Continued-fraction representation of the Kraus map for non-Markovian reservoir damping

TL;DR

This paper develops a continued-fraction approach to exactly solve the Kraus map for non-Markovian quantum reservoir damping, enabling perturbative analysis that preserves physical properties and applying it to the Jaynes-Cummings model.

Contribution

It introduces a continued-fraction representation of the Kraus map for non-Markovian dynamics and constructs a perturbation theory that maintains positivity and probability.

Findings

Exact solutions for Kraus matrices via continued fractions.

Perturbation theory conserving positivity and probability.

Application to the damped Jaynes-Cummings model showing prolonged maximum entropy states.

Abstract

Quantum dissipation is studied for a discrete system that linearly interacts with a reservoir of harmonic oscillators at thermal equilibrium. Initial correlations between system and reservoir are assumed to be absent. The dissipative dynamics as determined by the unitary evolution of system and reservoir is described by a Kraus map consisting of an infinite number of matrices. For all Laplace-transformed Kraus matrices exact solutions are constructed in terms of continued fractions that depend on the pair correlation functions of the reservoir. By performing factorizations in the Kraus map a perturbation theory is set up that conserves in arbitrary perturbative order both positivity and probability of the density matrix. The latter is determined by an integral equation for a bitemporal matrix and a finite hierarchy for Kraus matrices. In lowest perturbative order this hierarchy reduces to one equation for one Kraus matrix. Its solution is given by a continued fraction of a much simpler structure as compared to the non-perturbative case. In lowest perturbative order our non-Markovian evolution equations are applied to the damped Jaynes-Cummings model. From the solution for the atomic density matrix it is found that the atom may remain in the state of maximum entropy for a significant time span that depends on the initial energy of the radiation field.