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On the Weak-coupling Limit and Complete Positivity

Fabio Benatti, Roberto Floreanini B

Published Feb 23, 2000• 8 citations
OpenAlex

Abstract

We consider two non-interacting systems embedded in a heat bath. If theyremain dynamically independent, physical inconsistencies are avoided only ifthe single-system reduced dynamics is completely positive also beyond theweak-coupling limit.

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The generalized quantum master equation provides an exact equation of motion for the reduced density operator of a quantum system coupled to one or more baths, with all bath-induced effects encoded in a memory kernel superoperator. In this paper, we investigate whether the memory kernel for a system coupled simultaneously to multiple independent baths can be written as a sum of memory kernels associated with the corresponding single-bath problems. We show that such additivity holds in the weak system-bath coupling limit, where the second-order memory kernel separates into independent bath contributions. Beyond this limit, however, the baths become dynamically correlated through their mutual coupling to the system, and the exact multi-bath memory kernel is no longer equal to the sum of exact single-bath memory kernels. We demonstrate this non-additivity for a two-level system linearly coupled to hot and cold harmonic baths, using the hierarchical equations of motion method to construct the memory kernels. The results show that non-additivity increases with system-bath coupling strength and is process-dependent: coherence damping remains comparatively well described by an additive kernel, whereas population-coherence and coherence-coherence transfer exhibit pronounced non-additive behavior. These findings clarify the limitations of additive multi-bath descriptions and highlight the ambiguity of decomposing heat flow into independent bath-resolved contributions beyond weak system-bath coupling.
We calculate the reduced density matrix for a system of coupled harmonic oscillators in a bosonic heat bath using the Born-Markov approximation and show that the expectation values of position and momentum evolve like classical quantities. We consider the cases where every oscillator is coupled to its own reservoir and where all oscillators are coupled to the same heat bath, pointing out the differences between these two models. In particular, for systems coupled to a common reservoir, we present an uncommon dissipation mechanism, which disappears, when the dynamics of the system are gouverned by the low frequency modes of the environment (Caldeira-Leggett limit). We also address the interesting phenomenon of reservoir-induced interactions.
Two non-directly interacting qubits with equal frequencies can become entangled via a Markovian, dissipative dynamics through the action of a weakly coupled Ohmic heat bath. In the standard weak-coupling limit derivation, this purely dissipative effect disappears if the frequencies are different because of the “ergodic average” used by this approach. However, there are physical situations where this technique is too rough to capture all the relevant aspects of the dissipative dynamics. In these cases, in order to better describe the physical behavior of the open system, it is necessary to go beyond the “ergodic average”. We show that, in this more refined framework, the entanglement capability of the environment persists also in the case of different frequencies.