Transient Dynamics of the Quantum Stuart-Landau Oscillator

Abstract: We investigate the transient dynamics of the quantum Stuart-Landau oscillator, the paradigm of a quantum system exhibiting the quantum limit cycle and synchronization. By considering the energy dynamics, we determine a condition for the validity of the classical-regime approximation -- the classical regime eligibility. We also determine the condition for the limit cycle's classical regime and formulate a guess function that fits the 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 study the classical-like behavior embodied by the coherent state in the quantum phase space, comparing its evolution with or without the classical regime eligibility and attributing its lengthy evolution to the slowness of the neighboring-level coherence decay. Furthermore, we consider the evolution of the Wigner negativity as an indicator of quantumness. The Wigner negativity can increase in some cases due to the nonlinear dissipation. To study the evolution speed, we analyze the Liouvillian gap of the system and record the time to reach the steady state for the Fock, thermal, and coherent states. The steady-state time may behave differently from the Liouvillian gap, depending on the initial state. For the diagonal states, there are speedy parameters for which the steady-state time is significantly lowered.