Quantization of the mean decay time for non-Hermitian quantum systems

Abstract

We show that the mean time, which a quantum particle needs to escape from a system to the environment, is quantized and independent from most dynamical details of the system. In particular, we consider a quantum system with a general Hermitian Hamiltonian $\hat{H}$ and one decay channel, through which probability dissipates to the environment with rate $\Gamma$. When the system is initially prepared exactly in the decay state, the mean decay time $\langle T \rangle$ is quantized and equal to $w/(2\Gamma)$. $w$ is the number of distinct energy levels, i.e. eigenvalues of $\hat{H}$, that have overlap with the decay state, and is also the winding number of a transform of the resolvent in the complex plane. Apart from the integer $w$, $\langle T \rangle$ is completely independent of the system's dynamics. The complete decay time distribution can be obtained from an electrostatic analogy and features rare events of very large dissipation times for parameter choices close to critical points, where $w$ changes, e.g. when a degeneracy is lifted. Experiments of insufficient observation time may thus measure a too small value of $w$. We discuss our findings in a disordered tight-binding model and in the two-level atom in a continuous-wave field.