Niels Bohr InstituteCopenhagen University

Quantum communication and

distributed quantum networks

require

quantum information (state) transfer

between light and atoms

detectorAtoms Light outLight in

Light – atoms quantum interfaceAtomic quantum correlations createdand probed via interaction with light

Quantummeasurement

Quantum feedback

Teleportation, quantum memory, atomic squeezing,QND probing of strongly coupled systems, etc

Applications:

Atoms

entangled

Quantum interface via off-resonant scattering

† † †ˆ ˆ ˆ ˆˆ ˆ . . 2 L AH i a b i ab h c P Pχ χ χ= + + =

QuantumNondemolitionInteraction

a)

b)

Kuzmich,Bigelow,Mandel; EP, Cirac, Mabuchi, Jessen

b)

a)

† † . .H a b h cχ= +))

Light-Atoms Entanglement

Cirac, Zoller, EPKimble, Kuzmich, Lukin

b)

a)

† . .H ab h cχ= +))

Light-to-Atoms mapping (memory)

EP, Mølmer, Lukin

2

2

λσπ

=

Quantum Optics

Second quantization of

light is critical

†ˆ ˆ, 1a a =

Quantum Atom Optics

Second quantization of

atomic variables is critical

†ˆ ˆ, 1b b = Combine both worlds: light-atoms quantum

interface (with atomic ensembles)

†

† † † †

ˆ 1 1

ˆ ˆ ˆ, 1

a n n n

a a aa a a n a a

+ = +

= − = =

[ ] iPXaaPaaX i =−=+= ++ ˆ,ˆ)ˆˆ(ˆ),ˆˆ(ˆ22

1

Light-Matter interface for:

Quantum computing

with photons

Quantum buffer

memory for light

Long distance

quantum

communication

(repeaters)

Quantum Key storage

in quantum cryptography

Quantum networks

Ensemble approachEnsemble approach

Our alternative program (1997 -)

EIT (2000 -), Raman (2001-):

Propagating light pulses +

atomic ensembles

Cavity Q E DCavity Q E DCavity Q E DCavity Q E D

Strong couplingStrong couplingStrong couplingStrong coupling

to a single atom to a single atom to a single atom to a single atom ---- qubitqubitqubitqubit

Caltech Caltech Caltech Caltech –––– optical optical optical optical λλλλP aris P aris P aris P aris –––– m icrow ave m icrow ave m icrow ave m icrow ave

M PQ M PQ M PQ M PQ –––– optical optical optical optical

N IST, N IST, N IST, N IST, InnsruckInnsruckInnsruckInnsruck –––– ions ions ions ions

Stanford Stanford Stanford Stanford ---- solid statesolid statesolid statesolid state

…………

Energy levels with rf or

microwave separation - no

need for λ3 confinement

ei•δk•r —> 1

δω G round stateG round stateG round stateG round state

hfhfhfhf or Z eem anor Z eem anor Z eem anor Z eem an

sublevelssublevelssublevelssublevels

Ensemble approachEnsemble approach

Our alternative program (1997 -)

EIT (2000 -), Raman (2001-):

Propagating light pulses +

atomic ensembles

Cavity Q E DCavity Q E DCavity Q E DCavity Q E D

Strong couplingStrong couplingStrong couplingStrong coupling

to a single atom to a single atom to a single atom to a single atom ---- qubitqubitqubitqubit

Caltech Caltech Caltech Caltech –––– optical optical optical optical λλλλP aris P aris P aris P aris –––– m icrow ave m icrow ave m icrow ave m icrow ave

M PQ M PQ M PQ M PQ –––– optical optical optical optical

N IST, N IST, N IST, N IST, InnsruckInnsruckInnsruckInnsruck –––– ions ions ions ions

Stanford Stanford Stanford Stanford ---- solid statesolid statesolid statesolid state

…………

Energy levels with rf or

microwave separation - no

need for λ3 confinement

ei•δk•r —> 1

δω G round stateG round stateG round stateG round state

hfhfhfhf or Z eem anor Z eem anor Z eem anor Z eem an

sublevelssublevelssublevelssublevels

Collective = ensem ble Collective = ensem ble Collective = ensem ble Collective = ensem ble

quantum variablesquantum variablesquantum variablesquantum variables

•Canonical variables for light and an ensemble of atoms

•Interaction Hamiltonian for polarized light and a spin-polarized atomic ensemble

•Entanglement of two atomic objects (2001)

•Quantum memory for light (2004)

•Quantum teleportation between light and matter (2006)

Canonical quantum variables for light

•Complementarity : amplitude and phase of

light cannot be measured together

[ ] iPXaaPaaX i =−=+= ++ ˆ,ˆ)ˆˆ(ˆ),ˆˆ(ˆ22

1X

P

t

)sin(ˆ)cos(ˆˆ tPtXE ωω +∝

Pulse: ∫=T

TdttxX

0

1L )(ˆˆ

X

P

Various

states

1/ 2X Pδ δ ≥

2 2 1/ 2X Pδ δ= =

( )† 2 212

ˆ ˆˆ ˆ ˆ 1n a a X P= = + −

Linear polarizations

Polarization quantum variables – Light

Polarization – Stokes

parameters

[ ] 123ˆ,ˆ iSSS =

v

h

)(21

1 hhvv aaaaS ++ −=

45-45

2S 3S

horizontal

verticalProp

agation d

irection

Circular polarizations

x

EOMStrong field A(t)

Quantum field a -> X,PPolarizingcube

-450 450

PolarizingBeamsplitter 450/-450

=−− + )]ˆ()ˆ( aAaA XAaaA ˆ)(2

121 =++−++= + )ˆ()ˆ[(ˆ

41

2 aAaAS

λ/4

PAS ˆˆ2

13 =

Phase

Amplitude

Benchmark I: quantum noise of Coherent State of Light – ,VarX P N∝

•Photodetectors withq.e.>99% and darknoise << shot noise oflight

•Stabilization of phaseand amplitude noiseof light down to theSQL = shot noise

Mach-Zehnder Interferometer

Polarization variables for light

Polarization – Stokes

parameters

[ ] 123ˆ,ˆ iSSS =

v

h

hhvv aaaaS ++ −=1

45-45

2S3S

nSPnSXnS Li

L 21

1232

12

ˆ,ˆˆ,ˆˆ ===

ˆ ˆ,X P i =

ˆ ˆ ˆcos( ) sin( )E X t P tω ω∝ +

Canonical variables for atomic ensemble and light

( )( )

1

2

2

ˆ ˆ ˆ

ˆ ˆ ˆi

X a a

P a a

+

+

= +

= −

[ ]x

y

A

x

zAAA

J

JP

J

JXiPX

ˆ,

ˆˆˆ,ˆ ===

Jy~P

Jz~X

Jx

[ ] xyz iJJJ =ˆ,ˆ

Spin polarized

Cesium atoms

mF=4F=4

3

t

X

P

( ) ( ) 12L LVar X Var P= =

Coherent state:

P

LXSingle photon:

1

N

i

i

J j=

=∑

Ensemble of 1012 atoms

Experimental long-lived entanglement of two macroscopic objects. B. Julsgaard et al. Nature, 413, 400 (2001).

Experimental demonstration of quantum memory for light. B. Julsgaard et al Nature, 432, 482 (2004),

Quantum teleportation between light and matter. J. F. Sherson et al. Nature 443, 557 (2006).

Atoms: ground state Zeeman sublevels

2/36P

2/16S Ω4

3

tJtJJ

tJtJJ

zy

Lab

y

zy

Lab

z

Ω+Ω−=

Ω+Ω=

cosˆsinˆˆ

sinˆcosˆˆ

Rotating frame spin

( ) ( )( ) NNJ

iNJNJ

lab

x

lab

y

lab

z

=−=

−=+= ++

3,34,4

4,34,34,34,3

ˆˆ

ˆˆˆˆˆˆ

ρρ

ρρρρ

A tom ic operatorsA tom ic operatorsA tom ic operatorsA tom ic operators

[ ]NJJJ

iJJJ

Fxyz

xyz

22122

,

===

=

δδ

Macroscopic spin ensemble –coherent spin state

gas samplegas samplegas samplegas sampleat room Tat room Tat room Tat room T

6S , F=41/2

6P3/2

Cesium

∑= mmx mNJ ρ)4,...,4(−=m

Coherences determine yz JJ ,

State

preparation

(optical

pumping)

X z y

x21

2

−NF

Magnetic Shields

Special coating – 104 collisionswithout spin flips

Decoherence from straymagnetic fields

Example – gas of spin polarized atoms at room temperature

Optical pumping with circularly

polarized light

2

1−2

1

δInformation carrying

wavelengthmuch longer thanthe sample length

QI

cλδ

=

δc/δ>>L

ˆˆ ˆ ˆL A Z ZH P P S Jχ χ= = %QND Interaction

a)

Why is atomic motion not a problem?

Off-resonantinteraction:

∆ >> Doppler Pulse durationmuch longer thanthe transient time

All atoms couple to lightin the same way:symmetric mode

1

ˆ ˆN

z ziJ j=∑

Questions after the first lecture.

Q.: Does the interface work with cold atoms?A.: Yes, it does well. The atoms should either be standing still

(cold atoms or solid state), or be moving fast to average the

interaction, as in RT ensembles with transient time ~ 100

secand interaction time ~ 1msec

Q.: What approximation is used to introduce canonical variablesfor polarized atomic ensembles?A.: We treat one of the projections of the collective spin, say Jx ,

as a large constant classical number. Then the commutator is equal to

this number. This is called Holstein-Primakoff approximation.

Approximate

with the plane

X

P

,z y xJ J iJ =

Light / Atom - Hamiltonian

zx

y

zS

Jz

J

( )( )( )( )++−−−

+−+

++

++−−−+−+

++

++−+−−−+

++

−+−+−+

+++

+−−

+

+∝∝

ggggaaaa

ggggaaaa

ggaaggaa

egaegadEH

21

21~

~~

~

Dipole interaction

Hamiltonian:

12

g+ =

12

e− = − 12

e+ =

12

g− = −

+a −a

Hamiltonian:

zJaSH 3=Happer 1976

Kuzmich, Bigelow, Mandel 1998

Yabuzaki et al 1999

Off-resonant, perturbative:

Isotropic –

total number

of photons and atoms

Dynamics of light and atoms x

y

3S

Z

[ ]QHQ i ˆ,ˆˆh

& =

ALz PPJSaH ˆˆˆˆˆ3 ∝=

nSPnSXnS Li

L 21

1232

12

ˆ,ˆˆ,ˆˆ ===

NJPNJXNJ xAi

zAy 21

22

1 ,ˆˆ,ˆˆ ===

Example: 2 1ˆ ˆ ˆ ˆ

2z L A

ia iaS S J X NnP= ⇒ =& &

h h

x

Quantum field

-450 450

PolarizingBeamsplitter 450/-450

Physics behind the Hamiltonian: 1. Polarization rotation of light

A

in

L

out

L PXX ˆˆˆ κ+=

ALz PPJSaH ˆˆˆˆˆ3 ∝=

xStrong field A(t) Atoms

21

21−

( )aiA ˆ2

1 − ( )aiA ˆ2

1 +

y

Physics behind the Hamiltonian: 2. Dynamic Stark shift of atoms

21− 2

1

ϕie

Quantum field - a

L

in

A

mem

A PXX ˆˆˆ κ+=

ALz PPJSaH ˆˆˆˆˆ3 ∝=

Figure of merit for the quantum interface – optical depth

ηαγταγτγ

α 002

22 ==∆≈ ∆∆∆ pulsepulse ssk

1<<η

Probe depumping

parameter:

scat

phonat mA

NA

kσησηα === 0

2

α∆= α0γ2/∆2 optical depth of the atomic sample (absorption coefficient)

s∆= s0γ2/∆2 saturation parameter – the ratio of the Rabi frequency to

spontaneous decay rate γ

η probability of spontaneous emission caused by probe pulse

in

atoms

in

light

out

light PXX ˆˆˆ κ+=

• Einstein-Podolsky-Rosen paradox – entanglement; 1935

2 particles entangled in position/momentum

11ˆ,ˆ PX mVPX =22

ˆ,ˆ

LXX =− 21ˆˆ

L

0ˆˆ21 =+ PP

Simon (2000); Duan, Giedke, Cirac, Zoller (2000)

Necessary and sufficient condition for entanglement

2)()( 2

21

2

21 <++− PPXX δδ

1010101012121212 spins in each ensemblespins in each ensemblespins in each ensemblespins in each ensemble

y z

x

y z

xSpins which are “more parallel” than that

are entangled

Experimental

long-lived

entanglement

of two

macroscopic

objects.

21

~−

N

B. Julsgaard, A. Kozhekin and EP, Nature, 413, 400 (2001)

X

Z or Y

2nd

1st

If the two macroscopic spins are collinear they must beentangled:

21≥× px δδ

xzy JJJ21≥δδ

Compare

2)()( 2

21

2

21 <++− PPXX δδ

xyyzz JJJJJ 2)()( 2

21

2

21 <+++ δδCompare

11ˆ,ˆ PX

1012 atoms in each ensemble

2 gas samples2 gas samples2 gas samples2 gas samples

6S 1/2

Cesium

)4,...,4(−=m

)3,...,3(−=m

4

3

Total z and y components of two

ensembles with equal and oppositemacroscopic spins can be determined simultaneously with arbitrary accuracy

[ ] 0)()(ˆˆ,ˆˆ212121 =−=+=++ xxxxyyzz JJiJJiJJJJ

x x

yz z

Therefore entangled state with

( ) ( ) 0ˆˆˆˆ2

21

2

21 ⇒+++ yyzz JJJJ δδCan be created by a measurement

Top view:

Parallel

spins must be

entangled

How to measure the total spin projections?

•Send off-resonant light through two atomic samples

•Measure polarization state of light

Duan, Cirac, Zoller, EP 2000

σσσσ++++pump

σσσσ−−−−pump

Y

Z

Z

Y

Entangling

beam

Polarization

detection

Entangled state of

2 macroscopic

objects

J1

J2

B

B

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4 Atomic Quantum Noise

Atomic noise power [arb. units]

Atomic density [arb. units]

)(ˆ)(ˆ)(ˆ

)(ˆ)(ˆ

tStJtJ

tJtJ

z

lab

z

lab

y

lab

y

lab

z

β+Ω−=

Ω=&

&

Lab

z

in

y

out

y JSS ˆˆˆ α+=

)]sin(ˆ)cos(ˆ[)(ˆ)(ˆ tJtJtStS yz

in

y

out

y Ω+Ω+= α

Jr

yz )(ˆ tS y

xS

Experimental realization with magnetic field

0ˆˆˆˆ

)sin(ˆ2ˆ),cos(ˆ2ˆ

)sin(ˆ2ˆ),cos(ˆ2ˆ

2121

in

2

in

2

in

1

in

1

=+=+

Ω−=Ω−=

Ω=Ω=

yyzz

zxzzxy

zxzzxy

JJJJ

tSaJJtSaJJ

tSaJJtSaJJ

&&&&

&

)]sin()ˆˆ(

)cos()ˆˆ[()(ˆ)(ˆ

21

21

tJJ

tJJtStS

yy

zz

in

y

out

y

Ω+

+Ω++= α

xzzyy JJJJJ 2)()( 2

21

2

21 =+++ δδ

Establishing the entanglement bound

xzy JJJ21≥×δδ

11 , zy JJ22 , zy JJ

Two independent ensembles

xzzyy JJJ212

2,1

2

2,1 == δδMinimal symmetric

uncertainties

0

Jy1+ Jy2

NJJ

JJJJ

xx

yyyy

21

221

121

2

2

2

1

2

21 )(

=+=

=+=+ δδδ0

Jz1+ Jz2

NJJ zz 212

21 )( =+δ

Spe

ctr a

lvar

ianc

eof

t he

prob

epu

lse

Collective spin of the atomic sample

J x [10 ]

0 2 4 60

10

20

30

40

12

CSS

xyyzz JJJJJ 2)()( 2

21

2

21 <+++ δδ

Entanglement criterion:

=

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

faraday angle [°]

κ2 = a

tom

/ sh

ot

Benchmark II: quantum noise of Coherent Spin State –

N

NJVar yz ∝,

( ) ( )2

1== AA PVarXVar

62

1

10−−≈N

Tomography of a coherent spin state (uncorrelated spins) –thermal atoms in a cell

•Shot noise limiteddetection of light

PLUS

•Stabilization of theatomic noise down tothe projection noise

xzzyy JJJJJ 2)()( 2

21

2

21 <+++ δδ

0

Jy1+ Jy2

0

Jz1+ Jz2

NJJ zz 212

21 )( =+δ

Entangled state

<

Proving the entanglement condition:

B-field

PBS

Time

Verifyingpulse

Entanglingpulse

0.5 ms

m=4

700MHz

6S

6PEntangling andverifying beams

S

Entangling andverifying pulses

F=3

F=4 Ω = 325kHzm=4

1/2

3/2

youtx2

Opticalpumping

Pumpingbeams

Jσ

x1+

Jσ-

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

0,0

0,2

0,4

0,6

Atomic density [arb. units]

[ ][ ]21

21

ˆˆ)sin(

ˆˆ)cos()(ˆ)(ˆ

yy

zz

in

y

out

y

JJt

JJttStS

+Ω

++Ω+=

α

α

Entangled spin states

Create entangled state and measure the state variance

Nor

mal

ized

spec

tral

varia

n ce

Collective spin of the atomic sample12Fx [10 ]

Sy(1pulse)

CSS

2Fx

Sy(1pulse)Light (1pulse)

Atoms

0 2 4 60.0

0.5

1.0

1.5

2.0

Julsgaard, Kozhekin, EP. Nature 413, 400 (2001).

Material objects deterministically entangled at 0.5 m distance

Niels Bohr Institute

December 2003

0 2 4 6 8 10 12 14 16

0,70

0,75

0,80

0,85

0,90

0,95

1,00

10-12-2003/noise.opj

Ato

m/s

hot(

com

p) /

PN

Mean Faraday angle [deg]

2121 yyzz JJorJJ ++

Quantum uncertaintyxJ2