Transient dynamics of the quantum Stuart-Landau oscillator

We investigate the transient dynamics of the quantum Stuart-Landau oscillator, a paradigmatic quantum system exhibiting a quantum limit cycle and synchronization. From the energy dynamics, we determine a condition for the classical regime of transient dynamics and the limit cycle. Additionally, we formulate a guess function that fits the classical-regime steady-state Wigner function. The equation of motion for the Wigner function is derived and compared to the Kramers-Moyal equation for stochastic processes. We then characterize the classical-like behavior as the system evolves from a coherent state, noting the slow decay of neighboring-level coherence. We also study the evolution of the Wigner negativity as an indicator of nonclassicality, showing its temporary increase for some specific cases. To quantify the evolution speed, we examined the system's Lindbladian spectra, particularly the Liouvillian gap. Finally, we record the time it takes to reach the steady state for some Fock, thermal, and coherent states. The parameter dependence of the steady-state time may differ from the Liouvillian gap, and the limit-cycle attraction is significantly slower for coherent states compared to Fock or thermal states. For the diagonal states, there are fast convergence regimes for which the steady-state time is locally minimized. This study provides a deeper insight into the transient behavior of self-sustained quantum systems.