Ab initio construction of an analytically tractable Kraus map for non-Markovian quantum dissipation

Abstract

Starting from a prescribed Hamiltonian, we construct a non-Markovian evolution equation for a non-relativistic quantum system that exchanges energy with a large reservoir. In order to create sufficient mathematical freedom, the density operator is replaced by a more flexible entity that depends on two times. If these times are chosen equal, the density operator is recovered. In deriving a non-Markovian integral equation for our bitemporal operator, it is assumed that initially system and reservoir are completely uncorrelated. Furthermore, in employing Wick's theorem for factorization of reservoir correlation functions, only those Wick contractions between reservoir potentials are retained that belong to a generalized nearest-neighbour class. The latter is established by subjecting the set of plain nearest-neighbour contractions to any cyclic permutation of reservoir potentials. Through generalizing the notion of nearest-neighbour contraction, it is ensured that the trace of the density operator is conserved. By construction, our bitemporal evolution equation agrees with the Kraus map for quantum dissipation. Moreover, a sound Markovian limit exists that reproduces the complete van Hove–Davies theory. By making use of a rotating-wave approximation and Laplace transformation, the density operator of a damped N-level atom can be computed. For large times and moderate coupling to the reservoir, the atom ends up near the state of thermal equilibrium. At zero temperature, our non-Markovian integral equation gives an exact solution for the atomic density operator.