The quantum signature of chaos through the
dynamics of entanglement in
classically regular and chaotic systems
Lock Yue Chew and Ning Ning ChungDivision of Physics and Applied Physics
School of Physical and Mathematical Sciences
EntanglementEntanglement
An important resource in quantum information processing:
• superdense coding
• quantum teleportation
• quantum cryptography
quantum key distribution
Practical SystemsPractical Systems
• A micromechanical resonators strongly coupled to an optical cavity field. Such a system has been realized experimentally. [S. Gröblacher et al, Nature 460, 724 (2009)]
• Optomechanical oscillator strongly coupled to a trapped atom via a quantized light field in a laser driven cavity. [K. Hammerer et al, Phys. Rev. Lett. 103, 063005 (2009)]
Lasers
Atom
Mechanical Oscillator
OutlineOutline
•Linear Systems
Quantum-Classical Correspondence in terms of Entanglement Entropy:
Two-mode magnon system
Coupled harmonic oscillator system
•Nonlinear System Coupled quartic system
Entanglement DynamicsEntanglement Dynamics
2)()()( 22 tvtut
)(|,,||,)(|, 2121210 0
21
1 2
tnnnnHmmtmmdt
di
M
n
M
n
number basis of harmonic oscillator
Initial States : 21 ||)0(|
Coherent state with center located at .),,,( 2211 pxpx
)(ln)(Tr)( 11 tttSvN
)()()( 21 txtxtu
)()()( 21 tptptv
the quantum state is entangled.Duan’s criterion : ,0)( t
Numerical Computation :
Analytical Calculation :
Phys. Rev. A 76, 032113 (2007);Phys. Rev. A 80, 012103 (2009).
Two-Mode Magnon SystemTwo-Mode Magnon System
21†2
†1
2
1
†
2
1aaaaaaH
jjj
†2
1†2
1
a
a
ii
ii
a
a
dt
d
)0(sinh)0(sinhcosh)( †211 at
iat
itta
2
12121
22
2
1
2jj
j ppxxxp
H
)0(sinh)0(sinhcosh)( 1†2
†2 at
iat
itta
12
Quantum-Classical CorrespondenceQuantum-Classical Correspondence
22
21sinh1
1
4)(
tt
For
1Classical : Center with frequency
21 112cos
1
2)( 2
ttQuantum : Periodic entanglement dynamics
For
1
Classical : Saddle
Quantum : vNS diverges
Frequency Doubling!
Coupled Harmonic OscillatorsCoupled Harmonic Oscillators
2
121
22
2
1
2jj
j xxxp
H •Periodic or quasi-periodic dynamics •Periodic dynamics:
•Two-frequency periodic•One-frequency periodic (Cross) – initial conditions are in eigenspace of either one of the frequencies
Classical Dynamics:
Periodic: Quasi-periodic:
1Restrict
Poincaré surface of section 61/11 19.0
11
12
Classical frequencies :
ttt 22212
1
21 2cos11
2
12cos1
1
2
1)(
Dynamical Entanglement Dynamical Entanglement GenerationGeneration
•Frequency Doubling: 11 2 22 2and
•Periodic or quasi-periodic dynamics depends on the ratio: 21 /
•Independent of initial coherent states
•Entanglement dynamics depends solely on the global classical behavior and not on the local dynamical behavior.
•A periodic classical trajectory can give rise to a corresponding quasi-periodic entanglement dynamics upon quantization.
Coupled Quartic OscillatorsCoupled Quartic OscillatorsClassical Dynamics:
4.0Regular orbits Mixed regular and
chaotic orbits 8.0
Chaotic orbits 7.2
2
1
22
21
42
41
2
32j
j xxxxp
H
Quantum Regime Semi-classical Regime
Entanglement DynamicsEntanglement Dynamics
Phys. Rev. E 80, 016204 (2009).
Quantum Chaos via EntanglementQuantum Chaos via Entanglement DynamicsDynamics
•Entanglement entropy is much larger in the semi-classical regime.
•In both the quantum and semi-classical regime, the entanglement production rate is
•The highest in the pure chaos case,•Lower in the mixed case,•Lowest in the regular case.
•Identical results are obtained when different initial conditions are employed in the mixed case.
=> Entanglement dynamics depends entirely on the global dynamical regime and not on the local classical behavior.•Surprisingly, this result differs from:•S.-H. Zhang and Q.-L. Jie, Phys. Rev. A 77, 012312 (2008).•M. Novaes, Ann. Phys. (N.Y.) 318, 308 (2005)
•The frequency of oscillation increases as increases.
Thank You for your Thank You for your Attention!Attention!
SummarySummary•Dependence of entanglement dynamics on the global classical dynamical regime.
•This global dependence has the advantage of generating an encoding subspace that is stable against any errors in the preparation of the initial separable coherent states.
Such a feature will be physically significant in the design of robust quantum information processing protocols.
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