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 :

Entanglement DynamicsEntanglement Dynamics 2†

21†1

2

1

†

22

1aaaaaaH

jjj

Periodic

Quasi-Periodic

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.