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The Journal of Chemical Physics

Quantum dissipative master equations: Some exact results

Yang Zhao, G. H. Chen

Published Jun 22, 2001• 11 citations
Publisher

Abstract

With the help of quantum characteristic functions some exact results are found for two quantum dissipative master equations which contain dissipative Liouvillian operators of the Lindblad-type corresponding to two forms of dissipation mechanisms. Population relaxation is used to demonstrate how dynamic information can be retrieved from these solutions.

DOI
10.1063/1.1374537

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With the help of quantum characteristic functions some exact results are found for two quantum dissipative master equations which contain dissipative Liouvillian operators of the Lindblad-type corresponding to two forms of dissipation mechanisms. Population relaxation is used to demonstrate how dynamic information can be retrieved from these solutions.
We propose a family of master equations for local quantum dissipation. The master equations are constructed in the form of Lindblad generators, with the constraints that the dissipation be strictly linear (i.e., Ohmic), isotropic, and translationally invariant. The resulting master equations are given in both the Schr\"odinger and Heisenberg forms. We obtain fluctuation-dissipation relations, and discuss the relaxation of average kinetic energy to effective thermal equilibrium values. We compare our results for one dimension to the Dekker master equation [H. Dekker, Phys. Rep. 80, 1 (1981)], which can be interpreted as a low-length-scale approximation of our model, as well as the Caldeira-Leggett master equation [A.O. Caldeira and A. J. Leggett, Physica (Utrecht) A 121, 587 (1983)].
We present a general approach to derive Lindblad master equations for a subsystem whose dynamics is coupled to dissipative bosonic modes. The derivation relies on a Schrieffer-Wolff transformation which allows us to eliminate the bosonic degrees of freedom after self-consistently determining their state as a function of the coupled quantum system. We apply this formalism to the dissipative Dicke model and derive a Lindblad master equation for the atomic spins, which includes the coherent and dissipative interactions mediated by the bosonic mode. This master equation accurately predicts the Dicke phase transition and gives the correct steady state. In addition, we compare the dynamics using exact diagonalization and numerical integration of the master equation with the predictions of semiclassical trajectories. We finally test the performance of our formalism by studying the relaxation of a NOON state and show that the dynamics captures quantum metastability.