The general evolution equation for the density matrix is derived from the unitary Hamiltonian dynamics of the whole corresponding universe. Then, eliminating the bath variables, a reduced dynamics can be introduced by an effective evolution equation. This evolution is in analogy with a classical transport dynamics, which determines the evolution of any open system S
In the Markovian framework Kossakoswki and Lindblad (KL) have established the structure of the master equation (the semigroup generator) in order that the dynamics of the quantum system be well behaved. In this way the semigroup is a completely positive map and preserves trace, positivity, and hermiticity of the density matrix during all its time evolution. A completely positive semigroup is characterized by the KL generator. The map will be a completely positive semigroup if and only if the hermitian structure matrix is positive definite
Starting from a microscopic dynamics where a system S is coupled to some reservoir B, the Quantum Master Equation (QME) has been derived in several different ways. One of these derivations comes from a perturbative ordered expansion, which can be obtained by tracing-out the bath variables. If any evolution time of the system is much greater than the correlation time of the bath, and a separable structure for the density matrix is assumed, a Markovian dominant contribution can be found. Hence in a second order approximation (in the coupling parameter) the QME has a KL form, but in general, it is not completely positive. For systems with discrete levels of energy and in the weak coupling approximation, it is always possible to introduce an average formalism that leads to a bonafide KL generator. This general procedure is the Davies device; in the context of quantum optics there is an equivalent approach known as the Rotating Wave Approximation
The dynamics of an open system can alternatively be understood through the use of stochastic equations for the state vector of S. This point of view includes the (linear and non-linear) quantum state diffusion approach, the wave packet reduction, and the quantum jump processes. The Shrodinger-Langevin (SL) picture belongs to this formulation and its interest resides in giving an alternative way to obtain the QME with a KL form. This fact has allowed us to define a mapping between the SL and the tracing-out technique, therefore establishing a non-Markovian dynamics for the open system S that can reproduces (in the Markovian limit) the behavior of the QME obtained from tracing-out the bath variables. This kind of stochastic formalism has the additional advantage that can be applied to very different environments, then the influence of the quantum bath can be taking into account by using arbitrary random operators in the SL equation
The path integral method, due to Feynman, is another formalism for eliminating the bath variables, which starts from the total Hamiltonian of the system S plus B. We note that in general the kind of interactions and environments (infinite set harmonic oscillators) treated by the path integral formalisms give rise to a Gaussian influence functional. One of the pioneer work belongs to Caldeira and Leggett who obtained the QME in the high temperatures limit by using some Ohmic linear coupling between S and B. With these assumptions Caldeira and Leggett's QME has a kernel local in time. But other types of coupling with the environment can also be assumed
It is important to remark that in general a Markovian "approximation" does not satisfy the structural theorem, i.e. does not generate completely positive semigroups (this fact can lead to non-positive or non-physical density matrices during the early short-time evolution. Then many different approximations have been introduced to overcome this difficulty. For example in the context of the weak coupling approximation we could introduce some restrictions on the interaction Hamiltonian, in order to fulfil a necessary condition for the completely positivity character. Recently a path integral approach at intermediate temperatures has been considered, therefore obtaining a bonafide KL. Also a pure phenomenological description can be useful to build quantum dissipative semigroups. This approach consists in considering, ad-hoc, suitable interaction operators in the generator. In this case the generator does not have any problem of positivity, but it is necessary to postulate a suitable temperature dependence in the intracting operators, so that the evolution leads to the corresponding thermal equilibrium state of S…
In classical physics it is well accepted, within the Markovian stochastic description, that the master equation is an excellent approximation to describe fluctuation and dissipation at the mesoscopic level. In quantum mechanics the most general form of a Markovian evolution for the reduced density matrix, that give rise to irreversibility, is less popular. Unfortunately sometimes the epithet "Markov" is used with regrettable looseness. A Markov evolution (i.e.: a quantum semigroup) has to guarantee that the density matrix be hermitian positive definite with unit trace at all time, i.e.: von Neumann's conditions…
In general we are concerned with those mentioned formalisms. We study a general approach, using Terwiel's cumulants, to derive a perturbative QME from the total microscopic Hamiltonian. We are concerned with second and higher order approximations, and we are interested in the QME written in terms of the momentum and position operators in order to check its uncertain relations during the evolution.