Semiclassical phase reduction theory for quantum synchronization

We develop a general theoretical framework of semiclassical phase reduction for analyzing synchronization of quantum limit-cycle oscillators. The dynamics of quantum dissipative systems exhibiting limit-cycle oscillations are reduced to a simple, one-dimensional classical stochastic differential equation approximately describing the phase dynamics of the system under the semiclassical approximation. The density matrix and power spectrum of the original quantum system can be approximately reconstructed from the reduced phase equation. The developed framework enables us to analyze synchronization dynamics of quantum limit-cycle oscillators using the standard methods for classical limit-cycle oscillators in a quantitative way. As an example, we analyze synchronization of a quantum van der Pol oscillator under harmonic driving and squeezing, including the case that the squeezing is strong and the oscillation is asymmetric. The developed framework provides insights into the relation between quantum and classical synchronization and will facilitate systematic analysis and control of quantum nonlinear oscillators.