Master-Equation Formulations of Decoherence

Abstract

In this chapter, we will introduce the description of decoherence dynamics in terms of so-called master equations. Such master equations directly and conveniently yield the time evolution of the reduced density matrixˆρmatrixˆ matrixˆρ S (t) for the open quantum system S interacting with an environment E. They relieve us from the need of having to first determine the dynamics of the total system-environment combination and to then trace out the degrees of freedom of the environment. The master-equation approach is motivated by two issues. First, we are usually not interested in the dynamics of the environment or of the global system-environment combination. All we really care about is the influence of the environment on our system of interest. Second, it is often impossible to analytically determine the time evolution of the density matrix. In such cases, one can use approximation schemes that lead to master equations for the approximate evolution of the reduced density matrix. In contrast with the dynamics of the density matrix of a closed system, the evolution equation for the reduced density matrix will of course, in general, be nonunitary, since the interaction with the environment will typically change the amount of coherence present in the system. Put into more formal terms, since the trace operation used to obtain the reduced density matrix is a nonunitary operation, the master equation must in general also be nonunitary. This chapter is organized as follows. In Sect. 4.1, we will introduce the basic ideas and the general formalism of master equations. In Sect. 4.2, we will then discuss in some detail a very important type of master equation, namely, the so-called Born-Markov master equation. This master equation plays a key role in studies of decoherence. The derivation of this equation, given in Sect. 4.2.2, is inevitably somewhat technical, so some readers may prefer to just glance over this section or skip it altogether. In Sect. 4.3, we will discuss so-called Lindblad master equations, a special and more simplified form of the general Born-Markov master equation that is valid under an additional assumption. As we shall see, Lindblad master equations allow for a very intuitive representation of decoherence processes. Finally, in Sect. 4.4, we will briefly discuss the case of non-Markovian dynamics. We note that master equations also play an important role in the theory of so-called quantum dynamical semigroups. Since our focus here is on