niels bohr institute copenhagen university eugene polziklecture 5
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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
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
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