Quantum Maps with Memory from Generalized Lindblad Equation

Abstract

In this paper, we proposed the exactly solvable model of non-Markovian dynamics of open quantum systems. This model describes open quantum systems with memory and periodic sequence of kicks by environment. To describe these systems, the Lindblad equation for quantum observable is generalized by taking into account power-law fading memory. Dynamics of open quantum systems with power-law memory are considered. The proposed generalized Lindblad equations describe non-Markovian quantum dynamics. The quantum dynamics with power-law memory are described by using integrations and differentiation of non-integer orders, as well as fractional calculus. An example of a quantum oscillator with linear friction and power-law memory is considered. In this paper, discrete-time quantum maps with memory, which are derived from generalized Lindblad equations without any approximations, are suggested. These maps exactly correspond to the generalized Lindblad equations, which are fractional differential equations with the Caputo derivatives of non-integer orders and periodic sequence of kicks that are represented by the Dirac delta-functions. The solution of these equations for coordinates and momenta are derived. The solutions of the generalized Lindblad equations for coordinate and momentum operators are obtained for open quantum systems with memory and kicks. Using these solutions, linear and nonlinear quantum discrete-time maps are derived.

General Fractional Dynamics

“…Note that the proposed equations and mappings can be used to describe economic processes with memory [17,47], for non-Markovian quantum processes [48], processes in the dynamics of populations [49], and many other processes.…”

“…(3) Another way of describing general fractional dynamics with discrete time can be based on the use of discrete mappings obtained from exact solutions of general fractional differential and integral equations with periodic kicks. For the first time discrete mappings with nonlocality in time were derived from fractional differential equations in [43][44][45], in which the Riemann-Liouville and Caputo fractional derivatives were used (see also [2,[46][47][48][49]). The proposed approach allows us to derive discrete time mappings with nonlocality in time from integro-differential equations of non-integer orders without approximation.…”

“…, X 1 ), where the number of variables in the function F increases with the number n ∈ N. For the first time such discrete mappings were derived from fractional differential equations in [43]. Then, this approach was developed in [2,44,45], and it was applied in [17,[46][47][48][49][50]. We should emphasize that nonlocal mappings were derived from fractional differential equations without approximations.…”

Trends, Directions for Further Research, and Some Open Problems of Fractional Calculus

“…For the first time, the use of FC to take into account memory effects (non-Markovity) in open quantum systems was proposed in work [109] (see Chapter 20 in book [109, pp. 477-482] and [180,181,182]). The solutions of non-Markovian equations describing quantum systems with memory, which is based on FC, were proposed in these works.…”

General Non-Markovian Quantum Dynamics

“…For the first time, the use of fractional derivatives and integrals of non-integer order with respect to time to describe non-Markovian dynamics of OQS (in the form of powerlaw fading memory) was proposed in [37] (see Chapter 20 in book [37] (pp. 477-482) and [38][39][40]). Exact solutions of generalized Lindblad equations, which describe non-Markovian quantum dynamics, were derived in these works.…”