  The dynamics of a quantum self-sustained oscillator as it evolves toward its
limit cycle may be useful in solving related problems like those in quantum
synchronization, yet is inadequately studied. Here we investigate the evolution
of a quantum Rayleigh-van der Pol (RvdP) oscillator, the simplest form of a
self-sustained oscillator exhibiting a quasiharmonic limit cycle, starting from
Fock, thermal, and coherent states. We find that the phase-space dynamics
significantly differ depending on the initial state -- one evolution toward the
limit cycle may take much longer than another and a least-time parameter may be
present. We describe the resulting dynamics in terms of the coherence decay and
the redistribution of eigenstate occupation.
