Stochastic representation of the Lindblad time evolution

Abstract

In general it is possible to obtain a stochastic representation of a solution of a partial differential equation if the equation is of the second order. In Chapters 5–10 and Chapter 13 we have applied such a representation to the Hamiltonian time evolution. In the classical statistical physics the dissipative dynamics is often modelled by a Markov process. It is assumed that the environment has a short memory so that the Markovian approximation is justified. In general, the environment leads to a non-local dynamics of the reduced density matrix. Only in some approximations we can obtain closed equations for the reduced density matrix [361] [178]. If the closed reduced dynamics is to be consistent with the probabilistic interpretation of quantum mechanics then we must restrict ourselves to approximations in the Lindblad form. The Lindblad operators L in equation (12.10) can be expressed as (non-linear) functions of the position and momentum operators. Strictly speaking the Lindblad theorem on the form of the generator of a semi-group is rigorous only under the assumption that L are bounded operators. However, if the strength of the dissipation is to be comparable to the effect of the Hamiltonian evolution then L must be at least linear in the momentum. In subsequent chapters we restrict ourselves to Lindblad operators which are at most linear in the momentum operator. Such an assumption can be considered as a first order approximation to a non-linear analytic function of the momentum.