  Synchronization of quantum nonlinear oscillators has attracted much attention
recently. To characterize the quantum oscillatory dynamics, we recently
proposed a fully quantum-mechanical definition of the asymptotic phase, which
is a key quantity in the synchronization analysis of classical nonlinear
oscillators (Kato and Nakao 2022 Chaos 32 063133). In this work, we further
extend this theory and introduce multiple asymptotic phases using the
eigenoperators of the adjoint Liouville superoperator of the quantum nonlinear
oscillator associated with different fundamental frequencies. We analyze a
quantum van der Pol oscillator with Kerr effect in the strong quantum regime
and show that the system has several different fundamental frequencies. By
introducing order parameters and power spectra in terms of the associated
quantum asymptotic phases, we reveal that 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.
