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)