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 Entanglement An important resource in quantum information processing: • superdense coding • quantum teleportation • quantum cryptography quantum key distribution 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)] Mechanical Oscillator Lasers Atom Outline Quantum-Classical Correspondence in terms of Entanglement Entropy: •Linear Systems Two-mode magnon system Coupled harmonic oscillator system •Nonlinear System Coupled quartic system Entanglement Dynamics Numerical Computation : number basis of harmonic oscillator M M d i m1 , m2 | (t ) m1 , m2 | H | n1 , n2 n1 , n2 | (t ) dt n1 0 n2 0 Initial States : | (0) | 1 | 2 Coherent state with center located at ( x1 , p1 , x2 , p2 ) . SvN (t ) Tr1 (t ) ln 1 (t ) Analytical Calculation : u(t ) x1 (t ) x2 (t ) (t ) u(t ) v(t ) 2 2 2 v(t ) p1 (t ) p2 (t ) Duan’s criterion : (t ) 0 , the quantum state is entangled. Phys. Rev. A 76, 032113 (2007); Phys. Rev. A 80, 012103 (2009). Two-Mode Magnon System 2 p2 1 2 H x j x1 x2 p1 p2 j j 1 2 2 1 2 H a † a j a1† a2 a1a2 j † j 1 2 d a1 i i a1 † a i † dt 2 i a2 2 1 i a1 (t ) cosh t sinh t a1 (0) sinh t a2 (0) i † † i a2 (t ) cosh t sinh t a2 (0) sinh t a1 (0) † i Quantum-Classical Correspondence For 1 Classical : Saddle Quantum : S vN diverges 4 (t ) 2 1 1 sinh t 2 1 2 Frequency Doubling! For 1 Classical : Center with frequency 1 2 Quantum : Periodic entanglement dynamics (t ) 2 1 cos 2t 1 2 1 Coupled Harmonic Oscillators Classical Dynamics: Restrict 1 2 p2 1 2 H x j x1 x2 j •Periodic or quasi-periodic dynamics j 1 2 2 •Periodic dynamics: •Two-frequency periodic •One-frequency periodic (Cross) – initial conditions are in eigenspace of either one of the frequencies Classical frequencies : 1 1 2 1 Poincaré surface of section Periodic: 11 / 61 Quasi-periodic: 0.19 Entanglement Dynamics 1 † 2 H a † a j j a1 a1 a2 a2 † j 1 2 2 Periodic Quasi-Periodic Dynamical Entanglement Generation 1 12 1 (t ) 2 12 1 cos21 t 1 22 1 1 cos22 t 2 •Frequency Doubling: 1 21 and 2 22 •Periodic or quasi-periodic dynamics depends on the ratio: 1 / 2 •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 Oscillators Classical Dynamics: 2 p2 H 3x14 x2 x12 x2 j 4 2 j 1 2 Regular orbits Mixed regular and Chaotic orbits 0.4 chaotic orbits 2.7 0.8 Entanglement Dynamics Quantum Regime Semi-classical Regime Phys. Rev. E 80, 016204 (2009). 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. •The frequency of oscillation increases as increases. •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) 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!