Spectral Gaps and Midgap States in Random Quantum Master Equations

How nonequilibrium systems relax to steady states is a central topic in many-body dynamics. Of particular interest are universal features of such dynamics. In this talk I will discuss the decay rates of a chaotic quantum system coupled to noise. I will do so within a model where the Hamiltonian and (possibly) the system-noise coupling are described by random N x N Hermitian matrices. The focus will be on the spectral properties of the resulting Liouvillian superoperator that governs the time evolution of the system's density matrix. We find that the asymptotic decay rate to the steady state generically remains nonzero in the thermodynamic limit, i.e., the spectrum of the superoperator is gaped as N approaches infinity. However, the size of the gap depends nontrivially on the strength of the coupling to the environment. Initially, the gap increases with the dissipation strength but then reaches a maximum and declines upon additional strengthening of the coupling. Furthermore, a sharp spectral transition takes place: for dissipation beyond a critical value, the slowest decaying eigenvalues of the Liouvillian separate from the main cloud of eigenvalues and become isolated "midgap" states. I will discuss some of the implications of these findings.

link to recoreded talk