  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.
