The quantum fidelity for the time-periodic singular harmonic oscillator

2005/03/08 by Monique Combescure, M. Combescure
Mathematics · Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1063/1.2178153

published as Journal of Mathematical Physics 47 (2006) 032102

arxiv created 2005/03/08 · openalex publication_date 2006/03/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this paper we perform an exact study of “quantum fidelity” (also called Loschmidt echo) for the time-periodic quantum harmonic oscillator of the following Hamiltonian: Ĥg(t)≔(P2∕2)+f(t)(Q2∕2)+(g2∕Q2), when compared with the quantum evolution induced by Ĥ0(t) (g=0), in the case where f is a T-periodic function and g a real constant. The reference (initial) state is taken to be an arbitrary “generalized coherent state” in the sense of Perelomov. We show that, starting with a quadratic decrease in time in the neighborhood of t=0, this quantum fidelity may recur to its initial value 1 at an infinite sequence of times tk. We discuss the result when the classical motion induced by Hamiltonian Ĥ0(t) is assumed to be stable versus unstable.

Citations

• Quantum-classical correspondence in perturbed chaotic systems
• Quantum irreversibility, perturbation independent decay, and the parametric theory of the local density of states
• Decay of the Loschmidt Echo for Quantum States with Sub-Planck-Scale Structures
• Decay of the classical Loschmidt echo in integrable systems
• Stability of classical chaotic motion under a system’s perturbations