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Title: Experiments With Entangled Photons


1
Experiments With Entangled Photons
Paulo Henrique Souto Ribeiro Instituto de Física
- UFRJ
Summer School of Optics Concépcion January/2010
2
Quantum Optics Group at IF/UFRJ
3
Group members
Experiments Prof. Paulo Henrique Souto Ribeiro
Prof. Stephen Patrick Walborn Theory Prof.
Luiz Davidovich Prof. Nicim Zagury Prof. Ruynet
Matos Filho Prof. 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.
4
UFAL
UFMG
UFF
UFRJ
USP-SÃO PAULO
5
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 -Bells 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
6
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
7
Parametric Down-conversion
Espontaneous emission
Twin Photons
Stimulated emission
8
Parametric Down-conversion
9
Observation of simultaneity
10
Observation of simultaneity
11
Parametric down-conversion quantum state
Time evolution
Time evolution operator
Time integral
12
Simultaneity in parametric down-conversion
Quantum state for weak interaction
13
Simultaneity in parametric down-conversion
Quantum state including some approximations
14
Simultaneity in parametric down-conversion
Calculation of expectation values
Electric field operator
Intensity
Coincidence
15
Simultaneity in parametric down-conversion very
simple view
16
Simultaneity in parametric down-conversion very
simple view
Quantum state simple version
Electric field operator plane wave, almost
monochromatic
Coincidence
17
Simultaneity in parametric down-conversion very
simple view
Plane wave pumping field
18
Coincidence detection
19
Coincidence detection
20
Measurement of time delays
s168ps
s185ps
21
Simultaneity in parametric down-conversion very
simple view detection filters
Plane wave pumping field
22
Simultaneity in parametric down-conversion very
simple view detection filters
Interference filter typical Dl 10nm, Dw 3.8
x 1013 Hz,
Dt 82 fs ltlt 100ps
23
Simultaneity in parametric down-conversion very
simple view timing resolution
24
Localized one photon state
25
Localized one photon state
26
Violation of a classical inequality
27
Violation of a classical inequality
28
Hong, Ou and Mandel Interferometer
29
Hong, Ou and Mandel Interferometer single mode
approach
Beam splitter
Input-output relations
30
Hong, Ou and Mandel Interferometer single mode
approach
Two-photon input state
Beam splitter
Coincidence probability
31
Hong, Ou and Mandel Interferometer
32
Single-photon tunneling time
33
Part II - Polarization entanglement - Bells
inequalities - Entanglement measurement
34
Polarization entanglement generation
Kwiat et al. PRL 75, 4337 (1995)
35
Polarization entanglement generation
Kwiat et al. PRA 60, R773 (1999) White et al. PRL
83, 3103 (1999)
36
Polarization entanglement generation
Kwiat et al. PRA 60, R773 (1999) White et al. PRL
83, 3103 (1999)
37
Mixed states and entangled states
Mixed state
Pure entangled state
38
Detection of entanglementviolation of the Bell
inequality
39
Bell inequality and Bell states
Bell-CHSH inequality
40
Bell inequality and Bell states
Bell states for the photon polarization
Coincidence rate for f
41
Bell inequality and Bell states
Bell states for the photon polarization
Coincidence rate for f
42
Bell inequality and Bell states
Maximal violation
43
Bell inequality and Bell states
Maximal violation
44
Bell inequality and Bell states
Maximal violation
45
Bell inequality and entanglement
  • 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

46
Quantum state tomography
Take a set of measurements
Reconstruction of the density matrix
47
Quantum state tomography
48
Quantum state tomography
49
Quantum state tomography
With r12 one can compute all quantities related
to the system
50
Direct measurement of entanglement
Concurrency
51
Direct measurement of entanglement using copies
of states
Mintert, Kus, and Buchleitner, Phys. Rev. Lett.
95 260502 (2005).
52
Direct measurement of entanglementpure states
Pure state
Two copies
Maximally entangled state
Two copies
53
Experiment with entangled photons
54
Two copies of a state in a single photon
Polarization state
55
Two copies of a state in a single photon
Linear momentum state
56
Two copies of a state in a single photon
Simultaneous entanglement in polarization and
linear momentum
57
Bell state projection
Bell states combining momentum and polarization
58
C-NOT with a SAGNAC interferometer
59
Spatial rotations with cilyndrical lenses
60
Spatial rotations with cilyndrical lenses
61
Direct measurement of entangled with two copies
62
Direct measurement of entangled with two copies
S. P. Walborn, P. H. Souto Ribeiro, L.
Davidovich, F. Mintert, A. Buchleitner, Nature
440 1022 (2006)
63
Direct measurement of entangled with two copies
S. P. Walborn, P. H. Souto Ribeiro, L.
Davidovich, F. Mintert, A. Buchleitner, Nature
440 1022 (2006)
64
Part III -Entanglement dynamics -Kraus
operators -Entanglement sudden death -Process
tomography -Evolution of entanglement
65
Entanglement dynamics
T. Yu, J. H. Eberly, Phys. Rev. Lett. 93, 140404
(2004). T. Yu, J. H. Eberly, Phys. Rev. Lett.
97, 140403 (2006).
66
Quantum channel and Kraus map
Amplitude decay channel
67
Quantum channel and Kraus operators
Operadores de Kraus para o canal de amplitude
68
Amplitude decay channel for one photon
polarization
69
Amplitude decay channel for one photon
polarization
70
Amplitude decay channel for one photon
polarization
71
Amplitude decay channel for one photon
polarization
72
Amplitude decay channel for one photon
polarization
73
Amplitude decay channel for one photon
polarization
74
Amplitude decay channel for one photon
polarization
75
Amplitude decay channel for one photon
polarization
76
Amplitude decay channel for one photon
polarization
77
Amplitude decay channel for one photon
polarization
78
Amplitude decay channel for one photon
polarization
79
Amplitude decay channel for one photon
polarization
80
Amplitude decay channel for one photon
polarization
81
Amplitude decay channel for one photon
polarization
82
Amplitude decay channel for one photon
polarization
83
Amplitude decay channel for one photon
polarization
84
Amplitude decay channel for one photon
polarization
85
Amplitude decay channel for one photon
polarization
86
Amplitude decay channel for one photon
polarization
87
Amplitude decay channel for one photon
polarization
88
Polarization entangled state
Kwiat et al. PRA 60, R773 (1999) White et al. PRL
83, 3103 (1999)
89
Experimental observation of the entanglement
sudden death
M. P. Almeida et al., Science 316, 579 (2007)
90
Experimental observation of the entanglement
sudden death
M. P. Almeida et al., Science 316, 579 (2007)
91
Process tomography
92
Reconstruction of the Kraus operators
93
A dynamical law for the entanglement
T. Konrad et al., Nature Physics 4, 99 (2008).
94
A dynamical law for the entanglement
95
A dynamical law for the entanglement
96
A dynamical law for the entanglement
97
A dynamical law for the entanglement
98
A dynamical law for the entanglement
O. Farias et al., Science 324, 1414 (2009)
99
A dynamical law for the entanglement experimental
test
O. Farias et al., Science 324, 1414 (2009)
100
A dynamical law for the entanglement generalizati
on for mixed states
T. Konrad et al., Nature Physics 4, 99 (2008).
101
A dynamical law for the entanglement generalizati
on for mixed states
102
A dynamical law for the entanglement generalizati
on for mixed states
103
A dynamical law for the entanglement generalizati
on for mixed states
A. Jamiolkowski, Rep. Math. Phys. 3, 275 (1972)
How to find '
104
A dynamical law for the entanglement generalizati
on for mixed states experimental test
O. Farias et al., Science 324, 1414 (2009)
105
Part VI -Spatial correlations -The transfer of
the angular spectrum -Continuous variables
etanglement- EPR paradox -Non-gaussian
entanglement -Non-local optical vortex
106
Spatial correlations in the far field
107
Spatial correlations in the far field
108
Spatial correlations in the far field
109
Spatial correlations in the far field
110
Spatial anti-bunching non-classical behavior
Cauchy-Swartz inequality
Homogeneity and stationarity
111
Spatial anti-bunching non-classical behavior
112
Inseparability
Lu-Ming Duan, G. Giedke, J. I. Cirac, and P.
Zoller Phys. Rev. Lett. 84, 2722 (2000).
DGCZ criterion
S. Mancini, V. Giovannetti, D. Vitali, and P.
TombesiPhys. Rev. Lett. 88, 120401 (2002).
MGVT criterion
113
Inseparability
114
Inseparabilityproof
115
Inseparabilityproof
116
Inseparabilityproof
117
Inseparabilityproof
118
Inseparability criterion
DGCZ criterion
119
Inseparability
120
Inseparabilityproof
121
Inseparabilityproof
122
Inseparabilityproof
123
Inseparabilityproof
124
Inseparabilityproof
MGVT criterion
125
Inseparability
126
Inseparability
127
Inseparability
128
Inseparability
129
Non-gaussian entanglement
Gaussian states are completely characterized by
the second order momenta
Then, DGCZ, MGVT and other criteria based on
second order momenta are non optimal for
non-gaussian states.
130
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
a and b are annihillation operators for modes a
and b.
131
Higher order criterion
The state has a positive partial transpose, if
and only if all principal minors are non-negative.
E. Shchukin and W. Vogel Inseparability criteria
for continuous bipartite quantum states. Phys
Rev Lett. 95, 230502 (2005)
132
Gaussian and non-gaussian states
Production of a gaussian state with parametric
down-conversion
133
Gaussian and non-gaussian states
Production of a non-gaussian state with
parametric down-conversion
134
Higher order criterion
We found a non-gaussian state that does not
violate any second order criterion
According to R. Simon Phys. Rev. Lett. 84, 2726
(2000), if
is satisfied, no second order criterion is
violated.
For 0.57 lt s/t lt 1.73 Y satisfies the inequality.
135
Higher order criterion
However it gives the negative minor below for the
higher order criterion
136
Isomorphism between a multimode single photon
field and a single mode multiphoton field
The inequality is violated for r1/t and 0.68 lt
s/t lt 1.53
137
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)
138
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)
139
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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