Quantum Chaos in Driven Power-law Potentials: Generalized Coherent States to Wave Packet Evolution

Abstract

Quantum wave packets generally spread in their time evolution except in the special case of harmonic oscillator. In bounded Hamiltonian systems, the wave packet reconstructs itself under the condintion of phase matching and manifests the phenomena of quantum revivals and fractional revivals. In an earlier work, Gazeau and Klauder proposed a formalism to construct the quantum states (generalized coherent states) for general Hamiltonian systems with discrete and continuous spectra, and proclaimed them as ‘temporally stable’, i.e. they do not spread under time evolution. In this thesis we study, first, the dispersion of Gazeau-Klauder’s ‘temporally stable’ states, and then study how to overcome the dispersion and to build non-dispersive wave packets for nonlinear dynamical systems by the use of external periodic modulation. Gazeau-Klauder coherent states are developed for power-law potentials and their evolution in space and time is analyzed. We show that these states follow classical dynamics as long as the underlying energy spectrum is linear, otherwise they follow a classical-like evolution upto a few classical periods and disperse thereafter, despite their special construction. The auto-correlation function and probability density as a function of space and time explain the spatio-temporal behavior of these states. The analysis of the wave packet dynamics and resulting recurrence phe- nomena is extended to periodically driven power-law potentials. In the pres- ence of an external periodically modulating force, these potentials may ex- hibit classical and quantum chaos. We show that the dynamics of a quantum wave packet in the modulated power law potentials manifests quantum re- currences at various time scales. We develop general analytical relations for these times and discuss their parametric dependence. We use the recurrence phenomenon as a probe to find out the signatures of nondispersive wave packets in the periodically driven systems.