Quantum Origin of Limit Cycles, Fixed Points, and Critical Slowing Down

TL;DR

This paper explores how quantum systems can exhibit classical-like limit cycles and critical slowing down through spectral analysis of Markovian master equations, bridging quantum and classical dissipative dynamics.

Contribution

It demonstrates the emergence of classical limit cycles and algebraic decay in quantum systems near the classical limit using spectral analysis of the Liouville spectrum.

Findings

Limit cycles correspond to slow-decaying spectral branches with vanishing decoherence rates.

Power-law decay arises from spectral collapse at bifurcation points.

Quantum fixed points have gapped spectra, distinct from classical linearized dynamics.

Abstract

Among the most iconic features of classical dissipative dynamics are persistent limit-cycle oscillations and critical slowing down at the onset of such oscillations, where the system relaxes purely algebraically in time. On the other hand, quantum systems subject to generic Markovian dissipation decohere exponentially in time, approaching a unique steady state. Here we show how coherent limit-cycle oscillations and algebraic decay can emerge in a quantum system governed by a Markovian master equation as one approaches the classical limit, illustrating general mechanisms using a single-spin model and a two-site lossy Bose-Hubbard model. In particular, we demonstrate that the fingerprint of a limit cycle is a slow-decaying branch with vanishing decoherence rates in the Liouville spectrum, while a power-law decay is realized by a spectral collapse at the bifurcation point. We also show how these are distinct from the case of a classical fixed point, for which the quantum spectrum is gapped and can be generated from the linearized classical dynamics.