Generalized theory of pseudomodes for exact descriptions of non-Markovian quantum processes

Abstract

We develop an exact framework for describing the non-Markovian dynamics of an open quantum system interacting with an environment modeled by a generalized spectral density function. The approach relies on mapping the initial system onto an auxiliary configuration, comprising the original open system coupled to a small number of discrete modes, which in turn are each coupled to an independent Markovian reservoir. Based on the connection between the discrete modes and the poles of the spectral density function, we show how expanding the system using the discrete modes allows for the full inclusion non-Markovian effects within an enlarged open system whose dynamics is governed by an exact Lindblad master equation. Initially we apply this result to obtain a generalization of the pseudomode method [B. M. Garraway, Phys. Rev. A. 55, 2290 (1997)] in cases where the spectral density function has a Lorentzian structure. For many other types of spectral density function, we extend our proof to show that an open system dynamics may be modeled physically using discrete modes which admit a non-Hermitian coupling to the system and for such cases determine the equivalent master equation to no longer be of Lindblad form. For applications involving two discrete modes, we demonstrate how to convert between pathological and Lindblad forms of the master equation using the techniques of the pseudomode method.

On the connection between microscopic description and memory effects in open quantum system dynamics

“…While giving access to a much larger class of systems, this kind of approaches would however inevitably weaken the exact equivalence between the reduced dynamics of different microscopic models, which is instead one of the main motivations of our analysis. An alternative path could then be to compare the open system dynamics due to an overall unitary evolution with the dynamics induced by a non-unitary evolution, by means of the so-called Lindbladian embedding methods [51][52][53][54][55][56][64][65][66][67]. Here, the global system is complex enough to account for a large variety of realistic models and would possibly require the use of approximated or numerical techniques to evaluate global system-environment quantities; on the other hand, crucially, the equivalence between the reduced dynamics of different models would still be guaranteed a-priori in an exact way.…”

“…We stress that we did not assume any specific form of the initial states of the environments, so that the equivalence between the open system evolutions guaranteed by Eq. ( 9) does not follow from the recent equivalence theorems among different dynamics with initial Gaussian bosonic or fermionic environmental states [51][52][53][54][55][56].…”

Accurate Truncations of Chain Mapping Models for Open Quantum Systems

“…It is thus tempting to directly use the diagonal form Equation ( 23 ), in which the spectral density is written as a sum of simple poles (and the correlation function a sum of complex exponentials). However, in practice, this turns out to be difficult, as recently discussed in some detail in [ 25 ]. First, since is complex, this form of the spectral density cannot be directly mapped to a physical system.…”

“…We are not aware of any constructive algorithm to achieve this, apart from the one developed for the case in Ref. [ 25 ]. It is thus advantageous to perform the fitting in the original form, where the physicality can be guaranteed by constraining all to be nonnegative, and no additional work to find a physical system is necessary after the fit.…”

“…In particular, such approaches often are based on the general idea of enlarging the system by including one (or several) “reaction modes” of the environment within it in such a way that these modes contain the “memory” of the bath, and are then in turn coupled to a new residual bath that ideally has little or no memory. Among these approaches are the pseudomode method [ 22 , 23 , 24 , 25 ], effective Lindblad master equation procedure [ 26 ], and reaction coordinate mappings [ 27 , 28 , 29 , 30 ]. In such a mapping, an orthogonal transformation is applied on the bath modes such that the interaction between the system and the full environment is captured by a single mode, which itself interacts with the residual bath ( Figure 1 b).…”