Comparison of density matrix and state vector approaches to dissipative evolution of hyperfine levels coupled to optical and radio-frequency fields in the two-level approximation

Abstract

We consider a three-level atom interacting with two optical and one microwave fields in the adiabatic approximation resulting in a simplified description in the terms of a two-level system. Working within the reduced density operator framework, we assume this two-level system to be affected by two types of environment, described by some ad hoc non-Hermitian Hamiltonian and Gorini-Kossakowski-Sudarshan-Lindblad’s models. We compare the three types of dissipative evolution which can occur: driven by equations for a normalized density matrix, a non-normalized density matrix and a normalized state vector. Using the latter type, we derive an effective Hamiltonian, which encodes information about not only the Hamiltonian part of an original master equation but also its non-Hamiltonian (Liouvillian) part. The Hamiltonian turns out to be dependent on the wavefunction itself: the effects of above-mentioned environments induce, respectively, cubic and quintic nonlinear terms. For evolutions driven by density operators, we study various indicators of quantum purity. It is shown that if a trace of density operator is not conserved then conventional von Neumann entropy can no longer be used as a purity indicator; therefore we introduce the purity-normalized definition of quantum statistical entropy.