1. The quantum signature of chaos through the dynamics of entanglement in classically regular and chaotic systems Lock Yue Chew and Ning Ning Chung Division of Physics and Applied Physics School of Physical and Mathematical Sciences

2. Entanglement An important resource in quantum information processing: superdense coding quantum teleportation quantum cryptography  quantum key distribution

3. Practical 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

4. Outline Linear Systems Quantum-Classical Correspondence in terms of Entanglement Entropy:  Two-mode magnon system  Coupled harmonic oscillator system Nonlinear System  Coupled quartic system

5. Entanglement Dynamics number basis of harmonic oscillator Initial States : Coherent state with center located at the quantum state is entangled.Duan’s criterion : Numerical Computation : Analytical Calculation : Phys. Rev. A 76, 032113 (2007); Phys. Rev. A 80, 012103 (2009).

6. Two-Mode Magnon System

7. Quantum-Classical Correspondence For Classical : Center with frequency Quantum : Periodic entanglement dynamics For Classical : Saddle Quantum :diverges Frequency Doubling!

8. Coupled Harmonic Oscillators 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: Restrict Poincaré surface of section Classical frequencies :

9. Entanglement Dynamics Periodic Quasi-Periodic

10. Dynamical Entanglement Generation Frequency Doubling:and Periodic or quasi-periodic dynamics depends on the ratio: 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.

11. Coupled Quartic Oscillators Classical Dynamics: Regular orbitsMixed regular and chaotic orbits Chaotic orbits

12. Quantum Regime Semi-classical Regime Entanglement Dynamics Phys. Rev. E 80, 016204 (2009).

13. Quantum Chaos via Entanglement Dynamics 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 asincreases.

14. Thank You for your Attention! Summary 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.