The coarse-graining approach to deriving the quantum Markovian master equation is revisited, with close attention given to the underlying approximations. It is further argued that the time interval over which the coarse-graining is performed is a free parameter that can be given a physical measurement-based interpretation. In the case of the damping of composite systems to reservoirs of different temperatures, currently of much interest in the study of quantum thermal machines with regard to the validity of `local' and `global' forms of these equations, the coupling of the subsystems leads to a further timescale with respect to which the coarse-graining time interval can be chosen. Different choices lead to different master equations that correspond to the local and global forms. These can be then understood as having different physical interpretations based on the role of the coarse-graining, as well as different limitations in application.
The coarse-graining approach to deriving the quantum Markovian master equation is revisited, with close attention given to the underlying approximations. It is further argued that the time interval over which the coarse-graining is performed is a free parameter that can be given a physical measurement-based interpretation. In the case of the damping of composite systems to reservoirs of different temperatures, currently of much interest in the study of quantum thermal machines with regard to the validity of local' and global' forms of these equations, the coupling of the subsystems leads to a further timescale with respect to which the coarse-graining time interval can be chosen. Different choices lead to different master equations that correspond to the local and global forms. These can be then understood as having different physical interpretations based on the role of the coarse-graining, as well as different limitations in application.
We study the coarse-graining approach to derive a generator for the evolution of an open quantum system over a finite time interval. The approach does not require a secular approximation but nevertheless generally leads to a Lindblad-Gorini-Kossakowski-Sudarshan generator. By combining the formalism with full counting statistics, we can demonstrate a consistent thermodynamic framework, once the switching work required for the coupling and decoupling with the reservoir is included. Particularly, we can write the second law in standard form, with the only difference that heat currents must be defined with respect to the reservoir. We exemplify our findings with simple but pedagogical examples.
We present a derivation of the Markovian master equation using the time coarse graining method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then applying the renormalization group method, it is shown that dependence of the specific initial time in the solution can be removed and the Markovian semi-group master equation in Lindblad form is attained in the weak coupling limit.
We present a first-principles derivation of the Markovian semigroup master equation without invoking the rotating-wave approximation (RWA). Instead we use a time coarse-graining approach that leaves us with a free time-scale parameter, which we can optimize. Comparing this approach to the standard RWA-based Markovian master equation, we find that significantly better agreement is possible using the coarse-graining approach, for a three-level model coupled to a bath of oscillators, whose exact dynamics we can solve for at zero temperature. The model has the important feature that the RWA has a nontrivial effect on the dynamics of the populations. We show that the two different master equations can exhibit strong qualitative differences for the population of the energy eigenstates even for such a simple model. The RWA-based master equation misses an important feature which the coarse-graining-based scheme does not. By optimizing the coarse-graining time scale the latter scheme can be made to approach the exact solution much more closely than the RWA-based master equation.
The study of quantum thermal machines, and more generally of open quantum systems, often relies on master equations.Two approaches are mainly followed.On the one hand, there is the widely used, but often criticized, local approach, where machine sub-systems locally couple to thermal baths.On the other hand, in the more established global approach, thermal baths couple to global degrees of freedom of the machine.There has been debate as to which of these two conceptually different approaches should be used in situations out of thermal equilibrium.Here we compare the local and global approaches against an exact solution for a particular class of thermal machines.We consider thermodynamically relevant observables, such as heat currents, as well as the quantum state of the machine.Our results show that the use of a local master equation is generally well justified.In particular, for weak inter-system coupling, the local approach agrees with the exact solution, whereas the global approach fails for non-equilibrium situations.For intermediate coupling, the local and the global approach both agree with the exact solution and for strong coupling, the global approach is preferable.These results are backed by detailed derivations of the regimes of validity for the respective approaches.
The study of quantum thermal machines, and more generally of open quantum systems, often relies on master equations. Two approaches are mainly followed. On the one hand, there is the widely used, but often criticized, local approach, where machine sub-systems locally couple to thermal baths. On the other hand, in the more established global approach, thermal baths couple to global degrees of freedom of the machine. There has been debate as to which of these two conceptually different approaches should be used in situations out of thermal equilibrium. Here we compare the local and global approaches against an exact solution for a particular class of thermal machines. We consider thermodynamically relevant observables, such as heat currents, as well as the quantum state of the machine. Our results show that the use of a local master equation is generally well justified. In particular, for weak inter-system coupling, the local approach agrees with the exact solution, whereas the global approach fails for non-equilibrium situations. For intermediate coupling, the local and the global approach both agree with the exact solution and for strong coupling, the global approach is preferable. These results are backed by detailed derivations of the regimes of validity for the respective approaches.
Abstract Master equations are a useful tool to describe the evolution of open quantum systems. In order to characterize the mathematical features and the physical origin of the dynamics, it is often useful to consider different kinds of master equations for the same system. Here, we derive an exact connection between the time-local and the integro-differential descriptions, focusing on the class of commutative dynamics. The use of the damping-basis formalism allows us to devise a general procedure to go from one master equation to the other and vice versa, by working with functions of time and their Laplace transforms only. We further analyze the Lindbladian form of the time-local and the integro-differential master equations, where we account for the appearance of different sets of Lindbladian operators. In addition, we investigate a Redfield-like approximation, that transforms the exact integro-differential equation into a time-local one by means of a coarse graining in time. Besides relating the structure of the resulting master equation to those associated with the exact dynamics, we study the effects of the approximation on Markovianity. In particular, we show that, against expectation, the coarse graining in time can possibly introduce memory effects, leading to a violation of a divisibility property of the dynamics.