The double Caldeira-Leggett model: Derivation and solutions of the master equations, reservoir-induced interactions and decoherence

TL;DR

This paper investigates the dynamics of two interacting dissipative harmonic oscillators using the double Caldeira-Leggett model, deriving master equations, analyzing decoherence, and exploring reservoir-induced interactions under different coupling scenarios.

Contribution

It provides a comprehensive derivation and solution of master equations for two oscillators with various reservoir couplings, including analysis of decoherence and reservoir-induced interactions.

Findings

Derived a general decay function for off-diagonal density matrix peaks.

Identified reservoir-induced interactions leading to collective damping effects.

Showed that high-temperature regimes blur reservoir-induced interactions.

Abstract

In this paper we analyze the double Caldeira-Leggett model: the path integral approach to two interacting dissipative harmonic oscillators. Assuming a general form of the interaction between the oscillators, we consider two different situations: i) when each oscillator is coupled to its own reservoir, and ii) when both oscillators are coupled to a common reservoir. After deriving and solving the master equation for each case, we analyze the decoherence process of particular entanglements in the positional space of both oscillators. To analyze the decoherence mechanism we have derived a general decay function for the off-diagonal peaks of the density matrix, which applies both to a common and separate reservoirs. We have also identified the expected interaction between the two dissipative oscillators induced by their common reservoir. Such reservoir-induced interaction, which gives rise to interesting collective damping effects, such as the emergence of relaxation- and decoherence-free subspaces, is shown to be blurred by the high-temperature regime considered in this study. However, we find that different interactions between the dissipative oscillators, described by rotating or counter-rotating terms, result in different decay rates for the interference terms of the density matrix.