Lett. highfinesse 102, 030501 optical(2009)
cavity, the resulting entanglement is
q, 42.50.Xa
ed for
Control fields
Quantumstate tomography
atoms ter than 80% [Fig. 2(b)] [17]. Next, entantween
the atomic Zeeman state and the po
m ind,
are
of ation the emitted photon is created by driving
stimulated to an Raman adiabatic passage (vSTIπpolarized
laser pulse addressing the Stark
PRL 102, 030501 (2009) PHYSICAL REVIE
buted
wards
inherently deterministic [1,3]:
no
en
2 ↔reF atoms he F =1 ↔ F ′ =1 transition [Fig. 2(d)] [18].
′ =1 transition and the cavity frequency
j APi ¼ 1 tra
pffiffiffi
ðj1; 1ij þ i j1; þ1ij iÞ: (1) en
tom their trapped and coupled to the high nesse
2
to
, high
ity, the resulting entanglement is inherently
limit
PRL After 102, a userselected 030501 (2009) time interval, the PHYSICAL atomphoton entan
is converted into a photonphoton entanglement qu
REVIEW
tic g prom
in
inherently via a second deterministic vSTIRAP [1,3]: step with a polarized F ¼ 1 $ non tan
[1, 3]:
Deterministic atom delivery
3glement
FIG. 1: Individual
is enod
to 2 intersection of two standingwave dipole trap beams. Lin⊥lin
density ontomatrix. the polarization Thisffiffiffi
density ðj1; of a second 1ij matrix þ i emitted represents j1; þ1ij photon, the iÞ: resulting jΨ − (1) ens by
87 Rb atoms are trapped within the TEM 00
F 0 ¼ 1 laser pulse
Entanglement
j APi ¼ p 1 [Fig. 2(e)].
generation
This maps the atomic state trap Th
P〉 = √ 1 mode (j1; of −1〉jσ a high + 〉−j1; nesse optical +1〉jσ cavity − 〉): ( nesse≈ 3×10 (1)
4 )atthe FIG. 3: Real and imaginary parts of the measured twophoton
apped
PP
in an entangled photon 2
〉 Bell
polarized laser beams orthogonal to the cavity axis provide
pair:
toof
elecits
are and the creation of entangled photons are polarized along theAfter a userselected time interval, the atomphoton entan
T
state of the photons with a delity ofF =0: 902 0: 009. a
FIG. motional cooling, while additional beams for optical pumping
rselected 2 (color online). time interval, The experimental the atomphoton procedure. (a) When enis
converted cavity
atoms are first loaded into the cavity, State a 300mapping
ms laser pulse is applied for
glement is j converted PP i ¼ p 1 of
optical cooling. During this time, a camera images the cavity mode to confirm the presence of a single atom. (b–e) The entanglement
generation protocol
axis into
runs
and
at
aindependently aphotonphoton repetition rate
directed
of 50entangle
aF 2
onto
kHz.
the
(b)
atom.
Atomic
The
recooling. (c) A polarized ffiffiffi into ðj laser a þ ij resonant photonphoton i with j the ijF entanglement
þ iÞ: ¼ 2 $ (2) qua F
cavity output is coupled into an optical ber and directed
ntan
0 second ¼ 2 transition vSTIRAP together with step resonant withlasers a πpolarized
on the F ¼ 1 $ F via a second vSTIRAP step with a polarized F ¼ 1 $ tang inc
to the photonic state detection apparatus. Perpendicular 0 combination ¼ 1 and F ¼ 1 $ of F
to
0 four ¼ 2 transitions different optically polarization pump the atom bases, to we obtain
$ F 0 0 ¼Bell 1 laser transition generates atomphoton entanglement.
We1 characterize laser signals pulse of [Fig. our entanglement 2(e)]. This maps by measuring the atomic astate
Bell The of
the ′ opti
otons. After awithin time t, theapolarized trap. The displayed F ¼ 1 image $ F 0 shows ¼ 1 laser threemaps atoms the cou
quantum state of theViolation atom ontoof a second Bell inequality photon.
=1 jF ¼laser 2;m the F cavity pulse ¼ 0i axis, Zeeman [Fig. a CCD sublevel. 2(e)]. camera(d) This used A polarized maps to monitor the Fthe ¼atoms
2
(e)
te onto the polarization of a second emitted onto inequality the polarization violation of athe second two emitted photons photon, resulting [11]. The by E N 0
ith a pled to the mode of the cavity and aligned along the 1030 nm
ulting in inform an entangled of Bell inequality photon pair: violated here is based on the of cla
beam. an entangled SPCM: singlephoton counting pair:
S(0 ◦ ; 45 ◦ ; 22: 5 ◦ ; −22: 5 ◦ )=2: 46 0: 05 and
module, NPBS: nonpolarizing
beam splitter, PBS: polarizing beam splitter, 0305012 λ= 4: expectation value Eð;
F
], our
j PP i ¼ p 1 Þ of correlation measurements in of twt
, and
quarterwave plate, λ= 2: halfwave plate.
different bases [17]: ffiffiffi ðj þ ij i j ij þ iÞ: (2) Fca
on P〉 en = j1; 0〉⊗jΨ − ¼
PP 〉 S(22: 5 ◦ ; −22: 5 ◦ ;0 ◦ ; 45 ◦ )=2: 53 0: 05;
wi
3
A Single A single Atom atom Quantum quantum memory Memory
Nature, may 2011
One One atom atom in in a high high finesse
Nature, may 2011
cavity One cavity atom (strong in a high coupling) finesse
and of Nature, may 2011
Storage cavity (strong and readout coupling) of Storage a and retrieval of polarization qubit
weak weak coherent pulse pulse with with
Storage and readout of a Specht et al. Nature 473, 190 (2011)
arbitrary polarization state state
weak coherent pulse with
arbitrary polarization state
~9% ~9%
~9%
Quantum control of sound and light
Boozer, Boca, Miller, Northup & Kimble, Phys. Rev. Lett. 97, 083602 (2006)
•! First observation of cooling to the ground state for strongly coupled atomcavity system
•! Entering the quantum regime for all degrees of freedom in cavity QED
Internal – atomic dipole + cavity field
External – atomic center of mass
Quantum interface: Motional states (phonons) # Field states (photons)
Internal degrees of freedom –
Atomic dipole and cavity field
(t)
g
External degrees of freedom  Atomic motion (q, p)
A new paradigm for optical physics atombyatom and photonbyphoton
Quantum state exchange between motion and light
A. S. Parkins and H. J. Kimble, J. Opt. B: Quantum Semiclass. Opt. 1, 496 (1999); ...
The exciting new field of cavity optomechanics
Cavity QEDbased quantum computation
Singlephoton singleatom nonlinear phase shift (phasegate)
Turchette, Hood, Lange, Mabuchi, Kimble Phys. Rev. Lett. 90, 253601 (2003)
Probe
Pump
Heterodyne
Local
" a
oscillator
M 1
M 2
" b
Optical pumping
!/2
PBS
Cs beam # ,g
 
# ,g
+ +
Probe phase shift " a

19°
18°
17°
16°
15°
14°
13°
pump: ! 
pump: ! +
0.00 0.05 0.10 0.15 0.20
Number of intracavity pump photons m b
Entanglement generation between atoms in different cavities
Duan, Kimble, Phys. Rev. Lett. 90, 253601 (2003)
Faulttolerant quantum repeaters with singlephoton emitters
Childress, Taylor, Sorensen, Lukin, Phys. Rev. A 72, 052330 (2005)
Physical systems for cavity QED
O. Painter
J. Martinis, A. Cleland,
H. Mooij
Cavity QED with SPP: pushing beyond diffraction limit
!"#$%&'( )*%+,.+( .( %( $*%/( 0./'#$
Figure 1.3: Photograph of the most recent cavity constructed for use in the lab
1 experiment. 1D The metallic cavity systems mirrors, Surface with fabricated plasmons surface – on coupled plasmon BK7excitations substrates, polariton of are EM field onlyand 9.2 freec µm
apart; the •! Confinement mirror facesand are enhancement 1
density
mm in diameter,
waves
of a
guided
singlephoton withalong a 10a cm
conductordielectric
(evanescent radius of curvature, wave)
interface
and
coned so that they can be brought •! Infiniterange close together. interactions The substrates are held in BK7
Simplest example: SPs on a flat surface
vblocks glued to shearmode•!
piezoelectric Singlephoton transducers, transistor with a copper mounting block
beneath. (Lecture #3) Electric field / charge distribution
8
i i i
operate in this strong coupling regime, where coherent coupling dominates dissipative
rates.
states of the 6P
32 manifold are not
trapping times for Cesium atoms of
twocolor trap with wavelengths
blue
780 nm. Moreover, for two
total atom number N=2000, the
ptical depth = 8, corresponding to
ce per atom of = 0.65%.Inspired
have investigated trap designs that
ms for earlier implementations [35].
atoms at each node of U
trap near the
e despite the strong attraction from
s from the surface.
om the bluedetuned beam enable
d in (b) the plane. Specifically, U
trap
unterpropagating reddetuned beams,
ed beam in a so
as shown in the inset. The standing
e attractive reddetuned fields and the
al trap U
trap for Cesium atoms around
h is indicated by the gray shaded
erated by two evanescent fields that
nal confinement for the trapped atoms
diameter optical fiber, shown in (a)
shown along the axis of the fiber.
powers ~ 40mW, trap depths ~1mK
ievable.
toms can be trapped in an array of potential wells parallel to the z axis of
along the reddetuned standingwave by red
2 . Two sets of trapping wells
xis separated by an azimuthal angle of . For the case of linear polarization
0 (normal dielectric)
e.g., Drude model
[8]; an example cavity is shown in Figure 1.3. In order to meet the strong coupling
Surface plasmon
How to derive the surface plasmon modes:
criterion, we •! coupled want toexcitations maximize of g, EM thefield scalar and product freeelectron of thedensity atomicwaves
dipole and the
Guess solutions
•! guided along a conductordielectric interface
electric field within the cavity:
j
E Ee 0 ˆ
ˆ
Freespace cavity QED
strong coupling
j j,
r
2
g = ~µ k· ~E ~!a
j= ,
µ
j( / )
2✏ V , field confinement beyond (1.4)
cal traps
idual neutral atoms in a geometry compatible with the cavities to be used in
ll use nanowire traps, [34,35], based on far off resonance laser traps
Drude model:
e diffraction limit for conventional freespace optics. In this section, we
ed research with nanowire optical traps to enable interactions of arrays of
ith lithographically patterned optical resonators (e.g., zipper cavities) and
0nm will not be possible with ions due to charging effects and must be
utral atoms as proposed here. For our applications, the requirements for
near the surface are well beyond any existing capability for freespace
ing atom chips [33].
2 ( ) 1
We construct the optical cavities that we use in the lab from highfinesse mirrors
s for
red
, Fig. 5(b) displays the potential trap
U in the x z plane, with the
f atomic Cesium and a fused silica nanowire of diameter d 500 nm. As
ptical wavelength. The potential trap
U for such a nanowire trap is shown in
using the special properties of the optical modes for a dielectric rod of
to localize and manipulate atoms within the near fields of optical micro and
can in turn harness the power of such structures on spatial scales at and
Experimental realizations of
quantum memory and quantum interfaces
Part 2 –
Collective strong coupling
with atomic ensembles
References:
• “Optical Coherence and Quantum Optics” by L. Mandel and E. Wolf
• “Quantum Optics” by M.O. Scully and M.S. Zubairy
• “AtomPhoton Interactions” & “Photons and Atoms”
by C. CohenTannoudji, J. DupontRoc, G. Grynberg
Review articles:
• H.J. Kimble, The quantum internet, Nature 453, 1023 (2008)
• A.I. Lvovsky, B.C. Sanders & W. Tittel, Optical quantum memory.
Nature Photon. 3, 706 (2009).
• K. Hammerer, A.S. Sorensen, E.S. Polzik, Quantum interface between light and matter
Rev. Mod. Phys. 82, 1041 (2010)
• N. Sangouard, C. Simon, H. de Riedmatten and N. Gisin,
Quantum repeaters based on atomic ensembles and linear optics,
Rev. Mod. Phys. 83, 33 (2011)
ˆk) as
sion processes of the individual atoms. where Afterh.c. a delay is a hermitian t 0 , the initial spontaneously emitted photons
e coherences and the collective Collective amonginversion the atoms, operator leading enhancement: a superradiant pulse. a From historical Eq. 2.3, we writeaccount
the multiry
of the Dicke Hamiltonian (r < 0 ) (in the electric Ŝ z ' X constant,
Ŝ 2 = 1 2.2.2 Collective spin states
18
2 (Ŝ+ 0 Ŝ0 + Ŝ0Ŝ 2 Ŝ+ 0
= 1 conjugate of the term ~g ~k Ŝ 0 + â~
~ckd
k
, ~g ~k = i
2 0
2✏ 0 V ~✏ · ~✏ a is t
photon coupling )+Ŝz. (2.8)
Collective spin states dipole
2 (Ŝ+ ~✏
0 Ŝ0 ,a are + the Ŝ0 polarization Ŝ+ 0 )+Ŝz. vectors of the photon and the atom
coherence volume. S, Here, ˆzi
mi . we and for used the rotating collective maximum wave lowering angular approximations) (2.7) q
Cooperative where decay: h.c. issuperradiance
a hermitian conjugate of the term ~g ~k Ŝ 0 +
e atomic states (labeled a) as a system and electromagnetic modes (labeled ) as a Markovian
â~
~ckd
k
, ~g ~k = i
2 0
i
ng Eqs. 2.4–2.6, we assumed the subwavelength condition e i~ k·~r 2✏ 0 V ~✏
i
' e i~ k·~r 0
for 8 i (Ŝ~ k
' P and momentum raising operators S = N/2 are g
i ˆi ),
n writing
In addition,
Eqs.
we define
2.4–2.6,
the totalwe angular R.H. assumed Dicke, momentum
the Phys. photon operator
subwavelength coupling Rev. (also 93, constant, known 99 (1954). ~✏ as ,a
condition s
the are the length polarization ofethe i~ k·~r i
Bloch vectors '
vector of e i~ k·~r 0
the photon forand
8 i
to the introduction of collective symmetric states S, mi of
icke Hamiltonian is given by
Ŝz and Ŝ~ c k
. Ŝ ~k
= X (S e i~ k·~ri
+ m)! ˆi '
coherence volume. S, Here, mi we Ŝ0 = X
~Ŝk Dicke ) as
used
model for a collection of N A twolevel atoms localized in a N!(S collective
region rm)! lowering
0 (with (Ŝ0 )S and m ˆi
i
raising e ···ei, operators
eading to the introduction of collective symmetric states S, mi of RWA)
Ĥ Dicke = ~! 0 Ŝ z + X 18
Ŝ 2 = 1 Ŝz and Ŝ~ c i
k
.
~! ~k â † ~ â k ~k + X ⌘
"
2 (Ŝ+ 0 Ŝ0 ⇣~g + Ŝ0
~k Ŝ Ŝ+ 0 + 0 q
)+Ŝz. Ŝ + ~
= X e i~ k·~ri
ˆ+
Collective spin states
k â~ k
+ h.c. , (2.4)
where h.c. is a hermitian conjugate of the term ~g ~k Ŝ + ~ 0 k ~ â~
~ckd
k
, ~g ~k = i
2 0
2✏ k 0 V ~✏ · ~✏ Ŝ ~k
= X i ' Ŝ+ 0 = X ˆ+
(2.8)
i ,
e i~ k·~ri
ˆi '
a is the singleatom singlephoton
coupling constant,
Ŝ0 = X i
i
with S apple m apple S. The collective state N A /2,mi in Eq. 2.9
ˆi
i
In writing Eqs. 2.4–2.6, 
~✏
{z we}
assumed  {z the subwavelength }  condition {z e i~ k·~r i
} ' e i~ k·~r 0
for
,a are the polarization vectors
Ĥ a Ĥ
18 of the photon and the atomic Ŝ + ~
dipole, = and X8 i (Ŝ~
eV is i~ k·~ri
the ˆ+
Ĥ k i ' Ŝ+ 0 = X k
' P represents a fully
ve .2.2 spin states Collective S, mi for the maximum spin(Nstates
angular momentum S = N/2 are given by (ref. 43 i ˆi ),
A
and /2+m) the collective atomsinversion are the operator excited state ei and ) (N A /2 m) atoms are in
ˆ+
i , t
leading to the introduction of collective s
a i
i
coherence volume. Here, we used collective collective symmetric
lowering spinand states states
raising S, S, mi mi
operators qare ofsimultaneous Ŝz and Ŝ~ c k
. eigenstates Ŝ z ' X ˆzi
of.
Eqs. 2.7–2.8 with the19
(S + m)!
licitly re h.c. include is a hermitian the polarization conjugate ✏ by absorbing of the the term notation ~g ~k Ŝ ( ~ k, 0 + â~ ✏) k
,! ~g ~ ~ckd
k. ~k = i
2 0
•! Singleatom single S, mi photon = coupling constant:
2✏ 0 V ~✏ · ~✏ a is the singleatom singleton
coupling constant, ~✏ ,a are the Ŝ ~k
N!(S
otable examples of transient cooperative effects include optical free induction decay and photon echo.
polarization = X m)! (Ŝ0 and )S the m i
ollective spin states S, mi for 19 the maximume collective ···ei, angular
e i~ k·~ri vectors of the photon and the atomic dipole, and V is the
ˆi ' Ŝ0 = X inversion momentum operator S = (2.9) N/2 are given by (
In addition, we define the total angular The collective momentum spin operators follow (also
2.2.2 •!
Collective operators: spin states
•! Inversion operator:
ˆi Ŝ z (2.5) ' X known the commutator as the leng re
19 19
spin operators follow the commutator Ŝ z S, = mS, mi ˆzi
.
19relations
~Ŝk ) as s
i
ors he i
i
rence volume. Here, we used collective lowering and raising operators
Ŝ + ~
= X e i~ k·~ri
ˆ+
k i ' Ŝ+ 0 = X i
applecommutator follow m the S. commutator The relations collective relations state N
Collective spin states S, mi for the A /2,mi Eq. 2.9 represents a fully symmetric state whereby
hŜ+
ve spin operators follow the commutator relations imaximum angular momentum (S + m)! S
In addition, we define ˆ+
i , = N/2 are given by (ref.
theŜ 2 S, mi = S(S
total angular momentum 43 + (2.6) ) 1)S, mi.
operator (also known as
+m) atoms are in the excited state ei
Ŝ ~k
= X and S, (Nmi i A /2=
2 m) atoms
e i~ k·~ri
ˆi ' Ŝ0 = X are in the ground state gi. The
i
s ~Ŝk ) as N!(S m)! (Ŝ0 = 1 2 )S (Ŝ+ 0 Ŝ0 m e + ···ei,
Ŝ0 Ŝ+ 0 )+Ŝz.
0 , Ŝ0
i
i i
hŜ+ 0 , Ŝ0 = 2Ŝz (2.13)
hŜz ,
hŜ+ 0 ,
hŜ+
e spin states Ŝ0 = 0 ,
S, mi are Ŝ0 2Ŝz = 2Ŝz ii
(2.13) (2.13)
Ŝ± 0
simultaneous eigenstates (S of + m)! 2.7–2.8 with ˆi (2.5)
and the•!
collective Collective inversion spin algebra: operator S, Similarly,
In writing the collective
Eqs. 2.4–2.6,
raising
we
Figure 2.2: Generating and retrieving collecti
i
i
collective excitations. A weak write pulse illum
+ photon, called field 1. The detection of a single p
~
= X e i~ k·~ri Ŝ ˆ+
k i ' Ŝ+ 0 = X z ' X Ŝ 2 = 1 2 (Ŝ+ 0 Ŝ0 + Ŝ0 Ŝ+ 0
N!(S m)! (Ŝ0 )S m assumed
and
the following relations
e ···ei, lowering
the subwavelength
operators Ŝ± condition
0 acting on
e i~ k·~r i
(2.9) S, )+Ŝz. mi
' e
are
i~ k·~r
i i
0
hŜ+
hŜz , Ŝ± 0 =
hŜz , ±Ŝ± 0 . (2.14)
Ŝ± 0 = ±Ŝ± 0 . 0 , hŜz , 0 = Ŝ0
±Ŝ± 0 . = 2Ŝz
leading to the introduction (2.14) of collective (2.14)
(2.13)
i
symmetric states S, mi of
ith S apple m apple S. The collective state ˆzi
N. ˆ+
i , (2.7)
Ŝ In writing Eqs. 2.4–2.6, we assumed (2.6)
collective excitation subwavelength si in thecondition ensemble. eb, i~ k·~r z S, mi = mS, mi A /2,mi in Eq.
(2.10)
i
Retr
i
i
with S apple m apple S. The collective state N i
A /2,mi in Eq. 2.9
Ŝ
represents 0 ± S, mi = p 2.9 represents a fully Ŝz and Ŝ~
symmet
c
k
.
hŜz , Ŝ± 0 = ±Ŝ± 0 . We will use (2.14)
the language of collective spin algebra in th
to study the thermal behavior of entanglement in quantu
eof spin language collective algebraof spin collective the algebra context inspin the of quantum algebra context of in manybody quantum the context manybody theory of quantum in chapter theory manybody 9in chapter theory 9 a fully
in (Schapter leading to the introduction of collective
storage symmetric
⌥ m)(S 9 ±
time
symmetric
⌧, a strong state
m +
states
read whereby
1)S, m ± 1i.
S,
pulse
mi
maps
of
the coll
In
Naddition, A /2+m)
we define
atoms
the total
are Ŝangular 2 S, inmi momentum 2.2.2 = excited S(S Collective operator + 1)S, state mi. (also
ei spin known
and states as
(N
Since A length
/2
the Dicke
emission. ofm) Hamiltonian
the Bloch
atoms (2.11) vector
are in the Ŝzgrou
and Ŝ~ k
nglement ermal the
ior
language
the(N of entanglement behavior
collective A /2+m) in quantum
of collective of
inversion atoms
entanglement spin quantum
operator are models.
spin algebra
in spin the excited models. quantum the context
state
spin of
ei
models. quantum manybody theory in chapter 9
ĤDicke in Eq. 2.4 and (N A /2 m) atoms are in ground state gi. The
~Ŝk ollective ) as •! Note
collective spin that states states S, mi S, are mi simultaneous are Collective simultaneous
Ŝ is expressed terms of the normalized slowlyv
z ' X motion. On the other hand, [Ŝz,
iltonian Dicke
thermal in Eq. Hamiltonian
behavior ĤDicke 2.4 commutes
of
Eq.
entanglement ĤDicke 2.4 with commutes in the Eq.
in
operator
quantum
2.4 withcommutes the
spin
ˆzi
. (2.7)
Ŝ 2 = 1 Ŝ2 operator ,. Thus, models.
eigenstates
with spin states the is a operator constant
2 (Ŝ+ 0 Ŝ0 + 2.2.2 of eigenstates S,
i
Ŝ0 Ŝ+ 0 )+Ŝz. Collective Eqs. mi 2.7–2.8 for Ŝ2 of the , motion maximum
spin hŜ2 with of i isEqs. states the
a constant
following angular 2.7–2.8 of momentum relations with ĤDicke] 6= 0. Thus, a
hŜ2 i is Ŝ2 a , constant hŜ2 i is of a constant of
S the = N/2 followin are given
that the inverted atomic system ( (t = 0)i = e ···e
y, e the collective raising and lowering operators Ŝ± 0 acting on S, mi are
he icke] d,
Dicke
other [Ŝz, 6= 0. ĤDicke] Hamiltonian
hand, Thus, 6= [Ŝz, as0. we ĤDicke] Thus, Lightmatter will discuss as
in
6= we
Eq.
0. will Thus, in
2.4 interaction the discuss
commutes
next as we section, will the preserves with
next discuss we
the
section, can
operator the in expect symmetry the we Ŝ2 next can
, hŜ2 expect
i
section, of isthe a constant atomic of
we s can system expect (2.8)
t ystem the Collective spin states S, mi for the maximum ~Ê+ 1 (~r, t) =i
ddition, In writing weEqs. define 2.4–2.6, the total we assumed angular momentum the subwavelength operator condition (also known e i~ k·~r i ' the e i~ k·~r length 0
for 8 i of(Ŝ~ k
' Bloch P angular momentum S = N/2
Ŝ z S, mi = mS, mi vector
i ˆi ), (2.10)
Ŝ 0 ± S, mi p (S + m)!
ed= other
atomic 0)i
•!
( = (t hand, Note
system e = ···ei) 0)i [Ŝz,
( = ĤDicke]
atomic state confined in a ladder formed by (2S +1) eq
undergoes e (t···ei) 6= 0.
= 0)i aundergoes .
series Thus, The system
= e ···ei) of ascascade awe series
undergoes will
(S c undergoes of emissions discuss cascade
a inseries a series with the emissions next the
of symmetric section,
of cascade with S, the wecollective can expect damping process
mi emissions = with the
⌥For m)(S r < ± 0 m in+ the 1)S, optical m ± regime, 1i. one cannot neglect N!(S them)! effect (2.12)
(Ŝ0 of )S van m with an
atomic cascade confined in a ladder formed by (2S+1) equidistant collective energy levels states S, mi, shown analogous in e ···ei,
der Fig. to Waals 2.1a, a force analogous ⇠ 1/
erted atomic system ( (t = 0)i = e ···ei) undergoes a series of cascade emissions with the
Ŝ z S, mi = mS, mi
ladder ed by (2S formed +1) spontaneous by equidistant (2S +1) emission equidistant energy levels of a energy discussions single E m = levels spin m~! of of Enonideal 0m angular of= the m~! symmetric superradiance momentum 0 of the symmetric inS.
the presence of dipoledipole s coupling between the atoms.
'
angular momentum S.
angular momentum S. ✓
◆ ✓
as (Eq. 2.16)
20
0 t
Cooperative emission from a subwavelength ˆ⇢
2.2.3 Superradiant emission a (t + t) ' 1
for an atomic ensemble 2
sample
Ŝ+ 0 Ŝ0
0
ˆ⇢
in a (t) 1
subwave2
! 2.2.3 Superradiant 2.2.3 ✓ Superradiant ◆ ✓
emission foremission an atomic 2.2.3
for " ensemble an ◆
Superradiant
atomicin ensemble a emission subwavelength in for
a subwavelength {z
an atomic ensemble i
Since the systemreservoir Hamiltonian is Ĥa = P volume volu
0 t
ˆ⇢ a (t + t) ' 1
~ k
(~g ~k Ŝ 0 + â~ k
e i(! 0 !
2 Ŝ+ 0 Ŝ0
0 t
ˆ⇢ a (t) 1
Systemreservoir 2 Ŝ+ 0 Ŝ0 + 0
coupling
k )t Since the systemreservoir Hamiltonian is Ĥa = P + h.c.) in
Since the systemreservoir Hamiltonian
~ k
(~g ~k Ŝ 0 + â~ k
e i(! 0
Since the systemreservoir Hamiltonian (Eq. is 2.4), Ĥa we= P is
can write ~ the real part d k
(~g Ĥa ~k Ŝ + = P tŜ0 ˆ⇢ a(t)Ŝ+ 0 +O( t 2 ), (2.17)
“no” photon loss
 {z } 
of the master equation (in BornMarkov
⇣
0 â~ k
e i(! ~ k
(~g {z
0 ~k ! Ŝ + k )t }
0 â~ + k
e
h.c.) i(! 0 ! k )t + h.c.) in the interaction
“no” photon loss
with “yes” the photon twoloss
terms corresponding the interaction to the conditional pictureden
(Eq. 2.4), we can write d
Real (Eq. part 2.4), of the we can write the real with part d of master approximation
equation (in th
dt ˆ⇢ 1 R t
(Eq. 2.4), we can write the real parta(t) real =
~
Tr 2 0 dt0 [Ĥa (t), [Ĥa ⇣
(t 0 ), ˆ⇢ a (t) ⌦ ˆ⇢ (0)]]⌘
d the real part
of the master d of the master equation (in the BornMarkov following approximation
⇣ equation photons, (in the respectively. d
dt ˆ⇢ 1 R
BornMarkov Since approximation the collective jump e ) the with operato stand
with the two terms corresponding
d ⇣ t
d
as
a(t) real =
~
Tr
2 0 dt0 [Ĥa (t), [Ĥa (t 0 ), ˆ⇢ a (t) ⌦ ˆ⇢ (0)]]⌘
fol
dt ˆ⇢ 1 R
a(t) real =
dt ˆ⇢ to the conditional
1 R t
density matrices
a(t)
~
Tr real t = 2 0 dt0 ~
Tr
[Ĥa (t), [Ĥa (t 0 ), ˆ⇢ a (t) ⌦ ˆ⇢ (0)]]⌘
2 0 dt0 [Ĥa (t), [Ĥa (t 0 for zero and single spontaneous emitted
Figure 2.1: Superradiant states and atomic Fresnel number. ), momentum ˆ⇢ a (t) ⌦a, ˆ⇢ S) Energy (0)]]⌘
of ˆ⇢
following the a
following (t), levels the timeevolution forthe thestandard collective ofprocedure
standard procedures 164–166 ˆ⇢ spin a (t) fro
photons, respectively. Since the collective jump operators
states. A ladder of symmetric collective spin states as
Ŝ± 0 cannot alter the symmetry (and the total angular
of maximal angular momentum S as
A /2 is shown for
inversion will remain in the S = N
asmomentum m 2 S) { of S, ˆ⇢ a (t), S +1, the··· timeevolution ,S 1,S}. N c is the normalization d
dt ˆ⇢ constant. 0b, a(t) real =
d
Figure 2.1: Superradiant states and atomic Fresnel number. a, Energy levels for the collective spin
states. A ladder of symmetric collective spin states of maximal angular momentum S = N A /2 is shown for
dt ˆ⇢ 2 n Pencilshaped (Ŝ0 Ŝ+ 0 ˆ⇢ atomic
a 2Ŝ+ 0 ˆ⇢ ensemble.
aŜ0 +ˆ⇢ The
geometric angle is given by
aŜ0 Ŝ+ 0
Dynamics ✓ )
g = p of ˆ⇢ a (t) from the initially symmetric state  (t = 0)i with
⇡wof 0 2 symmetric collective damping
d
/L, whereas the diffraction angle is ✓ d = 0 / p total A /2 manifold with a
by p(S, mi !S, m 1i) =
inversion will remain in the S = N ⇡w
0
a(t) real =
2 n 0 2.
A /2 manifold with a transition probability from S, mi to S, m 1i given 0 thŜ+ 0 i = 0 t(
0
(Ŝ0 Ŝ+ 0 ˆ⇢ a 2Ŝ+ 0 ˆ⇢ aŜ0
d
+
2 (n + 1)(Ŝ+ 0 Ŝ0 ˆ⇢ a 2Ŝ0 ˆ⇢ aŜ+ 0 +ˆ⇢ aŜ+ 0 Ŝ0
m 2 { S, S +1, ··· ,Sdt ˆ⇢ 0
a(t) real = dt ˆ⇢ 0
by p(S, mi !S, m 1i) = a(t)
1,S}. N c is the normalization constant. b, Pencilshaped atomic ensemble. The
geometric angle is given by ✓ g = p 2 n real =
(Ŝ0 Ŝ+ 0 ˆ⇢ a2 n (Ŝ0 2Ŝ+ 0 ˆ⇢ Ŝ+ 0 ˆ⇢ a
aŜ0 +ˆ⇢ 2Ŝ+ 0 ˆ⇢ aŜ0
aŜ0 Ŝ+
⇡w0 2/L, whereas the diffraction angle is ✓ d = 0 / p 0 )
+ˆ⇢ aŜ0 Ŝ+ 0 )
0 thŜ+ 0 Ŝ0 i = find a collectively enhanced emission of p(S, 0i !S
0 t(S + m)(S m + 1). In particular, for m =0, we
0
find a collectively inverted state enhanced  (t emission = 0)i = ofS, p(S, Si 0i (e !S, ···ei) 1i) to ' lower 0
⇡w0 2.
2 (n + 1)(Ŝ+ 0 Ŝ0 ˆ⇢ a 2Ŝ0 ⇢
0
where 0 = k 3 d 2 0/(3⇡✏ 0 ~) is the singleatom spontaneous emission rate in the Wigne
2 (n + 1)(Ŝ+ 2 (n 0 Ŝ0 ˆ⇢ + symmetric collective 1)(Ŝ+ 0 Ŝ0
a 2Ŝ0 ˆ⇢ ˆ⇢ states
a aŜ+ 0 +ˆ⇢ 2Ŝ0 ˆ⇢ aŜ+
aŜ+ 0
0 Ŝ0 ), +ˆ⇢ S, mi (progressively
0 tN 2 A0 4
,
tN
relative A for
to
a collection
the transition
of independent
probability
atoms ( 0 t fo
aŜ+ 0 Ŝ0 ),
decaying from m = N A /2 to m = N A /2) in the subspace of S = N A /2 (Fig. 2.1a).
(2.16)
0 tN A for a collection of independent atoms ( 0 t for single atoms).
The equation of motion for the collective spin oper
spontaneous decay.
inverted state  (t = 0)i = S, Si (e ···ei) to lower symmetric where
collective 0 = k 3 d
states 2 0/(3⇡✏
S, 0 ~)
mi is
(progressively
the singleatom spontaneous emission r
The equation Indeed, ofinmotion the quantum for the
where
decaying where from m = N A /2 to m = N A /2) the Tosubspace describeof superradiance spontaneous
S = N A /2 (Fig. decay.
the 2.1a). optical domain, we may approximate reservoi
0 = k 3 d 2 0 = k 3 collective
d 2 jump picture, spin operators we can write
(with
0/(3⇡✏ 0 ~) from is theWignerWeisskopf singleatom {Ŝ± the shorttime ( t) evolution of the atomic state ˆ⇢
spontaneous emission rate in the WignerWeisskopf a (t)
0 (t), the Ŝz(t)} master canequation be solved(Eq. analytically 2.16) in the fromsemiclassical app
theory of spontaneous decay)
th
the master (Eq. equation 2.16) 0/(3⇡✏ (Eq. 2.16) 0 ~) in is the semiclassical singleatomapproximation. spontaneous emission Using 2.13–2.14), the commutator ratewe inobtain the relationships WignerWeisskopf the following (Eqs. differential theoryequatio
of
spontaneous decay. Shorttime
Indeed, in the quantum jump picture, westates evolution
can write withthezero shorttime mean
in the Tothermal ( describe quantumjump
t) evolution occupation superradiance picture
of the atomic (n
in =0). state the optical Then, ˆ⇢ a (t)
the domain, surviving we may term approxim in this
spontaneous 2.13–2.14), we decay. obtain the following ✓ differential◆equations ✓ (Eq. 2.16) ◆
To describe superradiance 0
as (Eq. 2.16)
term)
t
describes
in the
a states symmetric
optical withdomain, zero collective mean we thermal may
damping
approximate occupation process for
the (n the
reservoir =0). system, Then, modes
cascading the survi as
ˆ⇢ f
To describe a (t + t) ' 1
superradiance in the optical domain, we may approximate the reservoir modes as vacuum
✓states with zero ◆ mean
2 Ŝ+ 0 Ŝ0
0 t
d
ˆ⇢ a (t) 1
✓ thermal occupation term) ◆
describes (n
2 Ŝ+ 0 Ŝ0 + 0 tŜ0 ˆ⇢ a(t)Ŝ+ 0 +O( t 2 ), dt hŜ0 (2.17)
i =
d
 {z =0). a symmetric } Then,  the collective surviving {z damping } term in process this master for theequat
syst
0 t
dt hŜ0 d Note i that
ˆ⇢ a (t + t) ' 1
2 Ŝ+ 0 Ŝ0
0 t
= the dispersive 0hŜzŜ+ imaginary 0 i part of the master equation gives rise to (2.18) d
collective Lamb shift and v
states with zero mean thermal occupation “no”
ˆ⇢ (t) Namely, 1 we(n
photon
find =0).
2 Ŝ+ 0 Ŝ0 loss
+ Then, 0 tŜ0 the ˆ⇢ surviving a(t)Ŝ+ “yes”
0 +O( t 2 ), term photon inloss
(2.17) this master equation dt hŜzi =
(2
term) describes a symmetric collective damping process for the d
 {z }
d 2 system,
Note
Note
 that
that
{z the
the
dispersive
dispersive
} imaginary
imaginary
part
part
of
of
the
the
master
master
equation
equation
gives
gives
rise
rise to
collect
collect
d
Namely, we find
“no” photon loss
Namely, “yes” find photon loss
2
dt ˆ⇢ a(t) imaginary = id2 0 4 X " cascading from # the initia 3
a symmetric collective damping process for the system, cascading from the 1initial totally 3(~✏ a · ~r ij )
term) describes a symmetric collective dt hŜzi =
damping process 0hŜ+ 0 Ŝ0 i. (2.19)
for the system, cascading 2
4⇡✏ 0 r 3 1from the initial ˆ+
d
with the two terms corresponding to the conditional density matrices for zero and single spontaneous dt ˆ⇢ a(t) imaginary
emitted
id2 3(~✏ ~r ij dt ˆ⇢ a(t) imaginary id2 0 4 X " ˆ+ totally
i>j ij
rij
2 i j , ˆ⇢ a5 with the two
.
inverted terms d corresponding
Note that
state
the
!(t
dispersive
= 0)! to
imaginary
= the S,S! conditional (e···e!)
part of
to
the
lower density
master
symmetric matrices
equation
In the semiclassical
gives
collective for zero
rise to collective
states and approximation single S,m! spontaneous emitted
Lamb shift and
(i.e.,
1van taking
der Waals
operat
3(~✏ a · ~r inte ij )
d Namely, (progressively we find decaying from m = N
4⇡✏
4⇡✏ 0 r 3 1
Note that the dispersive imaginary partThe of the superradiance master equation for A /2 to m = "N
Eq. 2.16 gives occurs A /2) in the subspace of S = N
rise2
to because collective of theLamb indistinguishability A /2.
In the semiclassical photons, respectively. approximation Since(i.e., the collective taking operators jump operators as cnumbers), Ŝ± (Eqs. 0 cannot we 2.18–2.19) solve shift and vaninder the emission Waals pathways among
i>j ijinteractionr i>j ij
44 ij
ij
2 .
van der Waals interaction (Eq. 2.15) has a characteristic dipoledipole coupling g
photons, respectively. Since the collective jump operators Ŝ± 0 cannot The superradiance
The superradiance alter the symmetry for Eq. 2.16
for Eq. 2.16 (andoccurs the total because
because angular of the indistinguishability vdW ' d 0
Namely, we find
d
2
4⇡✏ 0 inr ij
3 , where
2
the emiss
⇣ ⌘ 3. of the indistinguishability in the emiss
d
is g vdW
' 1 van 0 der Waals interaction (Eq. 2.15) has a characteristic dipoledipole coupling g
momentum S) of ˆ⇢ a (t), the timeevolution of ˆ⇢ a (t) from the initially van der Waals symmetric interaction state (Eq.  (t 2.15) = 0)i has with characteristic total dipoledipole coupling vdW
0 10⇡ r For r ⌧ 0 , the frequency shifts of this dipoledipole interaction may break
ij
⇣ ⌘ vdW
dt ˆ⇢ a(t) imaginary = id2 0 4 X "
# 3
dt ˆ⇢ a(t) imaginary = id2 0 4 X " alterthe the and equations symmetry obtain # of hŜz(t)i (and motions 3the ' total S tanh( angular 0 S(t
1 3(~✏ a · ~r ij )
(Eqs. momentum S) of ˆ⇢ Semiclassical 2.18–2.19) solution: and obtain
1
4⇡✏ 0 3(~✏ i>j a ·
r
~r ij
3 1
ˆ+ ˆ+
a hŜz(t)i (t), the' timeevolution S tanh( 0 S(t of ˆ⇢ t a d (t) )). from This leads the of Iinitially c to=
a superradiant symmetric ) 2 r 2 emission state i  intensity
j (t , ˆ⇢ a= 5
0 dhŜzi
.
dt
= N 2 A 0
4
sech 2 N A 0
20)i(t witht d total ) .
superradiance 4⇡✏as 0 discussed r 3 1
ˆ+ ˆ+ ij
of I
here. The full analysis r 2 i j , ˆ⇢ c =
a5 inversion 0 dhŜzi
will . (2.15)
dt
= N
The
remain 2
A 0
superradiance
4
sech in 2 the N A 0
forS 2
Eq.
= (t
2.16
N A
t/2 d )
occurs
manifold .
becausewith of thea indistinguishability
transition 3
probability
including the emission
from S,
van derCollective Waals pathways
mi to
dephasing enhancement
among
S, mis out the1i of atoms.
given
scope for Theth
Extended sample: The role of spatial phases
radiation is emitted spontaneously by a
Superradiance from a subwavelength sample
f. NDicke A atoms calculated as a single thequantum rate at which system, radiation is emitted spontaneously by a
an Bycooperatively considering the decay entire into collection the ground of N A atoms as a single quantum system,
ch ditions faster the than atoms their inincoherent the excitedemission
state can cooperatively decay into the ground
Collective enhancement in superradiant fluorescence
hanced gle mode with at aI coh
rate/ 1/⌧ N A c ~! / N 0 /⌧ Ac
0 / much NA 2 , faster than their incoherent emission
21
ntensity A. Indeed, I coh the is thereby initial investigations collectively
vs.
enhanced of with I coh / N A ~! 0 /⌧ c / NA 2 ,
N A independent atoms
2.2.4 Superradiance for extended atomic ensembles
ion withintensity the studies I inc of /‘superradiance’ N A ~! 0 /⌧ 0 / Nfor
A . Indeed, the initial investigations of
Superradiance from an extended sample
llective spontaneous The dynamics emissions of multimode began with superradiance the studies for of extended ‘superradiance’ samples 167,168 for is more complex than the cla
•!
Nonsymmetric collective damping
Quantum analogue of Bragg reflection of light
from atomic phase grating
For subwavelength samples r < % 0 , where the initial spontaneous emission of an inverted
articular ngth region mode example (r atomic could < of 0
system ) be Dicke 42,44 enhanced . leads superradiance to (suample
r ~
a phase 42 coherence in sectionbetween 2.2.3, as the the atomic master dipoles equation due to involves the intrinsic various spatial ph
indistinguishability in the emission process.
icke predicted k·~ri •! The that (thus, 0, depending superradiant radiation the geometry into upon emission of a particular theof atomic an extended sample) modesample could as well is bealso asenhanced a associated secondorder (subradiance)
s manifested Bloch
with propagation the classical equation (i.e., Maxw
constructive interference of the wavelets produced by periodically located scattering sites in
for bya the aequation, spatially quantum “forward” extended direction
see analogue also
set
Eq.
sample of by
2.39)
the sample
through
r geometry.
the atomic sample of length L 0 (see Fig. 2.1b). Fo
0, depending upon the
ms
for 44,155 subwavelength current •! Quasi discussion, 1dimensional
. In this case,
samples it
superradiance
r suffices < approximation
is 0
to , say that for if atomic the Fresnel number F
manifested by a quantum analogue a = ⇡w
of 0/L 2 0 is '(endfire 1 for the mode) atomic sam
Phase arrayed antenna
leads to a phase (F ' coherence 1 for our experimental between theparameters, see section 2.3.2.2), the propagation equations of the field fo
atomic phase grating. Unlike the case for subwavelength samples r < 0 ,
n process, the ‘pencil superradiant shaped’ sample emission can of be well approximated to a onedimensional model 44,70,167,168 . The superrad
mission Superradiance: of an inverted An essay atomic on the system theory leads of to a phase coherence between the
nterference emission of the wavelets takes place produced along the by elongated direction ~ Spatial, temporal coherence of
collective spontaneous emission.
k 0 of the sample superradiant (socalled “endfire Raman scattering mode”) 161 , for w
sicGross, indistinguishability Haroche. Phys. Rep. in the 93, 301 emission (1982). process, the superradiant emission of
the samplethe geometry. collectiveSuch variables collective S ~ ~k = P Raymer et al. Phys. Rev. A 32, 332 (1985)
Superradiance
i ei~ k·~r i
~ i are “phasematched.” Superradiant In this case, Rayleigh the socalled scattering ‘shape from BEC funct
ciated Rehler, with Eberly, the classical constructive interference of the wavelets produced by
a wide variety f( ~ Phys.
k, of ~ k 0 Rev.
) physical = 1 A 3, P1735 NA
systems, i,j exp[i(~ (1971).
in k ~ k 0 Innouye et al. Science 285, 571 (1999)
)(~r j ~r i )] determines the phasematching condition from the sam
N 2 A
Strong coupling regime & Superradiance
Superradiance is a transient coherent process involving a
collective mode of all the N A atoms in the sample.
•!
In the collective mode, correlation between the atomic dipoles arise through spontaneous emission in an
inverted system, due to the intrinsic indistinguishability of the emission processes of the individual atoms
Small sample limit
superradiance!
Cooperative decay: superradiance
R.H. Dicke, Phys. Rev. 93, 99 (1954).
The Coherence Brightened Laser
Dicke, Columbia University Press.
Cooperative phenomena in resonant electromagnetic propagation
Arecchi & Courtens, Phys. Rev. A 2 (1970).
Maser oscillation and microwave superradiance
in small systems of Rydberg atoms
Gross, Goy, Fabre, Haroche, Raimond, Phys. Rev. Lett. 43, 343 (1979).
Led to the phenomenal development of
microwave cavity QED in the late 80s !
Observation of Dicke superradiance in optically pumped HF gas
Skribanowitz, Herman, MacGillivray, Feld, Phys. Rev. Lett. 30, 309 (1973).
Singlepulse superfluorescence in Cesium.
Gibbs, Vrehen, Hikspoors, Phys. Rev. Lett. 39, 547 (1977).
Large collective dipole moment may lead to a strong coupling regime in free space!
Figure 1  Overview of the experiment. a, Quantum interfaces for
Strong matterlight c interaction
ct the optical cavities that we use in the lab from highfinesse c 2
mirrors
cavity
d
•! Transport is shown in and Figure communication: 1.3. In orderPhotonics
meet the strong
d
coupling
2
ant to maximize g, the scalar product of the atomic dipole Atom and the c
•! Coherent storage and processing:
thin the
Control
cavity:
logic D
two or three level h system (Atom)
OUT
r
Read
g = ~µ · ~E ~!a
= µ , (1.4)
2✏ 0 V m IN
Write
cavity mode volume, Atomic is ensemble proportional to the cavity length and to the
strongly coupled single collective
to modern atomic ensemble
excitation and one photon
ˆ () experiments
ode waist. Thus, we should minimize the mode volume by building
IM W
nd using mirrors write
with a small
Quantum
radius ofnodes
curvature. However, Emissive the fullimum
(FWHM) linewidth of a cavity is given by the ratio of its free
QM
• DuanLukinCiracZoller approach
a Quantum
Reversible
channels
QM
b
• Dynamic EIT quantum memories
c
•! OffresonantRamanbased
quantum memories
d
• Photonecho quantum memories
b 2
From superradiance of extended atomic samples
g
Fields 2
g
e
porbital
sorbital
p 1000
p 0100
Matterlight quantum interface
What’s inside here?
•! Ensemble of ~ 10 6 Cs atoms released from MOT
•! Utilize strong interaction
of singlephotons and
collective spin excitations (in the singleexcitation regime)
•! Inputoutput coupling k(t) usercontrolled externally by
lasers
Noncollinear geometry
Harris group (2005)
Writing and reading collective spin waves
Readout processes
EIT, offresonant Raman,
photon echo (CRIB/AFC..)
atomic ensembles in CV regime : Hammerer, Sorensen, Polzik arxiv0807.3358 (2009)
.1.1 Interaction Hamiltonian
n the weak depletion limit h , where Z the h Rabi frequency Hamiltonian ⌦
Parametric atomlight
24 w (~r, t) is constant over z, we can write the interacon
Hamiltonian in thes (par) = d~rn ~g
2.3.1.1 Ĥ Interaction Hamiltonian
interaction
the weak depletion limit rotating h wave approximation,
A p Ê(~r){~ 1 (~r, t)e i~ k 1·~r wˆee (~r, ˆes t) (~r, t)+~⌦ Ĥ(par) s in Eq. 2.22, we denoted the i collective atomic vari
~ w1ˆss w (~r, t)e i~ k w·~r ˆeg (~r, t)+h.c. }, (2.22)
, where the Rabi frequency ⌦ w (~r, t) is constant over z, we can write the interacn
Hamiltonian In theinweak rotating depletion Z wavelimit approximation,
~g h p , Êwhere 1 (~r, t)ethe i~ k 1·~r Rabi ˆes
.1.1 Interaction Hamiltonian h over a small volume i containing N ~r 1 atoms) i
the continuum limit
q(~r, frequency t)+~⌦ w ⌦(~r, w (~r, t)et) i~ k w·~r is constant ˆeg (~r, t)+h.c. over z, we }, can write (2.22) the interac
Pencilshaped atomic ensemble
•! where Spontaneous n Raman interaction
the weak depletion Ĥ A
Hamiltonian (par) (~r) is
s limit =
the h atomic
, where in d~rn the the A (~r){~
density, g
Rabi frequency p
rotating wˆee
wave (~r,
=
approximation,
t)
d
wes
⌦ es w (~r, ~ 2~✏
Z
w1ˆss t) 0 is V 1 constant (~r,
is
t)
the atomphoton coupling constant with dipole
over z, we can write the interac
Consider an atomic ensemble consisting of N A atoms in a &level system
Hamiltonian
matrixĤ element h
i
s (par)
in the=
rotating
d
q
where n A (~r) is the es = d~rn wave
he
atomic A (~r){~ ˆdsi. approximation,
We wˆee
take
~g p Ê 1 (~r, Zt)e i~ k 1·~r (~r, t) the quantization
density, g p = d ~ w1ˆss wes (~r, t) volume V
ˆes (~r, es t)+~⌦ w (~r, t)e i~ k
2~✏ w·~r 0 V 1
is the atomphoton 1 as the sample ˆµ⌫ volume. (~r, t) = In 1 XN ~r
writing ˆ(i) the
ˆeg (~r, t)+h.c.
coupling
},
constant
(2.22)
with N dipoleµ⌫e iw µ⌫t ,
~r
Hamiltonian In the weak depletion h limit (with RWA)
i
matrix element Ĥ(par) d ZĤ (par) es =
Ĥ s (par)
s ~g he p Ê ˆdsi.
i
s in Eq. 2.22, we
1 (~r, = t)e i~ k 1·~r denoted the collective
d~rn ˆes A (~r, (~r){~ t)+~⌦ wˆee (~r, t) ~ w1ˆss (~r, q w (~r, t)e i~ k w·~r atomic variables defined locally at ~r (evaluated
We take the quantization volume V ˆeg (~r, 1 as t)+h.c. the sample }, volume. (2.22) In writing the
over a small volume
hereHamiltonian n
= d~rn A (~r){~ wˆee h (~r, t) ~ w1ˆss (~r, t)
A (~r) is the Ĥ(par) atomic density, g p = d
wes
s in i containing N
Eq. 2.22, we ~r 1 atoms) in the continuum limit ( P ! R d~rn
denoted es
i
2~✏
h
~g i "
~g p Ê 1 (~r, t)e i~ k p Ê 1 (~r,
0
the
V
t)e
1
collective i~ is the atomphoton atomic variables couplingdefined k 1·~r ˆes (~r, t)+~⌦
1·~r ˆes (~r, t)+~⌦ w (~r, t)e i~ k w (~r, t)e i~ constant locally A (~r)) of
with dipole
k w·~r ˆeg (~r, t)+h.c. }, (2.22)
atrix element d es = he ˆdsi.
q with singleatom operator ˆ(i)
at ~r (evaluated
ere n A (~r) is the atomic density, Weg p take = the d
wes
over a small volume i containing N w·~r ~r es quantization 12~✏ atoms) 0 V 1
isinthe volume theatomphoton continuum V 1 as ˆeg the limit coupling (~r, sample ( t)+h.c.
P constant ! volume. R d~rn }, with A In (~r)) writing dipole (2.22) of the
trix amiltonian element q
where Ĥ(par) d
s es = inhe Eq. ˆdsi. 2.22, We we takedenoted the quantization ˆµ⌫ the (~r, collective t) volume = 1 XN ~r
atomic V 1 as ˆ(i)
variables the sample defined volume. locally In writing at ~r (evaluated the
n q A (~r) is the atomic density,
ere n A (~r) is the atomic density, g p = d
wes
g p = d
wes
with atomfield coupling const.
es
es 2~✏ 0 V 1
is the atomphoton 2~✏ 0 V 1
is the atomphoton coupling constant with dipole
ver miltonian a small volume i containing N ~r
coupling constant with dipole
matrix element
trix element d es = he ˆdsi. d es = he
We take the ˆdsi.
1 atoms) in the continuum limit ( P ! R Ĥ(par) s in Eq. 2.22, we denoted the collective atomic variables definedd~rn locally A (~r)) at of ~r (evaluated
ˆµ⌫ (~r, t) = 1 N
µ⌫e iw µ⌫t , (2.23)
~r N
[ˆ↵ (~r, t), ˆµ⌫ (~r 0 ,t)] = V 1
(~r ~r 0 )( µˆ↵⌫ (~r, t)
X ~r i
N ~r ˆ(i)
er a small volume i Collective containingatomic N ~r variable 1 atoms) We take the quantization volume V
quantization in in the continuum N
µ⌫e
volume limit V 1 as ( P iw µ⌫t the ! R , (2.23)
~r
sample volume. 1 as the sample volume. In writing the
i
d~rn A (~r)) of
with singleatom operator ˆ(i)
In writing the
µ⌫ = µi i h⌫. The In
miltonianHamiltonian Ĥ(par) s in Eq. Ĥ(par) 2.22, s we in denoted Eq. 2.22, thewe collective denotedatomic the collective variables defined atomic locally variables at ~r defined (evaluated locally at ~r (evaluated
ˆµ⌫ (~r, t) = 1 particular, collective
XN ~r
the variables hyperfine follow groundstate the commutation coherence relations, {ˆgs , ˆsg } follows th
ˆ(i)
r a small over volume a small i containing volume N i N
µ⌫e iw µ⌫t , containing 1 atoms) N in the continuum limit ( P ! R ~r 1 atoms) in the continuum d~rn limit A (~r)) ( P of ! R (2.23)
~r
ˆµ⌫ (~r, t) = 1 XN with singleatom operator ˆ(i)
~r
µ⌫ = µi i h⌫. The collective ˆ(i) i
d~rn A (~r)) of
N
µ⌫e iw µ⌫t variables follow the commutation relations,
, (2.23)
Note the
[ˆ↵ (~r, t),
commutator
ˆµ⌫ (~r
relationship
0 ,t)] ~r
= V 1
(~r ~r 0 )( µˆ↵⌫ (~r, t) ⌫↵ˆµ (~r, t)). (2.24)
i
ith singleatom operator ˆ(i)
µ⌫ = µi i h⌫. The collective
ˆµ⌫ (~r, t) = 1 X
~r
variables
ˆµ⌫ (~r, ˆ(i)
N
µ⌫e t) = iw µ⌫t 1 follow
XN ~r
the commutation relations,
th singleatom operator ˆ(i)
[ˆ↵ (~r, t), ˆµ⌫ (~r 0 ,t)] = V N ~r
[ˆsg (~r, t), ˆ†sg(~r 0 ,t)] ' V 1
25
1
The total Hamiltonian including the respective (~r ~r
µ⌫ = µi i h⌫. The collective N ~r
reservoir 0 )( µˆ↵⌫ modes (~r, t) for ⌫↵ˆµ the atomic (~r, t)). coherences (2.24)
variables follow , the ˆ(i) commutation (2.23)
Figure 2.2: Generating and retrieving collective ex
~r
i
N
µ⌫e iw µ⌫t µ⌫ is
In particular, the hyperfine groundstate coherence {ˆgs , ˆsg } follows the Bosonic , relations, commutator relations (2.23)
The hyperfine ground state coherences ~r
i
collective excitations. A weak write pulse illuminate
[ˆ↵ (~r, t), ˆµ⌫ (~r 0 ,t)] = V in in the the weak excitation limit 1
In particular, the hyperfine groundstate (~r ~r 0 gg ' 1 ee, ss.
)( µˆ↵⌫ (~r, t) ⌫↵ˆµ (~r, t)). (2.24)
N photon, called field 1. The detection of a single photo
[ˆ↵ (~r, t),
collective excitation si in the ensemble. b, Retrievin
h singleatom operator ˆ(i) ˆµ⌫ (~r 0 ,t)] = V coherence
1
~r 0 {ˆgs , ˆsg } follows the Bosonic commutator relations
)(
µ⌫
with singleatom
=
operator
µi i h⌫. ˆ(i)
µˆ↵⌫ (~r, t) ⌫↵ˆµ (~r, t)). (2.24)
[ˆsg (~r, t),
The N ˆ†sg(~r 0
~r
,t)] ' V ✓ ◆
Ĥ tot = 1 Ĥ(par) s +
(~r ~r 0 Ĥr + X Ĥ sr (µ⌫) . 1
(2.28)
)ˆgg (~r, t)+O
collective variables follow the commutation relations,
µ⌫ = µi i h⌫. The collective variables follow storage the time commutation ⌧, a strong read relations, pulse maps the collectiv
n particular, the hyperfine groundstate coherence {ˆgs , ˆsg } follows the Bosonic commutator emission. relations
[ˆsg (~r, t), ˆ†sg(~r 0 ,t)] ' V N ~r ✓ N◆
2 (2.25)
25
µ,⌫
2.3.1.2 HeisenbergLanvegin equations
1
1
(~r ~r 0 ~r
)ˆgg (~r, t)+O
particular, The total theHamiltonian hyperfine groundstate including Collective
coherence the spinwave respectiveexcitations [ˆ↵ (~r, t), ˆµ⌫ (~r 0 ,t)] = V reservoir {ˆgs ,
1
(~r ~r 0 ˆsg N } ~r follows modes exhibit
the for quasibosonic
Bosonic the atomic commutator coherences statistics
relations
)( µˆ↵⌫ (~r, t) ⌫↵ˆµ (~r, t)). (2.24)
[ˆ↵ (~r, t), N ˆµ⌫ ~r (~r 0 ,t)] = V N~r
2 (2.25)
µ⌫ is
in the weakInexcitation the HeisenbergLangevin limit gg ' 1 approach
ee, ss.
1
(~r ~r 0 )( µˆ↵⌫ (~r, t) is expressed
⌫↵ˆµ
in(~r, terms t)). of the normalized slowlyvaryin (2.24)
[ˆsg (~r, t), ˆ†sg(~r 0 ,t)] ' V In addition, 143,164–166 , we can describe the dynamics of the atomic operators
•! Dynamics of open quantum system: HeisenbergLangevin the systemequation ✓ ◆
1
1
(~r ~r
N N 0 )ˆgg ~r
(~r, t)+O
N~r
2 (2.25)
[ˆsg (~r, t), ˆ†sg(~r 0 ,t)] ' V ✓ Ĥ(par) s interacts of motion with a thermal reservoir (Markov
◆
in the weak (fromexcitation Eq. 2.28) limit by a set
gg ' of1selfconsistent equations
1ee, 1
Ĥ (~r
ss.
~r 0 tot = )ˆgg (~r, t)+O
N
particular, the hyperfine groundstate coherence ~r N ~
{ˆgs , ˆsg } follows the Bosonic ~r
2 (2.25)
2.3.1.2 HeisenbergLanvegin equations
Ĥ(par) s + Ĥr + X of motions
Ĥ (µ⌫) (ref. 143 )
System + environment:
sr .
commutator relations
~Ê+
Ĥ 1 (~r, t) =ir
In particular, the hyperfine groundstate coherence
2✏
the weak excitation limit gg ' 1 ee, ss.
{ˆgs , ˆsg } follows the Bosonic commutator r = X (2.28)
µ,⌫
i
~w j ˆr † relations ~ ˆr k,j ~k,j
the weak 2.3.1.2
In addition, excitation HeisenbergLanvegin
the limit system gg ' 1 equations @ tˆµ⌫ = Ĥ(par)
µ⌫ ˆµ⌫ [ˆµ⌫, Ĥ(par)
interacts ee, ss. with 143,164–166 a thermal reservoir (Markovian ✓ ◆s ]+
~ bath)
ˆF
Langevin noise forces
Resulting atomic dynamics:
µ⌫ . Fluctuationdissipation (2.29)
~ k,j
s
µ⌫ = µi i h⌫. The collective variables follow
N ~r
(~r ~r 0 )ˆgg (~r, t)+
V 1
In the following, we solve the steadystate solution for HeisenbergLangevin equation of motion (Eqs. 2.30–
nian (neglecting the noise terms and assuming constant atomic distribution n A (~r) =N A /
Parametric atomlight interaction
2.3.1.3 Adiabatic elimination of excited state
Z
2.32). If we assume the far offresonant limit w se, eg and the narrowbandwidth w w ⌧ w of the
where Ŝ(~r, write t) laser, =p we N A can e adiabatically i(~ k w
~ k1 )·~r ˆgs eliminate (~r, t) isthe the excited phasematched state ei andslowlyvarying obtain the steadystate spinwave solutions amplitude, for the and
p Ĥ In (par)
⌦
eff the following, = N A
wed~r
solve
⇢~ the steadystate solution for HeisenbergLangevin equation of m
•! p(~r, Steadystate t; optical w, w1) coherences solution ' g to (i.e., the @ tˆse HeisenbergLangevin ⇤ p NA
w (~r,t)
ˆee (~r, t) ~ ˆss (~r, t)+ ~⌦ w(~r, t) 2
ˆgg i ~⌦
= @ tˆeg =0). Namely, equation of motion for collective atomic variables
w w
(adiabatic elimination 2.32). of the
gs the
V
w(~r,2
1
effective parametric coupling constant. Here,
w
the collective w
If excited we assume state the in the far offresonant limit w se, eg)
and the narrowbandwidth
enhancement ( p N A ) is manifested not ✓
◆
⌦
by the + 1 Z
o
increased d~r
n~
w
ˆse ' write laser, we 1+ can w1 adiabatically + i
emission p (~r, t; rate w, ofw1)Ê1(~r, V
the Raman t)Ŝ(~r, scattered t)+h.c. photon, , but by
1 se
e i~ k w·~r eliminate the excited state ei and obtain
ˆsg (2.33) the steadysta
the increased quantum correlation between w w gs w w gs
optical
⌦ ⇤ ✓
field
coherences ✓
1
◆◆
and collective excitation
(i.e., @ tˆse = @ tˆeg
✓
(section
=0). Namely, ✓ ◆◆
2.4).
26
w
eg
ˆeg ' 1 i e i~ k w·~r i
eg
The first term of Eq. where 2.35 Ŝ(~r, includes t) =p theNbarestate atomic ˆgg Hamiltonian, 1 i ˆF eg . (2.34)
w
w
w ✓
light shift
w ◆
(⇠ ~⌦ w 2
A e i(~ k w
~ k1 )·~r ˆgs (~r, t) (neglecting is the phasematched the noise terms slowlyvarying and assuming ), and the constant spinwa a
w
p
26 ⌦ w
ˆse '
1+ w1 + i se
e i~ k
population loss of w·~r ˆgg due to optical pumping (⇠ i~⌦ ⌦
p(~r, t; w, w1) ' g w ⇤ 2
p NA
w (~r,t)
eg
w2
w gs ). isFor theour effective experiments, parametric we can ˆsg neglect the
w
By substituting these solutions (Eqs. 2.33–2.34) Eq. 2.22, w we w gs obtain the effective w w gs interaction Hamiltoj
This is a reasonable V
nian (neglecting the noise terms and
Parametric
assuming constant
atomlight
atomic
Hamiltonian
distribution
⌦ ⇤ ✓ Ĥ ✓n (par)
eff ◆◆
✓ ✓ ◆◆
later two effects (optical pumping and light shift), as the intensity
w
I A (~r) =N = N Z ⇢ coupling constant. H
26
A
enhancement ( p N w for egthe write A /V 1 ) d~r ~ ˆee (~r, t) ~ ˆss (~r, t)
(EPR correlation between collective A ) is manifested not by the increased emission
hyperfine excitation & Raman scattered V
ˆeg ' 1 i e i~ k 1 w·~r laser is rate well i of below the Raman the scatte
photon)
eg
nian (neglecting the noise terms ˆgg 1 i ˆF
w
w
w
w
saturation intensity I sat with a typical saturation parameter s ⌘ I w /I sat ⌘ 2⌦ w  / 2 ⌧ 10 4 (weak
Ĥ (par)
eff
= N Z
A
d~r
⇢~ ˆee (~r, t) ~ ˆss (~r, t)+ ~⌦ w(~r, t) 2
ˆgg i ~⌦ w(~r, + t) 1 and Z assuming n constant atomic d
the increased quantum correlation between field 1 and collective 2 egd~r
excitation ~ p (~r, t; (section
w, w1)Ê1(~r, 2.4). t
V
2 1 ˆgg
approximation given that optical transitions correspond to a temperature scale > 3, 000 K, relative roomtemperature
excitation limit).
300
The 1
w
w
K.
second term, however, By substituting corresponds these solutions to a nondegenerate (Eqs. 2.33–2.34) parametric to Eq. 2.22, amplification. we obtain the This effective in
+ 1 Z
Ĥ o
d~r
n~ (par)
parametric matterlight V 1 interaction, p (~r, t; w, w1)Ê1(~r,
denoted as
t)Ŝ(~r, t)+h.c. eff
= N Z ⇢
The first term of Eq. 2.35 includes the barestate A atomic Hamiltonian, light shift
d~r ~ ˆee (~r, t) ~ ˆss (~r, t)+ ~⌦ (⇠
where , Ŝ(~r, V t) =p N A e i(~ k w
~ k1
(2.35) )·~r population loss of 1
ˆgg due to optical pumping ˆgs (~r, t) is the phase
p ⌦
with twomode squeezing interaction Hamiltonian
p(~r, t; w, w1) ' g ⇤ p NA
w
+ 1 (⇠Z
i~⌦ w 2
eg
n2
). For our experiments, w
w
later two effects (optical pumping and light shift), (~r,t)
w w gs
is the effective p
where Ŝ(~r, t) =p N A e i(~ k w
~ k1 )·~r j This is a reasonable
⇣
V 1
asd~r
ˆgs approximation given that optical transitions correspond to a temperature scale > 3, 0
temperature Ĥ (par) (~r, t) is the phasematched
int
(t) 300 =~ K. p(t)Ê1Ŝ + slowlyvarying ⇤ enhancement
p(t)Ê 1Ŝ†⌘ † spinwave ( p the ~ intensity p (~r, t; I w, w1)Ê1(~r, t)Ŝ(~r, t
w for the write laser
N amplitude, and
p , A ) is manifested not by the increased
saturation intensity I (2.36)
⌦
p(~r, t; w, w1) ' g ⇤ p NA
w (~r,t)
sat with a typical saturation parameter s ⌘ I w /I sat ⌘ 2⌦ w  2 / e
2
w w gs
is the effective parametric coupling constant. Here, the collective
enhancement ( p where Ŝ(~r, t) the =p increased N A e quantum i(~ k w
~ k1 correlation )·~r between field 1 and co
excitation limit). The second term, however, corresponds to a nondegenerate
ˆgs (~r, t) is the phasematch
parametric a
N A ) is manifested not by the increased emission rate of The the Raman first term pscattered of⌦Eq. photon, 2.35 includes but by the barestate a
can generate a twomode entangled state between the field p(~r, 1t;
andw, thew1) collective ' g ⇤ p atomic NA
w
Effective parametric mode (~r,t)
w w via squeezing
the increased
operation ˆD
quantum ⇣ correlation R between field ⌘ 1 and collective excitation (section 2.4).
gs
is the effective paramet
parametric coupling matterlight constant: interaction, denoted
i 1
= exp
~
dt 0 Ĥ (par)
population loss of
The first term of Eq. 02.35 includes int
(t the 0 ˆgg due to optical pumping (⇠ i~⌦
) barestate (section atomic 2.4). enhancement ( p as
N A ) is manifested not by the increased emissio
*Note that the collective enhancement ( ) is manifested Hamiltonian, light shift (⇠ ~⌦ w 2
later not two by ⇣the effects increased (opticalemission
pumping ), and the
w and light shift), as th
rate of the Raman scattered photon, but by
population loss of ˆgg due to optical pumping (⇠ i~⌦ the the w increased 2
quantum
egĤ correlations between field
1 and collective
2 ). (par)
int For (t) our =~ experiments, we can neglect the
1 and collective excitation
w saturation intensity
p(t)Ê1Ŝ I + ⇤ sat with p(t)Ê a typical 1Ŝ†⌘
† ,
saturation param
2.3.2 later two Threedimensional effects (optical pumpingtheory and light of shift), spontaneous as the intensity The first Raman Iexcitation w
term for the ofscattering
write Eq. 2.35
limit). laser includes
Theis second well below the barestate
term, however, the atomic H
corresponds
saturation intensity I sat with a typical saturation parameter population s ⌘ I w loss
parametric
/I sat of⌘ ˆgg 2⌦due matterlight w  2 / toeg 2 optical ⌧
interaction,
10 pumping 4 (weak (⇠ i~⌦ w 2
eg
can generate Generation a twomode of matterlight entangledentanglement
state between the field 1 and the collective atomic2mode
denoted as w
Here, we derive a threedimensional quantum theory of spontaneous Raman scattering by expanding the
excitation limit). The second operation term, however, ˆD
⇣ R ⌘
corresponds
i 1
= exp later to a two nondegenerate effects (optical parametric pumping amplification. and light This shift), as the inten
~
dt 0 Ĥ (par)
0 int
(t 0 ) (section 2.4).
equations of motions in terms of the HermiteGaussian modes with mode indices (l, m). Under ˆ(par) certain⇣
ˆ
⇣ p
with the squeezing parameter given by p (z,t; Ĥ (par)
⌦
w, w1) ' g ⇤ p NA
w (z,t)
w gs
. Additionally, we
Twomode squeezed The initial state atomfield as quantum state g int
(t) =~ p(t)Ê1Ŝ resource 29 for DLCZ + ⇤ p(t)Ê 1Ŝ†⌘ † ,
2.4 Twomode a , 0 1
i in the Schrödinger’sw
picture evolves protocol
 i
R squeezed state as a quantu a
⇣
1
R ⌘
simplicity
2.4 Twomode
p (t 0 )= ˆD 1 L
i 1
L= 0exp
dz p(z,t
squeezed 0 ).
state as a quantum resource for D
can generate Generation a twomode ~ dtFor 0 Ĥ (par) a rigorous
0 int
(t 0 ) withtreatment the parametric of dissipation interactionand Hamiltonian propagation ef Ĥ(par int
of matterlight entangledentanglement
needs to solve the
state between the field 1 and the collective atomic mod
section selfconsistent 2.3.1.3.
operation ˆD
The HeisenbergLangevin ⇣col
final R ⌘
i 1
atomfield state equations (t !1) isingiven Eqs. by2.47–2.48, a twomodefrom squeezed whic
= exp
~
dt 0 Ĥ (par)
0 int
(t 0 ) (section 2.4).
correlationcol
functions could be evaluated from Einstein’s relations 143 .
•!
The initial atomfield  state i g a , 0 1
i in the Schrödinger’s pictur
2.3.2 Threedimensional ⇣
1 a = p 1 ⇠ X Here, we make several further remarks:
⇠ n/2 n
R theory of spontaneous ⌘
1
,n a i,
Raman scattering
The initial atomfield state g a , 0 1
i in the i 1
ˆD = exp Schrödinger’s picture evolves to  i a 1
via the
⇣ R ⌘ ~
dt 0 Ĥ (par) Our beloved EPR pairs
0 int
(t 0 ) with the parametric interactio
1. The meaniphoton ˆD = exp Here, where 1
we number derive threedimensional quantum theory of spontaneous Raman scattering
~
dt 0 n
Ĥ (par)
0 int section (t 0 1
i (n in a
field i) are1 the is given numberstates by n
) with the parametric 1 = for photons ⇠ (â
2.3.1.3. The final atomfield interaction state Hamiltonian (t !1) † ) n 1 a h ⇣ ˆn 1  i 0 1
i Ĥ(par) (collective ex
is given int by (Eq. a
equations in field 1 of(atomic motions ensemble), in terms of and the⇠ HermiteGaussian = tanh 2 i R 1 a = ⇠
1 ⇠ (= sinh(i R 1
d
0
1 R
dt 0 modes 1 L
Thus, the excitation probability ⇠ =
0 L 0 with mode indices (l,
section 2.3.1.3. The final atomfield state (t !1) is given by a twomode dz p(z,t )⌘
n
1+n
follows the familiar thermal distribution. 0 ⌧When
1 is th
squeezed state
circumstance, with the squeezing we showparameter that the 3D given theory by reduces an effective 1D model of a nondeg
amplifier between a singlemode (l, m) in field 1 and a single collective atomic mode (l, m
 i 1 a = p 1 ⇠ X  i 1 a = p ⌦
1 ⇠ X p (z,t; w, w1) ' g ⇤ p NA
w
is traced over, Entangled the remaining atomic counterpart Twomode issqueezing equivalent parameter to a thermal state (z,t)
where
w w
⇠ n/2 gs
. Add the
23 R n
simplicity p (t 0 )= 1 L
L 0
dz p(z,t 0 ). For a rigorous
⇠ n/2 treatment of dissipation and p
"
exhibits superPoissonian spinwave statistics, g
!
(2) (⌧) •! Quantum = h:ˆn a(t)ˆn a (t+⌧):i
correlation
n
hˆn a i
=2(for ⌧ =0).
1
,n 2 a i,
Evolution of the atomlight state:
needs to solve the selfconsistent HeisenbergLangevin
in the photon number
equations
basis
in Eqs. 2.47–2.4
2. For multiple
where n
2.3.2.1 Propagation equations 1
i (n a i) are the numberstates for the photons ⇠ (â
of quantum fields and collective atomic variables
†
correlation
ensembles
functions
and fields
could
1 (with
be evaluated
the ensemble 0 photon
from Einstein’s
field 1 system0 atom
relations
labeled ⇣143 in field 1 (atomic ensemble), and ⇠
We start by deriving the equation of motion for theRfield Ê ~ = tanh 2 i R .
by ↵ 2 {a, b
1 R
dt 0 1 L
1 photon 1 atom
the overall state after Here, the weparametric make several Raman furtherinteraction remarks: is ideally
1 + (~r,  i t) traveling 0 along L ~ 0
k 1 k ẑ in
envelope approximation 143,170 . The wave0 equation
dt 0 1 L
L for 0 Ê dz ~ 1 (~r, p(z,t
t) = 0 Ê
)⌘
dz
tot = Q ↵  i(↵) 1a, where
p P n/2
2 photon 2
with the squeezing parameter given by p (z,t; ~ atom
1 ⇠↵ ⇠
1 + (~r, ⌧ t)e w, 1 iw is w1) 1t the' + Ê ~ exc g
↵ n
1 (~r p
1
,n a i ↵ .
1. The mean photon number R
in a nearresonant simplicity atomic medium p (t 0 in
)= is given 1 field L 1
L 0 by dz is given by
p(z,t 0 n
). 1 =
For
1 a h ˆn
a rigorous 1  i 1 a = ⇠
w w gs
. Additional treatmen
1 ⇠ (=
3. In the ideal case, the Thus, conditional the excitation atomic probability state•!
upon Heralded ⇠ = a photoelectric n single follows excitation the detection familiar source of thermal a single distribu field
where n 1
i (n a i) are the numberstates for the photons ⇠ (â † ) n 0 1
i (collective excitatio
in field 1 (atomic ensemble), and ⇠ = tanh 2 ⇣ i R 1
with the squeezing parameter given by p (z,t; w, w1) ' g p
p
NA
⌦ ⇤ w (z,t)
simplicity
p (t 0 )= 1 L
R L
0
needs dz p(z,t to solve 0 ). For thea selfconsistent rigorous treatment HeisenbergLangevin of dissipation and propaga equatio
on the mode â traced over, the remaining
[@ t 2 •! atomic
c 2 Synchronization
r ~ 2 counterpart
] E ~ is of
1 (~r, t) = 1 equivalent independent
1,↵ is given by ˆ⇢ c = Tr 1 (â † @ t 2 P
✏ ~ a thermal 1,↵â1,↵ ˆ⇢ 1 a), where the initial
(~r,
atomphoton
t).
sta
needs to solve the selfconsistent correlation HeisenbergLangevin functions could single
evaluated
photon
equations from
source
in 0 Eqs. Einstein’s 2.47–2.48, relation
exhibits superPoissonian spinwave statistics, g (2) (⌧) = h:ˆn a(t)ˆn a (t+⌧):i fro
projection by â † hˆn a i
=2(f
1,↵â1,↵ is ˆ⇢ 2
1 a =  i (↵)
1ah .
•! Mediated nonlocal
correlation functions could beHere, evaluated we make fromseveral Einstein’s further relations remarks: 143 interaction among
Figure 2.2: Generating and retrieving collective excitations to photons. atomic a, Generating systems via and storing . single
collective excitations. A weak write
2.
pulse
For
illuminates
multiple ensembles
the cold atomic
andsample, fields
generating
1 (with the
a Raman
ensemble
scattered
field 1 system labeled
4. The mode operators can be transformed nonlocally to â
photon, called Here, field 1. we Themake detection several of a single further 1. photon The remarks:
0 1,↵ = P unitary transformation
↵
U
in
the overall statemean field 1 heralds
after the photon the
parametric number generation
Raman in of a
interaction field correlated 1 0 ↵,↵ 0â 1,↵
0 where U ↵,↵
issingle
given ideally by n 1 i = Q 0
a h 
1+n
...
Twomode squeezed state: correlated 29
2.4 Twomode squeezed photon state pairs as a quantu
2.4 Twomode squeezed 4. Thestate mode operators as a quantum can be transformed resource nonlocally for to â 0 1,↵ D=
Kuzmich, Bowen, Boozer, Boca, col Chou, Duan, Kimble, Nature 423,
a unitary transformation
30 731 (2003).
of the mode operators â 1,↵ 0. A pho
“Atomic memory for correlated photon states”
col
Laboratory of Physics 1233, California Institute of Technology, Pasadena, California 91125, USA
port signi cant improvements in the retrieval efficiency of a single excitation stored in an
ensemble and in the subsequent generation of strongly correlated pairs of photons. A 50%
lity of transforming the stored excitation into one photon in a wellde ned spatiotemporal
t the output of the ensemble is demonstrated. These improvements are illustrated by the
ion of highquality heralded single photons with a suppression of the twophoton component
% of the value for a coherent state. A broad characterization of our system is performed for
t parameters in order to provide input for the future design of realistic quantum networks.
p
Van der Wal, ⌘2 âEisaman, 2 + p 1 Andre, ⌘ 2ˆv 2
Walsworth, and a state mode Phillips, label â 0 Zibrov,
2, respectively. Here, we account for the ret
1,↵ leadsLukin, to anScience effective 301, interaction 196 (2003). among the ↵ 0 syst
The initial atomfield state g a , 0 1
i in the Schrödinger’s pictur
loss in the propagation and the detection of field ⌘
The initial atomfield state g a , 0 1
i in the Schrödinger’s dt 0 Ĥ (par) Our beloved 2 with EPR a transmission pairs efficiency ⌘ 2
In section 2.5, we show picture evolves to  i a 1
via the
int
(t 0 that, ) with after the a delay parametric ⌧, the collective interactio ex
transformation, where ˆv 2 is a vacuum mode operator k . Thus, ideally, we can transfer the
section another 2.3.1.3. withquantum the The parametric final field, atomfield called interaction 2 state (with, Hamiltonian (t ideally, !1) unit is Ĥ(par) probability) given int by (Eq. a
state between an ensemble and field 1 to an equivalent state between fields 1 and 2,
Transfer
section 2.3.1.3. The final atomfield
2.2b),
state
with to lightlight
(t
the
!1)
darkstate entanglement
is given
polariton
by a 86 ˆ
twomode d(z,t) =cos✓(t)Ê2(z,t)
squeezed state
evolution. When the atomic
 i 1 a = p 1 ⇠ X  i 1 a = p 1 ⇠ X ⇠ n/2 n
 i 1 a 7!  i 1 2
= p state is traced
1 over, the matterlight t
⇠ X
⇠ n/2 n 1
,n 2
i.
the collective operators Ŝ and
⇠ n/2 the state label, a, (indicating the a
n 1
,n a i,
nt for long distance quantum communication is the ability to efficiently interface atoms and light.
ralded storage of light in atomic systems is essential for guaranteeing the scalability of protocols
m entanglement over large distances, such as in the quantum repeater scheme
⇣
[1]. RIn 2001, a
ards the realization of a quantum repeater was the proposal by Duan, Lukin, Cirac, i 1
•! Evolution of the atomlight state: ˆD = exp and Zoller
native design involving atomic ensembles, ⇣ linear optics, and single photon detectors [2]. The
is roadmap is a large ensemble of identicalR ⌘ ~ 0
i atoms 1 with a Λtype level con guration as sketched
write pulse induces
ˆD spontaneous = exp Raman scattering of a photon in eld 1, transferring an atom
the initially empty
Quantum
js〉 ground state.
correlation ~
dt 0 Ĥ (par)
0 int
(t 0 )
For a low enough write power, such that two excitations
tate are unlikely in tothe occur, photon the detection number of the basis eld1 photon heralds the storage of a single
ributed among the whole ensemble. A classical read pulse can later, after a userde ned delay,
excitation into another 0 photon photonic mode ( eld0 2). atom These scattering events are collectively enhanced
tom interference effect and can result in a high signaltonoise ratio [3]. By following this line,
tions [4, 5, 6, 7] and 1 photon entanglement [8] have been 1 atom observed between pairs of photons emitted by
mble. By combining the output of two different ensembles, as originally suggested in the DLCZ
ntanglement between two remote ensembles has been recently demonstrated [9], paving the way for
2 photon 2 atom
mentations of DLCZ schemes. A posteriori (probabilistic) polarization entanglement between two
as also been demonstrated recently [10], which does not lead to scalable capabilities for quantum
In practice, we where control n the 1
i (n excitation a i) are parameter the numberstates ⇠ = tanh 2 for (i R the photons ⇠ (â †
⇣ 1
dt
where n 1
i (n a i) are the numberstates field 1 (atomic forensemble), the photons ⇠ (â † ) n
⇣ and ⇠ = 0 tanh 1
i 2 (collective i R 0
0 p(t 0 )) with t
1 R
Laurat et al. Opt. Express 14, 6912 dt (2005). 0 excitatio
1 L
modify the spinwave statistics. For ⇠
1000
in field 1 (atomic ensemble), and ⇠ = tanh 2 i R 1, the two modes contain significant 0 L continu 0
1 R L
0 dz p(z,t )⌘
dz
glement, whereaswith in the theregime squeezing of weak parameter excitationgiven ⇠ ⌧by 1, the 0 p (z,t; twomode ⌧w, 1 is squeezed w1) the' exc g p s
...
projection by â † 1,↵â1,↵ is ˆ⇢ 1 a =  i (↵)
1ah .
“Generation of nonclassical photon pairs for scalable quantum communication with atomic ensemble”
dt 0 1 0 L
100
R p
with the squeezing parameter simplicity
⌦
given by p (z,t; w, w1) ' g ⇤ p NA
w (z,t)
p (t 0 )= 1 L
L 0
dz p(z,t 0 ). For a rigorous
w w gs
. Additional treatmen
R 10
simplicitynonclassical p (t 0 )= 1 correlations, L
as demonstrated experimentally in refs.
L Violation 0 of CauchySchwarz inequality (a)
72,73 , and be used a
needs to solve for quantum the selfconsistent information correlation processing HeisenbergLangevin functions andcould communication 1
evaluated equations 4 . Here, from in Eqs. weEinstein’s calculate 2.47–2.48, various relation froi
strong quantum correlations in the numberstate basis. The field 2 and the field 1 can, in
g 12
needs dz p(z,t to solve 0 ). For thea selfconsistent rigorous treatment HeisenbergLangevin of dissipation and propaga equatio
Here, we make several 0.75 further remarks:
correlation
between
functions
the fields
could
1
be
and
evaluated
2, and obtain
from
important
Einstein’s
benchmark
relations
(b)
143 parameters
.
(used througho
characterize our experiments.
Manifestly quantum correlation 0.50
Here, we make several further 1. The remarks: mean photon number in field 1 is given by n 1 = a h
c
Twomode squeezed state: heralded 29
2.4 Twomode squeezed single photons state as a quantu
2.4 Twomode squeezed 4. Thestate mode operators as a quantum can be transformed resource nonlocally for to â 0 1,↵ D=
“SinglePhoton Generation from col Stored Excitation in Atomic
a unitary transformation
30 Ensemble”
of the mode operators â 1,↵ 0. A pho
Chou, Polyakov, Kuzmich, Kimble, Phys. Rev. Lett. 92, 213601 (2004).
col
Laboratory of Physics 1233, California Institute of Technology, Pasadena, California 91125, USA
port signi cant improvements in the retrieval efficiency of a single excitation stored in an
ensemble and in the subsequent generation of strongly correlated pairs of photons. A 50%
lity of transforming the stored excitation into one photon in a wellde ned spatiotemporal
t the output of the ensemble is demonstrated. These improvements are illustrated by the
ion of highquality heralded single photons with a suppression of the twophoton component
% of the value for a coherent state. A broad characterization of our system is performed for
t parameters in order to provide input for the future design of realistic quantum networks.
p
⌘2 â 2 + p 1 ⌘ 2ˆv 2 and a state modelabel â 0 2, respectively. Here, we account for the ret
1,↵ leads to an effective interaction among the ↵ 0 syst
The initial atomfield state g a , 0 1
i in the Schrödinger’s pictur
loss in the propagation and the detection of field
The initial atomfield state g a , 0 1
i in the Schrödinger’s dt 0 Ĥ (par) Our beloved 2⌘with EPR a transmission pairs efficiency ⌘ 2
In section 2.5, we show picture evolves to  i a 1
via the
int
(t 0 that, ) with after the a delay parametric ⌧, the collective interactio ex
transformation, where ˆv 2 is a vacuum mode operator k . Thus, ideally, we can transfer the
section another 2.3.1.3. withquantum the The parametric final field, atomfield called interaction 2 state (with, Hamiltonian (t ideally, !1) unit is Ĥ(par) probability) given int by (Eq. a
state between an ensemble and field 1 to an equivalent state between fields 1 and 2,
Transfer
section 2.3.1.3. The final atomfield
2.2b),
state
with to lightlight
(t
the
!1)
darkstate entanglement
is given
polariton
by a 86 ˆ
twomode d(z,t) =cos✓(t)Ê2(z,t)
squeezed state
evolution. When the atomic
 i 1 a = p 1 ⇠ X  i 1 a = p 1 ⇠ X ⇠ n/2 n
 i 1 a 7!  i 1 2
= p state is traced
1 over, the matterlight t
⇠ X
⇠ n/2 n 1
,n 2
i.
the collective operators Ŝ and
⇠ n/2 the state label, a, (indicating the a
n 1
,n a i,
nt for long distance quantum communication is the ability to efficiently interface atoms and light.
ralded storage of light in atomic systems is essential for guaranteeing the scalability of protocols
m entanglement over large distances, such as in the quantum repeater scheme
⇣
[1]. RIn 2001, a
ards the realization of a quantum repeater was the proposal by Duan, Lukin, Cirac, i 1
•! Evolution of the atomlight state: ˆD = exp and Zoller
native design involving atomic ensembles, ⇣ linear optics, and single photon detectors [2]. The
is roadmap is a large ensemble of identicalR ⌘ ~ 0
i atoms 1 with a Λtype level con guration as sketched
write pulse induces
ˆD spontaneous = exp Raman
~
scattering dt 0 ofĤ (par)
0
a photon int
(t 0 in )
eld 1, transferring an atom
the initially empty js〉 ground state. For a low enough write power, such that two excitations
tate are unlikely to occur, the detection of the eld1 photon heralds the storage of a single
ributed among the whole ensemble. A classical read pulse can later, after a userde ned delay,
excitation into another photonic mode ( eld 2). These scattering events are collectively enhanced
tom interference effect Heralded and cansingle result inexcitation a high signaltonoise source ratio [3]. By following this line,
tions [4, 5,
Detection
6, 7] and entanglement
of a single
[8]
photon
have been
in
observed
field 1
between
projects
pairs of photons emitted by
mble. By combining the output of two different ensembles, as originally suggested in the DLCZ
ntanglement the between atomic twostate remoteto ensembles single has excitation. been recently demonstrated [9], paving the way for
mentations Subsequent of DLCZ schemes. retrieval A posteriori maps (probabilistic) the single polarization excitation entanglement between two
as also been demonstrated recently [10], which does not lead to scalable capabilities for quantum
projection by â † 1,↵â1,↵ is ˆ⇢ 1 a =  i (↵)
1ah .
where n 1
i (n a i) 50% are retrieval the numberstates efficiency into a
In practice, we control the excitation parameter ⇠ = tanh 2 for single
(i R the photons ⇠ (â †
⇣ 1mode dt
where n 1
i (n a i) are the numberstates field 1 (atomic forensemble), the photons ⇠ (â † ) n
⇣ and ⇠ = 0 tanh 1
i 2 (collective i R 0 fiber
0 p(t 0 )) with t
Laurat et al. Opt. Express 14, 6912 (2005). 1 R
dt 0 excitatio
1 L
modify the spinwave statistics. For ⇠
in field 1 (atomic ensemble), and ⇠ = tanh 2 i R 11, the two modes contain significant 0 L continu 0
1 R L
0 dz p(z,t )⌘
dz
glement, whereaswith in the theregime squeezing of weak parameter excitationgiven ⇠ ⌧by 1, the 0 p (z,t; twomode ⌧w, 1 is squeezed w1) the' exc g p s
to single photon in a triggered fashion
dt 0 1 0 L
R L
0.10
p ⌦
with the squeezing parameter given by p (z,t; w, w1) ' g ⇤ p NA
w
strong quantum correlations simplicity in (z,t)
p the (t 0 )= numberstate 1 L
dz basis. p(z,t The 0 ). field For2a and rigorous the
w w gs
. field Additional treatmen 1 can, in
R
simplicitynonclassical p (t 0 )= 1 correlations, L
needs
L 0
dz p(z,t to as solve 0 ). demonstrated For thea selfconsistent rigorous experimentally treatment HeisenbergLangevin of in dissipation refs. 72,73 , and andbepropaga
equatio used a
Photon antibunching
needs to solve for quantum the selfconsistent information correlation processing HeisenbergLangevin functions and0.01
could communication evaluated equations 4 . Here, from in Eqs. weEinstein’s calculate 2.47–2.48, various relation froi
w
“Good source of heralded single photons”
correlation
between
functions
the fields
could
1
be
and Here, evaluated
2, and we make obtain
fromseveral important
Einstein’s further benchmark
relations remarks: 143 parameters
.
(used througho
10 100
characterize our experiments.
Manifestly singlephoton statistics
Here, we make several further 1. The remarks: mean photon number in field
g 12 1 is given by n 1 = a h
Synchronization NATURE PHYSICS DOI: of 10.1038/NPHYS1152 independent singlephoton sources
Conditional control of heralded singlephoton sources
Felinto et al. Nature Phys. 2, 844 (2006).
Synchronized Independent NarrowBand Single Photons and Efficient Generation of Photonic Entanglement
Yuan, Chen, Chen, Zhao, Koch, Strassel, Zhao, Zhu, Schmiedmayer, Pan, Phys. Rev. Lett. 98, 180503 (2007)
Synchronizing heralded single photons
outputs of the fiber beam splitter are connected to detectors
2 D28and D3. Electronic pulses from the detectors are gated
derived 31 .
with 120 ns (D1) and 100 ns (D2 and D3) windows
Perpendicular
and the Table 1  Measured values of ↵, measured efficiency ⌘ and centered on times determined by theHongOuMandel
write read light
Parallel
2
n. Within intrinsic efficiency ⌘ int (see the Methods section). pulses, respectively. Subsequently, the electronic pulses
interference
thermally
from D1, D2, and D3 are fed into a timeinterval analyzer
lized with
T (ms) ↵ ⌘(%) ⌘ int (%) which 6 records photoelectric detection events with a 2 ns
e formula
time resolution.
0.0012 0.02±0.01 6.3 25
The transfer of atomic excitation to the detected idler
(7,5) ms 1 0.12±0.04 2.8 11 field at either Dk (k 2, 3) is given by a linear optics
model of 4 0.17±0.07 1.3 5
p
p
relation ^a k i =2 ^A 1 i =2 ^k , where ^a
ogeneous
k
16 0.10±0.10 1.1 4.5 depends 4 parametrically on and 1 corresponds to a mode
here ⇠ is
with an associated temporal envelope t , normalized so
effective
that R 1
0
dtj t j 2 1, and ^k is a bosonic operator
sulting in
which accounts for coupling to degrees of freedom other
than those detected. The efficiency i =2 is the probability
that a single atomic excitation stored for results in a
optically Table FIG. 2  1 Measured (color online). values Schematic of g (2)
2
D
(0), of experimental measuredsetup, deterministic with
visibility singlephoton the inset showing source the efficiency atomic level ✏scheme and intrinsic (see text). source photoelectric event at Dk, and includes the effects of idler
lock state efficiency ✏ int (see the Methods section).
retrieval and propagation losses, 0symmetric beam splitter
[1]. The scattered light with polarization orthogonal to the (factor of 1=2) and nonunit detector ñ4 0efficiency. We ñ2 0start
±2 ⇡ 0. In
0
20 40
dephasing 0t p (ms) write pulse is collected g (2)
by a singlemode fiber and directed from the elementary probability density Q
0
kj1 t c for a
D (0) ✏ (%) ✏ int (%)
τ (ns)
a damped 0 onto a singlephoton 5 detector 10 D1, with overall 15 propagation 20 count at time t c and no other counts in the interval 0;t c ,
4 0.06±0.04
and detection efficiency s. N 1.9 8
Starting with the correlated Q kj1 t c j t c j 2 h: ^a y k ^a R
k exp
tc
pping by
0
dtj t j 2 ^a y k ^a k Conditional
:i [28].
control of
5 state of signal 0±0.06 field and atomic excitation, 1.6 we project out6
Using Eq. (1), we then calculate probability p
ed in the
R10
the vacuum from the state produced by the write pulse
Figure 3 Conditionalheralded kj1
dtQ joint detection singlephoton probability p c kj1 t that detector Dk registers at least one photoelectric
detection event.
22
(τ ) of sources
ith Fig. 2,
recording events in
Zhao using thet projection al. Nature operator Physics : ^1 e ^d y 5, ^d : , 100 where(2009)
^d
Figure 2 Probabilities 11 and p 1122 of coincidence detection as functions of the
both DWe 2a and
similarly
D 2b Felinto , once
calculate
the
the et twoproba
bility p 23j1 of at least one photoelectric event occurring at
along the
al. ensembles Nature are Phys. ready to 2, 844 re, as(2006).
a function of the
p p
s ^as 1 s
^s, ^a s is the detected signal mode, and
ic motion give the measured values of ↵, the main results of this paper,
number N of trials waited between the independent preparation of the two
time difference τ between the two detections. The lled squares (open circles)
^ both detectors. These probabilities are given by
is a bosonic operator accounting for degrees of freedom
p 1122
(×10 7 )
Deterministic single photon source via conditional control
Matsukevich, Jenkins, Kennedy, Chapman, Kuzmich, Phys. Rev. Lett. 97, 013601 (2006).
optically pumped atoms in an optical lattice. a, Diamonds, U 0 = 80 µK; circles,
with T c = 7.2±0.25 ms (blue) and T c = 5.0±0.1 ms (red) 31 . b, Shorttime oscillations
id line fit gives B
ARTICLES
0 = 0.43 G and p 0 ⇡ 0.85 (see the Methods
PRL 97, 013601 (2006)
PHYSICAL section), for U 0 = REVIEW 60 µK. LETTERS week ending
7 JULY 2006
, U 0 = 60 µK. Error bars Deterministic represent ±1 standard single deviationphoton based photoelectron source
p 11
(×10 5 )
p 22
(×10 4 )
c
Collective atomlight interaction
Coherent energy exchange
between the two oscillators
Collective atomic oscillator
+
Signal mode
Coupled quantum dynamics
(i) Atomic system driven by N A &level atoms dressed by applied laser fields
(ii) Optical response of the coherent atomic medium.
Closely related to classical coherence phenomena of dark resonances
Coherent population trapping (CPT)
Stimulated Raman adiabatic transfer (STIRAP)
•!
Darkstate polariton
Halfphoton and halfmatter quasiparticle excitation
•! Quantum description of lowlight level
Electromagnetically Induced Transparency
Optimal control theory
Optimizing mapping efficiencies of quantum states
c
c
signal field
twophoton transition detuning gi ei), of called each of thethese signal atoms field, is coupled whereastothe a slowlyvarying transition si quantized ei is resonantl radiation
by
Collective
the Hamiltonian Ês. The signal field
atomlight Ês propagates within the EIT window of the ensemble, provided by ⌦
interaction
c (z,t).
classical twophoton control Ĥ(map) s
field
in
detuning of
the
Rabi
rotatingwave Here, we consider
), called frequency the signal ⌦
approximation a collection
c (with field, detuning
(following of ⇤level
whereas the c). transition The
the atoms
dynamics
effective interacting
si ei of this
onedime with
2.5.1 Interaction Hamiltonian and formation of dark c). The states dynamics of this system is described
resonant system
proximation by thein classical Hamiltonian sectioncontrol 2.3.2.2 Ĥ(map) field and
s of inneglecting transition gi
Rabi the rotatingwave frequency the transverse ei of each
⌦ c approximation (with detuning profiles), these atoms
(following with is coupled to a slowlytwophoton
detuning ), called the signal field, whereas the tran
2.5.1 Interaction s in the Hamiltonian rotatingwave approximation and formation(following of dark states the effective
c). The
onedimensional
dynamics the effective of this
ap
Here, we consider a collection of ⇤level atoms interacting with the two singlemode optical fields. The onedim syste
transition by the Hamiltonian Ĥ(map) s in the rotatingwave approximation (following the effective oned
Here, gi weei proximation
consider of eachaof collection these in section atoms of Z 2.3.2.2
⇤level is coupled and
atoms neglecting a slowlyvarying interacting
the transverse profiles), with
XN A ⇣ withquantized the two singlemode radiation mode
⌘ optical Ês (with fields. The
proximation Ĥ (map)
classical
in section 2.3.2.2 and neglecting the transverse profiles), with
s = dw~wâ
Z
† control field of Rabi frequency
wâ w + ~w esˆ(i)
N
ss + ~w ˆ(i) ⌦ c (with detuning
twophoton
transition
detuning
gi
),
c).
ei Zcalled of each
the
of
signal
these
field,
atoms
whereas
is coupled
the
to
transition
a slowlyvarying
si ei
X A ⇣
quantized
is resonantly eg ggradiation driven
⌘
mode
by a
XN A ⇣
⌘
Ês (with
classical Ĥ s (map) by the Hamiltonian
= dw~wâ Z
† i=1
twophoton detuning ), called the signal field, whereas wâ w + the transition Ĥ(map) s in the rotatingwave
XN A
~w esˆ(i)
⇣ ss + ~w ˆ(i) approximation (fo
Ĥ control field of Rabi frequency ⌦ c (with detuning c). The dynamics of this system is described
s (map) = dw~wâ † wâ w +
esˆ(i)
ss + ~w ˆ(i)
eg gg si ei is eg resonantly gg ⌘ driven by a
Ĥ s (map) XN A ⇣
= dw~wâ † wâ w
i=1 + ~w esˆ(i)
ss + ~w
~⌦ ˆ(i)
cˆ(i)
eg gg
XN A ⇣es e i(kk c z i w c t) + d ˆ(i)
eg
i=1 eg ~✏ a · ~Ê
⌘
by the Hamiltonian
classical control Ĥ(map) s in the rotatingwave i=1approximation proximation(following sectionthe 2.3.2.2 effective andonedimensional neglecting
s + e i(k the
sz i wap
proximation
transverse
s t) profil
field of Rabi frequency ⌦ c (with detuning c). The dynamics of this system is described + h.c. ,
Consider a collection
in section
of Λlevel
2.3.2.2
XN atoms A ⇣
and neglecting
interacting
XN ~⌦ A ⇣ cˆ(i)
es e i(kk c z i w c t) + d ˆ(i)
eg
~⌦ cˆ(i)
es e i(kk c z i w c t) eg ~✏ a · ~Ê
⌘
i=1
the
with
transverse
the two
profiles),
singlemode
with
optical fields.
The dynamics + + d ˆ(i) s
eg eg ~✏ a · ~Ê
e i(k sz i w
by the is described Hamiltonian by Ĥ(map)
s
s ~⌦ in cˆ(i)
theesrotatingwave 1D e i(kk c
approximation z i w c t) approximation + d(with ˆ(i)
t) eg eg RWA) ~✏ a (following · ~Ê
⌘
{z the effective onedimensional+ aph.cproximation
in section 2.3.2.2 and neglecting  the transverse (map)
s + e i(k Zsz i w s t) + h.c. XN , A ⇣ (2.68)
} ⌘,
s e i(k sz i w s
i=1 Ĥ t) + h.c.
i=1
 Z
s =
Ĥ (map) dw~wâ
profiles), with int {z † wâ w + ~w esˆ(i)
ss + ~w
XN A ⇣ {z ⌘
}
}
Ĥ
i=1
 Ĥ (map)
i=1
s (map) = dw~wâ † wâ w + ~w esˆ(i)
ss + Ĥ (map) ~w ˆ(i)
eg gg
{z }
int
int
Ĥ (map)
int
where Ê ~ q
s
+ ~w
= i s
RZ
i=1
where Ê ~ q
+ 2✏ 0 V dwâw
where Ê ~ ~wq
s = i s
R e iwz/c XN A ⇣
⌘ XN A ⇣
Ĥ ~✏ s is the positive frequency component of the signal field, and
s (map) = dw~wâ † wâ w + ~w esˆ(i)
s
+ ~w
= i s
R
2✏ 0 V dwâw e iwz/c ss + ~w ˆ(i) ~⌦ cˆ(i)
N
eg
es e i(kk c z i w c
q
X
t) + d ˆ(i)
A ⇣
eg eg
gg
~⌦ cˆ(i)
es e i(kk c z i w c t) + d
~✏
2✏ 0 V dwâw e iwz/c s is ˆ(i)
eg
i=1 the eg ~✏ a ·
positive ~Ê
⌘
s + e i(k sz i w
~w s
= i
frequency i=1
t) s
R
+ h.c.
component of the signal field, a
is the longitudinal projection of the wavevector ~✏ s is along the positive ẑ (also, frequency k c
? =  ~ , (2.68)
2✏ 0 V dwâw e iwz/c ~✏ s is the positive frequency component of the  signal field, and kc k =
kcomponent c · of the signal field,
is the longitudinal projection of the wavevector along ẑ (also, k
is the longitudinal projection of the wavevector along ẑ (also,
c
? =
k c
? 
= ~ (ˆx, ŷ) ' 0). We ~ k
assum
{z
i=1
c·ẑ
 XN A ⇣
{z
k c
 ~ · (ˆx, ŷ) ' 0). We ass
signal field propagates along the
~⌦ cˆ(i)
quantization es e i(kk c z i w c t) axis
+ dẑ eg of ˆ(i)
eg the
~✏ a ·
system ~Ê
} ⌘
=  ~ k
s + e i(k sz i w s
(section t) Ĥ (map)
int
Ĥ (map)
c · (ˆx, ŷ) ' 0).
+
We
2.3.2.2).
h.c.
assumed
k c · (ˆx, ,
that
ŷ) '(2.68)
the
int
0). We as
signal
signal
field propagates i=1
field propagates  along
along
the quantization
the quantization
axis {z axis
ẑ of
ẑ
the
of the
system
system
(section
(section
2.3.2.2). } 2.3.2.2).
A simple explanation for the formation where ~ q
Ê + ~w
s = of idarkstate s
R
where ~ q
Ê
Positive + frequency ~w component of signal field:
polariton is the existence of a fami
Ĥ
A simple A simple explanation explanation for the for formation the formation of (map) 2✏
int 0 V dwâw e iwz/c ~✏ s is the positive frequency co
s = i s
R
2✏ 0 V dwâw e iwz/c ~✏ s is the positive frequency component of the signal field, and kc darkstate of darkstate polariton polaritonis is the k = ~ k
A simple explanation for the formation of darkstate polariton is the existence of a family theexistence c·ẑ
existence of dark of ofaafaf
is the longitudinal eigenstates D, mi for the interaction is the Hamiltonian longitudinal
Existence of a family eigenstates of dark eigenstates D, mi D, for mithe forinteraction the interaction Hamiltonian Hamiltonian Ĥ(map) projection
int
(ref. of 87 ). the Ĥ(map) int Ĥ(map) (ref.
int
(ref. 87 In wavevector particular, along singleexci ẑ (also, k
). 87 ). In In particular, the thesingleex
where Ê
D, m ~ projection q of the wavevector along ẑ (also, k c
? c
? = 
s =1i + ~w
= i s
R ~ k c · (ˆx, ŷ) ' 0). We assumed that the
is 2✏ (ref.
D, m D, =1i 87 0 V dwâw e
)
iwz/c ~✏ s the signal positive (ref. 87 ). In particular, the singleexcitation
fieldfrequency propagates component along the of the quantization signal field, axis and ẑk k
m is =1i (ref. is 87 c of = the ~ state
signal field propagates along the quantization axis ẑ of the system (section 2.3.2.2).
k c·ẑ system
(ref. )
87 )
A simple is the longitudinal explanationprojection for the
Singleexcitation
formation of the wavevector of darkstate
darkstate along ẑ (also, k c
? = 
D, 1i
D,
= Acos simple polariton
polariton
1i D, =
✓1i d cos
(t)g explanation is the existence
= cos a (t)g
,
✓1 d s (t)g
i ~ k c · (ˆx, ŷ) ' 0). We assumed that the
1
sin for
a , i1 s
✓the of
isin d (t)s formation a family of
sin ✓ d (t)s ✓ a d
,(t)s 0 s i, ofdark
darkstate polar
a , 0 a , s 0i,
s i,
eigenstates signal D, field mipropagates for the interaction D, along 1i = the Hamiltonian cos quantization ✓ d (t)g axis ẑ of the system (section 2.3.2.2).
eigenstates Ĥ(map) a , 1 s i
int
(ref. sin
D, 87 ). ✓ d In (t)s particular, a , 0 s i, the singleexcitation state (2.69)
mi for the interaction
p p p
qHamiltonian qq Ĥ(map) int
(ref. 87
D, m =1i where Ais simple (ref. where tan ✓ d = g d NA /⌦ c defines the mixing angle, g d = id
ws
Mixing tan where 87 angle: ✓)
explanation d tan = g✓ d eg 2~✏ 0 V (~✏ = NA for
g d
/⌦ the NA c
formation defines /⌦ c defines the of darkstate mixing the mixing angle, polariton
angle, g d
is = the
g d
id existence ws
Coupling constant: = eg
of
id
ws a
eg 2~✏ eg · ~✏ s ) is 2~✏ 0 V (~✏ 0 V (~✏ family
eg · ~✏ of
eg s ·) dark
D, m =1i is (ref. 87 )
~✏ is s ) the is the si
eigenstates D, mi for the D, interaction 1i = cos ✓Hamiltonian Coherent superposition Ĥ(map) int
(ref. 87 2~✏ d (t)g a , 1 s i sin ✓ d (t)s a , 0 s ). i, In
0 V (~✏ eg · ~✏ s ) is the single atomparticular,
the singleexcitation (2.69) state
state
D, 1i = cos ✓ d (t)g a , 1 s i sin ✓ d
D, m =1ibetween is p(ref. 87 )
q
singlephoton excitation and single collective excitation
where tan ✓ d = g d NA /⌦ c defines the mixing angle, g d = id
ws
eg 2~✏ p 0 V (~✏ eg · ~✏ s ) is the single atomclassical
control field of Rabi frequency ⌦ c (with detuning
by the Hamiltonian Ĥ(map)
proximation in section 2.3.2.2 and neglecting the transverse profiles), with
where ~ Ê + s
is the longitudinal projection of the wavevector along ẑ (also, k ? c
signal field propagates along the quantization axis ẑ of the system (section 2.3.2.2).
eigenstates D, mi for the interaction Hamiltonian Ĥ(map)
D, m =1i is (ref. 87 )
where tan ✓ d = g d
p
NA /⌦ c defines the mixing angle, g d = id eg
q
ws
+ i c)ˆge + Zi⌦ Lc
e i(k c
kIn Following the
Collective
s )z the adiabatic
ˆgs + ig d Êcondition s (ˆgg ˆee )+( where ˆF
erning 0 =
HeisenbergLangevin ge + i
atomlight
⌦c
ge , ⇠ tc
⌧
approach (Eq. 2.29), we obtain
interaction
ge ˜d0 (2.73) (L) with resonant optical depth given by
a set of differential equations governing
the atomic evolutions (assuming 86,190,191 ( @ t⌦ c
Ĥ (map)
R
the z
c.
atomic evolutions (assuming weak signal field approximation g d ⌧ ⌦ c and n s ⌧ N A )
s = dzn A (z){~ cˆee (z,t) ~ ˆss (z,t)
In the 0 adiabatic
0 dz0 2g 2 d n A(z 0 )z
), we perturbatively
condition
expand Eq. 2.76 to the order of @
gec
⌦
h
weak signal field approximation gi
d ⌧ ⌦ c and n s ⌧ N A )
for the quantum field
• HeisenbergLangevin ~g d Ê Ês(z,t)
c
⇠ 1 t Ô ⇠
t c
⌧ ge ˜d0 (L) with resonant optical depth given by Ô/ ˜d 0 (z) t c, and we o
=
R z
s (z,t)e
@ equation ik insz an effective
ˆeg (z,t)+~⌦ onedimension
of motion for the continuum atomic collective variables
tˆse = ( se + i( c (z,t)e
c w ik cz (Eq.
ˆes (z,t)+h.c. 2.39),
0 dz0 2g 2 d n A(z lowestorder 0 )z perturbation ˆgs '
g dÊs
), we perturbatively expand Eq. ⌦2.76 c
e
to i(kk c
k s
the order )z . Thus, we obtain
of @
gec t Ô}, ⇠
gs ) i )ˆse + i⌦ c e i(kk c
k Ô/ s )z t the adiabatic equation of motio
c, (2.70) and we obtain the
(ˆss ˆee )+ig d Ê sˆge + ˆF se
lowestorder  quantum
perturbation
field Ês(z,t) {z }
ˆgs '
g dÊs
⌦
@ tˆse = ( se + i( c w c
e i(kk c
k s )z . Thus, we obtain
gs ) i )ˆse + i⌦ c e i(kk c
k s
@ )z the adiabatic equation
(ˆss ˆee )+ig d Ê sˆge + ˆF se (2.71)
tˆgs = gsˆgs + i⌦ ⇤ ce i(kk c
k s )z ˆge ig d Ê sˆes + ˆF
of motion ! for the
Ĥ (map)
int
(@ gs
t + quantum c@ z ) Ês(z,t) field Ês(z,t) =ig d n A (z)Lˆge (z,t).(@ t + c@ z ) Ês(z,t) ' g2 d n A(z)L @ Ê s (z,t)
@ tˆgs = gsˆgs + i⌦ ⇤ ce i(kk c
k s )z ˆge ig d Ê sˆes + ˆF
(2.74)
gs (2.72)
@ tˆge = ( ge + i c)ˆge + i⌦ c e i(kk c
k s )z s the linear atomic density ( R (*)
dzn ˆgs + ig d Ê s (ˆgg ˆee )+ ˆF A (z) (@ t =N + c@ z A )).
Ês(z,t) ' g2 d n ⌦ ⇤ !
.
A(z)L @ Ê s (z,t)
c(z,t) @t ⌦ c (z,t)
ge ,
⌦
@ tˆge = ( ge + i c)ˆge + i⌦ c e i(kk c
k s )z ⇤ . (2.77)
We note that the characteristic pulse widths c(z,t) t
Langevin the HeisenbergLangevin operators for theapproach atomic operators (Eq. 2.29), ˆµ⌫ we , as obtain described aˆgs set+ of in c
@t ' 10 ⌦ c ns (z,t) of the control
igdifferential d Ê s (ˆgg ˆee equations )+ ˆF laser (or the read laser) in o
section 2.3. ge govic
evolutions (assuming
, (2.73)
We note thatiments the characteristic are on thepulse samewidths
order of t
and a propagation
weak signal
equation for the quantum c magnitude ' 10 ns of the as
g
field Ês(z,t) the control adiabatic laser (or criteria the read 1/ laser) t
in an effective onedimension c ' in ge our ˜d0 experiments
are on
(L), where the
(Eq. 2.39),
Weak field approximation d ⌧ ⌦ c and n s ⌧ N A )
and a propagationtransmission the same order
equation for(absent of magnitude
the quantum the control as the
field Ês(z,t) laser) adiabatic isindefined criteria
an effective as1/ T
ximation and adiabatic condition
0 t c
onedimension = ' e ˜d ge 0 ˜d0 (L) (L), . Thus, where
(Eq. instead the resonant
2.39), of the simplifi
Coupled dynamics between matter and light
transmission
=
•
(
Propagation equation (absent the
equation (Eq. control
for the 2.77), laser)
signal wefield
numerically is defined as
(@ t + c@ z ) Ês(z,t) =ig d n A (z)Lˆge (z,t).
se + i( c w gs ) i )ˆse + i⌦ c e i(kk c
k s )z solve T 0 = the e coupled ˜d 0 (L) . Thus, differential instead
37 (ˆss ˆee )+ig d Ê sˆge + ˆF
equations of the simplified of motions wave(Eqs. 2.71
se (2.71)
oximationequation with g d (Eq. ( chapter gg 2.77), ' 1we 6. numerically ee,
(@
ss, solve es '
t + c@ z ) Ês(z,t) the0) coupled and with differential negligible equations spinwave of motions (Eqs. 2.71–2.74) in
=ig d n A (z)Lˆge (z,t). (2.74)
=
2.72)
gsˆgs +
and
i⌦ ⇤ obtain
ce i(kk
Here, ˆF
cthek s coupled )z equations
ˆge ig d Ê
of
sˆes +
motions ˆF ⇣ (by substituting ⌘
ˆge into Eq. 2.74,
gs (2.72)
and using Eq. 2.73)
e interaction Adiabatic chapter time condition 6. t c , we
µ⌫ are
approximate
the quantum
ˆge
Langevin
= i
operators
e i(kk c
k
From Eq. (*),
s
for )z /⌦
the ⇤ catomic @ tˆgs
operators
(Eq.
ˆµ⌫ , as described in section
= Here, ( ˆF ge µ⌫
+ are i the c)ˆge 2.5.2.2
quantum + i⌦ c Coherent
Langevin e i(kk c
k s )z operators
ˆgs + ig d for Ê s (ˆgg the atomic
ˆee )+ operators ˆF ge ,
ˆµ⌫ , as described (2.73)
(@ t + c@ z ) Ês(z,t) atomic medium
' g dn A and (z)LEIT
in section 2.3.
⌦ ⇤ c(z,t) ei(kk c
k s )z @ tˆgs (2.75)
2.5.2.2 Coherent atomic medium and EIT
2.5.2.1In Eq. Weak 2.77, field we recover approximation the usual wave and adiabatic condition
ion2.5.2.1 equationWeak for thefield quantum approximation field Ês(z,t) g d Ê s
equation with slowlight phenomena in static EIT (with stati
ˆgs
and in adiabatic an'
effective e
condition onedimension i(kk c
k s )z
In Eq. 2.77, we recover the usual wave equation with slowlight phenomena (Eq. in2.39),
static EIT (with static control
field ⌦ ⌦ c
In the weak signal c (z,t) =⌦
field c ) with modified group velocity v
approximation with g d ( gg ' 1 g = c cos
ee, 2 ✓ d . Furthermore, if there is very lit
field ⌦ c (z,t) =⌦ c ) with modified group velocity v g = c cos 2 ✓ d ss, es ' 0) and with negligible sp
0
⇣
⌘
In the weak signal dephasing field approximation gs ' 0 over the withinteraction g d ( gg ' time 1 t c
ee,
, wess, approximate ' 0) and with negligible spinwave
⇣ˆge = i e i(kk c
k
(@
s
⌘
)z /⌦ ⇤ t + c@ z ) Ês(z,t) =ig d n A (z)Lˆge ⌦(z,t). c  2 @ 1. Furthermore,
tˆgs
⌦ c  2 @2 t ˆgs +i e if there i(kk c
k s )z is very little population
ˆss and ˆse , the control field ⌦ c (z,t) ' ⌦ ⌦ c (2.74)
c @ t
polarization vectors for the lation atomic indipole ˆss and (gi ˆse ei , the transition) control and fieldthe⌦ signal c (z,t) field. ' ⌦ c (t z/c) propagatesˆFge according , to (2.76) the freespa
 c (t z/c) propagates according to the freespace wave
{z }
dephasing gs '
equation
0 over the
((@
interaction t + c@ z ) ⌦ c (z,t)
time
=0).
t c , In
approximate
this case, we obtain
ˆge =
a
i
wave
e
equation i(kk c
k
(Nonadiabatic terms)
s
equation ((@ )z /⌦
with ⇤
t + c@ z ) ⌦ c (z,t) =0). In this case, we obtain a wave equation with variable groupc
velocity @
variable
tˆgs (Eq.
group
v
the quantum v g (z,t);
where Langevin namely, g (z,t); namely, wave equation with variable
0 = operators ge + i for the atomic operators✓ group velocity
ˆµ⌫ , as described in section 2.3.
c. ✓ @ @
In the adiabatic condition 86,190,191 ( @ t⌦
@t + v c g(z,t) @ ◆
⌦ c
⇠ 1 Ê
t c
@z ⌧@t + s (z,t) v g(z,t) @ ◆
=0,
Ê s (z,t) =0,
(2.78)
ge ˜d0 (L) with @z resonant optical depth given by ˜d 0 (z) =
field approximation R z and adiabatic condition
0 dz0 2g 2
l d
~✏ n A(z 0 )z
where the group eg
where
gec and velocity ~✏ the s
), are we group the vperturbatively respective polarization expandvectors Eq. 2.76 for the toatomic the order dipole of (gi @ t Ô ei ⇠ transition) Ô/ t c, and and we signal obtainfield.
the
g (z,t) velocity =c cos v 2 g (z,t) ✓ d (z,t) =c is dynamically cos 2 ✓ d (z,t) controlled is dynamically by thecontrolled Rabi frequency by the ⌦ c
Rabi (z,t) frequency
k
ed =0. line) with field ⌦a c
control (z,t) =⌦ laser s c ) with Rabi modified frequency group ⌦ c velocity / medium ge = v g
1, = (black as c cos well line) ✓ d
as . Furthermore, with for ⌦ c bare =0. ifatomic
there Dynamic is verycontrol little populationClassical
ˆss and ˆse , the control phenomena
of the group v
s(⌦ line) c , with )). =0) b, ⌦allows Real c =0. part shapepreserving Dynamic of the Im( susceptibility ˆss control s )
and
' 0acceleration/deceleration ˆse of
at
, the function =0.
group control Re( =0) velocity field s (⌦ allows ⌦ c ,(z,t) (i.e., )). shapepreserving ofv We '
field ⌦ c (z,t) ' ⌦ c (t of z/c) dark
g
the ⌦= show c
signal (t the z/c) field dispersions c
acceleration/deceleration propagates in the ofaccording the to the freespa
1+(w
propagates according resonances
eg )dn/d
atpresence transparency of the sign
to the freespace wave
shapepreserving edium Im( (red s ) line) ' 0acceleration/deceleration with=0.
a control laser Rabi
equation ((@ t + c@ z ) ⌦
of frequency
c (z,t)
the signal Im( ⌦
=0).
field c / ge = 1, as well as for the bare atomic
s ) '
In this
in 0 at the =0.
case,
presence
we obtain
of transparency susceptibility s of th
a wavec
um equation with variable group
of=0.
the (black signal equation line) field with ((@ in⌦ t c a+ =0. homogeneous c@ z ) ⌦Dynamic c (z,t) =0). control In this of the case, group we obtain velocity a wave (i.e., equation v g =
1+(w with eg variable )dn/dgroup at velocity
) allows shapepreserving v
v g (z,t); namely, g (z,t); acceleration/deceleration
susceptibility
EIT medium (defined as P(z,t) =✏ 0 s E s (z,t)) for a
namely, s of the
wave equation of
signal
with the variable signal
field
field
in a
✓ group in
homogeneous
the velocity presence of
EIT
transparency
medium resonant (defined control as P(z,t) field
field ✓ @ @
@t + v g(z,t) @ ◆
Ê s =0, (2.78)
@z@t + v g(z,t) @ ◆
s) ' 0susceptibility ( at c =0) =0. is given s ofby the
resonant
(refs. signal 94,143 field
control
) in a
field
homogeneous
( susceptibility EIT s medium of the signal (defined fieldas in P(z,t) a homogeneous =✏ 0 s EITE s (z,t)) medium for(d
a
c =0) is given by (refs.
Ê 94,143 s (z,t)
)
=0,
s of the signal field in a homogeneous EIT medium (defined as P(z,t) =✏
resonant control field (@z 0
c =0) s E s (z,t)) for a
resonant control field ( c =0) is given by (refs. 94,143 )
is given by (refs. 94,143 )
ptibility l field ( c s of =0) theissignal given
where the group where field by (refs.
velocity the in agroup homogeneous 94,143 )
s v g
= 2g2 d
(z,t) velocity N A
=c cos vEIT 2 g (z,t) medium
✓ d (z,t) =c is dynamically cos (defined 2 ✓ d (z,t) as P(z,t)
controlled is dynamically =✏
by 0
the s Econtrolled Rabi s (z,t)) for
frequency by a the ⌦ c
Rabi (z,t) frequency
w
s, (2.79)
s
ant control ⌦
of field the ( control c =0) oflaser. the is given control Here, by the laser. (refs. mixing 94,143 Here, angle ) theismixing given by angle cos is ✓ d given = p by c
⌦
cos ✓ for constant density n g
2 d = p c s = 2g2 d N A
for constant A densi
Susceptibility of the signal field in a homogeneous EIT medium for a resonant
d
Ncontrol A +⌦ 2 field
c g
2
d
N A +⌦ w
s,
2
c s
s = 2g2 d N A
s = 2g2 d N s = 2g2 d N A
A
w
s,
s
N A /L. N A /L. w
s, w
s, (2.79)
s (2.79)
2 2 i s
ge
is the normalized susceptibility function
s = 2g2 d N and P(z,t) = p where s = ⌦c  2 2
where N Aˆge is the atomic
s = ⌦c  2 2 Ai where ge
is the normalized susceptibility function
We now briefly We turn nowtobriefly a moreturn classic to w asituation more s, ;
(2.79)
classic encountered
s = ⌦c 
situation 2 in 2 EIT iencountered ge
is the normalized susceptibility function
where (see also in chapter EIT (see 6). For alsoa chapter resonant
control nant field with controlc field =0, with the EIT medium behaves as a nonabsorbing dispersive media within the
6). Fo
c 2 2 i ge
is s =
the normalized susceptibility function and P(z,t) = p ⌦c  2 2 i ge
is normalized susceptibility function and P(z,t) = p polarization. and P(z,t) Im( = p
s (
( s (⌦ c , )) describes thepolarization. transparency for the signal field = 0 with the transmis N
N
polarization. Im(
c =0, the EIT medium s (⌦ Aˆge is the atomic Aˆge is the atomic
⇣
Im( s (⌦ c , ))
⌘
describes the transparency for thesion signal given field byat T (⌦=
c ,
behaves a nonabsorbing dispersive media s = ⌦c transparency  2 2 i ge
is normalized susceptibility function and P(z,t) = p )) describes ⇣ the transparency ⌘
c ,
for the sign
⇣
m( ⌦ c , polarization.
s (⌦ )=exp( c , )) describes kIm( s LIm( the s (⌦ sion transparency s c ,)))) = describes exp for ˜d0 the the Im( signal transparency s ) field (Fig. at for 2.4a), = the0whereas signal with Nthe field Re(
transparency window givenwindow by ⌦ c at given the twophoton sion given
by ⌦ resonance by T (⌦ c , =0shown )=exp( Aˆge
transmis at = 0 with the transmission
, )=exp( given index by kT n ss (⌦ LIm( ( c ), = )=exp( p given by T (⌦ c , )=exp( k s LIm( s )) = exp ˜d0 Im( s ) (Fig. 2.4a), w
⇣ ⌘ ⇣ ⌘
s (⌦ c , ))
the atomic
inkFig. s LIm( 2.4. s )) The = adiabatic exp ˜d0 Im(
c at the twophoton resonance =0shown in Fig. 2.4. The s )
ization. T e (⌦ refractive
Im(
approximation s (⌦ c , )) describes the transparency for 38 the signal field at = 0 with the transmisgiven
the refractive aby T
approximation
in section 2.5.2.1 section
in essence
2.5.2.1 ⇣ compares contributes
in essence
the ⌘
topulse the refractive
compares
bandwidth index
the pulse
to the n
bandwidth
EIT s ( window = p c contributes s )) = 1+Re( ˜d0 ) (Fig. 2.4a), whereas s (⌦ c , ))
to
⌦1+Re( the c .
EIT
If the s ) for the
window ⌦
1.0
0.50
pulse
(⌦ c , index
bandwidth
)=exp( n s ( ) = p tok s the LIm( s ) refractive for s )) the= exp ˜d0 Im( s ) (Fig. 2.4a), whereas Re( s (⌦ c , ))
contributes to the refractive
w
k s s '
LIm( 1+Re( index
2⇡/ s
t
))
c '
= n s s )
⌦
exp ( for ) = p signal index 38 field n s ( )(group = p contributes to the ref
velocity 1+Re( given s ) for ; by the signal field
c
v g =
(gro
1+(w eg )dn/d )
1.0
the
c , higherorder ˜d0 bIm( signal 1+Re(
dispersion s ) field (Fig. c (group
must
2.4a),
be taken
whereas
into
Re(
account. s (⌦ c
Specifically,
, ))
the
pulse bandwidth w s ' 2⇡/ t c ' ⌦ c , higherorder dispersion must be taken into account. Specifi
ibutes to the refractive index n s ( ) = p v g =
s ) for 0.50
1+(w eg )dn/d ) (Fig. the velocity signalgiven field by (group velocity given by
n/d ) (Fig. 2.4b)m a.
c
v g =
b
1+(w eg )dn/d ) (Fig. 2.4b)m .
⇣ 2.4b)m . ⌘
Perturbatively exp
0.8
)dn/d ) (Fig. 2.4b)m c
v g = .
1+(w 1+Re( s ) for0.25
eg )dn/d ) (Fig. ⇣
0.8
expanding s (Eq. 2.79) around 2.4b)m .
Perturbatively the signal ⇣ field 0.25
⌦c
⌧ 1, we find s ' 2g2 d N A
w s expanding ⌦ c  (group + i 2 eg
Perturbatively expanding 2 ⌦ s
velocity (Eq. c 
+ 2.79) O( given ⌘ around by
⇣
3 s (Eq. 2.79) around 4 ⌦c
⌧ 1, we find ) , s '
⌦c
⌧ 2g2 d
1, N A
where the linear we find
c0.6
ely expanding
1+(w eg )dn/d ) s (Fig. (Eq. 2.79) 2.4b)m around .
where linear dispersion
⇣
gives v g = c cos 2 ⌦c
⌧ 1, we find s ' 2g2 d N w
A
0.00
w s ⌦ 
+ i 2 s
dispe ⌦⌘
c 
0.6
eg
Perturbatively expanding 2 ⌦ c 
+ O( 3 s (Eq. 2.79) around ) ,
4
⌦c
⌧ 1, we find s ' 2g2 d N A
w s ⌦ c  ✓ d
+
⌘
. In i 2 eg
dispersion gives v 2 ⌦addition, c 
+ O( we 3 g = c cos where 2 ✓ d . the In addition, linear dispersion we findgives the bandwidth v g = c cos find ) ,
0.00 of 2 ✓the d . In EIT addition, mediumweT via find ' exp( the 4 bandwidth 2 / w o
0.4
erturbatively r dispersion expanding gives v s (Eq. 2.79) around ⌦c
⌧ 1, we find s ' 2g2 d N A
s ⌦ c 
+ i 2 g = c cos 2 EIT
2
0.4
✓ d . In addition, we find the bandwidth eg
w where the linear dispersion gives v T ' exp( 2 of
/ w 2 the 2 ⌦ c 
+ O( 3 ) ,
EIT ), where EIT medium the EIT 4 bandwidth via
is w EIT = ⌦ c
egp 2
g = c cos 2 ✓ d . In addition, we find the bandwidth of the EIT medium via
EIT 2 ), where the EIT bandwidth is w EIT = ⌦ c
T ' exp( 2 / w p 2
0.25
0.2
˜d
the wEIT 2 linear ), where dispersion the EIT gives bandwidth v g = c cos is 2 ✓ w EIT 2 ), where . This the EIT leads bandwidth to adiabatic is w
d . In= addition, ⌦ pc 2
0.25
EIT condition, = ⌦ c
where we . This findleads the initial the to adiabatic
signal pulse’s thecondition,
T ' exp( 2 0.2
/ w 2 bandwidth EIT medium wvia
EIT ), where the EIT bandwidth eg ˜d0
where p 2
.
eg ˜d0 is w EIT = ⌦ pc 2
eg ˜d0
the This initial leadssign
to a
signal pulse’s bandwidthwhere w . This leads to an adiabatic s must becondition,
smaller exp( 2 0.0
5
/ wEIT 2 4 ), 3 where 0.0
s the must initial be smaller signal pulse’s than the bandwidth
wof 2 the 1 EIT 0 bandwidth 1 2 3 is 4 w 0.50
al signal pulse’s bandwidth w 5
EIT = ⌦ pc 5 2
0.50 s must EIT be smaller medium
eg ˜d0
wthan EIT : i.e., the bandwidth w s <
5 4 3 s must 2 1be 0smaller 1 2 than 3 the 4 bandwidth 54 3 2 1 0 1 2 3 4 5
EIT : i.e., . This leads of 5
tothe 4 EIT 3
adiabatic medium
2 1 0 1 2 3 4 5
where the initial signal pulse’s bandwidth w condition,
< w EIT . In addition, the adiabatic w passage of the s darkstate must
eg ˜d0 bew polariton smaller s < w EIT than 192 EIT : i.e., w s < w EIT . In addition, the adiabatic . Insets the addition, passage abandwidth limit the to ofadiabatic thepassage EIT medium of the d
rotation the darkstate speed polarito of the m
s the < initial w EIT signal . In addition, pulse’s the bandwidth adiabatic passage w s mustofbe the
rotation smaller darkstate
speed thanpolariton the of the bandwidth mixing 192 sets
angle of a limit the✓ d
EIT to the
of the medium
the mixing w polariton ˆ d(z,t) n EIT : angle i.e., ✓w s < w EIT . In addition, . Intr
of
T: the i.e., mixing w s < angle w EIT
✓. d In ofaddition, the polariton the adiabatic ˆ the adiabatic passage of the darkstate polariton 192 d of the rotation polariton speed ˆ d(z,t) of the n mixing . Introducing angle ✓a characteristic timescale t sets a limit to the
d(z,t) passage n d of the polariton ˆ d(z,t) n . Introducing c , a charact
Electromagnetically induced transparency . Introducing
we of obtain the darkstate a characteristic
the criteria polariton timescale
t 192 c > eg sets v g a limit to the
Figure 2.4: Susceptibility
Harris, Phys. Today, 50, 36 (1997)
on speed of the mixing angle ✓ d of the polariton
Lukin & Fleishhauer, ˆ g 2
d(z,t)
Phys. n d N Ac for adiabatic t c ,
rotation speed Figure following 86 . Finall
teria t s of EIT medium. a, Imaginary part of the susceptibility function,
riteria Im( t s c (⌦> c , )). eg v g
of
g 2 b, Real part of the susceptibility function
the presence . Re( Introducing a characteristic timescale t
Rev. s (⌦
of c ,
Lett. Zeeman
)). We
84, 5094 population
show the dispersions
(2000) and assumed
of the c ,
d an ideal ⇤level syst
EIT medium N Ac for the 2.4: adiabatic mixing Susceptibility following angle ✓ d of the polariton ˆ Slowlight propagation we obtain the criteria
d(z,t) n . Introducing a characteristic timescale t 86 s of EIT medium. a, Imaginary part of the susceptibility function,
c > eg v g
c ,
gd 2N Ac Im( for s (⌦ adiabatic we obtain
c , )). b, Real following the criteria 86 . Finally, t c > I eg v
note
g
that we have so far neglected
part of . the Finally, susceptibility I note gd 2N Ac
function that for we adiabatic Re( have so far following 86 . Finally,
neglected the presence I noteof that Zeema we
we obtain the (redcriteria line) witht c a > control eg v g
2 laserfor Rabi adiabatic frequencyfollowing ⌦ c / ge = 86 s (⌦ c , )). We show the dispersions of the
EIT medium the(red presence line) with of Zeeman a control population laser Rabi frequency and assumed . 1, Finally, ⌦ c well / ge an as I= note for ideal 1, the asthat ⇤level well bare we as atomic have for system. thesobare far Inatomic
fact, neglected the d
eeman population and assumed an ideal ⇤level system. In fact, the distribution of Zeeman
+
+
Origin of transparency – Heuristic approach
ElectromagneticallyInduced Transparency
'()
'(#
'(%
signal
c
(0)
'('
!" !# !$ !% !& ' & % $ # "
38
KramersKronig relations
Anomaly in absorption spectrum is accompanied
by an anomaly in the dispersion
38
! &('
+
+
" '("'
amplitudes of optical transitions in atomic
'(*
medium '(%" can lead to strong modifications of
"
'("'
+
Strong
absorption at
resonance
'(''
Quantum interference effects in the
its optical properties.
Quantum interference in the transition amplitudes leads
!'(%"
to strong modifications of its optical properties.
!'("'
!" !# !$ !% !& ' & % $ # "
Figure 2.4: Susceptibility s of EIT medium. a, Imaginary part of the susceptibility function,
'(%"
Im( s (⌦ c , )). b, Real part of the susceptibility function Re( s (⌦ c , )). We show the dispersions of the
EIT medium (red line) with a control laser Rabi frequency ⌦ c / ge = 1, as well as for the bare atomic
'(''
medium (black c
c
line) with ⌦ c =0. Dynamic control of the group velocity (i.e., v g =
1+(w eg )dn/d
at
=0) allows shapepreserving acceleration/deceleration of the signal field in the presence of transparency
!'(%"
Im( s ) ' 0 at =0.
"
!'("'
!" !# !$ !% !& ' & % $ # "
susceptibility s of the signal field in a homogeneous EIT medium (defined as P(z,t) =✏ 0 s E s (z,t)) for a
0
resonant control field ( c =0) is given by (refs. 94,143 )
Chanelière et al. Nature 438, 833 (2005)
. a, Imaginary part of the susceptibility function, Harris, Phys. Today, 50, 36 (1997)
Lukin & Fleishhauer, s = 2g2 d N A
function Re( s (⌦ c , )). We show the dispersions of the Phys. Rev. Lett. 84, 5094 (2000)
w
s, (2.79)
s
From Nature 438, 833 (2005)
As As discoveredby by Fleischhauer 2.5.3 and Lukin Darkstate
Darkstate polariton: 86 , we can polariton equivalently introduce aanew newset set of of slowlight polaritonic
{ ˆ Ĥ (map)
int
ˆ
ˆ d(z,t) = cos ✓ d (t)Ês(z,t) Quantum d(z,t) = cos ✓
of the sin ✓ d (t)Ŝ(z,t) approach
d (t)Ês(z,t) sin ✓
excitations { ˆ d d(z,t), ˆ approximation. Namely, we have
here ˆ R
d(z,t) = 2⇡ b(z,t)} as the normal modes of the system (Eqs.
As discovered by Fleischhauer and Lukin 86 (Eqs. 2.75–2.76) in inthethe weak weak (t)Ŝ(z,t) signal (2.80)
L dk ˆ d,k (t)e ikz and ˆ R
ˆ d(z,t) = cos b(z,t) ✓ = 2⇡ L dk ˆ b,k (t)e ikz d (t)Ês(z,t) sin ✓ d (t)Ŝ(z,t) .
(2.80)
signal
R
, we can equivalently introduce a new set o
dwâw approximation. e iwz/c approximation.
Namely,
Namely, ˆ
we
we ˆhave
ˆˆ ~✏
A new set of s is the positive frequency b(z,t)
slowlight polaritonic excitations
component = { ˆ sin✓of the d(z,t), ˆ signal b(z,t)} as
field, the
and normal
k k d (t)Ês(z,t) b(z,t)
+ d(z,t) =
cos ✓ d (t)Ŝ(z,t),
= cossin✓ ✓
c = modes ~ d (t)Ês(z,t) + cos ✓ d
k (t)Ŝ(z,t),
d (t)Ês(z,t) sin ✓
In the adiabatic limit, where ⌦ b(z,t) d
c·ẑ
(t)Ŝ(z,t)
cˆgs +g = d Êsin✓ s e i(kk c
k s )z ' 0 (Eq. 2.76), the brightstate polariton(2.81)
is ˆ d (t)Ês(z,t) + cos ✓ d (t)Ŝ(z,t), b '
of ofthe the system (Eqs. 2.75–2.7
ojection of the wavevector along ẑ approximation. (also, k c
?  ~ ˆ b(z,t) = sin✓ d (t)Ês(z,t) + cos ✓ d
n this limit, we can write the equation k c Namely, · (ˆx, ŷ) we ' have 0). We assumed that the
where Namely,
d(z,t) cos d (t)Ês(z,t) sin ✓ d (t)Ŝ(z,t) (2.80)
s along the quantization Ŝ(z,t) =p ˆ
N
axis A e
ẑ of i(kk c
where k s )z Ŝ(z,t) of motions
ˆgs is =p for the
the Nslowlyvarying A e i(kk c
k s
darkstate polariton ˆ (t)Ŝ(z,t),
d(z,t) = cos ✓ d (t)Ês(z,t) )z sin ✓ ˆgs phasematched (z,t) d (t)Ŝ(z,t) d with the perturbatio
where is the slowlyvarying collective spin operator, phasematched (2.80)
and co
where
p the system (section 2.3.2.2).
✓ d = arctan(g d NA /⌦ c ) is✓ ˆ Ŝ(z,t) p =p N A e i(kk c
k s )z ˆgs (z,t) is the slowlyvarying phasematched collective spin oper
ˆ(z,t) Ŝ(z,t) =p N A e i(kk c
k s )z ˆgs is the phasematched collective spin operator, and
'
g d Ê s
⌦ d the b(z,t) = b(z,t) mixing arctan(g angle. d
sin✓ NA d (t)Ês(z,t)
p c
(Eq.
p
✓ 2.77) as (ref. 86 d = arctan(g d NA /⌦ c ) is the mixing ), angle. These operators
✓ d = arctan(g
sin✓ d
These d /⌦
(t)Ês(z,t)
NA c ) operators is + arecos known ✓
ˆ + cosare d (t)Ŝ(z,t), as
✓ d(z,t) = d (t)Ŝ(z,t), known angle. the darkstate
asThese the darkstate operators (brightstate)
/⌦ c ) is the mixing angle. These operators are known (brightstate)
as are
cos ✓ d (t)Ês(z,t) sin (t)Ŝ(z,t)
the (2.81) known darkstate as (brig th
(2.81)
ation for polaritons the formation ˆ of darkstate polariton is the existence of a family of dark
d(z,t)
polaritons
r the interaction ˆ b(z,t) = sin✓ d (t)Ês(z,t) + cos ✓ d
where Ŝ(z,t) Hamiltonian ˆ ( ˆ b(z,t)),
d(z,t) ( ˆ in direct analogy polaritons
b(z,t)), polaritons in direct =p N Ĥ(map)
(t)Ŝ(z,t),
A e i(kk int
(ref. 87 analogy ˆ d(z,t) ˆ d(z,t)
with ( ˆ ( the ˆ b(z,t)), in direct analogy with the classic dark (bright) states di = cos ✓
classic in dark (bright) states withdi the= classic cos ✓ d
dark gi (bright) sin ✓ d sistates
d gi
✓
with the classic
◆
dark (bright) states di = cos ✓ d gi sin ✓ d si
Ŝ(z,t) =p N A e i(kk c
k s )z (bi where =sin✓ ˆgs (z,t) ). (bi In =sin✓ is is particular, the d gi slowlyvarying + costhe ✓ d si) singleexcitation observed phasematched coherent population state
collective trapping spin (chapter operator, operator. 6). These and polariton
d gi + cos ✓ d si) observed
c
k s )z in coherent @
(bi =sin✓ d gi + pcos ✓ d si) (bi observed =sin✓ ˆgs (z,t) d coherent gi is the + cos slowlyvarying population ✓ d si) observed trapping phasematched in (chapter coherentcollective 6). population These spin polaritons trapping operator, follow (chapter and 6
) ✓ d = Singleexcitation arctan(g d p NA /⌦ darkstate c ) is thepolariton
mixing @t + v population
@
ˆ
g
the quasibosonic commutation d(z,t)
trapping
=0.
(chapter 6). These polaritons follow
(2.8
angle. @z
relations
These operators are 86 (with the help of Eq. 2.25),
the quasibosonic commutation relations 86 (with the help of Eq. 2.25),
the ✓ d = quasibosonic arctan(g d NA
commutation /⌦ c ) is the mixing
where
relations Ŝ(z,t) 86 Bosonic
known
the quasibosonic angle. =p (withcommutation These operators
N
the
A e
help i(kk c
kof s
are known as the darkstate (brightstate)
)z Eq. relations 2.25),
86 quasipartite
as the darkstate
excitation
(brightstate)
(with the help of Eq. 2.25),
D, polaritons 1i =
ˆgs (z,t) is the slowlyvarying phasematched collec
polaritons ˆ cos ˆ d(z,t) ✓ d (t)g ( ˆ h
d(z,t) ( ˆ b(z,t)),
a , 1 s i insin direct ✓ d (t)s analogy a 37 , 0 with s i, the classic ˆ d,k (t),
dark (2.69)
b(z,t)), in direct analogy p with the classic dark ˆ i h † (bright) states di = cos ✓ d gi sin ✓ d si
d,k
(t 0 ) ' ˆ 0 b,k (t),
(bright) states ˆ i
†
b,k
(t 0 ) '
di 0 kk 0
h (t t 0 ),
hus, in the = cos ✓ d gi sin ✓ d si
(bi
adiabatic
=sin✓ h ✓ d = arctan(g q d NA /⌦ c ) is the mixing angle. These operators are known as the d
2.72) and obtain the coupled d gi
regime,
+ cos ✓
equations d
the
si)
darkstate
observed
of motions (by substituting
N A /⌦ c
(bi defines =sin✓ the d gi mixing + cosangle, ✓ d si) ˆ observed
polaritons g d = id ˆ eg
in coherent ws
d(z,t) ( ˆ population trapping (chapter 6). These polaritons follow
2~✏ 0 V b(z,t)), (~✏ ˆge into Eq. 2.74, and using Eq. 2.73)
d,k (t), ˆ in coherent
polariton
i h population h ˆ ˆ d,k (t), ˆ i h †
d(z,t) †
d,k
(t 0 ) ' ˆ 0 b,k (t),
eg · trapping
follows
i ˆ
~✏ †
in s b,k ) direct 0 is 0 the ) '
analogy single kk 0 (t
with atom t 0 d,k (t), ˆ (chapter ithehusual 6). These
wave †
d,k
(t 0 ) ' ˆ 0 b,k ),(t), ˆ polaritons
equation i follow
as in fre
d,k
(t 0 ) ' ˆ 0 b,k (t), ˆ i
†
where †
b,k
(t 0 ) ' (2.82)
the classic dark 0 kk 0 (t t
Adiabatic quasibosonic regime commutation relations 86 (with ˆ b,k
(t 0 ) '
R 0 kk 0 (t t 0 ), (2.82)
d(z,t) =
the 2⇡ L dk ˆ
help of
d,k (t)e
Eq. 2.25),
ikz and ˆ R
b(z,t) = 2⇡ L dk ˆ b,k (t)e ikz .
pace with the group velocity v g = c cos 2 ✓ d determined by the ‘amount’ of (bright) states di
the quasibosonic commutation relations 86 (with the help of Eq. 2.25),
(@ t + c@ z ) Ês(z,t) (bi ' =sin✓ g In the adiabatic limit, where ⌦ cˆgs +g d Absence Ê s e
dn A (z)L
i(kk c
k s
the photonic component (sign
of )z brightstate ' 0 (Eq. 2.76), polariton the brightstate polariton is
d gi + cos ✓ d si) observed in coherent population trapping (chapter 6). T
⌦
h ⇤ c(z,t) ei(kk c
k
eld Ês; where
i.e., ˆ R
d(z,t)
cos s )z @ tˆgs (2.75)
where ˆ 2 =
✓ d ) in 2⇡ L dk ˆ d,k (t)e
the Rpolariton ikz
d(z,t) = 2⇡ h
L dk ˆ and
d(z,t).
ˆ R
b(z,t) = 2⇡ (t)e ikz and ˆ R
where ˆ L dk ˆ
R b,k (t)e ikz .
d(z,t)
the quasibosonic commutation relations 86 (with the help of Eq. 2.25),
ˆ d,k (t), ˆ i b(z,t) = 2⇡ =
h 2⇡ L dk ˆ
L dk ˆ b,k (t)e †
d,k
(t 0 ) ' ˆ 0 b,k (t), ˆ i
ikz d,k (t)e ikz and ˆ R
ˆ . b(z,t) = 2⇡ L dk b,k (t)e ikz d,k (t), ˆ † In thisilimit, hwe can write
.
d,k
(t 0 ) ' ˆ 0 b,k (t), ˆ † the equation i Dissipationless of motions for the unitary darkstate transformation
polariton ˆ
b,k 0 ) ' 0 kk 0 (t t 0 d with the pert
In the adiabatic limit, where ⌦ cˆgs +g d Ê between ), matter and light (2.82)
Ŝ(z,t) s e i(kk c
k s )z
'
g dÊs
' 0 (Eq. 2.76), the brightstate polariton is ˆ
⌦
g †
d Ê
c
(Eq. 2.77) as (ref. 86 ),
b ' 0.
s
b,k
(t 0 ) ' 0 kk 0
In the adiabatic limit, (t t 0 ), (2.82)
ˆgs
where ' ⌦In e i(kk c
k s )z
cˆgs the+g adiabatic d Ê s e i(kk limit, c
k s
where )z ' 0 ⌦(Eq. 2.76), the brightstate polariton is ˆ cˆgs +g d Ê s e i(kk c
k
In this limit, we can write the equation of motions for the darkstate polariton s )z ' 0 (Eq. 2.76), b ' the 0. brigh
⌦ c h
ˆ d,k (t), ˆ i h †
d,k
(t 0 ) ' ˆ 0 b,k (t), ˆ i
Inwhere this limit, ˆ R
d(z,t) we= can 2⇡ L
†
b,k
(t 0 ) ' 0 kk 0 (t t 0 ),
where ˆ R
write dk ˆ
the In d,k (t)e
this equation ikz limit,
andof we ˆ ✓
R ˆ d◆
with the perturbation
motions b(z,t) for d(z,t) = 2⇡ L dk ˆ d,k (t)e ikz 0
and ˆ can write
= 2⇡ L the dk ˆ @
R
darkstate equation b,k
b(z,t) = 2⇡ L dk ˆ b,k (t)e ikz .
⌦ c  2 @ 1
tˆgs
⌦ c  2 @2 t ˆgs +i e (t)e ikz of polariton . ˆ i(kk @t + v @
.5.4
ˆ
Ŝ(z,t) Adiabatic '
g dÊs following of darkstate polariton
g
c
k s )z motions d(z,t) =0.
⌦ c
(Eq. 2.77) as (ref. 86 ),
@z dfor with the the darkstate perturbation polariton
In the adiabatic
ˆFge , (2.76)
Ŝ(z,t) '
g dÊs limit, where ⌦
⌦
⌦ c
(Eq. 2.77) as Ŝ(z,t) (ref. cˆgs 86 +g
'),
d g dÊs Ê s e
c
 ⌦ c
(Eq. i(kk c
k s
2.77) )z ' 0 (Eq.
as (ref.
{z
In the adiabatic limit, where ⌦ cˆgs
where ˆ +g d Ê s e R
d(z,t) = 2⇡ i(kk c
k s
L dk ˆ )z }
86 2.76), the brightstate polariton is
),
ˆ b ' 0.
Thus, in the adiabatic regime, the darkstate polariton ˆ d(z,t)
In this limit, we can write the equation of motions for the ' 0 darkstate (Eq.
d,k (t)e ikz 2.76),
and ˆ polariton the brightstate R
b(z,t) = ˆ follows the usual wave 2⇡ polariton
L dk ˆ
is
b,k (t)e ˆ equation as
he dark state polariton ˆ Wave equation for darkstate polariton
✓ ◆
d(z,t) = cos ✓ @ d (t)Ês(z,t) sin ✓ d (t)Ŝ(z,t) can be considered ikz b ' 0.
(Nonadiabatic terms)
d with the perturbation
as a beamsplitt
space with the group velocity v
✓ ◆
g = c cos 2 ✓ d determined by the ‘amount’ of the photonic . componen
In this limit, can write the equation of @ motions for the darkstate polariton
In the adiabatic limit, where ⌦ cˆgs +g d Ê s e ˆ ◆
Ŝ(z,t) '
g dÊs
ransformation between
⌦ c
a(Eq. signal 2.77) mode as (ref. Ês(z,t)
@t + v @
ˆ
g d(z,t) =0. (2.83)
@z
86 ), field i(kk c
k s )z d with the perturbation
where 0 = ge + i ' 0 (Eq. 2.76), the brightst
Ŝ(z,t) '
g c.
Ês; and i.e.,
dÊs
⌦ c
(Eq. 2.77) as (ref. 86 ),
@t + v cos a 2 @ spinwave ✓ d ) ˆthe polariton mode @
g d(z,t) =0. ˆ d(z,t). Ŝ(z,t),
(2.83)
@z
@t + v with @ the effective matterlig
Darkstate polariton follows the ˆ
g d(z,t) =0.
nteraction Thus, Hamiltonian in the adiabaticwritten regime, as the(Eq. darkstate✓ ◆
@z
In the adiabatic condition 86,190,191 In( @ this t⌦ limit, we can write the equation of motions for the darkstate polariton ˆ c
2.70) polariton ˆ usual wave equation as in freespace
with the group velocity determined d(z,t) follows by the ‘amount’ usual wave ( equation )
d
⌦ c
⇠ 2.5.4 1 t c
⌧ @ Adiabatic ge ˜d0 (L) with resonant optical depth given by ˜d
as in freespace
with the group velocity v g = c cos 2 photonic component the polariton
of the 0 (z) =
R z
Ŝ(z,t) '
✓ g dÊs
⌦ @ c
(Eq. 2.77)
◆
as (ref. 86 ),
0 dz0 2g Thus, 2 d n A(zin 0 )zthe ), we adiabatic perturbatively regime, Thus, expand the darkstate @t + inEq. the adiabatic 2.76polariton v @ following ˆ
g d(z,t)
@z
gec
@t + v the @ regime, order ˆ =0.
of darkstate polariton
✓ d determined by the ‘amount’ of the photonic component
d(z,t) of the@ ˆ t
follows darkstate Ô ⇠
g d(z,t) =0. Ô/ thepolariton t usual c, andwave we ˆ (signal(2.83)
obtain d(z,t) equation the follows as in freespaceperturbation
with the groupĤ @z
ˆgs
The dark state polariton ˆ d(z,t) = cos ✓ d (t)Ês(z,t) sin ✓ d (t)Ŝ(z,t) can be considered theas usual a beam
(2.83)
lowestorder velocity '
(map) g dÊsv ⌦ c
e g = i(kk c cos k
space with
transformation s
Thus, in the adiabaticint
= i ˙✓
⌘
field Ês; i.e., cos 2 ✓ d ) in the polariton ˆ d(z,t).
d (z,t)
⇣Ês
regime, the darkstate the )z 2 . ✓ d Thus, group determined
polariton (z,t)Ŝ† between
velocity obtain ˆ (z,t) aby signal vthe mode g
adiabatic = ‘amount’ Ê
c † cos ✓ 2 equation ✓ d
ofdetermined the photonic ◆ motion bycomponent for the the ‘amount’ (signal
d(z,t) follows s Ês(z,t) (z,t)Ŝ(z,t) and a spinwave . mode Ŝ(z,t), BS transformation
with the effective
the usual wave equation as in free
of the
(2.8 ma
ph
@ @
Origin of transparency – Semiclassical approach
2.5.3 Darkstate polariton
Dark resonance in classical optical phenomena
Coherent population trapping
Stimulated Raman adiabatic transfer
Electromagnetically induced transparency
Coherent population trapping – dark state pumping
39 !
!
olariton ce a new set
As
of
discovered
slowlight
by
polaritonic
Fleischhauer and Lukin 86 , we can equivalently introduce a new set of slowlight pola
⌦ s ⌦ c
=
Fanointerfarence
Eqs. 2.75–2.76)
excitations
in the
{ ˆ weak d(z,t),
signal ˆ b(z,t)} as the normal modes of the system (Eqs. 2.75–2.76) in the weak
hauer and Lukin 86 , we can equivalently introduce a new set of slowlight polaritonic
approximation. Namely, we have
b(z,t)} as the normal modes of the system (Eqs. 2.75–2.76) in the weak signal
we have
ˆ
)Ŝ(z,t) (2.80) d(z,t) = cos ✓ d (t)Ês(z,t) sin ✓ d (t)Ŝ(z,t)
Dark (bright)state polaritons
ˆ ˆ
)Ŝ(z,t), d(z,t) = cos ✓ d (t)Ês(z,t) (2.81) sin ✓ d (t)Ŝ(z,t) b(z,t) = sin✓ d (t)Ês(z,t)(2.80)
+ cos ✓ d (t)Ŝ(z,t),
First proposal and observation of EIT
ˆ •! Harris, Field, and Imamoglu, PRL 64, 1107 (1990)
b(z,t) = sin✓ d (t)Ês(z,t) + cos ✓ d (t)Ŝ(z,t), (2.81)
•! Boller, Imamoglu and Harris, PRL 66, 2593 (1991)
atched collective where Ŝ(z,t) spin operator, =p N A eand
i(kk c
k s )z ˆgs (z,t) is the slowlyvarying phasematched collective spin operat
p slow light with EIT
(k k c
k s )z nown asˆgs the ✓(z,t) d darkstate = is arctan(g the slowlyvarying (brightstate)
d NA /⌦ c ) phasematched is the mixingcollective angle. These spin
•!
operator, operators
Hau et al.,
and are known as the darkstate (brigh
Classical dark (bright)states
Nature 397, 594 (1999) – 17 m/s
•! Kash et al., PRL 82, 5229 (1999) – 90 m/s
c) ght) is states the mixing polaritons di = angle. cos ✓ ˆ These operators are known as the darkstate (brightstate)
d d(z,t) gi sin ( ˆ ✓b(z,t)), d si in direct analogy with the classic dark (bright) states di = cos ✓ d gi si
,t)), in direct analogy with the classic dark (bright) states di = cos ✓ d gi sin ✓ d si
(chapter 6). (bi These = sinpolaritons ✓ d gi + cos follow ✓ d si) observed in coherent population trapping (chapter 6). These polaritons
si) observed in coherent population trapping Quantum (chapter 6). 86 description: These polaritons Lukin & Fleishhauer, follow Phys. Rev. Lett. 84, 5094 (2000)
39
dynamic EIT – darkstate polariton
Darkstate polariton
•! Coherent mixture of matter and light
•! Adiabatic evolution (accelerate/decelerate DSP)
•! Rotate !()
t from 0 to ! /2for storage
•! Rotate !()
t from ! /2to 0 for retrieval
•! Group velocity given by v () t
2
c cos !()
t c
storage
g
ˆ!(z ,t ) = cos!(t )ˆ! Signal
" sin!(t ) N ˆ! gs
" () t
= = " + gN
2
C
2 2
C
() t
open transparency
window with control
laser
coherent darkstate
trapping (CPT) into ! #
convert ! from $ signal extinguish & c :
into collective spin phaseamplitude
excitation % gs#
stored in % gs#
!
c
() t signal !
c
() t signal !
c
() t
!()
t
QM
signal field escapes
ensemble
slow light propagation
through EIT window
convert ! from
% gs to $ signal
storage
retrieval
static EIT
adiabatic evolution of darkstate polariton
Experimentally, to avoid the dissipative absorption of the signal field E
e i(kk c
k s )z s (z,t) for our choice of polarization
ˆgs defined in section 2.3.1.3. The dynamics of the signal field Ês(z,t) and the spinwave mode
polarization), we optically pumped the atomic ensemble into a clock 6S
Mapping quantum states
1/2 , F =4,m
to and
F =0i with
from a quantum memory
efficiency. Initially, the strong control field ⌦ c (z,t) (resonant with 6S 1/2 ,F =3$ 6P 3/2 ,F =4
nsition with
Ŝ(z,t) is governed by a set of HeisenbergLangevin equations (Eqs. 2.71–2.74),
Nonadiabatic equation of motion (i.e., transition between dark and bright states)
(@ t + c@ z ) Ês(z,t) L
= ig d n A (z) p ˆP(z,t) (2.85)
NA
+ polarization) opens the transparency window ⌦ c (z,t) ' 24 MHz for the signal mode.
the wave packet Ês(z,t) of the signal field propagates through the ensemble, we extinguish the control
ds ⌦ c (z,t) in 20 ns, thereby coherently transforming the coherent state of the signal mode Ês,in(z,t) to
lective atomic excitation Ŝ(z,t). After ' 1.1 µs, the atomic state is converted back to the signal mode
@ t ˆP(z,t) = ( ge + i ) ˆP(z,t)+ig d
p
NA Ê s (z,t)+i⌦ c (z,t)Ŝ + p 2 ge ˆFP (2.86)
out(z,t) by switching on the control field ⌦ c (z,t). We measure the normalized crosscorrelation function
the input photonic state Ês,in(z,t) with g (2) =1.1±0.2, as well as for the output photonic state Ês,out(z,t)
@ t Ŝ(z,t) = gsŜ(z,t)+i⌦⇤ c(z,t) ˆP + p
in
2 gs ˆFS . (2.87)
h g (2)
out =1.0 ± 0.2, whereby we observe no degradation in the photon statistics.
In chapter 6, we discuss an experiment where we reversibly mapped a photonic entanglement into and
of quantum memories. We further examine the optimal control theory developed in ref. 188 , where we
oretically apply the principle of timereversal symmetry to optimize our reversible quantum interface.
Here, ˆF P and ˆF S are the respective correlated Langevin noise operators for ˆP(z,t) and Ŝ(z,t), with nonzero
terms h ˆF P (z,t) ˆF † P (z0 ,t 0 )i = L (z z 0 ) (t t 0 ) and h ˆF S (z,t) ˆF † S (z0 ,t 0 )i = L (z z 0 ) (t t 0 ). Since
ARTICLES
Mapping the normally coherent ordered states noise into operators and out of h ˆF collective † ˆF i i i =0with excitations i 2 {S, P} for vacuum Reading reservoirs, heralded we neglect single photons them in
2.5x10 5
1.0x10 1
5
0
80 40 0 40 80 1000 1040 1080 1120 1160
2 8
the numerical calculation of chapter 6 (see section 2.3.1.2).
30
25
We emphasize that the collective enhancement ( p 2.0x10 5
N A ) of single atomphoton coupling constant g d (Eqs.
20
6
2.85–2.87) 1.5x10 enables a strong collective matterlight interaction with an effective coupling constant g eff =
5
p
15
NA g d between a single spinwave of the ensemble and a single photon of the signal field. We are interested
p 1122 (×10 7 )
4
p 11 (×10 5 )
10
in the collectively enhanced storage (⌘ s ) and retrieval (⌘ r ) efficiency ⌘ sr = ⌘ s ⌘ r of the quantum field Ês(z,t),
which we define as the ratio of the number R 5.0x10 6
dzhÊ s † 5 (z,t)Ês(z,t)i of incoming photonic excitations in the signal
field to0.0the number of stored spinwave excitations dzhŜ† (z,t)Ŝ(z,t)i (and vice versa). Specifically,
2
R
for
an atomic ensemble with finite optical depth ˜d 0 , there is an optimal control field ⌦ c (z,t), which maximizes
0
0
τ (ns)
0 (Choi 5et al., unpublished 10 15 2007) 20
the transfer efficiency ⌘ sr , byNcompromising two competing goals 186 : (1) The characteristic Conditional control time variation of
Reversible
t c in the control laser ˙⌦ quantum interface
Figure 3 Conditional heralded jointdetection singlephoton probability p
for photonic c (z,t) entanglement must be slow relative z/c). The red to solid thecurve
two adiabatic criteria ( w s ' 2⇡ c sources
22
Felinto et al. Nature Phys. t c 2, < 844 w(2006).
EIT
ure 2.6: Reversible mapping of a coherent state to and from an atomic memory. The points around
0 ns (i.e., 40 to 20 ns) represent the leakage of the signal field due to the finite optical depth and length
the ensemble. The points beyond ⌧ ' 1 µs show the retrieved signal field. The blue solid line is the
(τ )ofrecordingeventsin
imated Rabi frequency ⌦ c (z,t) of the control pulse, where we assumed ⌦ c (t
rom a numerical Figure calculation 2 Probabilities solvingp 11 the and equation p 1122 of of coincidence motion of the detection signal as field functions in a coherently of the
both D 2a and D 2b ,oncethetwoensemblesarereadytofire,asafunctionofthe
dressed
v
time difference τ between the two detections. The filled squares (open circles)
p 22 (×10 4 )
c
2
1
0
–40 –20 0
Perpendicular
Parallel
20 40
advances with networks of quantum memories, including for
quantum repeaters. Our work thereby paves the way to scalable
quantum networks over distances much longer than set by
bre optic attenuation.
METHODS
EXPERIMENTAL DETAILS
6912ñ 6918 (2006).
Conditional control of remote quantum memories
Felinto, Chou, Laurat, Schomburg, Riedmatten, Kimble, Nature Phys. 2, 844 (2006)
Magneto optical traps are used to form the clouds of atoms, and are switched
off for 6 ms every 25 ms period. After waiting for the trap magnetic eld to
decay 24 , a train of write and read pulses excite the sample during the last 2 ms.
The write pulse is 10 MHz red detuned from the g → e transition. The
transverse waist of the write beam is 200 µm, and its peak power P write ≈ 2 µW.
We collect the light emitted by the ensemble in a PM bre, whose projected
mode on the ensemble corresponds to a beam with a 50 µm waist intersecting
the write beam direction at a three degree angle 15 . In the experiment, the read
pulse is delayed from the write pulse by 300 ns, leaving time for the pulses to be
gated off after the heralding signal, which occurs 100 ns after the write pulse
owing to propagation delays.
INCREASE Conditional IN PROBABILITY enhancement in the preparation probability
Assume Assume that p 1 that givespthe 1 gives probability the probability per trial of per storing trial of a collective storing a excitation,
and that collective it is possible excitation, to waitand up tothat N trials it is possible before reading to wait out up the to excitation N
and releasing trials before the corresponding reading out single the excitation photon. The and probability releasing the of having two
ensembles corresponding storing excitations single photon. in the same The trial probability is then of having two
ensembles storing excitations in the same trial is then
p 11 = p 1 {p 1 +2[(1− p 1 )p 1 + (1− p 1 ) 2 p 1 + +(1− p 1 ) N p 1 ]}
≈ (2N +1)p 2 1 , when p 1 ≪ 1.
15. Laurat, J. et al.Efficient retrieval of a single excitation stored in an
16. Blinov, B. B., Moehring, D. L., Duan, L. M. & Monroe, C. Observa
single trapped atom and a single photon. Nature 428, 153ñ 157 (200
17. Volz, J. et al. Observation of entanglement of a single photon with
030404 (2006).
18. Matsukevich, D. N. et al. Entanglement of a photon and a collective
95, 040405 (2005).
19. de Riedmatten, H. et al. Direct measurement of decoherence for en
stored atomic excitation. Phys. Rev. Lett. 97, 113603 (2006).
20. Matsukevich, D. N. et al. Entanglement of remote atomic qubits. P
21. Julsgaard, B., Kozhekin, A. & Polzik, E. S. Experimental long lived
objects. Nature 413, 400ñ 403 (2001).
22. Matsukevich, D. N. et al. Deterministic single photons via conditio
Lett. 97, 013601 (2006).
23. Chen, S. et al. A deterministic and storable single photon source ba
Rev. Lett. (in the press).
24. Felinto, D., Chou, C. W., de Riedmatten, H., Polyakov, S. V. & Kimb
the generation of photon pairs from atomic ensembles. Phys. Rev. A
25. Simon, C. & Irvine, W. T. M. Robust long distance entanglement a
ions and photons. Phys. Rev. Lett. 91, 110405 (2003).
26. Mandel, L. & Wolf, E. Optical Coherence and Quantum Optics (Ca
Cambridge, 1995).
27. Hong, C. K., Ou, Z. Y. & Mandel, L. Measurement of subpicosecon
Probabilistic process of singlephoton generation can be synchronized !
photons by interference. Phys. Rev. Lett. 59, 2044ñ 2046 (1987).
28. Santori, HongOuMandel C., Fattal, D., Vuckovi¥c, interference
J., Solomon, G. S. & Yamamoto, Y.
single photon device. Nature 419, 594ñ 597 (2002).
29. Legero, T., Wilk, T., Hennrich, M., Rempe, G. & Kuhn, A. Quantum
Phys. Rev. Lett. 93, 070503 (2004).
30. de Riedmatten, H., Marcikic, I., Tittel, W., Zbinden, H. & Gisin, N.
photon pairs created in spatially separated sources. Phys. Rev. A 67
31. Beugnon, J. et al. Quantum interference between two single photon
trapped atoms. Nature 440, 779ñ 782 (2006).
32. Duan, L. M., Cirac, J. I. & Zoller, P. Three dimensional theory for i
ensembles and free space light. Phys. Rev. A 66, 023818 (2002).
33. Chou, C. W., Polyakov, S. V., Kuzmich, A. & Kimble, H. J. Single ph
excitation in an atomic ensemble. Phys. Rev. Lett. 92, 213601 (2004
34. Legero, T., Wilk, T., Kuhn, A. & Rempe, G. Time resolved two pho
Phys. B 77, 797ñ 802 (2003).
Indistinguishability of single photon sources
The Observed factor of conditional two in the above enhancement expression with accounts decoherence for the two possible orders
in which the ensembles can be prepared.
Acknowledgements
This research is supported by the Disruptive Technologies Office (DT
Foundation. J.L. acknowledges nancial support from the European
D.F. acknowledges nancial support by CNPq (Brazilian agency).
Heralded capabilities for quantum communication
Entanglement generation
W
1
2
( 10 01 )
2
= +
Investigation of the relationship between
global dynamics of entanglement
and the decay of local quantum correlations
Phys. Rev. Lett. 99, 180504 (2007)
!!
•! Entanglement generated by quantum
interference in the measurement process
•! Heralded entanglement stored in the collective
excitations of atomic ensembles
•! Degree of entanglement stored in the
ensemble 1
•! C = 0.9 +/ 0.3
•! Asynchronous preparation – subexponential scaling 2
•! Functionality achieved by parallel operations
•! Quantum cryptography and teleportation
Conditional control of heralded entanglement
Science 316, 1316 (2007)
Entanglement connection
Initial work: Chou et al. Nature 438, 828 (2005).
1
Laurat, Choi et al. Phys. Rev. Lett. 99, 180504 (2007).
2
Chou et al. Science 316, 1316 (2007); Laurat et al. New J. Phys. 9, 207 (2007)
!!
!!
!!
Functional Quantum Nodes for Entanglement Distribution over
Scalable Quantum Networks
ChinWen Chou, et al.
Science 316, 316, 1316 1316 (2007) (2007);
DOI: 10.1126/science.1140300
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Conditional 68 control of heralded entanglement
! "
Asynchronous preparation
Preparation x35
Final state x20
Entanglement distribution
Violation of CHSH inequality
# $
Quantumkey distribution:
security against individual
eavesdropping attacks
duration at which the first entangled pair is stored before
retrieval
Figure 4.4: Measured CHSH parameters and the violation of Bell inequality. We measured the CHSH
Chou, Laurat, Deng, Choi, de Riematten, Felinto, Kimble. Science 316, 1316 (2007)
parameters S ± , for the two possible effective states in Eq. 4.3, as functions of duration M for which the first
DLCZbased Exeriments : Progress
DLCZbased Exeriments : Progress
Spectacular advances with DLCZbased experiments
Nature Phys. 3, 765 (2007)
Nature Phys. 3, 765 (2007)
Ensemble inside a cavity : readout>80%
Ensemble inside a cavity : readout>80%
Nature 454, 1098 (2008)
Nature 454, 1098 (2008)
Another example of elementary repeater segment
Another example of elementary repeater segment
Extension of storage time
Motional dephasing (decoherence of spatial phases)
⌘ = 95%
Memory time limited by
Thermal diffusion
Spinwave wavelength (momentum transfer)
collisions
Lene Hau, PRL (2009)
Storage in BEC
⌧ =1.5s
with efficiency
Experimentally,
e i(kk to
c
kavoid s )z the dissipative absorption of the signal field
ˆgs defined in section 2.3.1.3. The dynamics of the signal field Ês(z,t) and the spinwave mode
Mapping quantum states Ês(z,t) for our choice of polarization
polarization), we optically pumped the atomic ensemble into a clock 6S 1/2 , F =4,m
to and F =0i with
from a quantum memory
efficiency. Initially, the strong control field ⌦ c (z,t) (resonant with 6S 1/2 ,F =3$ 6P 3/2 ,F =4
Ŝ(z,t) is governed by a set of HeisenbergLangevin equations (Eqs. 2.71–2.74),
Nonadiabatic + polarization) equation opens the transparency of motion window (i.e., ⌦ c transition (z,t) ' 24 MHz between for the signal dark mode. and bright states)
(@ t + c@ z ) Ês(z,t) L
= ig d n A (z) p ˆP(z,t) (2.85)
NA
nsition with
the wave packet Ês(z,t) of the signal field propagates through the ensemble, we extinguish the control
ds ⌦ c (z,t) in 20 ns, thereby coherently transforming the coherent state of the signal mode Ês,in(z,t) to
lective atomic excitation Ŝ(z,t). After ' 1.1 µs, the atomic state is converted back to the signal mode
@ t ˆP(z,t) = ( ge + i ) ˆP(z,t)+ig d
p
NA Ê s (z,t)+i⌦ c (z,t)Ŝ + p 2 ge ˆFP (2.86)
out(z,t) by switching on the control field ⌦ c (z,t). We measure the normalized crosscorrelation function
the input photonic state Ês,in(z,t) with g (2)
@ t Ŝ(z,t) = gsŜ(z,t)+i⌦⇤ c(z,t) ˆP + p in
=1.1±0.2, as well as for the output photonic state Ês,out(z,t) 2 gs ˆFS . (2.87)
h g (2)
out =1.0 ± 0.2, whereby we observe no degradation in the photon statistics.
In chapter 6, we discuss an experiment where we reversibly mapped a photonic entanglement into and
of quantum memories. We further examine the optimal control theory developed in ref. 188 , where we
oretically apply the principle of timereversal symmetry to optimize our reversible quantum interface.
Here, ˆF P and ˆF S are the respective correlated Langevin noise operators for ˆP(z,t) and Ŝ(z,t), with nonzero
terms h ˆF P (z,t) ˆF † P (z0 ,t 0 )i = L (z z 0 ) (t t 0 ) and h ˆF S (z,t) ˆF † S (z0 ,t 0 )i = L (z z 0 ) (t t 0 ). Since
ARTICLES
Mapping the normally coherent ordered states noise into operators and out of h ˆF collective † ˆF i i i =0with excitations i 2 {S, P} for vacuum Reading reservoirs, heralded we neglect single photons them in
2.5x10 2 1.0x10
1 30 8
6
4
0.0
0
the numerical calculation of chapter 6 (see section 2.3.1.2).
We emphasize that the collective enhancement ( p 25
2.0x10
N A ) of single atomphoton coupling constant g d (Eqs.
5
20
2.85–2.87) enables a strong collective matterlight interaction with an effective coupling constant g eff =
1.5x10
p 5
NA
15
g d between a single spinwave of the ensemble and a single photon of the signal field. We are interested
p 1122 (×10 7 )
p 11 (×10 5 )
in the collectively enhanced storage (⌘ s ) and retrieval 10 (⌘ r ) efficiency ⌘ sr = ⌘ s ⌘ r of the quantum field Ês(z,t),
which we define as the ratio of the number R 5.0x10 6
dzhÊ s † 5 (z,t)Ês(z,t)i of incoming photonic excitations in the signal
field to the number of stored spinwave excitations dzhŜ† (z,t)Ŝ(z,t)i (and vice versa). Specifically,
2
R
for
80 40 0 40 80 1000 1040 1080 1120 1160
an atomic ensemble with finite optical depth ˜d 0 , there is an optimal control field ⌦ c (z,t), which maximizes
0
0
τ (ns)
0 (Choi 5et al., unpublished 10 15 2007) 20
the transfer efficiency ⌘ sr , byNcompromising two competing goals 186 : (1) The characteristic Conditional control time variation of
Reversible
t c in the control laser ˙⌦ quantum interface
Figure 3 Conditional heralded jointdetection singlephoton probability p
c (z,t) must be slow relative to the two adiabatic criteria ( w s ' 2⇡ c sources
22
for photonic entanglement z/c). The red solid curve Felinto et al. Nature Phys. t c 2, < 844 w(2006).
EIT
ure 2.6: Reversible mapping of a coherent state to and from an atomic memory. The points around
0 ns (i.e., 40 to 20 ns) represent the leakage of the signal field due to the finite optical depth and length
the ensemble. The points beyond ⌧ ' 1 µs show the retrieved signal field. The blue solid line is the
(τ )ofrecordingeventsin
imated Rabi frequency ⌦ c (z,t) of the control pulse, where we assumed ⌦ c (t
Figure 2 Probabilities p 11 and p 1122 of coincidence detection as functions of the
both D 2a and D 2b ,oncethetwoensemblesarereadytofire,asafunctionofthe
rom a numerical calculation solving v the equation of motion of the signal field in a coherently dressed time difference τ between the two detections. The filled squares (open circles)
p 22 (×10 4 )
c
2
1
0
–40 –20 0
Perpendicular
Parallel
20 40
EIT Storage and Retrieval of SinglePhotons
EIT Storage EIT Singlephoton Storage and Retrieval and storage Retrieval of and SinglePhotons
of
retrieval
EIT Storage and Retrieval of SinglePhotons
Nature 438, 837 (2005)
Nature 438, 837 (2005)
Nature 438, 837 (2005)
Nature 438, 837 (2005)
Nature 438, 833 (2005)
Nature 438, 833 (2005)
Nature 438, 833 (2005)
Nature 438, 833 (2005)
Singlephoton storage and retrieval
Deterministic capabilities for quantumstate transfer
K. S. Choi, H. Deng, J. Laurat, H. J. Kimble. Nature 452, 67 (2008).
Coherent and reversible mapping via dynamic EIT
Photonic entanglement Atomic entanglement Photonic entanglement
94
94
Photonic entangler
#
#
#
#
"
!
"
!
"
!
"
!
Vol 452  6 March 2008  doi:10.1038/nature06670
calculations t 5 1 ms represents following the the retrieved methods pulse in ref. after 23. 1.1We ms use of storage. the fitted Overall, function
we find of the good input agreement signal field between (Fig. 2a) ouras measurements the initial condition and numerical with all
other calculations parameters following from independent the methods measurements in ref. 23. We use (seethe Methods). fitted functionan
of overall the input storage signal and field retrieval (Fig. 2a) efficiency as the initial of g r 5condition 17 6 1%, with also all in
We
find
agreement other parameters with the from simulation independent of g theory r measurements 5 18%. (see Methods). We
find With an overall these results storageinand hand retrieval for the efficiency individual of gL ra 5,R 17 a ensembles, 6 1%, alsowe
in
next agreement turn to with the the question simulation of verification of g theory r 5 18%. of entanglement for the
optical Withmodes these results of L in ,R in hand Lfor out ,Rthe out . individual We followL the a ,R a protocol ensembles, introduced
next turn in ref. to5the by (1) question reconstructing of verification a reduced of entanglement density matrixfor r con
the
we
strained optical modes to a subspace of L in ,Rcontaining and L out ,R no out more . Wethan follow onethe excitation protocolinintro
duced and in ref. (2) 5assuming by (1) reconstructing that all offdiagonal a reduced elements densitybetween matrix rstates
each
mode, con
with strained different to a subspace numbers containing of photons novanish, more than thereby oneobtaining excitationainlower
each
Mapping bound mode, Entanglement and for any (2) purported assuming that entanglement. all offdiagonal Into the elements photonnumber and betweenOut
states basis
jn with L ,mdifferent R æ with {n,m} numbers 5 {0,1}, of photons the reduced vanish, density thereby matrix obtaining r is written a lower as
K. S. Choi 1 , H. Deng 1 , J. Laurat 1 { & H. J. Kimble 5
bound 1 for anyNature purported0
entanglement. 452, 67 In the(2008)
photonnumber 1
basis
C~ ence 1 Pbetween max 0,2modes j d j{2 L p k ,R00p k , 11 with , which k g {in, is out}. a monotone The degree function of enta
entanglement, of rranging can be quantified in terms of concurren
C~ 1 P
max 0,2 j d j{2 p ffiffiffiffiffiffiffiffiffiffiffiffi from
0 for a separable state to 1 for a ma
imally entangled state 29 p.
00 p 11 , which is a monotone function
entanglement, We first perform ranging tomography from 0 for onathe separable input modes state to L in 1,Rfor in toa ver m
that imally they entangled are indeed state entangled. 29 . To this end, we remove the memo
ensembles We first to perform transmit tomography the signalon fields the directly input modes into the L in ,Rverificati
in to stage, that they following
LETTERS
are indeed ourentangled. protocol of Tocomplementary this end, we remove measurements the memo
described ensemblesintoFig. transmit 1d. The theinterference signal fieldsfringes directly between into the theverificati
two inp
modes stage, following are shownour in protocol Fig. 3a. From of complementary the independently measurements determin
propagation described in Fig. and1d. detection The interference efficiencies fringes (see Methods), between thewe two use inp
measurements modes are shown at Din 1 ,DFig. 2 to3a. infer Fromthethe quantum independently state fordetermin
the inp
modes propagation L in ,R in and entering detection the faces efficiencies of L a ,R(see a (ref. Methods), 5), with we theuse
reco
structed measurements densityat matrix D 1 ,D 2
r in togiven inferinthe Fig. quantum 3a. The concurrence state for thederiv
inp
from modesr in L in
is ,R in
C in entering 5 0.10 6the 0.02, faces soof theL a
fields ,R a (ref. for5), L in with ,R in are the inde reco
entangled. structed density The value matrix ofrthe in given concurrence in Fig. 3a. isThe in good concurrence agreement deriv w
from r in is C in 5 0.10 6 0.02, so the fields for theory L in ,R5 in
0.10 are 6inde
0.
which entangled. depends The on value theof quality the concurrence of the single is photon in goodand agreement the vacuu w
component (that is, the overall efficiency) 17 . Given theory a heralding 5 0.10 6 0. cl
from whichour depends singlephoton on the quality source, of the single probability photon ofand having the vacuu a sin
photon component at the (that faceis, ofthe either overall memory efficiency) ensemble 17 . Given is 15%, a heralding leadingcl
t
vacuum from our component singlephoton of 85%. source, We also the independently probability of characterize having a sin
suppression photon at the w face of the oftwophoton either memory component ensemblerelative is 15%, toleading a cohere t
vacuum component of 85%. We also independently characterize
suppression w of the twophoton component relative to a coher
Mapping photonic entanglement into and out of a
quantum memory
p
Idea
00 0 0 0
jn L ,m R æ with {n,m} 5 {0,1}, the reduced density matrix r is written as 5 the independently derived expectation of C in
r~ 1 0 0 p 10 d 0
1
B p 00 0 0 P
r~ 1 0 d C
ð2Þ
@ p 01 0 A
the independently derived expectation of C
Mapping photonic entanglement
in
p 10 d 0
B 0 0 0 p 11
P @ 0 d C
ð2Þ
p 01
into and out of a
0 A
Developments in quantum information science 1 rely Here, critically p ij is the probability on of finding The iseparation photons in mode 20% entanglement transfer !
0 0 0 p 11
of L k and processes j in for the generation of entanglement
mode R k , d < KV(p 10 1 p 01 ) is the coherence between j1 L 0 R æ k and
quantum entanglement—a fundamental memory
aspect of quantum mechanics Here, p ij is the that probabilityand of finding fori photons its storage in modeenables L k and j in this drawback to be overcome. Here, we
causes parts of a composite system to show correlations
mode R k
stronger
, d a< KV(p 10 1 p 01 ) is the coherence between j1 L 0 R æ k and
Entanglement demonstrate V = 0.93 this ± 0.04mapping
by reversible mapping of an entangled state into a
Singlephoton storage and retrieval
K.
than
S.
can
Choi
be 1 ,
explained
H. Dengclassically 1 , J. Laurat 2 . In 1 {
particular,
& H. J. Kimble
scalable
input
1 quantum a quantum memory. The mapping is obtained by using adiabatic passage
based on dynamic electromagnetically induced transparency
400
V = 0.93 ± 0.04
1
networks require the capability
g (2) to create, store and distribute entanglement
among distant matter nodes by means of photonic chan
(EIT) 20–23 (see Methods). Storage and retrieval of optical pulses have
=0.1
400
0.1 1
nels 3 . Atomicensemblescanplaythe roleof suchnodes 4 .Sofar,inthe been demonstrated previously, for both classical pulses 24,25 and0.01
0.1sin
glephoton The separation pulses 26,27 of processes . Adiabaticfor transfer the generation of a collective of entanglement
excitation 0.001 0.01has
entanglement—a ensembles has been fundamental successfullyaspect demonstrated of quantum through mechanics probabilistic 200 that and beenfordemonstrated its storage enables between this two drawback ensembles
200
Developments photoncounting quantum regime, heralded information entanglement science 1 rely between critically atomic on
00to be coupled overcome. byHere, a cavity
11 we
0.001
protocols 5,6 . But an inherent drawback of this approach is the compromise
between the amount of entanglement and its preparation 0
10
10
01
causes parts of a composite system to show correlations stronger demonstrate mode 28 , which thiscan by provide reversible a mapping suitable approach of an entangled for generating state intoon
demand entanglement memory. The over mapping shortisdistances. obtained by However, using adiabatic to assist
a
00
01
10
11
11 00
than can explained classically 2 . In particular, scalable quantum quantum 01 passagcientbased
distribution
effi
–400
–300
–200
–100
0
01
10
networks probability, require leading the to capability intrinsically to create, low count store and rates distribute for highentan
glement. Here we report a protocol where entanglement between –1
–400
f
0
rel (degrees)
dynamic of entanglement electromagnetically over quantum induced networks,
C 11 00
in = (1.0±0.2) × 10
transparency reversible
mapping 20–23
two
–300
–200
–100
0
among distant matter nodes by means of photonic channels
atomic 3 . Atomicensemblescanplay is created by the coherent roleofmapping suchnodes of 4 .Sofar,inthe
an entangled been illustrated demonstrated in Fig. 1a, previously, has not been for both addressed classical until pulses now. 24,25 –1
and sin
(EIT)
f rel
(see
(degrees)
Methods). of an entangled Storagestate and retrieval betweenof matter opticaland pulses light, have as
C in = (1.0±0.2) × 10
state of light. By splitting a single photon 7–9 and performing output
photoncounting regime, heralded entanglement between atomic
b subsequent
state hastransfer, been successfully we separate demonstrated the generation through of entanglement
probabilistic L
glephoton In our experiment, pulses 26,27 . Adiabatic entanglement transfer between of a collective two atomic excitation ensembles has
400
V = 0.91 ± 0.03
ensembles b been a ,R a demonstrated is created by first between splitting two aensembles single photon coupled intoby twoa cavity modes
1
and its storage 10 . After a programmable delay, the stored entanglement
is mapped
400 L in ,R in toV = generate 0.91 ± 0.03
protocols 5,6 . But an inherent drawback of this approach is the compromise
mode 28 , which can an provide entangled a suitable state of approach light 7–9 . This for generating entangled field
0.1 ondemand
1
between
back
the amount
into photonic
of entanglement
modes with
and
overall
its preparation
efficiency of state is then
entanglement
coherently
over
mapped
short
to
distances.
an entangled
However,
matter state
to assist
for L
efficient
distribution of entanglement over quantum networks, revers
a ,R a .
0.01 0.1
probability,
17%. Together
leading
with
to
improvements
intrinsically low
in singlephoton
count rates for
sources
high entanglement.
Here we report a protocol where entanglement between two ible mapping of an
11 , our On demand, the stored atomic entanglement for L a ,R a is converted back
200
protocol will allow ‘ondemand’ entanglement of atomic ensembles, into entangled photonic modes L out ,R out . As opposed to the original 0.001 0.01
a powerful resource for quantum information science.
200 scheme of Duan et al. 4 entangled state between matter and light, as
, our approach is inherently deterministic, suffering
principally from the finite efficiency of mapping
00
110.001
atomic ensembles is created by coherent mapping of an entangled illustrated in Fig. 1a, has not been addressed until now.
In the quest to achieve quantum networks over long distances 3 ,
10
single excitations
01
to
state of light. By splitting a single photon 7–9 and performing subsequent
state transfer, we separate the generation of entanglement L
In our experiment, entanglement between 00 two 01 atomic 10 ensembles 11
there has been considerable interest in the interaction of light with and from an atomic memory. An efficiency10of about 11 50% 00 has
0
01 been
–400
–300
–200
–100
0
01
10
atomic ensembles consisting of a large collection of identical atoms 4 . achieved a ,R a is created by first splitting a single photon into two modes
(see Methods). Moreover, the contamination of entanglement
f 11 00
and its storage 0
rel (degrees)
C out = (1.9±0.4) × 10
In the regime of 10 . After a programmable delay, the stored entangle L
continuous variables, a notable advance has been the for in ,R
L in to generate an entangled state of light
,R from processes involving two excitations 7–9 . This entangled field
can be –2
arbitrarily
Counts Counts (D 1,2 ) (D 1,2 )
Counts Counts (D 1,2 ) (D 1,2 )
A new frontier of EIT in quantum regime
Vacuuminduced transparency
Tanji, Chen, Landig, Simon, Vuletic, Science 333, 1266 (2011)
Vacuuminduced transparency (VIT) spectra
shown in the following, has a photon number distribution
the num
that is an exact copy of that of the input probe field in
times ne
the limit of large atom number. The characteristic length
in the ca
of the probe pulse T is typically large compared to the
upper level relaxation time ( T 1) and thus the time
derivative in eq. (9) can be neglected. If the spectrum of
the probe pulse lies within the EIT transparency window
! EIT , i.e. if furthermore
ˆn(t) n
T
1
=
! p EIT G
2
OD
OD ⌘
L ,
l abs
(11)
In the c
of the pr
number
f (t) describes the shape of the probe field before entering
the medium. abs = c /g
InThis order is t
where l
Note N is the resonant absorption length
that this function is the same for componen In the
all
of
Fock
the medium
components
in the
corresponding
absence of EIT,
to a
and
single
OD
(pulsed)
the optical
depth,
time number ⌧ 1
mode. After
ˆge
propagating
can be adiabatically
✓ through the
eliminated,
medium ◆ the
and
state
the
(11) pulse and in(
Langevin noiseP
is  (L, t)i =
1 operators canL
be disregarded [23, 24]. In
single the grou pho
this adiabatic limit ↵ n the f tatomic dynamics
n=1 l abs (n + 1)G 2 is ni. governed Thus by
bev gr satisfie T . In
di↵erent
the equations
components of the probe will be spatially separated
for the d
(Fig.2). This separation is larger for Fock compo
It increa
nents with
Gâˆgs + gÊ =0, (12)
smaller number of photons. Specifically the of simpl
delay @ between components with m and m + 1 photons
after @t propagating =iG ⇤ âˆge â † @â @â
+âˆgs
over distance L is@t given by @t â† âˆgs (13)
highest
is smalle
and thus
Photonnumber
⌧ m =
L
cavity is
Combining eqs. (12), (13) selective and 1 (2) onegroup finally delay
l abs (m + 1)(m + 2) G 2 . arrives (20) at required.
pulse ca
the following cavity propagation induced equation transparency
of the probe field Experim
pulse ha
G ⇡ 10M
Nikoghosyan, ! Fleischhauer
@Ê
art techno
Phys. Rev. Lett. 105, 013601 (2010). the cavity
@t + G2
g 2 N â @Ê
@t + c@Ê â † Ê @â
@z @t â† + @â
ity field
@t â† Ê =0 photons
cloud wit
(14) experimen Now by
The first terms in eq.(14) describe a probe field propagation
with a cavity dependenr group velocity. The last can in pr
should lowingbe
s
two f (t) terms describes describe the shape a dynamical of the probe reduction field of before the probe entering
polar In orde mol
the medium. if the cavity Note that field this changes function in time, is the i.e. same during for compon In the p
amplitude
the all Fock periods components of enteringcorresponding and leaving the to amedium. single (pulsed) Since oftime
a weak
we mode. are not After interested propagating ✓these through transients the medium and◆
in the order state to Raman
(11) an
re
simplify P
is  (L, t)i
the
=
discussion 1 L
↵ n f
we
t
will disregard these We
single
have
p
n=1 l abs (n + 1)G 2 ni.
terms
Thus
in
the following. Taking into account that cavity and probe ofOne be
the
satis sees pro
operator di↵erent commute components we of arrive the at probe an operatorvalued will be spatiallygroup
separated
(Fig.2). This separation is larger for Fock compo
the Fock limit com
cavity the initi mo
velocity
nents with Quantized smaller number group of photons. velocity Specifically the
FIG. 2: (color online) Spatial separation of an initial probe large di↵erent opt
delay between components G with m and m + 1 photons
pulse into Fock state components.
(ˆn + 1)
the cavity
after propagating ˆv gr over = c
Gdistance 2 (ˆn +1)+g L is given N . by To be
number initially and thu o
tions Important practical limitations of the present scheme
⌧ be descri
result from dissipation m =
L 1
i.e.
l abs in(m the+ form 1)(m + of2) cavity G 2 . damping
required the
ˆv gr depends on the number of photons ˆn =â † â in the
 (t)i Expe=
cavity. On the other hand the cavity photon number is group vel
and spontaneous emission. Cavity damping comes into
G ⇡ 10
ferent art tech Foc
equations equations of of motion motion for for the the complex complex amplitudes amplitudes (OMIT). (OMIT). The The role roleof of the the control control laserís laserís Rabi Rabi S1) S1) measuring measuring the the phase phase quadrature quadrature of of the the fi
emerging emerging from fromthe the cavity cavity (25). (25). This This allows allow
ð−iD ′ þ k=2ÞA − ¼ −iGaX þ
pffiffiffiffiffiffiffi
tracting tracting the the parameters parameters of of the the device device used used in in th
x
h c k ds in ð3Þ zpf ¼ ħ=2m eff W m designates the spread of
which are given by (m eff ,
ð−iD ′ þ k=2ÞA − p
¼ −iGaX þ
x
h c k ds in ð3Þ zpf ¼ ħ=2m eff W m designates the spread of experiments, which are given by (m eff , G/
the the groundstate groundstate wave wave function function of of the the mechanical mechanical G m
G/2 m /2 p, p, W m
W/2 m /2 p, p, k/2 k/2 p) p) ≈ (20 ≈ (20 ng, ng, −12 −12 GHz/n
Optomechanically induced oscillator. transparency For W (OMIT)
2m eff W m ð−iD ′ þ G m =2ÞX ¼ −iħGaA − oscillator. For W c > G m , k the system enters the 41 kHz, 51.8 MHz, 15 MHz), placing it we
2m eff W m ð−iD ′ þ G ð4Þ strong coupling regime (22, 23) investigated re the resolved sideband regime (25). To prob
Weis m =2ÞX ¼ −iħGaA
et al., Science −
c > G m , k the system enters the 41 kHz, 51.8 MHz, 15 MHz), placing it well i
ð4Þ strong
330,
coupling
1520
regime
(2010)
(22, 23) investigated re the resolved sideband regime (25). To probe
cavity cavity transmission transmission spectrum spectrumin in the the presence
a control beam, the Ti:sapphire control la
A
C
a control beam, the Ti:sapphire control lase
A
C
frequency frequencymodulated modulatedat at frequency frequencyW Wusing
Control Control
broadband broadband phase phase modulator, modulator, creating creating two two si
Probe Probe in in field field
A
Optomechanically − A − and X, which require in the steady state (SOM
induced transparency
Eqs.
and X, frequency an atomic system is taken by the op
S26 which and S27)
require in the steady state (SOM frequency an
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coupling atomic system rate W is taken
Eqs. S26 and S27)
c ¼ 2aGx by the zpf , where
optomechanical
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coupling rate W c ¼ 2aGx zpf , where
RESEARCH
Probe Probe out out
LETTER
x(t) x(t)
B
D
1
1
B
D
Cavity RESEARCH LETTER
Cavity c
a
b
mode mode
REPORTS
x(t)
+ â x(t)
Δ
OMIT ñ d
SC
Δ
1.5
atomic and supercond
OC â
spectra
ñ
Δ
Fig. 3. Observation of A
ñ b d
a
OC
formation of hybrid qu
m
i
Probe i
B
OMIT. (A) Theoretically expected
intracavity probe
2 1 0 +1 +2Probe
Slow light with OMIT
Signal
laser
1
laser
s
METHODS SUMMA
G, c
0.98 1.00 1.021
0.98 1.00 1.02
0
1
power, oscillation amplitude
X, normalized probe probe power
t r  2
m
Intracavity e , ∆/2π = 69.1 MHz SafaviNaeini et al. Nature 472, 2 μm 69 (2011). Fabrication. The nanobeam
s
0
1
b om = C i
1.0
3.0
b
0.7 0.8 0.9 1.0 1.1 1.2 1.3
wafer from SOITEC (resisti
0.7 0.8 0.9d
1.0 1.1 1.2 1.3
Probe laser offset frequency Ω/Ω t r  2
power transmission jt′ p j 2 2.5
buriedoxide layer thickne
,
Optical frequency
m
48 μW
Optical frequency
i
Probe laser offset frequency Ω/Ω
2.0 m 0
beam lithography followed
and the normalized homodyne
signal jt′ hom j 2 1.5
0
Optical
Optical bath
1
transfer the pattern throug
as a
Fig. 1. waveguide Optomechanically induced / waveguide
Fig. 1. Optomechanically induced transparency.
transparency. ∆/2π = (A) 57.6 (A)
A generic
AMHz
generic
optomechanical
optomechanical
system
system
consists
consists
of
of
an
an 24 μW
1
Probe then frequency undercut using an HF
optical cavity with a movable boundary, illustrated here as a FabryPerotñ type resonator in which one
Probe frequency
function of the modulation
0.5
1.0
Mechanical optical cavity with a movable boundary, illustrated here as a FabryPerotñ type resonator in which one
cleaned using a piranha et
frequency W/2p (top to
mirror acts like a massonaspring movable along x. The cavity has an intrinsic photon loss 0.5
oscillation mirror acts like a massonaspring movable along x. The cavity has an intrinsic photon loss rate
rate
k
k 0 and is
0 and is 13 μW
nanobeam will be discussed
0
bottom panels). The first
coupled to an external propagating mode at the rate k ex . Through the external mode, the 0
amplitude coupled to an external propagating mode at the rate k
resonator is
ex . Through the external mode, the resonator is1
two panels have additionally
been normalized to ‘levels’ that represent systemthe is optical probedmode by a â, weak the mechanical probe fieldmode system is probed by a weak probe field sent
Figure
0
1  Optomechanical populated with system. a control a, Leveldiagram field (only intracavity picture, ∆/2π showing field = 51.8 isthree
shown). MHz
populated with a control field (only intracavity field is shown). The
The
response
response
of this this
driven
driven
optomechanical
optomechanical
Fig.
Fig.
2.
2.
Optomechanical
Optomechanical
system.
system.
(Top)
(Top)
A toro
A to
sent
toward
toward ^b
and ^b
Experimental setup. We
d . c, Series of Optomechanica scanning electron micrographs, crystal showing large array of optically pumped system at
3.746 3.748
and the the ‘bath’ cavity, optomechanical the transmission crystal of which nanocavities (i.e., the (topleft returned panel), microcavity zoomedin image is used of toSupplementary demonstrate Information
1
the cavity, the transmission 0 of which (i.e., the returned
Δ 3.750 3.752
SC (GHz)
microcavity is used to demonstrate OMIT:
OMIT:
The
Th
unity. When the twophoton
resonance condi
signal and control beams are indicated by blue and red doubleheaded arrows, cavity device (topright panel). d, From top to0
bottom: onator scanning is coupled electron to the sideband control atand frequency probevfie
resonance
of optical waveguide modes. The transitions between modes driven by the device array
field
field
ì Probe
ì Probe
outî
outî
) is
)
analyzed
is analyzed
here.
here.
(B)
(B)
The
The
frequency
frequency
of
of
the
the
control
control
field
field c (bottomleft panel), and zoomedin image of topview of single an electrooptical modulato
is detuned
is detuned
by
by
D from
D from
the
the
cavity
cavity onator is coupled to the control and probe
resonance
frequency,
frequency,
where
where
a detuning
a detuning
close
close
to
to
the
the
lower
lower
mechanical
mechanical 50
sideband,
sideband,
D ≈−W
D ≈−W m , is chosen. The using a tapered fiber. The optical mode s
c
tion D′ = 0 is met, the respectively. Wavy black arrows indicate decay from the different modes. See micrograph of a zoomedin m , is chosen. The1
probe
probe
laserís
laserís
frequency
frequency
is offset
is offset
by
by
the
the
tunable
tunable
radio
radio
frequency
frequency
W
W
from
from
the
the
control
control
laser.
laser. ∆/2π
region
=
The
The 44.6
showing
dynamics
dynamics MHz
the OMC usingdefect a tapered region; finiteelementmethod
40 (FEM) simulation results for thethrough optical field radiation showing pressure thejD OC force j?k, to it isthe effectively mechan fi
fiber. vThe LI /2poptical 5 89 kHz. mode Since coup t
of
of through radiation pressure force to the mech
mechanical oscillator is
text for definitions of symbols in a and b. b, The control beam at v c drives the
transition 0 between interest
interest the occur optical occur
when
when andthe mechanical the
probe
probe
laser mode, laser
is tuned
is dressing tuned
over
over the the optical the
optical
optical andresonance resonance electrical offield the the intensity cavity,
cavity,
which E(r) which ; has FEMsimulated has
a linewidth
a linewidth mechanical radial
radial
breathing mode breathing with mode the mode total of signal of
the
the beam structure.
structure. at v In
In
this
th
excited, giving rise to
c 6 D SC (wr
mechanical
of
of
k =
k
k
= k 0 + k ex .(C) Level scheme of the optomechanical system. The control field is tuned close to redsideband
0 + k
geometry, the cavity transmission, defined b
destructive interference 1 modes, resulting in the dressed state picture with dressed modes â
transitions, ex .(C) Level scheme of the optomechanical system. d displacement 30 Q(r)  shown.
The control field is tuned close to redsideband
transitions, in
optical cavity and is reflecte
geometry, the cavity transmission, defined by
in
which
which
a mechanical
a mechanical
excitation
excitation
quantum
quantum
is annihilated
is annihilated
(mechanical
(mechanical
occupation
occupation 1
ratio
ratio
of
of
the
the
returned
returnedNew probefield
probefield Focus PIN
amplitude
amplitude photodio
divi
d
of excitation pathways for
∆/2π = 35.4 MHz
an intracavity probe field. cooperativity n
nis m → n m − 1) when a photon is added to the cavity (optical occupation n p → n p + 1), therefore coupling the
the incoming probe field is simply given
m →defined n
corresponding m − 1) as when C ; a4G photon 2 20
/kc i for is an added optical to the cavity cavity decay (optical rate of occupation In order n
energy eigenstates. The probe field probes transitions p →to n characterize
in which p + 1), therefore coupling nearresonance the byoptical the incoming reflection probe of the amplifier where the compo
field is simply given by
The probe transmission k, and an intrinsic corresponding mechanical energy resonance eigenstates. damping The probe rate of field c i . probes transitions cavity10
in which the
the
mechanical
mechanical
oscillator
oscillator
transmission
transmission
through
through
the
the amplified
system, a sideband of the control beam created using electrooptic
power
tapered
taperedand fiber.
fiber. sent
(Bottom)
(Bottom) to an osc
the twophoton detuning Un
therefore exhibits an inverted
dip, which can be studies as an a
0 The drivedependent occupation
occupation
is loss unchanged.
is unchanged. rate c om (D) has (D)
Transmission been Transmission viewed in ofmost the
the
probe previous probe
laser
laser
power through modulation through
the
the
optomechanical (see optomechanical Methods system and system Supplementary 0
the
the
chosen
chosen
waveguidetoroid Information), waveguidetoroid
coupling
coupling
conditions,
conditions
20 40 the case of a60critically coupled 80 cavity k 0 = k ex 20
a function
30
of
40 0
normalized
50
probe
60
laser
70
frequency
80
Additionally, by using a
offset, is a nonzero probe power transmission t r  2 thl
in incoherent, the case of aquantumlimited critically coupledloss cavity channel, k
when the control field is off (blue lines) 0 = kand and ex aswas a function used of
forming0 normalized
a weak 50
probe
signal
laser
beam 100
frequency
with
offset,
tunable 150 frequency 200
is a nonzero
v250
s .
probe
The results
power transmission t
on (green lines). Modulation Dashed Frequency and full lines (MHz) Control power (μW)
sidebands relative to
correspond to the onance. The control field induces an additional r  2 the atc
easily identified in the in recent Modulation experiments Frequency (MHz)
when the control
to cool
field
the
is
mechanical
off (blue lines)
resonator
and on
close
(green
to
lines).
its of
Dashed
measurements
and full lines
performed
correspond
at a cryogenic
to the onance.
temperature
The control
of 8.7
field
K are of induces the group an additional delay imparte tra
homodyne signal. (B) Experimentally
observed normalized the dressed homodyne state view traces of EIT when 7 Δ
quantum ground models based on the full (Eq. 1) and approximative (Eq. 5) calculations, respectively.
parency window with a contrast up to 1 − t r 
models
state
based 25 .
on
In
the
the
full
dressed
(Eq. 1)
mode
and
picture,
approximative
by analogy
(Eq. 5)
to
calculations,
shown
d
15 μW
in
respectively.
Fig. 2. Here, the control e beam laser
parency
power
window
was varied
with
from
a contrast
cavity.
up to 1 − t 2
r  2 .
, it thebecomes probe clear beamthat sentato coherent the cavity is 0.5 6 mW(Æn in c æthese 5 25) measurements. = to ñ OC m
nearly 250The mW(Æn middle c æ 5panel
1,040). The frequencies of
10
frequency is scanned by sweeping cancellation the phase of the modulator loss channels frequency in the dressed W for optical showsand the operating mechanical conditions both where the control the control and signal beam is beams tunedare to the swept lower in order to map out the Received 8 December 2010
Control laser
Control laser
Probe Probe
field field
Probe power transmission t p
 2
Probe power transmission t p
 2
Reflection advance (μs)
Transmission delay (ns)
Probe
Probe
Control
Control
Reflection advance (μs)
ity
Control Control field field
Probe Probe
field field
Probe in
Probe in
Probe out
Probe out
Published online 16 March
Solid State Atomic Ensembles
te Atomic Ensembles
Photon echo storage techniques
s doped into
crystals.
raseodymium Rareearth atomic ensembles
An ensemble of rareearth ions doped inorganic crystals
Large number of atoms
Ions trapped in crystalline structure
Excellent coherence properties
at cryogenic temperature (T 1s
ed d Reversible Inhomogenous Broadening
Nature 465, 1052 (2010)
Controlled Reversible
Inhomogenous
Broadening (CRIB)
PHYSICAL REVIEW A 79, 052329 2009
Multimode quantum memory based on atomic frequency combs
Mikael Afzelius,* Christoph Simon, Hugues de Riedmatten, and Nicolas Gisin
Group of Applied Physics, University of Geneva, CH1211 Geneva 4, Switzerland
Received 9 June 2008; published 21 May 2009
An efficient multimode quantum memory is a crucial resource for longdistance quantum communication
Atomic frequency comb
Photon echo technique
demonstrated in quantum regime
olid State
General Ensembles
Photon echo
Idea storage
: PhotonEcho techniques
Techniques
State Atomic Ensembles
te Atomic Ensembles
oped into
stals. Making use of static Inhomogeneous inhomogeneous dephasing
seodymium
•!
Natural inhomogeneous dephasing: Nuclear spin bath
•! Large QM bandwidth
Collective state :
Classical 2pulse photon echo
•!
2pulse photon echo In the Bloch sphere
Not a
:
good QM strategyÖ
 Strong optical pulse in the
quantum channel
Rephasing of the atomic dipoles triggers collective emission of the absorbed signal
Very fast decay of the alignement of the atomic dipole
Needs a controlled rephasing process
animation
 Contamination of the singlephoton
echo by unavoidable
fluorescence
Rephasing at a certain How time to of avoid all these dipoles triggers
the reemission contaminations of the absorbed ? signal
2 ways : CRIB and AFC
Ruggiero, Gouet, Simon, Chaneliere, Phys. Rev. A 79, 053851 (2009).
Illustrations from W. Tittel et al., Laser Photonics Review 4, 244 (2009)
Sangouard et al., Phys. Rev. A 81, 062333 (2010).
•!
A detailed analysis:
Twopulse photon echo is not suitable for quantum storage
•! Strong optical pulse in the quantum channel
Contamination of the singlephoton echo by unavoidable fluorescence
CRIB
AFC
Controlled Reversible Inhomogenous Broadening
Controlled Reversible Inhomogeneous Broadening (CRIB)
Controlled Reversible Inhomogenous Broadening
Preparation
CRIB
CRIB
Controlled Reversible Inhomogenous Broadening
CRIB
Reversal
Singlephoton absorption
Collective rephasing
Singlephoton retrieval
Illustrations from W. Tittel et al.,
Laser Photonics Review 4, 244 (2009)
Illustrations from W. Tittel et al.,
Laser Photonics Review 4, 244 (2009)
Illustrations from W. Tittel et al.,
Laser Photonics Review 4, 244 (2009)
Coherent optical pulse sequencer for quantum applications
Hosseini et al. Nature 461, 241 (2009)
Illustrations from W. Tittel et al.,
Laser Photonics Review 4, 244 (2009)
Illustrations from W. Tittel et al.,
Laser Photonics Review 4, 244 (2009)
Atomic Frequency Comb (AFC)
PHYSICAL REVIEW A 79, 052329 2009
AFZELIUS et al.
Multimode quantum memory based on atomic frequency combs
PHY
Mikael (a) Afzelius,* Christoph Simon, Hugues de Riedmatten, and Nicolas Gisin
Group of Applied Physics, University of Geneva, e CH1211 Geneva 4, Switzerland
PHYSICAL REVIEW A 79, 052329 2009
Received 9 June 2008; published 21 May 2009
An efficient multimode quantum memory is a crucial resource for longdistance quantum communication
(a)
based on e quantum repeaters. We propose a quantumtial memory
mode
based
defined
on spectral
by
shaping
the direction
of an inhomogeneously
of propagation of the input
broadened optical transition into an atomic frequency field, comb j AFC. the detuning The spectral ofwidth the of atom the AFC withallows
respect to the laser
efficient storage of multiple temporal modes without frequency, the need to increase and the theamplitudes absorption depth c j ofdepend the storage on the frequency
material, in contrast to previously Longlived known quantum storage memories. and on the Efficient spatial readout position is possible thanks to rephasing
Preparation of the particular of atom j.
of the atomic dipoles due to the AFC structure. Longtime storage and ondemand readout is achieved by use
Atomic The Frequency collective
of spin states in a lambdatype configuration. We show that an AFCSpectral state s
quantum memory hole
canComb be
realized burning
understood solids (optical (AFC)
as a
doped pumping)
coherent ex
modes or
with rareearthmetal ions could store hundreds ofcitation of
more
a large
with close
number
to unitof efficiency,
AFC modes
for material
by a single photon.
s
with a modelocked laser
parameters achievable today.
These modes are initially
Atomic Frequency Comb (AFC)
g
at t=0 in phase with respect to
In the frequency domain, preparation
the spatial mode k. But the collective state will rapidly
Atomic DOI: g 10.1103/PhysRevA.79.052329 Frequency Comb PACS numbers: 03.67.Hk, 42.50.Gy, 42.50.Md
dephase(AFC)
of periodic absorption lines
into a noncollective state that does not lead to a
strong aux collective emission since each term acquires a phase
aux I. INTRODUCTION
of atoms on an optical transition which is inhomogeneously
expi
The distribution of entanglement between remote locations
is critical for future longdistance quantum networks
broadened. j t depending on the detuning
Ensembles are in general very j of each excited atom.
Atomic detuning
attractive as QMs
Atomic detuning
If we due consider to strong collective an AFC enhancement having very of the sharp lightmatter peaks, coupling
2. For j Storage are longlived approximately of single photons a using discrete EIT stopped set such light that
then the
(AFC)
detunings
and extended tests of quantum nonlocality. It is likely rely
ï Collective state t=0 j
(b) on quantum repeaters 1,2,
(b)
which require quantumAtomic memories
QMs that can store entanglement between distant net
=m j , haswhere been Frequency demonstrated with cold alkali atoms 6,8 which
storage m j are integers. ItComb follows that(AFC)
the collective
can be treated as homogeneous ensembles of identical atoms.
Input
Output
state is reestablished after a time 2/, which leads to a
Control fields Input
work nodes 3,4. Recent experimental achievements in
Here we will instead
Output
mode
mode
consider ensembles of rareearthmetal
coherent Control fields
quantum state storage
For longlived mode
photonecho
5–8 demonstrate that currently investigated
QMs can store a single mode. Yet, longdistance
RE ions in solids, mode 25–27 type reemission in the forward
which are inhomogeneously broadened.
For longlived ï Collective state at
Stopped direction.
t=0
ï Collective state lightThe at with t=0 efficiency storage times of up thisto process 1 s has been seedemon
strated caninreach REdoped 54% solids in the22,23, forwardwhere direction an approximate limited only
Sec. III for
storage Time
quantum 2 / T
details
repeaters having QMs that are only capable of storing
one mode would only generate very limited entangle
by reabsorption. homogeneous ensemble But if the was ï created reemission
0
T T storage
s
0
Time After bya spectrally time
is forced
t (dephasing)
2 / T
isolating to propa
narrow in the absorption backwardpeak direction, through optical by a proper pumping. phase In contrast matching
a
0
T s
T0
FIG. 1. Color ment online generation Therates principles 9. To achieve of the proposed useful ratesAFC
some waygate
of multiplexing the QM will be required 10,11. By using
to this approach, the quantum memory proposed here uses
quantum memory. a An inhomogeneously
FIG.
broadened
1. Color
opticalonline transition
ï •! Collective ge is shaped
The
operation
time 11, spatial 10,12,13, or multiplexing to
the principles
see below,
inhomogeneous of the
the
broadening proposed
process
as a resource AFC
can reach 100% efficiency.
in order to
Initial state into
store collective at t=0 an AFC by frequencyselective optical
single photons state quantum in many modes memory. N,
ï
the
After a entanglement An a time ï inhomogeneously After t (dephasing)
achieve a time better t (dephasing) For multimode performance. For that one needs to
The
generation rate can be increased by a corresponding factor N
coherently processcontrol described
broadened longlived
the dephasing so far
optical
only whichimplements transition
pumping to the aux level. The peaks in the AFC have width
is caused by athe QM inhomogeneous
storage by frequencyselective storage
with
FWHM and
11.
are
Here
separated
we consider
by ,
multimode
wherege we
QMs
define is shaped
capable
the combinto an
of storing Na fixed AFC time. In order distribution to allowoptical
offor theondemand atoms. The
ciency.
key readout
finesse as F=/. temporally b The distinguishable input mode pumping is modes, completely which to the absorbed is aaux natural andform level. of ofThe the feature stored peaks of our field inproposal which the AFC is to is achieve a have necessary this width control requirement by a specific for use
coherently excites sending the AFC information modes, also which usedwill in today’s dephase telecommunications
and then in quantum shaping of repeaters this distribution and longterm into ï Rephasing an atomic storage, frequency after a
the comb. time
FWHM and are separated by , where we define the comb single collective
rephase after a time networks. 2/, Time resulting multiplexing in a photonecho is extremely type challenging coherent when
The paper is organized in the following way. In Sec. II we
finesse as F=/. b The input
using current QM protocols such as stopped light based on
give mode excitation
an overview is completely in
of
e
the
is
proposal
transferred absorbed using a
toand
simplified
a groundstate
physical
spin
emission. ï After Aa pair time oft control (dephasing) fields on es allows for longtime level
electromagnetically induced transparency EIT 14, photon
picture. s. This In Sec. canIIIbe we done show results by applying from an analytical optical analy
onof es, the after physics for a time instance of atomica frequency short pulse. combs, The which excitation is furcontrol
storage as a collective spin wave coherently s, and ondemand excites readout the AFC modes, which will dephase and then
ï Rephasing fieldsis is
y Comb (AFC)
d
AFZELIUS et al.
Atomic density
Intensity
Input mode
Output mode
Control fields
Atomic density
Intensity
Input mode
tial mode defined by the d
field, j the detuning of
frequency, and the ampl
and on the spatial positio
The collective state ca
citation of a large numbe
These modes are initially
the spatial mode k. Bu
dephase into a noncolle
strong collective emissio
expi t depending on th
detunings j are approxi
=m j , where m j are inte
state is reestablished af
coherent photonecho 2
ward direction. The effici
details can reach 54% in
gate in the backward dir
ï
j
If we consider an AFC
by reabsorption. But if
•! Timeevolution •! Collective rephasing state operation t=0 with see delay below,
Output mode
Control fields
The process described
a fixed storage time. In or
of the stored field which
in quantum repeaters an
ï After a time t (dephasing)
5 using a more conservative esndix.
Both estimations yield a
ter than 0 by at least one stanthat
entanglement was indeed
crystals. This measurement reod
in which two threefold coin
The prohibitively long integraprevented
us from attempting
ers (i.e. for lower probability of
pair). Hence, to study how the
pump power, we used a second
coincidences, which we now de
, p 11 is estimated using a sup
(see the Appendix for details).
ted by the results obtained in
e assume that all the observed
twomode squeezedstate, and
Ideally,
ime crosscorrelation ḡ s,i can be
p, where p ⌧ 1 and p 2 are inilities
of creating one and two
ly. We then proceed as follows
st measure the zerotime crossdetections
in the idler mode and
h mode B blocked). Then we
way (with mode A blocked) and
e calculate the average of gs,i
A
d estimate p 11 using
= 4p 10p 01
ḡ s,i 1 . (2)
ide justifications and additional
ort our assumption and give evwer
bound on the concurrence.
red the secondorder autocorre
Heralded quantum entanglement (AFC)
a
1.0
Heralded quantum 0.9 entanglement between two crystals
800
Usmani, Clausen, Bussieres, Sangouard, Afzelius, Gisin
0.8
600
arXiv:1109.0440 (2011)
b
c
Visibility
"
ḡ s,i
Concurrence (10 4 )
0.7
0.6
0.5
0.4
30
25
20
15
10
5
0
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Coincidences
400
200
0
0.25 0.30 0.35 0.40 0.45
Phase (a.u.)
0 2 4 6 8 10 12 14 16
Pump Power (mW)
Entanglement between two crystals only exists transiently.
Nonetheless, FIG. significant 2. Results. advances a, Visibility with photon as a function echo of technique pump power. in the quantum regime
‘Quantum’ storage benchmarks
Quantum vs. classical recording?
⇢ = ⇢ 0 + ⇢ ent
F = h ⇢ ent  i 2
F = h ⇢ i 2
Fidelity for measureconstruct strategy
(~ teleportation fidelity)
For qubits,
For coherent states,