Adapted projection operator technique for the treatment of initial correlations

Abstract

Standard theoretical descriptions of the dynamics of open quantum systems rely on the assumption that the correlations with the environment can be neglected at some reference (initial) time. While being reasonable in specific instances, such as when the coupling between the system and the environment is weak or when the interaction starts at a distinguished time, the use of initially uncorrelated states is questionable if one wants to deal with general models, taking into account the mutual influence that the open-system and environmental evolutions perform on each other. Here we introduce a perturbative method that can be applied to any microscopic modeling of the system-environment interaction, including fully general initial correlations. Extending the standard technique based on projection operators that single out the relevant part of the global dynamics, we define a family of projections adapted to a convenient decomposition of the initial state, which involves a convex mixture of product operators with proper environmental states. This leads us to characterize the open-system dynamics via an uncoupled system of differential equations, which are homogeneous and whose number is limited by the dimensionality of the open system, for any kind of initial correlations. Our method is further illustrated by means of two case studies, for which it reproduces the expected dynamical behavior in the long-time regime more consistently than the standard projection technique.

Quantum non-Markovianity: Overview and recent developments

“…Alipour et al (2020) proposed a technique that makes use of the correlation parent operator that allows one to write a master equation with the initial correlation within a weak coupling regime. A technique based on adapted projection operators was introduced by Trevisan et al (2021), in which they applied a perturbative method to model a global S-E evolution that incorporates fully general initial correlations. For other recent attempts that have been made to accommodate initial S-E correlations into a valid theory of open systems, without sacrificing linearity and complete positivity, see Paz-Silva et al (2019), Pollock et al (2018b), andPollock et al (2018a), who specifically aimed to characterize open system evolution operationally and allow a quantum stochastic process to have an appropriate classical limit.…”

Open quantum system dynamics and the mean force Gibbs state

“…The initial SB state, ρ SB (0), is usually assumed to be of product form (45), although the evolution of correlated or entangled initial states is also relevant and studied 145,[147][148][149][150][151][152] . The state of S at time t is the reduced density matrix, cf.…”

“…Refs. 87,145,147,148,150,151,[153][154][155][156] . With very few exceptions, a master equation for ρ(t) cannot be derived without recourse to approximations.…”

Initial correlations in open quantum systems: constructing linear dynamical maps and master equations

“…For example, the treatment of the dynamics of open systems including correlations in the initial state leads to a general method for the local detection of correlations between the system and an inaccessible environment [12,13] which has been realized experimentally both in trapped ion and in photonic systems [14][15][16]. Several efforts have thus been made to extend the theory to allow for correlated initial states, both in terms of examining conditions for which the resulting dynamical map is completely positive (CP) [17][18][19][20][21][22][23] and of finding alternative methods for the dynamical description of the reduced system [24][25][26][27][28]. The approaches are numerous and varied, and the wide array of papers on the subject paints the picture of a complicated subject matter.…”