Relaxation of interacting open quantum systems
a Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, ul. Izhorskaya 13/19, Moscow, 127412, Russian Federation
b Dukhov Research Institute of Automatics, ul. Sushchevskaya 22, Moscow, 119017, Russian Federation
c Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow Region, 141700, Russian Federation
d Department of Physics, Queens College of the City University of New York, Flushing, New York, USA
e The Graduate Center of the City University of New York, Fifth Avenue 365, New York, NY, 10016-4309, USA
We consider a transition from the description of a closed quantum system that includes an open quantum system and a reservoir to the description of an open quantum system alone by eliminating reservoir degrees of freedom by averaging over them. An approach based on the Lindblad master equation for the density matrix is used. A general scheme for deriving the Lindblad superoperator that emerges after averaging the von Neumann equation over the reservoir degrees of freedom is developed. This scheme is illustrated by the cases of radiation of a two-level atom into free space and the dynamics of the transition of a two-level atom from the pure state to the mixed state due to the interaction with a dephasing reservoir. Special attention is paid to the open system consisting of several subsystems each of which independently interacts with the reservoir. In the case of noninteracting subsystems, the density matrix is a tensor product of the subsystem density matrices, and the Lindblad superoperator of the system is a sum of Lindblad superoperators of those subsystems. The interaction between the subsystems results not only in the emergence of the corresponding term in the Hamiltonian of the combined system but also in non-additivity of the Lindblad superoperators. The latter is often overlooked in modern literature possibly leading, as it is shown in the present methodological note, to serious errors; for example, the second law of thermodynamics could be violated.