Asymptotic phase function for quantum nonlinear oscillators reveals signatures of quantum synchronization

We propose the asymptotic phase function for quantum nonlinear oscillators. We introduce the asymptotic phase function of the system in terms of the eigenoperator of the adjoint Liouville superoperator associated with the fundamental frequency. This quantum asymptotic phase function yields appropriate phase values of the system even in the strong quantum regime and reproducing the conventional asymptotic phase in the semiclassical regime. We analyze a quantum van der Pol oscillator with Kerr effect and show that there are several dominant eigenoperators with different fundamental frequencies in the strong quantum regime. The quantum asymptotic phase functions with respective fundamental frequencies reveal that the multiple phase locking of the system with a harmonic drive at several different frequencies, an explicit quantum signature observed only in the strong quantum regime, can be interpreted as synchronization on a torus rather than a simple limit cycle.