Atom - KIAS

newton.kias.re.kr

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: Model-independent verification

of non-classicality 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

ight-Matter 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 off-diagonal 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 – atom-light interaction in the quantum regime

•!

Single atoms and photons are very simple systems

•! A hydrogen atom

2p

1s

•!

Harnessing atom-light interaction in the quantum regime allows us to examine

non-trivial quantum behaviors such as Schrodinger cat states and entanglement

for open quantum systems

•! Strong matter-light 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

Spin-boson 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

Spin-boson 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 matter-light interaction

ct the optical cavities that we use in the lab from high-finesse 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

p-orbital

Control logic D h

IN

s-orbital

Write

Quantum no

a

r

g = ~µ · ~E ~!a

= µ , (1.4)

2✏ 0 V m

cavity Cavity modeQED volume, – Atom-cavity 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

• “Atom-Photon Interactions” & “Photons and Atoms”

by C. Cohen-Tannoudji, J. Dupont-Roc, 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 – Atom-cavity molecule

Quantum regime of

strongly coupled single atom and one photon

P in

P out

INPUT

OUTPUT


The electric Jaynes-Cummings field The within atom-cavity

interaction the system:

Hamiltonian cavity: describes the coupling of a two-level

Jaynes-Cummings coherent coupling model: coherent coupling

atom to a single cavity mode [3]:

cavity field

The atom-cavity system:

coherent coupling

field raising operator

Atom-cavity 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 Jaynes-Cummings Hamiltonian, field written here in the reference frame of the

product product basis where basis|g, where ni and |g, ni |e, and n |e, 1i nare 1i n-excitation are n-excitation states states with an with atom an atom in thein th

probe:

ground ground (excited) (excited) state and Jaynes-Cummings 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 n-excitation pair of n-excitation states, states, leading leading to

+∆)+✏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 +∆),

(1.1)

atomic lowering

square of the mode waist. Thus, we should minimize th

width half-maximum (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 n|e, n n|e, 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


Jaynes-Cummings Hamiltonian

model:

into a master

steady-state

equation ˙⇢ = L⇢ for the

spectra

density matrix

system, where L is the Liouvillian superoperator [6]:

System: atom-cavity 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).

Steady-state 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 steady-state 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 vacuum-Rabi splitting — is well-resolved. 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 cavity-mediated 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

NATURE|Vol 436|7 July 2005

Single-photon 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 normal-mode 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 time-resolved 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

centre-of-mass 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 radio-frequency 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 time-evolution

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 atom-cavity detuning AC = F A =(! F ! C )

Figure 1.2: Steady-state transmission of the atom-cavity s

and

but

apply

a fixed

(2.21):

atomic resonance frequency. The resulting signal - a so-called “avoided crossing”

Figure 1.2: Steady-state and apply transmission (2.21): detuning

Time-domain of theRabi atom-cavity 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 atom-cavity 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 afrequency-domain want dynamics master

to look of strong the normal-mode coupling temporal regime evolution splitting, which of thewas system calculated where the

Not surprisingly, we findsystem, equation

the frequency-domain where ˙⇢ = LL⇢ normal-mode splitting, which was calcu

above, population corresponds is initially to a time-domain placed in g 0

aRabi superposition ( oscillation , apple,Tof of 1 is for the the Liouvillian density matrix superoperator of the [6]:

atom-cavity population ). between eigenstates states (i.e., |e, nentirely 1i in

system, where Labove, is thecorresponds Liouvilliantosuperoperator a time-domain 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 atom-cavity

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

nes-Cummings model, and because the coherent dynamics of the atom-field 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 re-written

rates, i.e.,

using the so-called

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

Fabry-Perot 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

•! Quantum-classical 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. Stamper-Kurn, 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


Quantum-state 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

Time-reversal 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)


Quantum-state 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)


Single-photon 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 two-photon

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 phase-referenced 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 vacuum-stimulated Raman adiabatic

Deterministic photon-photon 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 cavity-QED and

one atom is ing the Stark-shifted F ¼ 2 $ F 0 ¼ 1 transition and the

trapped by measuring a perfect photon

tocol inherently deterministic | an essential

Photon-Photon 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. high-finesse 102, 030501 optical(2009)

cavity, the resulting entanglement is

q, 42.50.Xa

ed for

Control fields

Quantum-state 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 (vSTIπ-polarized

laser pulse addressing the Stark-

PRL 102, 030501 (2009) PHYSICAL REVIE

buted

wards

inherently deterministic [1,3]:

no

en

2 ↔re-F 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 user-selected 030501 (2009) time interval, the PHYSICAL atom-photon entan-

is converted into a photon-photon 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 standing-wave 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 two-photon

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 user-selected time interval, the atom-photon entan-

T

state of the photons with a delity ofF =0: 902 0: 009. a

FIG. motional cooling, while additional beams for optical pumping

r-selected 2 (color online). time interval, The experimental the atom-photon 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 aphoton-photon 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 photon-photon 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 π-polarized

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 atom-photon 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, thea-polarized 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, 030501-2 λ= 4: expectation value Eð;

F

], our

j PP i ¼ p 1 Þ of correlation measurements in of twt

, and

quarter-wave plate, λ= 2: half-wave 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 read-out coupling) of Storage a and retrieval of polarization qubit

weak weak coherent pulse pulse with with

Storage and read-out 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 atom-cavity 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 atom-by-atom and photon-by-photon

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 QED-based quantum computation

Single-photon single-atom nonlinear phase shift (phase-gate)

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)

Fault-tolerant quantum repeaters with single-photon 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 free-c µm

apart; the •! Confinement mirror facesand are enhancement 1

density

mm in diameter,

waves

of a

guided

single-photon withalong a 10a cm

conductor-dielectric

(evanescent radius of curvature, wave)

interface

and

coned so that they can be brought •! Infinite-range close together. interactions The substrates are held in BK7

Simplest example: SPs on a flat surface

v-blocks glued to shear-mode•!

piezoelectric Single-photon 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

two-color 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 blue-detuned beam enable

d in (b) the plane. Specifically, U

trap

unter-propagating red-detuned beams,

ed beam in a so-

as shown in the inset. The standing

e attractive red-detuned 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 red-detuned standing-wave 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 free-electron of thedensity atomicwaves

dipole and the

Guess solutions

•! guided along a conductor-dielectric interface

electric field within the cavity:

j

E Ee 0 ˆ

ˆ

Free-space 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 nano-wire traps, [34,35], based on far off resonance laser traps

Drude model:

e diffraction limit for conventional free-space optics. In this section, we

ed research with nano-wire 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 free-space

ing atom chips [33].

2 ( ) 1

We construct the optical cavities that we use in the lab from high-finesse 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 nano-wire of diameter d 500 nm. As

ptical wavelength. The potential trap

U for such a nano-wire 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

• “Atom-Photon Interactions” & “Photons and Atoms”

by C. Cohen-Tannoudji, J. Dupont-Roc, 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 sub-wavelength 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

sub-wavelength 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 two-level 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 single-atom 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 sub-wavelength } | 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

•! Single-atom single |S, mi photon = coupling constant:

2✏ 0 V ~✏ · ~✏ a is the single-atom 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 maximum|e 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, = m|S, 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 sub-wavelength

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 sub-wavelength |si in thecondition ensemble. eb, i~ k·~r z |S, mi = m|S, 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 many-body quantum the context many-body theory of quantum in chapter theory many-body 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 many-body 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 slowly-v

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, Light-matter 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 sub-wavelength 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 = m|S, 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 = m|S, 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 Enon-ideal 0m angular of= the m~! symmetric superradiance momentum 0 of the symmetric inS.

the presence of dipole-dipole 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

sub-wave2

! 2.2.3 Superradiant 2.2.3 ✓ Superradiant ◆ ✓

emission foremission an atomic 2.2.3

for " ensemble an ◆

Superradiant

atomicin ensemble| a emission sub-wavelength in for

a sub-wavelength {z

an atomic ensemble i

Since the system-reservoir 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

System-reservoir 2 Ŝ+ 0 Ŝ0 + 0

coupling

k )t Since the system-reservoir Hamiltonian is Ĥa = P + h.c.) in

Since the system-reservoir Hamiltonian

~ k

(~g ~k Ŝ 0 + â~ k

e i(! 0

Since the system-reservoir 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 Born-Markov


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 Born-Markov following approximation

⇣ equation photons, (in the respectively. d

dt ˆ⇢ 1 R

Born-Markov 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 time-evolution 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··· time-evolution ,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 Pencil-shaped (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, Pencil-shaped 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 = of|S, 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 single-atom 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 single-atom 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 theWigner-Weisskopf single-atom {Ŝ± the short-time ( t) evolution of the atomic state ˆ⇢

spontaneous emission rate in the Wigner-Weisskopf a (t)

0 (t), the Ŝz(t)} master canequation be solved(Eq. analytically 2.16) in the fromsemi-classical app

theory of spontaneous decay)

th

the master (Eq. equation 2.16) 0/(3⇡✏ (Eq. 2.16) 0 ~) in is the semi-classical single-atomapproximation. spontaneous emission Using 2.13–2.14), the commutator ratewe inobtain the relationships Wigner-Weisskopf the following (Eqs. differential theoryequatio

of

spontaneous decay. Short-time

Indeed, in the quantum jump picture, westates evolution

can write withthezero short-time mean

in the Tothermal ( describe quantum-jump

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 semi-classical

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 semi-classical photons, respectively. approximation Since(i.e., the collective taking operators jump operators as c-numbers), 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 dipole-dipole 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 dipole-dipole coupling g

momentum S) of ˆ⇢ a (t), the time-evolution of ˆ⇢ a (t) from the initially van der Waals symmetric interaction state (Eq. | (t 2.15) = 0)i has with characteristic total dipole-dipole coupling vdW

0 10⇡ r For |r| ⌧ 0 , the frequency shifts of this dipole-dipole 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⇢ Semi-classical 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' time-evolution 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

•!

Non-symmetric collective damping

Quantum analogue of Bragg reflection of light

from atomic phase grating

For sub-wavelength 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 second-order (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 sub-wavelength current •! Quasi discussion, 1-dimensional

. 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 '(end-fire 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 sub-wavelength samples |r| < 0 ,

n process, the ‘pencil superradiant shaped’ sample emission can of be well approximated to a one-dimensional 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 (so-called “end-fire 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 “phase-matched.” Superradiant In this case, Rayleigh the so-called 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 phase-matching 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).

Single-pulse 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 matter-light c interaction

ct the optical cavities that we use in the lab from high-finesse 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

• Duan-Lukin-Cirac-Zoller approach

a Quantum

Reversible

channels

QM

b

• Dynamic EIT quantum memories

c

•! Off-resonant-Raman-based

quantum memories

d

• Photon-echo quantum memories

b 2

From superradiance of extended atomic samples

g

Fields 2

g

e

p-orbital

s-orbital

p 1000

p 0100


Matter-light quantum interface

What’s inside here?

•! Ensemble of ~ 10 6 Cs atoms released from MOT

•! Utilize strong interaction

of single-photons and

collective spin excitations (in the single-excitation regime)

•! Input-output coupling k(t) user-controlled externally by

lasers

Non-collinear geometry

Harris group (2005)

Writing and reading collective spin waves

Read-out processes

EIT, off-resonant 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 atom-light

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-

Pencil-shaped 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 atom-photon 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){~ ˆd|si. 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 atom-photon 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 ˆd|si.

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 atom-photon 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| ˆd|si.

q with single-atom 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 theatom-photon 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. ˆd|si. 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 atom-field coupling const.

es

es 2~✏ 0 V 1

is the atom-photon 2~✏ 0 V 1

is the atom-photon coupling constant with dipole

ver miltonian a small volume i containing N ~r

coupling constant with dipole

matrix element

trix element d es = he| ˆd|si. d es = he|

We take the ˆd|si.

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 single-atom 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 ground-state 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 single-atom 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 single-atom 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 single-atom 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 ground-state 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 ground-state (~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 single-atom operator ˆ(i) ˆµ⌫ (~r 0 ,t)] = V coherence

1

~r 0 {ˆgs , ˆsg } follows the Bosonic commutator relations

)(

µ⌫

with single-atom

=

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 ground-state 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 Heisenberg-Lanvegin equations

1

1

(~r ~r 0 ~r

)ˆgg (~r, t)+O

particular, The total theHamiltonian hyperfine ground-state including Collective

coherence the spin-wave 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 Heisenberg-Langevin limit gg ' 1 approach

ee, ss.

1

(~r ~r 0 )( µˆ↵⌫ (~r, t) is expressed

⌫↵ˆµ

in(~r, terms t)). of the normalized slowly-varyin (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: Heisenberg-Langevin 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 ' of1self-consistent equations

1ee, 1

Ĥ (~r

ss.

~r 0 tot = )ˆgg (~r, t)+O

N

particular, the hyperfine ground-state coherence ~r N ~

{ˆgs , ˆsg } follows the Bosonic ~r

2 (2.25)

2.3.1.2 Heisenberg-Lanvegin 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 ground-state 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 Heisenberg-Lanvegin

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⌫ . Fluctuation-dissipation (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 steady-state solution for Heisenberg-Langevin equation of motion (Eqs. 2.30–

nian (neglecting the noise terms and assuming constant atomic distribution n A (~r) =N A /

Parametric atom-light interaction

2.3.1.3 Adiabatic elimination of excited state

Z

2.32). If we assume the far off-resonant limit w se, eg and the narrow-bandwidth 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 phase-matched state |ei andslowly-varying obtain the steady-state spin-wave solutions amplitude, for the and

p Ĥ In (par)


eff the following, = N A

wed~r

solve

⇢~ the steady-state solution for Heisenberg-Langevin equation of m

•! p(~r, Steady-state t; optical w, w1) coherences solution ' g to (i.e., the @ tˆse Heisenberg-Langevin ⇤ 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 off-resonant limit w se, eg)

and the narrow-bandwidth

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 steady-sta

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 theNbare-state 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 phase-matched the noise terms slowly-varying and assuming ), and the constant spin-wa 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

atom-light

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 non-degenerate (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 matter-light 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 bare-state 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 two-mode 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 phase-matched

int

(t) 300 =~ K. p(t)Ê1Ŝ + slowly-varying ⇤ enhancement

p(t)Ê 1Ŝ†⌘ † spin-wave ( 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 non-degenerate

ˆgs (~r, t) is the phase-match

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 bare-state a

can generate a two-mode 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 matter-light 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~|⌦

) bare-state (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 Three-dimensional 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 bare-state

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 matter-light w | 2 / toeg 2 optical ⌧

interaction,

10 pumping 4 (weak (⇠ i~|⌦ w| 2

eg

can generate Generation a two-mode of matter-light entangledentanglement

state between the field 1 and the collective atomic2mode

denoted as w

Here, we derive a three-dimensional 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 non-degenerate 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 Hermite-Gaussian 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

Two-mode squeezed The initial state atom-field as quantum state |g int

(t) =~ p(t)Ê1Ŝ resource 29 for DLCZ + ⇤ p(t)Ê 1Ŝ†⌘ † ,

2.4 Two-mode a , 0 1

i in the Schrödinger’sw

picture evolves protocol

| i

R squeezed state as a quantu a


1

R ⌘

simplicity

2.4 Two-mode

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 two-mode ~ dtFor 0 Ĥ (par) a rigorous

0 int

(t 0 ) withtreatment the parametric of dissipation interactionand Hamiltonian propagation ef Ĥ(par int

of matter-light entangledentanglement

needs to solve the

state between the field 1 and the collective atomic mod

section self-consistent 2.3.1.3.

operation ˆD

The Heisenberg-Langevin ⇣col

final R ⌘

i 1

atom-field state equations (t !1) isingiven Eqs. by2.47–2.48, a two-modefrom 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 atom-field | state i |g a , 0 1

i in the Schrödinger’s pictur

2.3.2 Three-dimensional ⇣

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 atom-field 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 three-dimensional 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 number-states by n

) with the parametric 1 = for photons ⇠ (â

2.3.1.3. The final atom-field 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⇠ Hermite-Gaussian = 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 atom-field state (t !1) is given by a two-mode 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 non-deg

amplifier between a single-mode (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 Two-mode 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 super-Poissonian spin-wave 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 atom-light state:

needs to solve the self-consistent Heisenberg-Langevin

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 number-states 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 near-resonant 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 number-states 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 self-consistent rigorous treatment Heisenberg-Langevin 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,

atom-photon

t).

sta

needs to solve the self-consistent correlation Heisenberg-Langevin functions could single

evaluated

photon

equations from

source

in 0 Eqs. Einstein’s 2.47–2.48, relation

exhibits super-Poissonian spin-wave 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

...


Two-mode squeezed state: correlated 29

2.4 Two-mode squeezed photon state pairs as a quantu

2.4 Two-mode 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 12-33, 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 well-de ned spatio-temporal

t the output of the ensemble is demonstrated. These improvements are illustrated by the

ion of high-quality heralded single photons with a suppression of the two-photon 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 atom-field state |g a , 0 1

i in the Schrödinger’s pictur

loss in the propagation and the detection of field ⌘

The initial atom-field 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, atom-field 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 atom-field

2.2b),

state

with to light-light

(t

the

!1)

dark-state entanglement

is given

polariton

by a 86 ˆ

two-mode 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 matter-light 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 atom-light 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 Λ-type 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 eld-1 photon heralds the storage of a single

ributed among the whole ensemble. A classical read pulse can later, after a user-de 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 signal-to-noise 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 number-states ⇠ = tanh 2 for (i R the photons ⇠ (â †

⇣ 1

dt

where |n 1

i (|n a i) are the number-states 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 spin-wave 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; two-mode ⌧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

simplicitynon-classical p (t 0 )= 1 correlations, L

as demonstrated experimentally in refs.

L Violation 0 of Cauchy-Schwarz inequality (a)

72,73 , and be used a

needs to solve for quantum the self-consistent information correlation processing Heisenberg-Langevin 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 number-state basis. The field 2 and the field 1 can, in

g 12

needs dz p(z,t to solve 0 ). For thea self-consistent rigorous treatment Heisenberg-Langevin 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


Two-mode squeezed state: heralded 29

2.4 Two-mode squeezed single photons state as a quantu

2.4 Two-mode squeezed 4. Thestate mode operators as a quantum can be transformed resource nonlocally for to â 0 1,↵ D=

“Single-Photon 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 12-33, 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 well-de ned spatio-temporal

t the output of the ensemble is demonstrated. These improvements are illustrated by the

ion of high-quality heralded single photons with a suppression of the two-photon 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 atom-field state |g a , 0 1

i in the Schrödinger’s pictur

loss in the propagation and the detection of field

The initial atom-field 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, atom-field 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 atom-field

2.2b),

state

with to light-light

(t

the

!1)

dark-state entanglement

is given

polariton

by a 86 ˆ

two-mode 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 matter-light 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 atom-light 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 Λ-type 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 eld-1 photon heralds the storage of a single

ributed among the whole ensemble. A classical read pulse can later, after a user-de 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 signal-to-noise 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 number-states 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 number-states 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 spin-wave 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; two-mode ⌧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 )= number-state 1 L

dz basis. p(z,t The 0 ). field For2a and rigorous the

w w gs

. field Additional treatmen 1 can, in

R

simplicitynon-classical p (t 0 )= 1 correlations, L

needs

L 0

dz p(z,t to as solve 0 ). demonstrated For thea self-consistent rigorous experimentally treatment Heisenberg-Langevin of in dissipation refs. 72,73 , and andbepropaga

equatio used a

Photon antibunching

needs to solve for quantum the self-consistent information correlation processing Heisenberg-Langevin 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 single-photon 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 single-photon sources

Conditional control of heralded single-photon sources

Felinto et al. Nature Phys. 2, 844 (2006).

Synchronized Independent Narrow-Band 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 theHong-Ou-Mandel

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 time-interval 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 single-photon 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 single-mode 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 single-photon 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 single-photon 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, Short-time 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 atom-light 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)

•!

Dark-state polariton

Half-photon and half-matter quasi-particle excitation

•! Quantum description of low-light level

Electromagnetically Induced Transparency

Optimal control theory

Optimizing mapping efficiencies of quantum states


c

c

signal field

two-photon transition detuning |gi |ei), of called each of thethese signal atoms field, is coupled whereastothe a slowly-varying transition |si quantized |ei is resonantl radiation

by

Collective

the Hamiltonian Ês. The signal field

atom-light Ês propagates within the EIT window of the ensemble, provided by ⌦

interaction

c (z,t).

classical two-photon control Ĥ(map) s

field

in

detuning of

the

Rabi

rotating-wave 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

one-dime 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 rotating-wave frequency the transverse |ei of each

⌦ c approximation (with detuning profiles), these atoms

(following with is coupled to a slowlytwo-photon

detuning ), called the signal field, whereas the tran

2.5.1 Interaction s in the Hamiltonian rotating-wave approximation and formation(following of dark states the effective

c). The

one-dimensional

dynamics the effective of this

ap-

Here, we consider a collection of ⇤-level atoms interacting with the two single-mode optical fields. The one-dim syste

transition by the Hamiltonian Ĥ(map) s in the rotating-wave approximation (following the effective one-d

Here, |gi we|ei proximation

consider of eachaof collection these in section atoms of Z 2.3.2.2

⇤-level is coupled and

atoms neglecting a slowly-varying interacting

the transverse profiles), with

XN A ⇣ withquantized the two single-mode 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

two-photon

transition

detuning

|gi

),

c).

|ei Zcalled of each

the

of

signal

these

field,

atoms

whereas

is coupled

the

to

transition

a slowly-varying

|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

two-photon detuning ), called the signal field, whereas wâ w + the transition Ĥ(map) s in the rotating-wave

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 rotating-wave i=1approximation proximation(following sectionthe 2.3.2.2 effective andone-dimensional 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 Λ-level

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),

single-mode

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)

theesrotating-wave 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 one-dimensional+ ap-h.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 wave-vector ~✏ 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 wave-vector along ẑ (also, k

is the longitudinal projection of the wave-vector 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


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 idark-state 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 dark-state of dark-state polariton polaritonis is the k = ~ k

A simple explanation for the formation of dark-state 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 wave-vector particular, along single-exci ẑ (also, k

). 87 ). In In particular, the thesingle-ex

where Ê

|D, m ~ projection q of the wave-vector 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 single-excitation

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

Single-excitation

formation of the wave-vector of dark-state

dark-state 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

dark-state 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 single-excitation 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 dark-state 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 single-excitation (2.69) state

state

|D, 1i = cos ✓ d (t)|g a , 1 s i sin ✓ d

|D, m =1ibetween is p(ref. 87 )

q

single-photon 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 wave-vector 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 =

Heisenberg-Langevin ge + i

atom-light

⌦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

• Heisenberg-Langevin ~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)+~⌦ one-dimension

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 lowest-order 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

lowest-order | 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 Heisenberg-Langevin 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 one-dimension 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

one-dimension = ' 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 spin-wave 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 slow-light phenomena in static EIT (with stati

ˆgs

and in adiabatic an'

effective e

condition one-dimension i(kk c

k s )z

In Eq. 2.77, we recover the usual wave equation with slow-light 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 spin-wave

⇣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 free-spa

| c (t z/c) propagates according to the free-space 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

(Non-adiabatic 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 shape-preserving 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., )). shape-preserving 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 free-spa

1+(w

propagates according resonances

eg )dn/d

atpresence transparency of the sign

to the free-space wave

shape-preserving 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 shape-preserving 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 non-absorbing 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 non-absorbing 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 two-photon 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 two-photon 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 , higher-order ˜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 , higher-order 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 5-4 -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 dark-state 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 dark-state 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 dark-state

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 dark-state polariton 192 d of the rotation polariton speed ˆ d(z,t) of the n mixing . Introducing angle ✓a characteristic time-scale 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 dark-state a characteristic

the criteria polariton time-scale

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 time-scale 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 Slow-light propagation we obtain the criteria

d(z,t) n . Introducing a characteristic time-scale 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

Electromagnetically-Induced Transparency

'()

'(#

'(%

signal

c

(0)

'('

!" !# !$ !% !& ' & % $ # "

38

Kramers-Kronig 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 shape-preserving 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 Dark-state

Dark-state polariton: 86 , we can polariton equivalently introduce aanew newset set of of slow-light 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)

slow-light 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 bright-state 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 wave-vector 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 Nslowly-varying A e i(kk c

k s

dark-state polariton ˆ (t)Ŝ(z,t),

d(z,t) = cos ✓ d (t)Ês(z,t) )z sin ✓ ˆgs phase-matched (z,t) d (t)Ŝ(z,t) d with the perturbatio

where is the slowly-varying collective spin operator, phase-matched (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 slowly-varying phase-matched collective spin oper

ˆ(z,t) Ŝ(z,t) =p N A e i(kk c

k s )z ˆgs is the phase-matched 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 dark-state

asThese the dark-state operators (bright-state)

/⌦ c ) is the mixing angle. These operators are known (bright-state)

as are

cos ✓ d (t)Ês(z,t) sin (t)Ŝ(z,t)

the (2.81) known dark-state as (brig th

(2.81)

ation for polaritons the formation ˆ of dark-state 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 with|di 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 slowly-varying + costhe ✓ d |si) single-excitation observed phase-matched 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 slowly-varying population ✓ d |si) observed trapping phase-matched in (chapter coherentcollective 6). population These spin polaritons trapping operator, follow (chapter and 6

) ✓ d = Single-excitation arctan(g d p NA /⌦ dark-state c ) is thepolariton

mixing @t + v population

@

ˆ

g

the quasi-bosonic 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 quasi-bosonic commutation relations 86 (with the help of Eq. 2.25),

the ✓ d = quasi-bosonic arctan(g d NA

commutation /⌦ c ) is the mixing

where

relations Ŝ(z,t) 86 Bosonic

known

the quasi-bosonic angle. =p (withcommutation These operators

N

the

A e

help i(kk c

kof s

are known as the dark-state (bright-state)

)z Eq. relations 2.25),

86 quasipartite

as the dark-state

excitation

(bright-state)

(with the help of Eq. 2.25),

|D, polaritons 1i =

ˆgs (z,t) is the slowly-varying phase-matched 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)

dark-state

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 quasi-bosonic 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 quasi-bosonic 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 bright-state ' 0 (Eq. 2.76), polariton the bright-state 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 quasi-bosonic 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 dark-state 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 bright-state 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 bright-state 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 dark-state 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

dark-state 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 dark-state polariton

g

c

k s )z motions d(z,t) =0.

⌦ c

(Eq. 2.77) as (ref. 86 ),

@z dfor with the the dark-state 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 bright-state polariton is

),

ˆ b ' 0.

Thus, in the adiabatic regime, the dark-state polariton ˆ d(z,t)

In this limit, we can write the equation of motions for the ' 0 dark-state (Eq.

d,k (t)e ikz 2.76),

and ˆ polariton the bright-state 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 dark-state polariton

✓ ◆

d(z,t) = cos ✓ @ d (t)Ês(z,t) sin ✓ d (t)Ŝ(z,t) can be considered ikz b ' 0.

(Non-adiabatic 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 dark-state 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 bright-st

Ŝ(z,t) '

g c.

Ês; and i.e.,

dÊs

⌦ c

(Eq. 2.77) as (ref. 86 ),

@t + v cos a 2 @ spin-wave ✓ d ) ˆthe polariton mode @

g d(z,t) =0. ˆ d(z,t). Ŝ(z,t),

(2.83)

@z

@t + v with @ the effective matter-lig

Dark-state polariton follows the ˆ

g d(z,t) =0.

nteraction Thus, Hamiltonian in the adiabaticwritten regime, as the(Eq. dark-state✓ ◆

@z

In the adiabatic condition 86,190,191 In( @ this t⌦ limit, we can write the equation of motions for the dark-state polariton ˆ c

2.70) polariton ˆ usual wave equation as in free-space

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 dark-state @t + inEq. the adiabatic 2.76polariton v @ following ˆ

g d(z,t)

@z

gec

@t + v the @ regime, order ˆ =0.

of dark-state polariton

✓ d determined by the ‘amount’ of the photonic component

d(z,t) of the@ ˆ t

follows dark-state 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)

lowest-order 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 dark-state 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 spin-wave . 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 Dark-state 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

slow-light

by

polaritonic

Fleischhauer and Lukin 86 , we can equivalently introduce a new set of slow-light pola

⌦ s ⌦ c

=

Fano-interfarence

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 slow-light 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 slowly-varying phase-matched collective spin operat

p slow light with EIT

(k k c

k s )z nown asˆgs the ✓(z,t) d dark-state = is arctan(g the slowly-varying (bright-state)

d NA /⌦ c ) phase-matched is the mixingcollective angle. These spin

•!

operator, operators

Hau et al.,

and are known as the dark-state (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 dark-state (bright-state)

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 – dark-state polariton

Dark-state 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 dark-state

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 dark-state 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 spin-wave 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 Heisenberg-Langevin equations (Eqs. 2.71–2.74),

Non-adiabatic 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 cross-correlation 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 time-reversal 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 atom-photon coupling constant g d (Eqs.

20

6

2.85–2.87) 1.5x10 enables a strong collective matter-light interaction with an effective coupling constant g eff =

-5

p

15

NA g d between a single spin-wave 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 spin-wave 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 joint-detection single-photon 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 single-photon generation can be synchronized !

photons by interference. Phys. Rev. Lett. 59, 2044ñ 2046 (1987).

28. Santori, Hong-Ou-Mandel 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 – sub-exponential 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

Chin-Wen Chou, et al.

Science 316, 316, 1316 1316 (2007) (2007);

DOI: 10.1126/science.1140300

D

BS U

1a

D 1 b

Node “Right”

LU

Field 1

RU

D 2a

PBS Write

D

This copy is for your personal, non-commercial use only.

2c

PBS

Read

3 meters & Repumper

(!/2)

L

(!/2)

R

D 2b PBS L

PBS R

D 2d

LD

RD

clicking here.

D 1c

D

e

BS 1d

e

D

Node “Left”

Write

The following resources related to this article are available online at

www.sciencemag.org (this infomation is current Control as of Logic May 21, 2011 ):

g

C.-W. Chou, J. Laurat, H. Deng, K. S. Choi, H. de Riedmatten, D. Felinto, H. J. Kimble

If you wish to distribute this article to others, you can order high-quality copies for your

colleagues, clients, or customers by

Permission to republish or repurpose articles or portions of articles can be obtained by

following the guidelines here.

1a

1b

Field 1

s

Field 2

g

Read

s

Updated information and services, including high-resolution figures, can be found in the online

version of this article at:

http://www.sciencemag.org/content/316/5829/1316.full.html

“Effective” state giving one click on each side

Supporting Online Material can be found at:

http://www.sciencemag.org/content/suppl/2007/04/03/1140300.DC1.html

This article has been cited by 64 article(s) on the ISI Web of Science

This article has been cited by 1 articles hosted by HighWire Press; see:

http://www.sciencemag.org/content/316/5829/1316.full.html#related-urls


Conditional 68 control of heralded entanglement

! "

Asynchronous preparation

Preparation x35

Final state x20

Entanglement distribution

Violation of CHSH inequality

# $

Quantum-key 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


DLCZ-based Exeriments : Progress

DLCZ-based Exeriments : Progress

Spectacular advances with DLCZ-based 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

Spin-wave 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 spin-wave 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 Heisenberg-Langevin equations (Eqs. 2.71–2.74),

Non-adiabatic + 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 cross-correlation 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 time-reversal 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 atom-photon coupling constant g d (Eqs.

-5

20

2.85–2.87) enables a strong collective matter-light interaction with an effective coupling constant g eff =

1.5x10

p -5

NA

15

g d between a single spin-wave 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 spin-wave 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 joint-detection single-photon 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 Single-Photons

EIT Storage EIT Single-photon Storage and Retrieval and storage Retrieval of and Single-Photons

of

retrieval

EIT Storage and Retrieval of Single-Photons

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)

Single-photon storage and retrieval


Deterministic capabilities for quantum-state 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 off-diagonal 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 off-diagonal Into the elements photon-number 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)

photon-number 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 single-photon 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 single-photon of 85%. source, We also the independently probability of characterize having a sin

suppression photon at the w face of the oftwo-photon either memory component ensemblerelative is 15%, toleading a cohere t

vacuum component of 85%. We also independently characterize

suppression w of the two-photon 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

Single-photon 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-

gle-photon 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 photon-counting 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

photon-counting regime, heralded entanglement between atomic

b subsequent

state hastransfer, been successfully we separate demonstrated the generation through of entanglement

probabilistic L

gle-photon 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 single-photon

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 ‘on-demand’ 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

Vacuum-induced transparency

Tanji, Chen, Landig, Simon, Vuletic, Science 333, 1266 (2011)

Vacuum-induced 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 ↠âˆgs (13)

highest

is smalle

and thus

Photon-number

⌧ 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


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

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 operator-valued 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 ground-state ground-state 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 Safavi-Naeini 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

buried-oxide 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 Fabry-Perotñ 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 Fabry-Perotñ type resonator in which one

cleaned using a piranha et

frequency W/2p (top to

mirror acts like a mass-on-a-spring movable along x. The cavity has an intrinsic photon loss 0.5

oscillation mirror acts like a mass-on-a-spring 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, Level-diagram 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 set-up. 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 (top-left returned panel), microcavity zoomed-in 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 double-headed arrows, cavity device (top-right 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 (bottom-left panel), and zoomed-in image of top-view of single an electro-optical 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 zoomed-in 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; finiteelement-method

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 FEM-simulated 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 red-sideband

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 probe-field

probe-field Focus PIN

amplitude

amplitude photo-dio

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 near-resonance 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 two-photon detuning Un

therefore exhibits an inverted

dip, which can be studies as an a

0 The drive-dependent 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

waveguide-toroid Information), waveguide-toroid

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 aquantum-limited 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 Rare-earth atomic ensembles

An ensemble of rare-earth 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, CH-1211 Geneva 4, Switzerland

Received 9 June 2008; published 21 May 2009

An efficient multimode quantum memory is a crucial resource for long-distance quantum communication

Atomic frequency comb

Photon echo technique

demonstrated in quantum regime


olid State

General Ensembles

Photon echo

Idea storage

: Photon-Echo 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 2-pulse photon echo

•!

2-pulse 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 re-emission 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:

Two-pulse photon echo is not suitable for quantum storage

•! Strong optical pulse in the quantum channel

Contamination of the single-photon 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

Single-photon absorption

Collective rephasing

Single-photon 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 CH-1211 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 long-distance 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 Long-lived 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. Long-time storage and on-demand readout is achieved by use

Atomic The Frequency collective

of spin states in a lambda-type 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 rare-earth-metal 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 mode-locked 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 long-distance 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 light-matter peaks, coupling

2. For j Storage are long-lived 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 re-established 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 rare-earth-metal

coherent Control fields

quantum state storage

For long-lived mode

photon-echo

5–8 demonstrate that currently investigated

QMs can store a single mode. Yet, long-distance

RE ions in solids, mode 25–27 type re-emission in the forward

which are inhomogeneously broadened.

For long-lived 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 RE-doped 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 re-emission

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 g-e 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 frequency-selective 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 long-lived

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 frequency-selective storage

with

FWHM and

11.

are

Here

separated

we consider

by ,

multimode

whereg-e we

QMs

define is shaped

capable

the combinto an

of storing Na fixed AFC time. In order distribution to allowoptical

offor theon-demand 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 long-term 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 photon-echo 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 ground-state

physical

spin

emission. ï After Aa pair time oft control (dephasing) fields on e-s allows for long-time 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 e-s, 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 on-demand 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 re-established af

coherent photon-echo 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

•! Time-evolution •! 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

two-mode squeezed-state, and

Ideally,

ime cross-correlation 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 zero-time 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 second-order 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 measure-construct strategy

(~ teleportation fidelity)

For qubits,

For coherent states,

More magazines by this user
Similar magazines