Exact Non-Markovian Evolution with Several Reservoirs
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act Non-Markovian Evolution with Several Reservoirs A. E. Teretenkova, b, * a
Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, 119991 Russia b Moscow State University, Moscow, 119991 Russia *e-mail: [email protected] Received December 20, 2019; revised January 16, 2020; accepted January 29, 2020
Abstract—The model of a multilevel system interacting with several reservoirs is considered. The exact reduced evolution of a system’s density matrix can be obtained for this model without using the Markov approximation. Namely, this evolution is completely defined by the finite set of linear differential equations. The results obtained earlier for one Lorentz peak in the spectral density are generalized to the case of an arbitrary number of such peaks. The contribution of Ohmic spectral density is also considered. DOI: 10.1134/S1063779620040711
1. INTRODUCTION A rigorous derivation of the equations for the reduced evolution of system’s density matrix in the Markov approximation originates in the paper of Krylov and Bogolyubov [1]. The methods described there were developed in the context of the theory of stochastic limit, a modern exposition of which can be found in [2]. In this case, the kinetic equations of the density matrix of the system are written as the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL equation) [3, 4]. However, the problems of non-Markovian evolution have attracted increasing interest in recent times [5–12]. In particular, it is natural to ask when the nonMarkovian evolution of the system can be dilated to Markovian evolution for a finite-dimensional density matrix. In this paper, we consider a model of a multilevel system interacting with reservoirs at zero temperature, and the evolution of the density matrix of the system can be obtained for this model exactly in terms of the finite-dimensional Schrodinger equations with a non-Hermitian Hamiltonian. We analyze cases when the evolution of the reduced density matrix can be dilated to Markovian evolution with a larger but finite dimension. This work develops and generalizes the results obtained in [13] and [14]. In section Model, we describe the model under consideration and present the results of [14], which are necessary for this article. We refer the reader to [13] and [14] for a detailed discussion of how this model is related to other models known in the literature. We only mention that the model is closely related to the Friedrichs model [15], and the method we use in Propositions 1 and 3 is a development of the pseudo-mode method proposed in [16–18]. In section Model, we also consider the
case when the spectral density of the reservoir is a combination of Lorentz peaks, and we take into consideration the Ohmic contribution to the spectral density. Finally, in the Conclusions section, we summarize and outline possible directions for further research. 2. MODEL We consider evolution in a Hilbert space N
* ≡ (C ⊕ CN ) ⊗ ⊗Fb (+2(R)). i =1
Here, C ⊕ C is an (N + 1)-dimensional Hilbert space with a distinguished one-dimension
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