Copenhageninterpretation

Entanglement - qubits

2 quantum coins

2 spins ( spin “up” or spin “down”)

Entangled state many qubits:

Entangled state:

1100 ba

2 ensembles of quantum coins

Entanglement – collective variables

3 heads, 3 tails

4 heads, 2 tails

Now, imagine 1012 spins in each ensemble…

2 gas samples

•Only interactions/measurements of the collective spin of each ensemble are necessary•Atoms are indistinguishable - high symmetry of the system – - robustness against losses of spins•No free lunch: limited capabilities compared to ideal maximal entanglement

Outline

Continuous quantum variablesAtoms: Collective spin of the sample

Light: Stokes parameters of the pulse Hald, Sorensen, Schori, Polzik PRL 83, 1319 (1999)

Entangling atoms via interaction with lightTheory Kuzmich, Polzik PRL 85, 5639 (2000)

Duan,Cirac, Zoller, Polzik PRL 85, 5643 (2000)

Experiment: entangled state of two Cs gas samples – two macroscopic entangled objectsJulsgaard, Kozhekin, Polzik Nature 413, 400 (2001).

Quantum communication protocols withentangled atomic samples

Proposals Kuzmich, Polzik PRL 85, 5639 (2000)Duan,Cirac, Zoller, Polzik PRL 85, 5643 (2000)

t

tEvnhnhvE

nhvE

2 tvnntv

Quantum limits on the communication raten photons, frequency duration t

frequency multiplexing

Quantummemory

Quantum State (information) Processing

Quantum memory for light:

Write in the memory: map polarization state of light onto atomic spin

Entangled ensembles

Unknownquantum state

of light

Rotationsof spin

memory

Teleportation of atoms:

Entangled ensemblesSpinrotations

target

Light pulse

Ensembleto be

teleported

Memory read-out: map atomic spin state onto polarization of light

Entangled lightPolarization rotations

Output beammemory

Teleportationof light

InnsbruckRome

Caltech-Aarhus

Why use ensembles of atoms?•Quantum information processing often requires efficient interaction between light and atoms•Entangled (squeezed) states of atomic ensembles are required in applications such as frequency standards

Basic light - matter interaction: EdH ˆˆˆ Must be large

Increase with cavity:EA photon gets many chances

to interact with the atom.

Caltech (H. J. Kimble)Munich (G. Rempe), ...

1

And increase dipole moment d:Atoms in Rydberg states n=50, 51 arelarge and easy to hit with a photonParis ( S. Haroche et al)

2

Use ensemble of atoms:3Aarhus, since 1997

Tren d s in "en sem b le p h y sic s"

S to p p in g an d s to rin g lig h t p u lse s :

gs

e

E

E g s

is s to red inin a to m s C o n tro l p u lse

A to m s E

C o ld g as , L en e V. H au .

C o a ted ce ll w ith ru b id iu m , M . L u k in , A . F le isch h au e r, ...

Io n s w ith a lo w fin e sse cav ity, o n g o in g p ro jec t, io n trap lab .

S p in sq u eez in g :

sq u eezedlig h t

J . H a ld , . .Q u an tu m o p tic s lab .

e t a l

sp in sq u eezed a to m sS S aS Fy y x z

ou t in = +

A to m s

M easu re F z

sp in sq u eezed a to m s

A . K u zm ich , ...

M acro sco p ic A to m ic S p in sS p in v a riab le s :

F = f ( )ii= 1

N

[ ] , = F F iFy z x

Var( )Var( ) /F F F 4y z x 2 F 4Nx

( )Fy

( )Fz

C o h e ren t sp in s ta te : Var( ) = Var( ) = /2F F Fy z x

E x p erim en ta l rea liza tio n :

f ( )i

P ara ffin co a ted ce ll

O p tica lp u m p in g

C o h e ren ce tim e o f sp in s .3 0 m s

R o u g h ly a to m s.1 0 1 2

Vap o r ce ll a t .ro o m tem p era tu re

F = 3

F = 4m = 4

6 sta te s o f ces iu mS 1 /2

>

Spin memory with Coherent Spin States

Quasi-continuous encoding

0

18090

270

NJJ yz 41

Indistinguishable coherent states

•Densely coded states are impossible to read but possible to transfer via teleportation

Entangled or inseparablecontinuous variable systems

• EPR example 19352 particles entangled in position/momentum

11ˆ,ˆ PX 22

ˆ,ˆ PX

Perfect EPR state

0ˆˆ

0ˆˆ

21

21

PP

XX

iPX ],[

Simon PRL (2000)Duan, Giedke, Cirac, Zoller PRL (2000)Necessary and sufficient condition for entanglement

2)()( 221

221 PPXX

• EPR state of light Ou, Pereira, Kimble 1992

Along x: all tails

Along y,z: random misbalance between heads and tails N

Coherent state of spin-½ atoms

yz

j=1/2j=1/2 j=1/2

y z+ + = = N/2Jx

Jz,y2=Jx/2=N/4

x

Uncorrelatedatoms

EPR state of two macro-spin systems [Jz,Jy] = iJx ipx ],[

PJ

JX

J

J

x

y

x

z ˆˆ

,ˆˆ

xyyzz JJJJJ 2ˆˆˆˆ 2

21

2

21

N and S condition for entanglement:

x

y

z

J1y

z

J2

Along x: all tails

Along y,z: ideally no misbalance between heads and tails of the twoensembles, or, at least, less than random misbalance N

y z

y z

Two samples oppositely polarized

Two entangled samples

-x

x

Total z and y components of twoensembles with equal and oppositemacroscopic spins can be determined simulteneously with arbitrary accuracy

0)()(ˆˆ,ˆˆ212121 xxxxyyzz JJiJJiJJJJ

x x

yz z

Therefore entangled state with

0ˆˆˆˆ 2

21

2

21 yyzz JJJJ Can be created by measurement

How to measure the total spin projections?

•Send off-resonant light through two atomic samples•Measure polarization of light (Faraday effect)

2 = in ten sity ( ) - in ten sity ( ) = -2 = in ten sity (4 5 ) - in ten sity (1 3 5 ) = -

2 = in ten sity ( ) - in ten sity ( ) = -

= +

S x y a a a aS a a a a

S a a a a

n a a a a

x x x y y

y

z

4 5 4 5 1 3 5 1 3 5

+ - + + - -

+ + - -

P B S

S to k es P a ram e te rs

Q u a n tized P o la r iza tio n o f L ig h t

C o m m u ta to rs

[ , ] = 1[ , ] = a a S S i S

y z x

x

y

M ea su rem en t o f fo r-p o la riz ed lig h t:

Sx

y

S ta tis tic s :

H eise n b e rg :

fo r u n co rre la ted lig h t

S = n /2x (c la ss ica l)

> /1 6 S S ny z

2 2 2

S S

S S ny z

y z

= = 0

= = /4 2 2

4 5

1 3 5

I -

S h o t n o ise ,p ro p o rtio n a l top h o to n n u m b er.

Continuousvariables

Bell measurement

Light / Atom - Interaction

Lu-Ming Duan, J. I. Cirac, P. Zoller, E. S. Polzik,

Phys. Rev. Lett., 85, 5643 (2000)A. Kuzmich and E. S. Polzik,Phys. Rev. Lett., 85, 5639 (2000)

Faraday effect:

Atomic spins rotate polarization of light

zx

y

ziny

outy JSS

6S , F=41/2

6P3/2

Cesium

mmz mNJ )4,...,4(m

Back action:

Light rotates spins of atoms

z

x

yziny

outy kSJJ

pump

pump

X

Y

Z

Z

Y

Entangling beam

Polarizationdetection

Entangled state of2 macroscopic

objects

J1

J2

300000 310000 320000 330000 3400000,0000

0,0002

0,0004

0,0006

0,0008

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,00,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

Shot noise level

Noi

se p

ower

[ar

b. u

nits

]

Frequency (Hz)

Density[arb. units]

----------------------------------1.00 ± 0.020.56 ± 0.010.21 ± 0.01

Atomic Quantum Noise

Ato

mic

noi

se p

ower

[ar

b. u

nits

]

Atomic density [arb. units]

,

Probe polarizationnoise spectrum

Larmorfrequency 320kHz

Detecting quantum fluctuations of the spin

Atomic density (a.u.)

1.00.5 0.2

RF frequency

z

Bprobe

)(ˆ)(ˆ)(ˆ

)(ˆ)(ˆ

tStJtJ

tJtJ

zzy

yz

inz

outz

zziny

outy

SS

JJSS

ˆˆ

)ˆˆ(ˆˆ21

y

21

21

ˆˆ)sin(

ˆˆ)cos()(ˆ)(ˆ

yy

zziny

outy

JJt

JJttStS

B-field

PBS

Time

Verifyingpulse

Entanglingpulse

0.5 ms

m=4

700MHz

6S

6P

Entangling andverifying beams

S

Entangling andverifying pulses

F=3

F=4 = 325kHzm=4

1/2

3/2

youtx2

Opticalpumping

Pumpingbeams

J

x1

+

J -

F o r th e c o h e ren t sp in s ta te :

Va r( + ) + Va r( + ) = 2F F F F Fy, y, z , z , x1 2 1 2

Va r( + ) + Va r( + ) < 2F F F F Fy, y, z , z , x1 2 1 2

E n tan g le m en t c rite rio n :

1 ) M easu re th e co h e re n t sp in s ta te lim it 2 F x

Spe

ctra

l var

ianc

e o

f the

pro

be p

ulse

Co llective spin of the atom ic sam ple

F x [10 ]

0 2 4 60

1 0

2 0

3 0

4 0

12

C S S

"Pro

babi

lity

" Distribution of CCS

Distribution of the createdentangled state after 0.5ms

Uncertainty of theverifying pulse

lightnoiseJJJJVarVar yyzz )()( 2121

Nor

mal

ized

sp

ectr

al v

aria

nce

Collective spin of the atomic sample12Fx [10 ]

Sy(1pulse)

Entangled spin state

2) Create entangled state and measure the state variance

CSS

2Fx

Sy(1pulse) Light (1pulse)

Atoms

0 2 4 60.0

0.5

1.0

1.5

2.0

Quantum memory

Quantum communication protocols withentangled atomic samples and tunable EPR light

Entangled atomic samples

Entangled (EPR) light source

Protocols (proposals): •Teleportation of atomic states•Light-to-atoms teleportation•Atom-to-light teleportation

02

)2(

Parametric downconversion in a resonator (OPO)

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

Frequency tunable entangled and squeezed light around 860nm800MHz

0

0

02 ,

)2( )2(

AOM

AOM

LO-

LO+

-

-

Cavity modes

PX ,

PX ,

107 photons per mode

22

ˆˆ aeAaeAi iLO

iLO

{ 0,2 XALO

2,2 PALO )(2 aeaeA ii

Classical field

-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]

2)()( 221

221 PPXX

1

1)( 221 PP

1)( 221 XX{OR

Degree ofentanglement0.6 – observed0.65 – corrected for detector noise(1- perfect )

)(ˆˆ)(

)(ˆ)(ˆ122

in

zzL

iny

outy SbPJ

aSXS

Quantum state of light stored in long lived spins

0 200 400 600 800 1000 1200 1400 16000,000

0,001

0,002

0,003

Noi

se p

ower

[V RM

S2 ]

Freq. [Hz]

S y<SQL S y=SQL Electronic noise

Larmor frequency

light noisereduced by squeezingof fluctuations

quantum noise of lightstored in atomic spin

200 Hz1.5 msec storage

Light withControlledX and P –controlledquantum state

+-

Ax

EPR source

x

p

Ax

EbEa

Ap

Actuators

EbAp

in

in

in

Classical channels

Quantum channel

Furusawa et al Science, 1998 Caltech-Aarhus-Bangor

0ˆˆ,ˆˆ inaina ppxx

ina xx ina pp

Quantum Teleportation of Light

Teleportation of an entangled atomic state

1

2 3

4

Pulse 1 Pulse 1*

21ˆˆ JJA

43ˆˆ JJB

Pulse 2

23ˆˆ JJC

•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 Spin 1 to cell 4:rotate spin 4 by A+B+C:

1324321444ˆˆˆˆˆˆˆˆˆˆ JJJJJJJJCBAJJ Tel

EPR spin Alice EPR spin Bob

Coherent pulse

Operation:Teleportation of atoms

Classical channel

Memory Bob

Distance limitations:•Losses of light – fiber (3 dB): 1 km at 850 nm

10 km at 1500 nm space: 100 km (diffraction)

OR•(Life time of ERP atoms)x(speed of their transport)Currently: (0.001sec)x(Boeing 747) = 30 cmWith 1 hour storage = 1000 km

MemoryAlice

Communication networks based on continuous spin variables

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

Operation:Storage of light and read-out from atomic memory

MemoryAlice

EPRpulses

EPR spins

Memory Bob

Light -Quantum channel

polarization rotation detection of light

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

Brian Julsgaard Christian Schori