A definition of the asymptotic phase for quantum nonlinear oscillators from the Koopman operator viewpoint

TL;DR

This paper introduces a new way to define the asymptotic phase for quantum nonlinear oscillators using Koopman operator theory, enabling better analysis of quantum oscillatory systems across different regimes.

Contribution

It provides the first explicit definition of the asymptotic phase for quantum nonlinear oscillators based on eigenoperators of the backward Liouville operator.

Findings

The proposed phase aligns with classical isochronous phase in semiclassical regimes.

It remains valid and meaningful in strong quantum regimes.

Application to quantum van der Pol oscillator with Kerr effect demonstrates effectiveness.

Abstract

We propose a definition of the asymptotic phase for quantum nonlinear oscillators from the viewpoint of the Koopman operator theory. The asymptotic phase is a fundamental quantity for the analysis of classical limit-cycle oscillators, but it has not been defined explicitly for quantum nonlinear oscillators. In this study, we define the asymptotic phase for quantum oscillatory systems by using the eigenoperator of the backward Liouville operator associated with the fundamental oscillation frequency. By using the quantum van der Pol oscillator with Kerr effect as an example, we illustrate that the proposed asymptotic phase appropriately yields isochronous phase values in both semiclassical and strong quantum regimes.