Semiclassical Phase Redcution Theory for Quantum Dissipative Nonlinear Oscillators*

The phase reduction theory is a framework for analyzing rhythmic dynamics of weakly perturbed classical limit-cycle oscillators. It has been widely used for analyzing synchronization properties of classical dissipative nonlinear oscillators, but it has not been formulated for quantum dissipative nonlinear oscillators in a general way. Thus, we formulate a phase reduction theory for quantum dissipatve oscillators. More specifically, we derive a semiclassical multi-dimensional Langevin equation from a general master equation for quantum dissipative systems exhibiting limit-cycle oscillations, and reduce it to an approximate one-dimensional classical stochastic differential equation describing phase dynamics of the oscillator. The density matrix and power spectrum of the oscillator can be explicitly reconstructed from the reduced phase equation. As an example, we analyze synchronization properties of a quantum van der Pol oscillator with harmonic driving and squeezing. The proposed framework allows us to analyze the dynamics of quantum dissipative nonlinear oscillators by using a simple classical stochastic differential equation under semiclassical approximation.