  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.
