The interplay between local and non-local master equations: exact and approximated dynamics

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.

Evolution Equations for Quantum Semi-Markov Dynamics

“…A powerful connection between the local and non-local generators can be obtained with the damping-basis representation [ 38 ], when one restricts to (diagonalisable) commutative dynamics, i.e., satisfying with being the commutator [ 39 , 40 ]. In [ 41 ] it was shown, that in this case the local and non-local generators can be written as where and are functions of time (the eigenvalues of the corresponding damping-basis decompositions) and are related by where denotes the Laplace transform of the function , while denotes the inverse Laplace transform. What is more, the maps in Equation ( 10 ) can be written with bi-orthogonal bases and of operators acting on the open-system Hilbert space (the damping bases of the generators), as and, because of the commutativity of the dynamics, they are time-independent.…”

“…Though the inverse Laplace transform in Equation ( 11 ) in general cannot be calculated, the above link between the two characterisations is not only a formal one. In [ 41 ] it was shown that it enables to understand the relations between the Lindblad operator form of the two generators, as well as the connection between the (non-)Markovianity of the original and the Redfield-like approximated dynamics.…”

“…Moreover in both cases a gauge freedom is available, so that the operator structure is not uniquely fixed. However, starting from the damping basis decomposition given by Equation ( 10 ) one can bring both generators in Lindblad operator form, as shown in [ 41 ]. With the Lindblad operator form of the evolution equation we mean the one directly generalising the well-known GKSL master equation, i.e., with where the Lindblad operators and damping rates are time-dependent.…”

“…As commonly one (or both) of these conditions is (are) violated, in general some Lindblad operators contained in the local description can be missing in the non-local one, and vice versa [ 41 ]. This makes the interpretation of the underlying physical origin of the dynamics more difficult.…”

“…To obtain the corresponding local generator, we use the results obtained via the damping bases in Section 2 . The damping bases and coincide in this instance, as the generator is self-adjoint [ 41 ], and they read . Moreover, Equations ( 11 ) and (12), which relate the eigenvalues of the non-local and local generator via (13), lead for the present case to the expressions …”