Abstract
A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested.
Highlights
For the description of non-Markovian quantum processes, we can take into account nonlocality in time, which means that the behavior of the quantum observable A(t) (or state ρ(t)) and its derivatives may depend on the history of previous changes of this operator
We proposed using general fractional calculus and general fractional derivatives (GFD) as mathematical tools to allow us to take into account the general nonlocality in time for non-Markovian quantum processes
The proposed general fractional integral (GFI) and GFD can be used to formulate non-Markovian dynamics in the general form, where the nonlocality in time is described by the kernel pairs that belong to the Sonin set S−1
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Most approaches to the formulation of non-Markovian quantum dynamics were not related to fractional calculus and mathematical theory of equations with derivatives and integrals of non-integer orders (see [26,27,28,29,30,31,32]). In the general approach to non-Markovian quantum dynamics, all research and results should be related to the general form of nonlocality, operator kernels of almost all types The general form of non-Markovian dynamics of open quantum systems is described by the equations with general fractional derivatives and integrals [58,59]. The exact solutions of these equations were derived by using the general operational calculus proposed in [60]
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