Lie Algebraic Quantum Phase Reduction

TL;DR

This paper develops a quantum phase reduction framework for nonlinear oscillators using quantum trajectory theory, enabling analysis of phase dynamics and response to perturbations in quantum systems.

Contribution

It introduces a Lie algebraic approach to quantum phase reduction, applicable to finite-level systems and incorporating measurement effects on phase behavior.

Findings

Measurement induces phase clustering in quantum oscillators.

The method captures phase dynamics via observable clusters.

Applicable to systems without classical analogs.

Abstract

We introduce a general framework of phase reduction theory for quantum nonlinear oscillators. By employing the quantum trajectory theory, we define the limit-cycle trajectory and the phase according to a stochastic Schr\"{o}dinger equation. Because a perturbation is represented by unitary transformation in quantum dynamics, we calculate phase response curves with respect to generators of a Lie algebra. Our method shows that the continuous measurement yields phase clusters and alters the phase response curves. The observable clusters capture the phase dynamics of individual quantum oscillators, unlike indirect indicators obtained from density operators. Furthermore, our method can be applied to finite-level systems that lack classical counterparts.