Title: Experiments With Entangled Photons
1Experiments With Entangled Photons
Paulo Henrique Souto Ribeiro Instituto de FÃsica
- UFRJ
Summer School of Optics Concépcion January/2010
2Quantum Optics Group at IF/UFRJ
3Group 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.
4UFAL
UFMG
UFF
UFRJ
USP-SÃO PAULO
5Outline
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
6Part 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
9Observation of simultaneity
10Observation of simultaneity
11Parametric down-conversion quantum state
Time evolution
Time evolution operator
Time integral
12Simultaneity in parametric down-conversion
Quantum state for weak interaction
13Simultaneity in parametric down-conversion
Quantum state including some approximations
14Simultaneity in parametric down-conversion
Calculation of expectation values
Electric field operator
Intensity
Coincidence
15Simultaneity in parametric down-conversion very
simple view
16Simultaneity in parametric down-conversion very
simple view
Quantum state simple version
Electric field operator plane wave, almost
monochromatic
Coincidence
17Simultaneity in parametric down-conversion very
simple view
Plane wave pumping field
18Coincidence detection
19Coincidence detection
20Measurement of time delays
s168ps
s185ps
21Simultaneity in parametric down-conversion very
simple view detection filters
Plane wave pumping field
22Simultaneity 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
23Simultaneity in parametric down-conversion very
simple view timing resolution
24Localized one photon state
25Localized one photon state
26Violation of a classical inequality
27Violation of a classical inequality
28Hong, Ou and Mandel Interferometer
29Hong, Ou and Mandel Interferometer single mode
approach
Beam splitter
Input-output relations
30Hong, Ou and Mandel Interferometer single mode
approach
Two-photon input state
Beam splitter
Coincidence probability
31Hong, Ou and Mandel Interferometer
32Single-photon tunneling time
33Part II - Polarization entanglement - Bells
inequalities - Entanglement measurement
34Polarization entanglement generation
Kwiat et al. PRL 75, 4337 (1995)
35Polarization entanglement generation
Kwiat et al. PRA 60, R773 (1999) White et al. PRL
83, 3103 (1999)
36Polarization entanglement generation
Kwiat et al. PRA 60, R773 (1999) White et al. PRL
83, 3103 (1999)
37Mixed states and entangled states
Mixed state
Pure entangled state
38Detection of entanglementviolation of the Bell
inequality
39Bell inequality and Bell states
Bell-CHSH inequality
40Bell inequality and Bell states
Bell states for the photon polarization
Coincidence rate for f
41Bell inequality and Bell states
Bell states for the photon polarization
Coincidence rate for f
42Bell inequality and Bell states
Maximal violation
43Bell inequality and Bell states
Maximal violation
44Bell inequality and Bell states
Maximal violation
45Bell 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
46Quantum state tomography
Take a set of measurements
Reconstruction of the density matrix
47Quantum state tomography
48Quantum state tomography
49Quantum state tomography
With r12 one can compute all quantities related
to the system
50Direct measurement of entanglement
Concurrency
51Direct measurement of entanglement using copies
of states
Mintert, Kus, and Buchleitner, Phys. Rev. Lett.
95 260502 (2005).
52Direct measurement of entanglementpure states
Pure state
Two copies
Maximally entangled state
Two copies
53Experiment with entangled photons
54Two copies of a state in a single photon
Polarization state
55Two copies of a state in a single photon
Linear momentum state
56Two copies of a state in a single photon
Simultaneous entanglement in polarization and
linear momentum
57Bell state projection
Bell states combining momentum and polarization
58C-NOT with a SAGNAC interferometer
59Spatial rotations with cilyndrical lenses
60Spatial rotations with cilyndrical lenses
61Direct measurement of entangled with two copies
62Direct measurement of entangled with two copies
S. P. Walborn, P. H. Souto Ribeiro, L.
Davidovich, F. Mintert, A. Buchleitner, Nature
440 1022 (2006)
63Direct measurement of entangled with two copies
S. P. Walborn, P. H. Souto Ribeiro, L.
Davidovich, F. Mintert, A. Buchleitner, Nature
440 1022 (2006)
64Part III -Entanglement dynamics -Kraus
operators -Entanglement sudden death -Process
tomography -Evolution of entanglement
65Entanglement dynamics
T. Yu, J. H. Eberly, Phys. Rev. Lett. 93, 140404
(2004). T. Yu, J. H. Eberly, Phys. Rev. Lett.
97, 140403 (2006).
66Quantum channel and Kraus map
Amplitude decay channel
67Quantum channel and Kraus operators
Operadores de Kraus para o canal de amplitude
68Amplitude decay channel for one photon
polarization
69Amplitude decay channel for one photon
polarization
70Amplitude decay channel for one photon
polarization
71Amplitude decay channel for one photon
polarization
72Amplitude decay channel for one photon
polarization
73Amplitude decay channel for one photon
polarization
74Amplitude decay channel for one photon
polarization
75Amplitude decay channel for one photon
polarization
76Amplitude decay channel for one photon
polarization
77Amplitude decay channel for one photon
polarization
78Amplitude decay channel for one photon
polarization
79Amplitude decay channel for one photon
polarization
80Amplitude decay channel for one photon
polarization
81Amplitude decay channel for one photon
polarization
82Amplitude decay channel for one photon
polarization
83Amplitude decay channel for one photon
polarization
84Amplitude decay channel for one photon
polarization
85Amplitude decay channel for one photon
polarization
86Amplitude decay channel for one photon
polarization
87Amplitude decay channel for one photon
polarization
88Polarization entangled state
Kwiat et al. PRA 60, R773 (1999) White et al. PRL
83, 3103 (1999)
89Experimental observation of the entanglement
sudden death
M. P. Almeida et al., Science 316, 579 (2007)
90Experimental observation of the entanglement
sudden death
M. P. Almeida et al., Science 316, 579 (2007)
91Process tomography
92Reconstruction of the Kraus operators
93A dynamical law for the entanglement
T. Konrad et al., Nature Physics 4, 99 (2008).
94A dynamical law for the entanglement
95A dynamical law for the entanglement
96A dynamical law for the entanglement
97A dynamical law for the entanglement
98A dynamical law for the entanglement
O. Farias et al., Science 324, 1414 (2009)
99A dynamical law for the entanglement experimental
test
O. Farias et al., Science 324, 1414 (2009)
100A dynamical law for the entanglement generalizati
on for mixed states
T. Konrad et al., Nature Physics 4, 99 (2008).
101A dynamical law for the entanglement generalizati
on for mixed states
102A dynamical law for the entanglement generalizati
on for mixed states
103A dynamical law for the entanglement generalizati
on for mixed states
A. Jamiolkowski, Rep. Math. Phys. 3, 275 (1972)
How to find '
104A dynamical law for the entanglement generalizati
on for mixed states experimental test
O. Farias et al., Science 324, 1414 (2009)
105Part VI -Spatial correlations -The transfer of
the angular spectrum -Continuous variables
etanglement- EPR paradox -Non-gaussian
entanglement -Non-local optical vortex
106Spatial correlations in the far field
107Spatial correlations in the far field
108Spatial correlations in the far field
109Spatial correlations in the far field
110Spatial anti-bunching non-classical behavior
Cauchy-Swartz inequality
Homogeneity and stationarity
111Spatial anti-bunching non-classical behavior
112Inseparability
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
113Inseparability
114Inseparabilityproof
115Inseparabilityproof
116Inseparabilityproof
117Inseparabilityproof
118Inseparability criterion
DGCZ criterion
119Inseparability
120Inseparabilityproof
121Inseparabilityproof
122Inseparabilityproof
123Inseparabilityproof
124Inseparabilityproof
MGVT criterion
125Inseparability
126Inseparability
127Inseparability
128Inseparability
129Non-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.
130Higher 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.
131Higher 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)
132Gaussian and non-gaussian states
Production of a gaussian state with parametric
down-conversion
133Gaussian and non-gaussian states
Production of a non-gaussian state with
parametric down-conversion
134Higher 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.
135Higher order criterion
However it gives the negative minor below for the
higher order criterion
136Isomorphism 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
137Experimental 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)
138Experimental 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)
139Experimental 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)