Revivals in Caldeira - Leggett Hamiltonian Dynamics

TL;DR

This paper studies the dynamics of a quantum oscillator coupled to a discrete spectrum reservoir, revealing recurrence cycles, partial revivals, and transition to stochastic behavior, contrasting with traditional dissipative decay in continuous baths.

Contribution

It generalizes Caldeira-Leggett results to discrete spectra, showing recurrence cycles and revivals, and connects these to oscillatory behavior of the effective mass in the Hamiltonian.

Findings

Recurrence cycles with partial revivals of initial states.

Transition from regular to stochastic-like evolution over cycles.

Initial cycle exhibits exponential growth of effective mass.

Abstract

In this work we reconsider the problem discussed by Caldeira and Leggett (CL), and generalize their results for a quantum oscillator coupled bilinearly to a reservoir with dense discrete spectrum of harmonic oscillators. We show that for such systems dynamic evolution of any state of the CL Hamiltonian consists of recurrence cycles with partial revivals of the initial state amplitude. This revival or Loschmidt echo appears in each cycle due to reverse (from the reservoir) transitions (not existing in the limit of a macroscopic bath). Width and number of the Loschmidt echo components increase with the recurrence cycle number, eventually leading to irregular, stochastic-like time evolution. Standard for continuous spectrum thermal bath CL dynamics dissipative dynamics (exponential decay) takes place only within the initial cycle. In terms of the effective CL Hamiltonian, where the reservoir degrees of freedom are integrated out, our results mean oscillatory (time periodic) behavior of the effective mass. Only in the initial cycle, we found the known for CL Hamiltonian exponential in time increase of the effective mass. We do believe that the basic ideas inspiring our work can be applied to a large variety of interesting nano-systems and molecular complexes.