Niels Bohr InstituteCopenhagen University

Eugene Polzik LECTURE 5

Light – to – light Entanglement resource – parametric downconversion process

Atoms – to – atoms Entanglement resource – measurement induced entanglement of two atomic ensembles

Light – atoms, etc

..ˆˆ chaaiH param Parametric Hamiltonian, no dissipation:

aadt

dˆˆ

Equations of motion for field operators:

02)2(

Hamiltonian commuteswith the photon numberdifference operator:

0

aaaa

In photon number basis:

)(1100

)(tanhcosh

1

2

0

O

nnn

n

1 Workhorse ofphoton entanglement experiments!

02)2(

threshA

AParameter:

More accurate description : field modes in an optical resonator

0

Entangled cavity modes

02

)2(

Parametric downconversion in a resonator (Optical Parametric Oscillator below threshold)

P=Im(E)=i( a+ - a)

E+

E-X = Re(E)= a+ + a

When the two fields are separatedcorrelations – entanglement are

observed: X- X+

P- P+

0 XX

0 PP

aa ˆˆ

Frequency tunable entangled light around 860nm800MHz

0

0

02 ,

)2( )2(

AOM

AOM

LO-

LO+

-

-

Cavity modes

PX ,

PX ,

107 photons per mode

Classical field

2

-1 0 1 2 3 4 5 6

-6

-4

-2

0

2

4

6

8

(X

+-X

-)2 [

dB

(2 S

QL

)]

Phase [ Radians]

0

02 ,

)2(

Entangled cavity modes

Narrowband tunable entangled beams

Sorensen, Schori, Polzik

PRA, 2002 Necessary and sufficient condition for entanglement

2)()( 221

221 PPXX

Degree ofentanglement

0.8 – observed

Teleportation principle (canonical variables)L.Vaidman

VV PX ˆ,ˆ

22ˆ,ˆ PX11

ˆ,ˆ PX

0,0 2121 PPXXEinstein-Podolsky-Rosen entangled state

XC PC VV PX ˆ,ˆXC PC

Demonstrated experimentally for light variables by Furusawa, Sørensen, Fuchs, Braunstein Kimble, Polzik. Science 1998

0],[,],[ 2121 PPXXiPX

Classical benchmark fidelity for transfer of coherent states

)ˆˆ(ˆ2

1 aaX

)ˆˆ(ˆ2

aaP i

Atoms

Best classical fidelity 50%

e.-m. vacuum

K. Hammerer, M.M. Wolf, E.S. Polzik, J.I. Cirac, Phys. Rev. Lett. 94,150503 (2005),

Alice

|vin

LOp

_

LOx

Dx Dp

Victor

_

LOV

DV

Mp

Mx

Bob

mBob

out

ip

OutIn

Victor

__

ix

cvacuum vacuum

XP

Classicalteleportation

Alice

OPOPump 2Pump 1

|vin

LOp

_

LOx

Dx Dp

Victor

_

LOV

DV

Mp

Mx

Bob

mBob

out

EPR

beams

a b

i ii

Classical Informationip

OutIn

Victor

__

ix

c

Furusawa et al, Science, Vol 282, Issue 5389, 706-709 , 23 October 1998

2 units ofVacuum =

4.8 dB

Quantumteleportation

Classical boundary

conditional rotation detection of light

Communication networks based on continuous spin variables

Continuous variables:• polarization state of light• spin state of atoms

Input-Output interaction: free space off-resonant dipole interaction

MemoryAliceEPR

pulses

MemoryBob

EPR spins

Quantum channel

MemoryAlice

MemoryBob

EPR spin Alice EPR spin Bob

Classical channel

Coherent pulse

Symbols :

Operations:Light-atom teleportation

Operation:Teleportation of atoms

Light-to-Atoms Teleportation

ziny

outy kSJJ z

x

y

ziny

outy kJSS

inatoms

inlight

outlight PXX ˆˆˆ

k=1

YVZV SS ,

11, YZ JJ 22 , YZ JJ

Kuzmich, EP 2000

Light pulse DetectorAtoms 1

Atoms 2

entangled

Proposals:Duan, Cirac, Zoller, EP 2000Kuzmich, EP. 2000

Atoms X

Classical signal

Teleported

Teleportation of atomic states

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

Ato

mic

noi

se p

ower

[ar

b. u

nits

]

Atomic density [arb. units]

]sin)ˆˆ(cos)ˆˆ[(ˆˆ1121 tJJtJJSSS yyzzx

iny

outy

y z)(ˆ tS yxS

Memory in rotating spin states - continuedx

)ˆˆ(cosˆcosˆ212

00

zzTS

Tiny

Touty JJdttSdttS x

)ˆˆ(sinˆsinˆ212

00

yyTS

Tiny

Touty JJdttSdttS x

Teleportation of an entangled atomic state

•Every measurement changes the single cellspin, BUT does not change the measured sum•Every pulse measures both y and z components of the sum – entanglement is created

To complete teleportation of entanglement onto cell 1 and cell 4:rotate spin 4 by A+B+C:

1324321444ˆˆˆˆˆˆˆˆˆˆ TelTel JJJJJJJJCBAJJ

3

2 1

4

Pulse A

Pulse B

Pulse T

Alice Bob

43ˆˆ JJB

23ˆˆ JJC

21ˆˆ JJA

Tripartite entanglement

Fan HY, Jiang NQ, Lu HLLance AM, Symul T, Bowen WP, et al.Van Look et al

For atomic ensembles via quantum measurement: simple step from 2 to 3

N atoms,spins up

N/2 atoms,spins down

N/2 atoms,spins down

0)(

)(

ˆˆˆ,ˆˆˆ

21

21

321

321321

xxx

xxx

yyyzzz

JJJi

JJJi

JJJJJJ

xyyyzzz JJJJJJJ 2

321

221

1

2

321

221

1ˆˆˆˆˆˆ

N and S condition for 3-party pairwise entanglement:

1

2

3

Coupling strength of the interface

Z xy

z

Initial coherent spin state: 2

21

atPe

degree of squeezing in Jz

21

1

2

1

k

Figure of merit for the quantum interface

Duan, Cirac, Zoller, EP PRL (2000)

results in distribution 2

21 )( out

photat XkPe ziny

outy KJSS

Measurement on light

inatoms

inphot

outphot PkXX ˆˆˆ

Spin squeezed state

Figure of merit for the quantum interface

002

22

pulsepulse ssk

1

Probe scatteringparameter:

scatphonat m

AN

Ak

02

Spontaneous emission – the fundamental limit

g

e e

g

a a

degree of entanglement

01

1

2

1

Figure of merit for the quantum interface

K. Hamerrer, K. Mølmer, E. S. Polzik, J. I. Cirac. PRA 2004, quant-ph/0312156

+

Spontaneous emission probability

0

optimal0.3

10 30 50

Single pass interaction

cavity enhanced interactioncavity enhanced interaction

enhanced phase shift

power build-up inside cavity

compensate with

smaller photon number

T

1

T

2

Tn1T

Tnn

2

T

1 n...D 2

i

cold atomic cloudcold atomic cloud

T: mirror transmission

: absorption