# Quantum Parametric Resonance of a dissipative oscillator: fading and persistent memory in the long-time evolution

## TL;DR

This paper investigates how a quantum oscillator with time-varying frequency and damping exhibits different long-term behaviors under parametric resonance, showing either classical energy limits or persistent quantum memory.

## Contribution

It reveals the conditions under which quantum oscillators lose or retain memory of initial states during parametric resonance, highlighting the interplay between quantum dynamics and external modulation.

## Key findings

- Energy converges to classical value in one case
- Quantum memory persists indefinitely in another case
- Long-time energy fluctuations depend on initial conditions

## Abstract

The evolution of a quantum oscillator, with periodically varying frequency and damping, is studied in the two cases of parametric resonance (PR) producing a limited, or unlimited stretching of the wave function. The different asymptotic behaviors of the energy distribution, show the non trivial interplay between PR and the initial quantum state. In the first case, the oscillator's mean energy tends asymptotically to a fully classical value, independent of the initial state, with vanishing relative quantum fluctuations. In the second case, the memory of the initial state persists over arbitrary long time scales, both in the mean value and in the large quantum fluctuations of the energy.