Quantum asymptotic amplitude for quantum oscillatory systems from the Koopman operator viewpoint

We have recently proposed a fully quantum-mechanical definition of the asymptotic phase for quantum nonlinear oscillators, which is also applicable in the strong quantum regime [Kato and Nakao 2022 Chaos 32 063133]. In this study, we propose a definition of the quantum asymptotic amplitude for quantum oscillatory systems, which naturally extends the asymptotic amplitude for classical nonlinear oscillators on the basis of the Koopman operator theory. We introduce the asymptotic amplitude for quantum oscillatory systems by using the eigenoperator of the backward Liouville operator associated with the largest non-zero real eigenvalue. Using examples of the quantum van der Pol oscillator with the quantum Kerr effect, which exhibits quantum limit-cycle oscillations, and the quantum van der Pol model with the quantum squeezing and degenerate parametric oscillator with nonlinear damping, which exhibit quantum noise-induced oscillations, we demonstrate that the proposed quantum asymptotic amplitude yields isostable amplitude values that decay exponentially with a constant rate. Furthermore, using the quantum asymptotic amplitude, we introduce effective quantum periodic orbits for quantum limit-cycle oscillations and quantum noise-induced oscillations.