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We discuss the decay rates of chaotic quantum systems coupled to noise. Wemodel both the Hamiltonian and the system-noise coupling by random $N \times N$Hermitian matrices, and study the spectral properties of the resulting Lindbladsuperoperator. We consider various random-matrix ensembles, and find that forall of them the asymptotic decay rate remains nonzero in the thermodynamiclimit, i.e., the spectrum of the superoperator is gapped as $N \rightarrow\infty$. A sharp spectral transition takes place as the dissipation strength isincreased: for weak dissipation, the non-zero eigenvalues of the masterequation form a continuum; whereas for strong dissipation, the asymptotic decayrate is an \emph{isolated eigenvalue}, i.e., a `mid-gap state' that is sharplyseparated from the continuous spectrum of the master equation. For finite $N$,the probability of finding a very small gap vanishes algebraically with ascaling exponent that is extensive in system size, and depends only on thesymmetry class of the random matrices and the number of independent decaychannels. We comment on experimental implications of our results.

Spectral gaps and mid-gap states in random quantum master equations