Lock Yue Chew and Ning Ning Chung Division of Physics and Applied Physics

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. Mechanical Oscillator Lasers Atom 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)]

4. Outline Quantum-Classical Correspondence in terms of Entanglement Entropy: • Linear Systems  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 Duan’s criterion : the quantum state is entangled. Numerical Computation : Analytical Calculation : Phys. Rev. A 80, 012103 (2009). Phys. Rev. A 76, 032113 (2007);

6. Two-Mode Magnon System

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

8. Coupled Harmonic Oscillators Restrict Classical frequencies : Poincaré surface of section Classical Dynamics: • 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 Quasi-periodic: Periodic:

9. Entanglement Dynamics Periodic Quasi-Periodic

10. Frequency Doubling: and • Periodic or quasi-periodic dynamics depends on the ratio: Dynamical Entanglement Generation • 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 Regular orbits Mixed regular and chaotic orbits Chaotic orbits Classical Dynamics:

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

13. The frequency of oscillation increases as increases. Quantum Chaos via EntanglementDynamics • 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)

14. 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. Thank You for your Attention!