Experiments With Entangled Photons

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Experiments With Entangled Photons
Paulo Henrique Souto Ribeiro Instituto de Física
- UFRJ
Summer School of Optics Concépcion January/2010
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Quantum Optics Group at IF/UFRJ
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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.
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UFAL
UFMG
UFF
UFRJ
USP-SÃO PAULO
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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 -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
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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
Twin Photons
Stimulated emission
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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
Calculation of expectation values
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
Quantum state simple version
Electric field operator plane wave, almost
monochromatic
Coincidence
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Simultaneity in parametric down-conversion very
simple view
Plane wave pumping field
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Coincidence detection
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Coincidence detection
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Measurement of time delays
s168ps
s185ps
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Simultaneity in parametric down-conversion very
simple view detection filters
Plane wave pumping field
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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
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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
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Hong, Ou and Mandel Interferometer single mode
approach
Two-photon input state
Beam splitter
Coincidence probability
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Hong, Ou and Mandel Interferometer
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Single-photon tunneling time
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Part II - Polarization entanglement - Bells
inequalities - Entanglement measurement
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Polarization entanglement generation
Kwiat et al. PRL 75, 4337 (1995)
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Polarization entanglement generation
Kwiat et al. PRA 60, R773 (1999) White et al. PRL
83, 3103 (1999)
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Polarization entanglement generation
Kwiat et al. PRA 60, R773 (1999) White et al. PRL
83, 3103 (1999)
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Mixed states and entangled states
Mixed state
Pure entangled state
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Detection of entanglementviolation of the Bell
inequality
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Bell inequality and Bell states
Bell-CHSH inequality
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Bell inequality and Bell states
Bell states for the photon polarization
Coincidence rate for f
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Bell inequality and Bell states
Bell states for the photon polarization
Coincidence rate for f
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Bell inequality and Bell states
Maximal violation
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Bell inequality and Bell states
Maximal violation
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Bell inequality and Bell states
Maximal violation
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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

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Quantum state tomography
Take a set of measurements
Reconstruction of the density matrix
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Quantum state tomography
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Quantum state tomography
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Quantum state tomography
With r12 one can compute all quantities related
to the system
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Direct measurement of entanglement
Concurrency
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Direct measurement of entanglement using copies
of states
Mintert, Kus, and Buchleitner, Phys. Rev. Lett.
95 260502 (2005).
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Direct measurement of entanglementpure states
Pure state
Two copies
Maximally entangled state
Two copies
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Experiment with entangled photons
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Two copies of a state in a single photon
Polarization state
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Two copies of a state in a single photon
Linear momentum state
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Two copies of a state in a single photon
Simultaneous entanglement in polarization and
linear momentum
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Bell state projection
Bell states combining momentum and polarization
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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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Direct measurement of entangled with two copies
S. P. Walborn, P. H. Souto Ribeiro, L.
Davidovich, F. Mintert, A. Buchleitner, Nature
440 1022 (2006)
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Direct measurement of entangled with two copies
S. P. Walborn, P. H. Souto Ribeiro, L.
Davidovich, F. Mintert, A. Buchleitner, Nature
440 1022 (2006)
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Part III -Entanglement dynamics -Kraus
operators -Entanglement sudden death -Process
tomography -Evolution of entanglement
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Entanglement dynamics
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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Quantum channel and Kraus map
Amplitude decay channel
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Quantum channel and Kraus operators
Operadores de Kraus para o canal de amplitude
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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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Amplitude decay channel for one photon
polarization
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Amplitude decay channel for one photon
polarization
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Polarization entangled state
Kwiat et al. PRA 60, R773 (1999) White et al. PRL
83, 3103 (1999)
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Experimental observation of the entanglement
sudden death
M. P. Almeida et al., Science 316, 579 (2007)
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Experimental observation of the entanglement
sudden death
M. P. Almeida et al., Science 316, 579 (2007)
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Process tomography
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Reconstruction of the Kraus operators
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A dynamical law for the entanglement
T. Konrad et al., Nature Physics 4, 99 (2008).
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A dynamical law for the entanglement
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A dynamical law for the entanglement
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A dynamical law for the entanglement
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A dynamical law for the entanglement
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A dynamical law for the entanglement
O. Farias et al., Science 324, 1414 (2009)
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A dynamical law for the entanglement experimental
test
O. Farias et al., Science 324, 1414 (2009)
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A dynamical law for the entanglement generalizati
on for mixed states
T. Konrad et al., Nature Physics 4, 99 (2008).
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A dynamical law for the entanglement generalizati
on for mixed states
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A dynamical law for the entanglement generalizati
on for mixed states
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A dynamical law for the entanglement generalizati
on for mixed states
A. Jamiolkowski, Rep. Math. Phys. 3, 275 (1972)
How to find '
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A dynamical law for the entanglement generalizati
on for mixed states experimental test
O. Farias et al., Science 324, 1414 (2009)
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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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Spatial anti-bunching non-classical behavior
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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
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Inseparability
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Inseparabilityproof
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Inseparabilityproof
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Inseparabilityproof
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Inseparabilityproof
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Inseparability criterion
DGCZ criterion
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Inseparability
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Inseparabilityproof
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Inseparabilityproof
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Inseparabilityproof
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Inseparabilityproof
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Inseparabilityproof
MGVT criterion
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Inseparability
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Inseparability
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Inseparability
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Inseparability
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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.
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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
a and b are annihillation operators for modes a
and b.
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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)
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Gaussian and non-gaussian states
Production of a gaussian state with parametric
down-conversion
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Gaussian and non-gaussian states
Production of a non-gaussian state with
parametric down-conversion
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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.
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Higher order criterion
However it gives the negative minor below for the
higher order criterion
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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
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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)
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)