Microscopic quantum generalization of classical Li\'{e}nard oscillators

TL;DR

This paper develops a quantum generalization of classical Lie9nard oscillators using a system-reservoir model, deriving quantum Langevin equations that reveal stable limit cycles and preserve dynamical stability under vacuum excitation.

Contribution

It introduces a microscopic quantum framework for Lie9nard oscillators, deriving quantum Langevin equations and quantum variants of classical models within a mean-field approach.

Findings

Quantum Langevin equations admit single or multiple limit cycles.

Dynamical stability is preserved under vacuum excitation.

Quantum versions of classical Lie9nard oscillators are derived.

Abstract

Based on a system-reservoir model and an appropriate choice of nonlinear coupling, we have explored the microscopic quantum generalization of classical Li\'{e}nard systems. Making use of oscillator coherent states and canonical thermal distributions of the associated c-numbers, we have derived the quantum Langevin equation of the reduced system which admits of single or multiple limit cycles. It has been shown that detailed balance in the form of fluctuation-dissipation relation preserves the dynamical stability of the attractors even in case of vacuum excitation. The quantum versions of Rayleigh, Van der Pol and several other variants of Li\'{e}nard oscillators are derived as special cases in our theoretical scheme within a mean-field description.