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Experiments With Entangled Photons
Paulo Henrique Souto RibeiroInstituto de Física - UFRJ
Summer School of OpticsConcépcion January/2010
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Quantum Optics Group at IF/UFRJ
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Group membersExperiments:Prof. Paulo Henrique Souto Ribeiro Prof. Stephen Patrick Walborn
Theory:Prof. Luiz Davidovich Prof. Nicim ZaguryProf. Ruynet Matos FilhoProf. Fabricio Toscano
Msc and PhD students: Adriana Auyuanet Larrieu, Adriano H. de Oliveira Aragão, Bruno de Moura Escher , Bruno Taketani, Daniel Schneider Tasca, Gabriel Horacio Aguilar, Osvaldo Jimenez farias, Gabriela Barreto Lemos, Rafael Chaves.
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UFRJ
UFMG
USP-SÃO PAULO
UFAL
UFF
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Outline:
Part I-Simultaneity in parametric down-conversion-Violation of a classical inequality-Consequences of simultaneity: i)localized one-photon state; ii)the Hong-Ou-Mandel interferometer iii) measurement of the tunneling time
Part II-Polarization entanglement-Bell’s inequalities -Entanglement measurement
Part III-Entanglement dynamics-Kraus operators-Entanglement sudden death-Process tomography-Evolution of entanglement
Part VI-Spatial correlations-The transfer of the angular spectrum-Continuous variables etanglement- EPR paradox-Non-gaussian entanglement-Non-local optical vortex
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Part I
- Simultaneity in parametric down-conversion
- Violation of a classical inequality
- Consequences of simultaneity:
i) localized one-photon state;
ii) the Hong-Ou-Mandel interferometer
iii) measurement of the tunneling time
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Parametric Down-conversion
Espontaneous emission
Stimulated emission
TwinPhotons
p i s
p i sk k k
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Parametric Down-conversion
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Observation of simultaneity
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Observation of simultaneity
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Parametric down-conversion: quantum state
Time evolution
Time evolution operator
Time integral
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Simultaneity in parametric down-conversion
Quantum state for weak interaction
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Simultaneity in parametric down-conversion
Quantum state including some approximations
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Simultaneity in parametric down-conversion
, , ,ˆ ˆ( ) ( )s i s i s iI t t E t E t t
ˆ ˆ ˆ ˆ, ( ) ( )i s s s i i i i s sC t t t E t E t E t E t t
Calculation of expectation values
.1ˆ,
i k r t
k kk
E r t l a e
Electric field operator
Intensity
Coincidence
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Simultaneity in parametric down-conversion:very simple view
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Simultaneity in parametric down-conversion:very simple view
0
1 2( ) 1 1i si t ti s i s i st c vac c d d v e
i tE t c d a e
2
, ( ) ( )
( )
i s s s i i i i s s
i i s s
C t t t E t E t E t E t t
E t E t t
Quantum state: simple version
Electric field operator: plane wave, almost monochromatic
Coincidence
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Simultaneity in parametric down-conversion:very simple view
0 02
,
1 1i i s s
i s
i t t t i t t ti s i s i s
C t t
d d v e e
2
, i sii s i sC t t d e
1 2
1 2
0
2
1 2,
1 1
i s
i s
i t i t
i s i t ti s i s i s
d a e d a eC t t
d d v e
Plane wave pumping field 0 i s i sv
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Coincidence detection
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Coincidence detection
0,0 0,5 1,0 1,5 2,0 2,5 3,00,0
0,2
0,4
0,6
0,8
1,0 = 370ps
even
ts (
norm
aliz
ed)
time delay (ns)
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Measurement of time delays
=168ps
=185ps
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Simultaneity in parametric down-conversion:very simple view + detection filters
0 02
( ) ( 1)
,
1
i i s s
i s
i s
i t t t i t t ti s i s i s
C t t
d d ef efv
22 2
( ), i sis si iC t t d ef F
1 2
1 2
0
1 2
2
1 2 )
1
(
1
),
(
i s
i s
i t i t
i s i t ti s i s i s
f fd a e d a eC t t
d d v e
Plane wave pumping field i s i sv
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Simultaneity in parametric down-conversion:very simple view + detection filters
( )f
0 5 10 15 20 25 30
0,0
0,2
0,4
0,6
0,8
1,0
= 3.8 x 1013 Hz
tran
smit
ance
(%)
frequency (Hz)
Interference filter: typical = 10nm, = 3.8 x 1013 Hz, t = 82 fs << 100ps
0 100 200 300 400 500 600
0,0
0,2
0,4
0,6
0,8
1,0
= 82 x 10-15 s
amplit
ude
time(fs)
( )f ( )tF
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Simultaneity in parametric down-conversion:very simple view + timing resolution
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Localized one photon state
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Localized one photon state
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Violation of a classical inequality
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Violation of a classical inequality
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Hong, Ou and Mandel Interferometer
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Hong, Ou and Mandel Interferometer:single mode approach
Beam splitter Input-output relations
122
211
ariatb
ariatb
21
221
22211
122121
aaraataairtaairt
ariatariatbb
trFor
221121 aaaairtbb
22
211
21221
212111
aaraataairtaairt
ariatariatbb
11
222
21221
121222
aaraataairtaairt
ariatariatbb
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Hong, Ou and Mandel Interferometer:single mode approach
Beam splitter Two-photon input state
Coincidence probability
2111 aa
011
11),(
2
2211
2
21
22121
21
21
aa
aa
aaaairt
bbbbbbC
011
11),(
2
222
112
1221
2
112
1111
21
21
aa
aa
aaraataairtaairt
bbbbbbC
011
11),(
21
21
112
222
1221
222222
aa
aa
aaraataairtaairt
bbbbbbC
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Hong, Ou and Mandel Interferometer
2.1
i s
C e
( )f
.c
( )f
2c
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Single-photon tunneling time
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Part II
- Polarization entanglement
- Bell’s inequalities
- Entanglement measurement
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Polarization entanglement:generation
Kwiat et al. PRL 75, 4337 (1995)
H V1
2HV ie
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HH1
2VVie
Kwiat et al. PRA 60, R773 (1999)White et al. PRL 83, 3103 (1999)
Polarization entanglement:generation
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V V1
2H Hie
Kwiat et al. PRA 60, R773 (1999)White et al. PRL 83, 3103 (1999)
Polarization entanglement:generation
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12 1 2 1 2
1
2 H H V V
Mixed state
12 1 2 1 2
1
2 H H V V
Pure entangled state
Mixed states and entangled states
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Detection of entanglement:violation of the Bell inequality
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Bell-CHSH inequality
1 1 2 2 2 1 1 2, , , , 2 S E E E E
, , , ,,
, , , ,
C C C CE
C C C C
Bell inequality and Bell states
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1,2 1 2 1 2
1
2H V V H 1,2 1 2 1 2
1
2H H V V
Bell states for the photon polarization
Coincidence rate for +: 2
, i sC E E
Bell inequality and Bell states
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1,2 1 2 1 2
1
2 H H V V
Bell states for the photon polarization
2
2
2
2
2
,
cos cos cos cos
cos cos sin sin
cos
i s
i s i s
C E E
a a H H V V
H H V V
H H H H
Bell inequality and Bell states
Coincidence rate for +:
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0 0 0 01 1 2 2
0 0 0 01 1 2 2
0 , 22,5 , 45 , 67,5
90 , 112,5 , 135 , 157,5
2
1 1 1 1
2
1 1 1 1
, , cos 22.5 0.854
, , cos 67.5 0.146
C C
C C
2
1 2 1 2
2
1 2 1 2
0.146
0.8
, , cos 67.5
, , cos 22 5 4. 5
C C
C C
Maximal violation
Bell inequality and Bell states
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2
2 1 2 1
2
2 1 2 1
, , cos 22.5 0.854
, , cos 67.5 0.146
C C
C C
2
2 2 2 2
2
2 2 2 2
, , cos 22.5 0.854
, , cos 67.5 0.146
C C
C C
Maximal violation
Bell inequality and Bell states
0 0 0 01 1 2 2
0 0 0 01 1 2 2
0 , 22,5 , 45 , 67,5
90 , 112,5 , 135 , 157,5
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1 2
0.146 0.146 0.854 0.854,
0.146 0.146 0.854 0 854
2
. 2
E
1 1
0.854 0.854 0.146 0.146 2,
0.854 0.854 0.146 0.146 2
E
2 1
0.854 0.854 0.146 0.146 2,
0.854 0.854 0.146 0.146 2
E
2 2
0.854 0.854 0.146 0.146 2,
0.854 0.854 0.146 0.146 2
E
1 1 2 2 2 1 1 2, , , 2, 2 2 .83 E E E ES
Maximal violation 0 0 0 01 1 2 2
0 0 0 01 1 2 2
0 , 22,5 , 45 , 67,5
90 , 112,5 , 135 , 157,5
Bell inequality and Bell states
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Violation of a Bell inequality
- Detects but does not quantify the entanglement properly - Some entangled states do not violate the Bell inequality- Valid for dichotomic or dichotomized systems
Bell inequality and entanglement
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Take a set of measurements :
(H,H); (H,V); (V,H); (V,V); (H,D); (H,L); (D,H); (R,H);
(D,D); (R,D); (R,L); (D,R); (D,V); (R,V); (V,D); (V,L)
C C C C C C C C
C C C C C C C C
Reconstruction of the density matrix
Quantum state tomography
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Quantum state tomography
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12
V V VV
V V V V V VV V
V
HH HH H HH H HH HH
HH H H H H H H
HH H H H H H H
HH H H
V V V V VV V
VV V VV V VV VV VV
Quantum state tomography
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12
V V VV
V V V V V VV V
V
HH HH H HH H HH HH
HH H H H H H H
HH H H H H H H
HH H H
V V V V VV V
VV V VV V VV VV VV
Quantum state tomography
With one can compute all quantities related to the system
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Concurrency:
0,
0
y y y
iC
i
Direct measurement of entanglement
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Mintert, Kus, and Buchleitner, Phys. Rev. Lett. 95 260502 (2005).
12 01 10
2C P
Direct measurement of entanglement using copies of states
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1 2
1 11 1 1 11 1
10;
20C
1
1 1
I / 2
1I / 4 ( )
41
14
P C
Direct measurement of entanglement:pure states
Pure state
Two copies
Maximally entangled state
Two copies
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Experiment with entangled photons
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1 21 2H2
VH1
Vie
Two copies of a state in a single photon
Polarization state
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11 22
1
2ia be ba
Linear momentum state
Two copies of a state in a single photon
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1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
1 1;
2 212
i iMOM POL
i i
a a e b b H H e V V
a a e b b H H e V V
Simultaneous entanglement in polarization and linear momentum
Two copies of a state in a single photon
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1 1
2 2aV bH aH bV
1
21
2
CNOT H V b b
CNOT H V a a
Bell state projection
Bell states combining momentum and polarization
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aH bH aV aV
bH aH bV bV
C-NOT with a SAGNAC interferometer
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Spatial rotations with cilyndrical lenses
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Spatial rotations with cilyndrical lenses
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Direct measurement of entangled with two copies
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S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich,
F. Mintert, A. Buchleitner, Nature 440 1022 (2006)
Direct measurement of entangled with two copies
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S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich,
F. Mintert, A. Buchleitner, Nature 440 1022 (2006)
Direct measurement of entangled with two copies
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Part III
-Entanglement dynamics
-Kraus operators
-Entanglement sudden death
-Process tomography
-Evolution of entanglement
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0,0
0,2
0,4
0,6
0,8
1,0
t
P1(e) e P
2(e)
0,0
0,2
0,4
0,6
0,8
1,0
?
t
Concurrency
Entanglement dynamics
1,2 1 2 1 2
1
2e g g e
1,2 1 2 1 2
in the computational basis
11 0 0 1
2
T. Yu, J. H. Eberly, Phys. Rev. Lett. 93, 140404 (2004). T. Yu, J. H. Eberly, Phys. Rev. Lett. 97, 140403 (2006).
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0,0
0,2
0,4
0,6
0,8
1,0
t
P(e)
Amplitude decay channel
0 0 0 0
1 0 1 1 0 0 1
E ES S
E E ES S Sp p
Quantum channel and Kraus map
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0,0
0,2
0,4
0,6
0,8
1,0
t
P(e)
Operadores de Kraus para o canal de amplitude
1 2 3 4
1 0 0 ˆ, , 00 1 0 0
pK K K K
p
†$( ) K K
Quantum channel and Kraus operators
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0 0 0 0
1 0 1 1 0 0
0 0
0 1 0 1
1
E ES S
E E ES S
E ES S
E E
S
ES S S
H H
V p
p
V p H
p
Amplitude decay channel for one photon polarization
HH
V1 pp V H
Environment
Environment
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Amplitude decay channel for one photon polarization
0 0 0 0
1 0 1 1 0 0
0 0
0 1 0 1
1
E ES S
E E ES S
E ES S
E E
S
ES S S
H H
V p
p
V p H
p
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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Amplitude decay channel for one photon polarization
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V V1
2H Hie
Kwiat et al. PRA 60, R773 (1999)White et al. PRL 83, 3103 (1999)
Polarization entangled state
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M. P. Almeida et al., Science 316, 579 (2007)
Experimental observation of theentanglement sudden death
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M. P. Almeida et al., Science 316, 579 (2007)
HH VV3
ie
HH 3 VVie
Experimental observation of theentanglement sudden death
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/ 2
/ 2
H
V
H V
R H i V
$ ,with ,
and , , ,
j j j j
j H V R
1 2 3 4
$ Kraus operators
, , e
j
K K K K
Process tomography
[( $) ]C I
![Page 92: Experiments With Entangled Photons](https://reader030.vdocument.in/reader030/viewer/2022033022/56814610550346895db31db9/html5/thumbnails/92.jpg)
Reconstruction of the Kraus operators
![Page 93: Experiments With Entangled Photons](https://reader030.vdocument.in/reader030/viewer/2022033022/56814610550346895db31db9/html5/thumbnails/93.jpg)
( )C
$
[( $) ]C I
'
[( $) ] [( $) ] ( )
For pure states
C I C I C
T. Konrad et al., Nature Physics 4, 99 (2008).
A dynamical law for the entanglement
[( $) ] [( $) ] ( )
For mixed states
C I C I C
![Page 94: Experiments With Entangled Photons](https://reader030.vdocument.in/reader030/viewer/2022033022/56814610550346895db31db9/html5/thumbnails/94.jpg)
$
'
( )C [( $) ]C I
[( $) ] [( $) ] ( ) C I C I C
A dynamical law for the entanglement
![Page 95: Experiments With Entangled Photons](https://reader030.vdocument.in/reader030/viewer/2022033022/56814610550346895db31db9/html5/thumbnails/95.jpg)
$
'
( )C [( $) ]C I
[( $) ] [( $) ] ( ) C I C I C
A dynamical law for the entanglement
![Page 96: Experiments With Entangled Photons](https://reader030.vdocument.in/reader030/viewer/2022033022/56814610550346895db31db9/html5/thumbnails/96.jpg)
$
'
$
[( $) ]C I
( )C [( $) ]C I
[( $) ] [( $) ] ( )C I C I C
A dynamical law for the entanglement
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A dynamical law for the entanglement
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O. Farias et al., Science 324, 1414 (2009)
A dynamical law for the entanglement
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O. Farias et al., Science 324, 1414 (2009)
[( $) ] ( )C I C
[( $) ]C I
A dynamical law for the entanglement:experimental test
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( ) mixed state C [( $) ]C I
Inequality
[( $) ] [( $) ] ( ) C I C I C
$
A dynamical law for the entanglement:generalization for mixed states
T. Konrad et al., Nature Physics 4, 99 (2008).
![Page 101: Experiments With Entangled Photons](https://reader030.vdocument.in/reader030/viewer/2022033022/56814610550346895db31db9/html5/thumbnails/101.jpg)
$'I
( )C [( $$') ]C I
[( $) ]C I
[( $) ] [( $$') ] ( )C I C I C
$' $
( )C $' $
[( $) ]C I
A dynamical law for the entanglement:generalization for mixed states
![Page 102: Experiments With Entangled Photons](https://reader030.vdocument.in/reader030/viewer/2022033022/56814610550346895db31db9/html5/thumbnails/102.jpg)
( )C [( $) ]C I
[( $) ] [( $$') ] ( )C I C I C
$' $
$
[( $ '$) ]C I $'
A dynamical law for the entanglement:generalization for mixed states
![Page 103: Experiments With Entangled Photons](https://reader030.vdocument.in/reader030/viewer/2022033022/56814610550346895db31db9/html5/thumbnails/103.jpg)
[( $') ]I
A. Jamiołkowski, Rep. Math. Phys. 3, 275 (1972)
$'
1 2tr 1 i i ii
r e e i i ii
r e f
How to find $'
1/ 2$' i j i j a i j babf f r r f e e f
A dynamical law for the entanglement:generalization for mixed states
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O. Farias et al., Science 324, 1414 (2009)
[( $ '$) ] ( )C I C
[( $) ]C I
A dynamical law for the entanglement:generalization for mixed states
experimental test
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Part VI
-Spatial correlations
-The transfer of the angular spectrum
-Continuous variables etanglement- EPR paradox
-Non-gaussian entanglement
-Non-local optical vortex
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Spatial correlations in the far field
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Spatial correlations in the far field
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Spatial correlations in the far field
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Spatial correlations in the far field
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Spatial anti-bunching:non-classical behavior
Cauchy-Swartz inequality
Homogeneity and stationarity
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0δ 2,22,2
t,,,, 222,2 ρIρIρρ 11
2ρρδ 1
For 0
222,2 ,, ρρρρ 11 C
-15 -10 -5 0 5 10 15
0,0
0,2
0,4
0,6
0,8
1,0
C(
)
2,2 C δ δ
Spatial anti-bunching:non-classical behavior
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S. Mancini, V. Giovannetti, D. Vitali, and P. TombesiPhys. Rev. Lett. 88, 120401 (2002).
S. Mancini, V. Giovannetti, D. Vitali, and P. TombesiPhys. Rev. Lett. 88, 120401 (2002).
2 2
2 1 2 1 1x x p p
Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. ZollerPhys. Rev. Lett. 84, 2722 (2000).
Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. ZollerPhys. Rev. Lett. 84, 2722 (2000).
2 2
2 1 2 1 2x x p p
Inseparability
DGCZ criterion
MGVT criterion
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Inseparability
2 2 22 1 2 1 2
2 2
2 1 2 1
1
. : 1 2
The state is inseparable if
x x p p aa
ex a x x p p
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Inseparability:proof
1 2 1 2
1 1;u a x x v a p p
a a
1 2 ii
p
2 22 2 2 2
2 2 2 2 2 21 2 1 22 2
1 2 1 2
2 2
1 1
2
i i ii
i i i i ii
i ii i i ii i
u v p u v u v
p a x x a p pa a
ap x x p p p
a
u v
![Page 115: Experiments With Entangled Photons](https://reader030.vdocument.in/reader030/viewer/2022033022/56814610550346895db31db9/html5/thumbnails/115.jpg)
Inseparability:proof
2 2
2 2 2 22 21 2 1 22 2
2 22 2
2 2 2 221 1 2 22
2 22 2
1 1
1
iì ì ì ìi
i i i iì i ì ii i i i
iì ì ì ìi
i i i iì i ì ii i i i
u v
p a x x a p pa a
p u p u p v p v
p a x p x pa
p u p u p v p v
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Inseparability:proof
2 2
1 1 1 1
2 2
2 2 2 2
1,
, 1
From the Heisenberg uncertainty principle:
ì ì
ì ì
x p x p
x p x p
2 2 22
2 22 2
1 11
ii
i i i iì i ì ii i i i
u v p aa
p u p u p v p v
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Inseparability:proof
22
22
1
From the Cauchy-Schwartz inequality:
and
i i i
ii
ìi i i
i i iìi i i
p p u p u
pp p v p v
2 2 22
2 2 2 2
11
i i i ii i i i
i i i i
u v aa
p u p u p v p v
0
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Inseparability criterion
2 2 22
1
u v a
a
DGCZ criterion
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Inseparability
22 2
2 1 2 1
2 2
,
. : , ,
The state is inseparable if
i i
i i i i
x x p p x p
ex x p i x p
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Inseparability:proof
1 2 ii
p
2 2
222 2 2
1 22
222 2 2
1 22
1
1
i i iì ii ii i i
i i iì ii ii i i
u v
p a x x p u p ua
p a p p p v p va
1 2 1 2
1 1;u a x x v a p p
a a
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Inseparability:proof
2 2
2 2
2 2 21 22
2 2
2 2 21 22
1
1
i i ii ii ii i i
i i ii ii ii i i
u v
p a x x p u p ua
p a p p p v p va
22
1i i i ii
ìi i i
p p v pp v
Using the Cauchy-Schwartz inequality:
and
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Inseparability:proof
2 2
2 2 21 22
2 2 21 22
01
10
i i ii
i i ii
u v
p a x xa
p a p pa
22
1i i i ii
ìi i i
p p v pp v
Using the Cauchy-Schwartz inequality:
and
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Inseparability:proof
2 2 2 2 2 21 2 1 22 2i ii i i i
i i
u v p x x p p p
2 2 2
Using the inequality:
Using again the Cauchy-Scwarz inequality:
2 2 2 21 2 1 2
212 2 2 2 4
1 2 1 2
i ii i i ii i
i i i i ii
p x x p p p
p x x p p
21
2 2 2 2 2 2 41 2 1 24 i i i i i
i
u v p x x p p
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Inseparability:proof
Using the uncertainty principle:
2 2
1 1 1 12 2 2 21 1 2 2
, ,
4 4and
i i i i
x p x px p x p
21
2 2 2 2 2 2 41 2 1 24 i i i i i
i
u v p x x p p
2 22 22 21 2 1 2 1 1, ,i iu v x p x x p p x p
MGVT criterion
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Inseparability
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2 2 21 2 1 2x x p p
1 1
2 2 22 0 2 0| | 0.01x px p
Inseparability
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Inseparability
2 2 21 2 1 2x x p p
1 1
2 2 22 0 2 0| | 0.01x px p
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Inseparability
1
1
2 2
2 0 1 2
2 2
2 0 1 2
|
|
x
p
x x x
p p p
I t is claimed that
Therefore the inequality is violated
1 1
1 1 1
1 1 1
2 0 2 0
22 22 0 2 2 0 2 2 2 0 2
22 22 0 2 2 0 2 2 2 0 2
( | ) ( | )
| ( | ) ( | )
| ( | ) ( | )
and is measured andx p
x x x
p p p
P x P p
x x P x dx x P x dx
p p P p dp p P p dp
1 1
2 2 22 0 2 0| | 0.01x px p
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Non-gaussian entanglement
Gaussian states are completely characterized by the secondorder momenta:
22 2 ( ) ( )x x P x dx x P x dx
Then, DGCZ, MGVT and other criteria based on second ordermomenta are non optimal for non-gaussian states.
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Higher order criterion
E. Shchukin and W. Vogel Inseparability criteria for continuous bipartite quantum states. Phys Rev Lett. 95, 230502 (2005)
To the second order:
† †
† † † † † † †
† †2
† †
† † † † † † †
1 a a b b
a a a a a a b a b
M a aa a a ab ab
b ab a b b b bb
b ab a b b b bb
a and b are annihillation operators for modes a and b.
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Higher order criterion
E. Shchukin and W. Vogel Inseparability criteria for continuous bipartite quantum states. Phys Rev Lett. 95, 230502 (2005)
† †
† † † † † † †
† †2
† †
† † † † † † †
1 a a b b
a a a a a a b a b
M a aa a a ab ab
b ab a b b b bb
b ab a b b b bb
The state has a positive partial transpose, if and only if all principal minors are non-negative.
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Gaussian and non-gaussian states
Production of a gaussian state with parametric down-conversion
2 2 2 2, exp / 4 exp / 4x x N x s x t
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Gaussian and non-gaussian states
Production of a non-gaussian state with parametric down-conversion
2 2 2 2, exp / 4 exp / 4x x N xx s x t
01 modeHG
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Higher order criterion
We found a non-gaussian state that does not violate any second order criterion:
2 2 2 2, exp / 4 exp / 4x x N xx s x t
According to R. Simon Phys. Rev. Lett. 84, 2726 (2000), if
0 ;
,
a b a b a b a bx x p p x p p x
x x x
with
is satisfied, no second order criterion is violated.
For 0.57 < s/t < 1.73 satisfies the inequality.
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Higher order criterion
†
† † †
1
HO
ab
D
a b a ab b
2 2 2 2, exp / 4 exp / 4x x N xx s x t
However it gives the negative minor below for the higher order criterion
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Isomorphism between a multimode singlephoton field and a single mode multiphoton field
† †1 1
2 2anda a
ir x a a p a a
r
4 2 2 2 24
2 2 2 2 22
11
12 2 0
a b a b a b a b
a b a b a b a b a b a b
r x x x p p x p pr
x p p x x x p p r x x p pr
The inequality is violated for r=1/t and 0.68 < s/t < 1.53
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Experimental observation of genuine non-gaussian entanglement
Quantum entanglement beyond Gaussian criteria R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro and S. P. WalbornProc. Nat. Acad. Sci. 106, 21517-21520(2009)
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Experimental observation of genuine non-gaussian entanglement
Quantum entanglement beyond Gaussian criteria R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro and S. P. WalbornProc. Nat. Acad. Sci. 106, 21517-21520(2009)
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Experimental observation of genuine non-gaussian entanglement
Quantum entanglement beyond Gaussian criteria R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro and S. P. WalbornProc. Nat. Acad. Sci. 106, 21517-21520(2009)