copenhagen interpretation entanglement - qubits 2 quantum coins 2 spins ( spin “up” or spin...
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Entanglement - qubits
2 quantum coins
2 spins ( spin “up” or spin “down”)
Entangled state many qubits:
Entangled state:
1100 ba
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
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
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