  Optimal entrainment of a quantum nonlinear oscillator to a periodically
modulated weak harmonic drive is studied in the semiclassical regime. By using
the semiclassical phase reduction theory recently developed for quantum
nonlinear oscillators, two types of optimization problems, one for the
stability and the other for the phase coherence of the entrained state, are
considered. The optimal waveforms of the periodic amplitude modulation can be
derived by applying the classical optimization methods to the semiclassical
phase equation that approximately describes the quantum limit-cycle dynamics.
Using a quantum van der Pol oscillator with squeezing and Kerr effects as an
example, the performance of optimization is numerically analyzed. It is shown
that the optimized waveform for the entrainment stability yields faster
entrainment to the driving signal than the case with a simple sinusoidal
waveform, while that for the phase coherence yields little improvement from the
sinusoidal case. These results are explained from the properties of the phase
sensitivity function.
