Quasi-Stationary States in Open Quantum Systems

Any open system will eventually end up in a steady state. Provided no symmetries are hidden in the evolution generator, the steady state is unique. The state of an open system is described with a positive element of some algebra: it is a probability vector or density (in classical physics) or a density matrix (in quantum physics). If the unique steady state is pure, evolution may take place is the bulk and might never rich the steady state during the physically meaningful time. Such evolution can be characterized with the asymptotic state of the bulk dynamics under the condition of non-adsorption to the steady state.

The concept of quasistationary states (QS) was developed in the framework of classical Markov processes more than forty years ago. It is different from a conventional approach based on the analysis of eigenmodes of the evolution generator, where so-called "metastable states" appear as the steady states "coated" with the long-living eigenmodes (neither of which is a state).

We generalize the concept of QS to open quantum systems with pure steady states. Our generalization leads to completely-positive non-trace-preserving quantum maps produced by Lindblad-type generators. We illustrate the idea by using the Heisenberg chain with a single-site spin-"leaking" dissipation.