Stability of the Grabert master equation

TL;DR

This paper analyzes the stability of the nonlinear Grabert master equation for finite-dimensional quantum systems, concluding that fixed points are stable due to non-negative eigenvalues, and explores conditions for possible instabilities.

Contribution

The study provides a stability analysis of the nonlinear Grabert master equation, showing fixed points are stable in finite-dimensional quantum systems.

Findings

Eigenvalues of the Jacobian are non-negative near fixed points

Fixed points of the equation are stable in finite dimensions

Raises questions about conditions for instability in quantum systems

Abstract

We study the dynamics of a quantum system having Hilbert space of finite dimension $d_{\mathrm{H}}$. Instabilities are possible provided that the master equation governing the system's dynamics contain nonlinear terms. Here we consider the nonlinear master equation derived by Grabert. The dynamics near a fixed point is analyzed by using the method of linearization, and by evaluating the eigenvalues of the Jacobian matrix. We find that all these eigenvalues are non-negative, and conclude that the fixed point is stable. This finding raises the question: under what conditions instability is possible in a quantum system having finite $d_{\mathrm{H}}$?