Niels Bohr InstituteCopenhagen University

Eugene Polzik LECTURE 3

Einstein-Podolsky-Rosen (EPR) entanglement

Experimental techniques for observing quantum noise and entanglement

Light•Photon statistics – dc measurements•Entanglement in

the spectral modes of light – ac measurements

AtomsAddition of magnetic field – ac measurements with atoms

Generation of Entangled state of two distant Atomic Objects

N

1

• 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)()( 221

221 PPXX

2 particles entangled in position/momentum

11ˆ,ˆ PX 22

ˆ,ˆ PX

L

What does it mean in practice?

2)()( 221

221 PPXX

1. Prepare many identical pairs of particles2. Measure X1- X2 on some of those pairs3. Measure P1+P2 on others 4. Plot statistical distributions of the results5. Measure the width of these distributions L

X1- X2

0

P1+ P2

(X1- X2)

(P1+ P2)

2)()( 221

221 PPXX

Why does it make sense?

21 px

11ˆ,ˆ PX

22ˆ,ˆ PX

Two independent particles

212

2,12

2,1 px Minimal symmetricuncertainties

0

P1+ P2

(P1+ P2)= 1

L

X1- X2

(X1- X2)= 1

<

Entangled state

sec10 3421 Jpx

Position – momentum uncertainty

The best optical interferometry measurement:

mx 1210

Macroscopic object – a mirror

Assume M=1mg

sec/10 19 mp

Instead of two EPR particles we use

Two beams of light 1998

Two atomic ensembles 2001

Coherent state of light. Poissonian photon statistics.Delta-correlated noise.

nn

en

!2

2

!!

22 2

n

ne

nen

n

nn

Probability of counting nphotons

Variance:

nn 2Compare to

123ˆ,ˆ iSSS

nSS 41

1212

2

For coherent state

12 14 16 18

4

2

0

2

4

X

(vacuum units)

Strong fie

ld A(t)

Polarizingcube

Polarizingcube 450/-450

XAaeeaAS ii ˆ)(ˆ

21

21

2

iea

0

X

coherentstate2

1

1

-1

12 14 16 18 20

4

2

0

2

4

Phase squeezed

Amplitudesqueezed

Quadrature operators. Squeezed light

)(radian

)( unitsvacuuminX

Single photon stateSingle photon state

XnetaetanS ii ˆ))(ˆ)(ˆ(ˆ

21

21

2

x

Polarization beamsplitter

Spectral measurements of Quadrature OperatorsHomodynedetector

RF spectrum analyzer

detXtXn

detitii

i

i

:)()(:

:)()(:

x

Polarization beamsplitter

Spectral measurements on vacuum state (coherent state)Homodynedetector

RF spectrum analyzer

vacuum

i

i

e

detXtXnS

)(

:)()(:2

Delta correlated noise

Flat spectrum of noiseFlat spectrum of noise2S

In real world

Flat spectrum of noiseFlat spectrum of noise

n

S 2

Technical (classical) noise

Quantum noise

2n

n

n > n > n > n

Spectral measurements of Quadrature Operators

Spectrumanalyzer

)(ˆ tS y)(ˆ yS

Advantage: getting rid of technical noise

T

yTS dtttSXx

0

2 )cos()(ˆˆ

T

zTS dtttSPx

0

2 )cos()(ˆˆ

iPX ]ˆ,ˆ[

Same for sine modes0 5 106 1 107 1.5 107 2 107 2.5107

FrequencyHz80

78

76

74

72

70

68

66

esioN

rewopmBd

0.8 mW

1.7 mW

3.0 mW

1 MHz

Optical Spectrum of the strong field and the quantum field

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]

)(ˆ)(ˆ)(ˆ

)(ˆ)(ˆ

tStJtJ

tJtJ

zlabz

laby

laby

labz

Labz

iny

outy JSS ˆˆˆ

)]sin(ˆ)cos(ˆ[)(ˆ)(ˆ tJtJtStS yziny

outy

J

yz )(ˆ tS y

xS

Measuring rotating spin states

300000 310000 320000 330000 3400000,0000

0,0002

0,0004

0,0006

0,0008

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

,

Probe polarizationnoise spectrum

Larmorfrequency 320kHz

Detecting quantum projections of the spin in rotating frame

Atomic density (a.u.)

1.00.5 0.2

RF frequency

z

Bprobe

y

• 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)()( 221

221 PPXX

1012 spins 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 xzy JJJ 2

1

Compare

2)()( 221

221 PPXX

xyyzz JJJJJ 2)()( 221

221

Compare

11ˆ,ˆ PX

Stern-Gerlach projectionon any axis to x:

1 21+2

J J J

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

1012 atoms in each ensemble

2 gas samples

6S 1/2

Cesium

)4,...,4(m

)3,...,3( m

4

3

Total z and y components of twoensembles with equal and oppositemacroscopic spins can be determined simultaneously with arbitrary accuracy 0)()(ˆˆ,ˆˆ

212121 xxxxyyzz JJiJJiJJJJx x

yz z

Therefore entangled state with

0ˆˆˆˆ 2

21

2

21 yyzz JJJJ Can be created by a measurement

Top view:

Parallelspins 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

pump

pump

Y

Z

Z

Y

Entangling beam

Polarizationdetection

Entangled state of2 macroscopic

objects

J1

J2

B

B

0ˆˆˆˆ

)sin(ˆ2ˆ),cos(ˆ2ˆ

)sin(ˆ2ˆ),cos(ˆ2ˆ

2121

in2

in2

in1

in1

yyzz

zxzzxy

zxzzxy

JJJJ

tSaJJtSaJJ

tSaJJtSaJJ

)]sin()ˆˆ(

)cos()ˆˆ[()(ˆ)(ˆ

21

21

tJJ

tJJtStS

yy

zziny

outy

xzzyy JJJJJ 2)()( 221

221

Establishing the entanglement bound

xzy JJJ 21

11, zy JJ22 , zy JJ

Two independent ensembles

xzzyy JJJ 212

2,12

2,1 Minimal symmetricuncertainties

0

Jy1+ Jy2

NJJ

JJJJ

xx

yyyy

21

221

121

22

21

221 )(

0

Jz1+ Jz2

NJJ zz 212

21 )(

Sp

ect

ral v

ari

an

ce o

f th

e p

rob

e p

uls

e

Collective spin of the atomic sample

J x [10 ]

0 2 4 60

10

20

30

40

12

CSS

xyyzz JJJJJ 2)()( 221

221

Entanglement criterion:

=

xzzyy JJJJJ 2)()( 221

221

0

Jy1+ Jy2

0

Jz1+ Jz2

NJJ zz 212

21 )(

Entangled state

<

Proving the entanglement condition:

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 -

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

zziny

outy

JJt

JJttStS

Entangled spin states

Create entangled state and measure the state variance

Nor

mal

ized

sp

ectr

al v

aria

nce

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 InstituteDecember 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

ho

t(co

mp

) /

PN

Mean Faraday angle [deg]

2121 yyzz JJorJJ

Quantum uncertaintyxJ2

Decoherence issues

•only collective spin states are entangled•particles are indistinguishable - high symmetry of the system – - robustness against losses. This is not a Schrodinger’s cat made of 1012 atoms! •no free lunch: limited capabilities compared to ideal maximal entanglement

Sources of decoherence:stray magnetic fields decoherence time 3 millisecondscollisions decoherence time 1-2 milliseconds

Realism – two noncommuting spin componentscannot be measured – therefore do not exist? But can be entangled

Non-locality – two entangled macroscopic objects can be used toviolate Bell-type inequalities via distillationAll entangled Gaussian two-mode states are distillable (Giedke et al)