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Niels Bohr InstituteCopenhagen University
Eugene Polzik LECTURE 3
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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
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• 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
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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)
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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
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sec10 3421 Jpx
Position – momentum uncertainty
The best optical interferometry measurement:
mx 1210
Macroscopic object – a mirror
Assume M=1mg
sec/10 19 mp
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Instead of two EPR particles we use
Two beams of light 1998
Two atomic ensembles 2001
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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
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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
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12 14 16 18 20
4
2
0
2
4
Phase squeezed
Amplitudesqueezed
Quadrature operators. Squeezed light
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)(radian
)( unitsvacuuminX
Single photon stateSingle photon state
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XnetaetanS ii ˆ))(ˆ)(ˆ(ˆ
21
21
2
x
Polarization beamsplitter
Spectral measurements of Quadrature OperatorsHomodynedetector
RF spectrum analyzer
detXtXn
detitii
i
i
:)()(:
:)()(:
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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
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In real world
Flat spectrum of noiseFlat spectrum of noise
n
S 2
Technical (classical) noise
Quantum noise
2n
n
n > n > n > n
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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
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Optical Spectrum of the strong field and the quantum field
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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
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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
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• 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
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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)
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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
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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
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1012 atoms in each ensemble
2 gas samples
6S 1/2
Cesium
)4,...,4(m
)3,...,3( m
4
3
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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
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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
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pump
pump
Y
Z
Z
Y
Entangling beam
Polarizationdetection
Entangled state of2 macroscopic
objects
J1
J2
B
B
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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
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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 )(
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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:
=
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xzzyy JJJJJ 2)()( 221
221
0
Jy1+ Jy2
0
Jz1+ Jz2
NJJ zz 212
21 )(
Entangled state
<
Proving the entanglement condition:
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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
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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).
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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
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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
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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)