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The Journal of Chemical Physics

Master equation for nonlinear dissipative systems

Gautam Gangopadhyay, Deb Shankar Ray

Published Mar 15, 1992• 13 citations
Publisher

Abstract

Based on the system-reservoir theory we have established a generalization of the master equation for a class of nonlinear oscillators within the Born–Markov approximation. We have made a detailed study of the two model systems, e.g., the Morse oscillator and the quartic oscillator as immediate application. Significant interplay of dissipation and nonlinearity has been demonstrated even in the restricted case of weak damping and low finite temperatures.

DOI
10.1063/1.462896

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Based on the system-reservoir theory we have established a generalization of the master equation for a class of nonlinear oscillators within the Born–Markov approximation. We have made a detailed study of the two model systems, e.g., the Morse oscillator and the quartic oscillator as immediate application. Significant interplay of dissipation and nonlinearity has been demonstrated even in the restricted case of weak damping and low finite temperatures.
Based on the standard system-heat-bath model, we have formulated a theory of the non-Markovian master equation for linear and nonlinear systems for a correlation time that is short but finite. We have applied the theory to linear systems, such as harmonic oscillators and two-level atoms and calculated the band-shape functions for transient optical absorption in the pump-probe experiments. The significant interplay of nonlinearity and non-Markovian decay has been illustrated by another application of the theory to a Morse oscillator. The theory is valid for arbitrary nonlinearity and finite temperatures but within the weak-coupling limit and has been derived without ad hoc implementation of a phenomenological memory function.
The different conditions that could be imposed on a Markovian master equation for a nonlinear oscillator weakly coupled to a thermal reservoir are formulated. They concern preservation of trace and positivity of a density matrix, return to a proper equilibrium state, and the detailed balance condition. It is shown that only one of the known master equations satisfies all of these conditions. Then the validity of the Markovian approximation is reanalyzed using certain non-Markovian weak-coupling approximations, and the existence of different stages of evolution associated with different time scales of the Hamiltonian dynamics is predicted. The consequences of these facts for the description of a damped nonlinear oscillator are discussed.
Markovian quantum master equations are generally thought to satisfy the following three conditions: (i) they describe the dynamics of time scales that are larger than the reservoir correlation time scale, (ii) these stationary solutions give the thermal equilibrium states with a reservoir, and (iii) they can be written in the Lindblad form. In fact, for single systems such as a single harmonic oscillator and a single two-level system, Markovian quantum master equations satisfying all these conditions can be obtained. However, for composite systems, which consist of more than one subsystem coupled with each other, such equations have not been obtained thus far. In this study, we found that we cannot use the Born–Markov approximation for obtaining quantum master equations for composite systems in the strong-coupling regime, no matter how short the reservoir correlation time is.
Driven nonlinear quantum oscillators are a central platform for quantum technologies, yet their dissipative dynamics are typically described using Lindblad or Caldeira-Leggett master equations derived under assumptions that exclude nonlinearities and driving. Here, we derive a generalized Caldeira-Leggett master equation for driven nonlinear oscillators by retaining the full nonlinear and time-dependent system dynamics in the construction of the dissipator. For position- and momentum-dependent system-bath coupling, the dissipator itself becomes dynamically dressed, generating nonlinear and drive-dependent dissipative channels beyond conventional fixed-dissipator approaches. This produces nonlinear damping together with dissipation-induced corrections to the effective drive. The resulting dissipative dynamics suppress large-amplitude excitations and reduce phase-space fluctuations. For a driven Kerr oscillator, this leads to the suppression of bistability, asymmetric resonance responses, and strongly modified fluctuation distributions. More broadly, our results establish a microscopic framework in which nonlinear dynamics and driving directly reshape the dissipative sector of driven open quantum systems.