Variational functions in degenerate open quantum systems

Abstract

We have derived a Lyapunov functional for a degenerate open atomic system. This functional develops monotonically towards its stationary state. The open system is described by a Lindblad-type master equation. For the construction of the variational functional it is necessary that the Lindblad operator can be diagonalized. Since the generator of motion is non-Hermitian, diagonalization is, in general, only possible if the eigenvalues are nondegenerate. In this paper, we propose that in a physical system the biorthogonal eigenbasis of the Lindblad operator remains complete even when degeneracy is present. Thus diagonalization of the Lindblad operator, and consequently the construction of the variational functional, is still possible. We discuss the reasons and illustrate the theory of the variational functional for a driven $\ensuremath{\Lambda}$-type three-level atom with degenerate ground state. The degeneracy has interesting effects on the variational functional in the steady state with respect to its interpretation as an entropic quantity. In case of the driven three-level atom, the dark state turns out to be an isentropic state.