System-environment correlations and Markovian embedding of quantum non-Markovian dynamics

Abstract

We study the dynamics of a quantum system whose interaction with an environment is described by a collision model, i.e. the open dynamics is modelled through sequences of unitary interactions between the system and the individual constituents of the environment, termed "ancillas", which are subsequently traced out. In this setting non-Markovianity is introduced by allowing for additional unitary interactions between the ancillas. For this model, we identify the relevant system-environment correlations that lead to a non-Markovian evolution. Through an equivalent picture of the open dynamics, we introduce the notion of "memory depth" where these correlations are established between the system and a suitably sized memory rendering the overall system+memory evolution Markovian. We extend our analysis to show that while most system-environment correlations are irrelevant for the dynamical characterization of the process, they generally play an important role in the thermodynamic description. Finally, we show that under an energy-preserving system-environment interaction, a non-monotonic time behaviour of the heat flux serves as an indicator of non-Markovian behaviour.

Precursors of non-Markovianity

“…Indeed as shown in section 3.3, in the collision model framework it appears that already the 'minimal approximation' for the environment, treated as being a single ancilla, actually provides quite reasonable bounds, indicating that the contributions arising from the system interacting with this particular ancilla are the most relevant ones for characterizing the dynamics. This is complementary to [30] from which we know that the incoming single ancilla is the only really relevant player in dictating the non-Markovianity of the evolution in the corresponding collision model. We remark that our choice of the Bures distance over other metrics goes beyond its comparative computational simplicity and applicability to infinite dimensional settings already mentioned.…”

“…Let us begin by outlining the basic collision model framework [30,33,34,[47][48][49][50][51][52] which provides a versatile tool for exploring the emergence of non-Markovianity and, due to their construction, serves as the ideal testbed for studying the precursors of non-Markovianity captured in equation (2) by exploiting equations (3) and (4). Following [30,47], the environment is composed of an array of individual ancillae, E i , initially factorized and all with the same initial state. The time evolution is discretized such that at 'time-step' n the system, S, collides with ancilla, E n , after which we retain all correlations established by this system-ancilla (SA) interaction while E n subsequently collides with E n+1 .…”

“…Due to the intra-environment ancilla-ancilla (AA) collisions, i.e. E n -E n+1 , S may already share some correlations with E n+1 before they interact [30,47]. Furthermore, due to the AA collision the incoming state of the ancilla, E n+1 , that the system interacts with is not typically its initialized state.…”

“…We exactly recover the standard evolution according to the Markovian master equation when there are no AA collisions, under the condition that the unitary  is an energy preserving exchange interaction and all constituents have the same free Hamiltonian terms [47,53]. Despite the framework is independent of the dimensionality of system and ancilla, most studies restrict to the discrete variable (DV) qubit states [30,[47][48][49], with only a few exceptions [50]. Therefore in this work, we are interested in examining any effect that dimensionality may have on the properties of the non-Markovian dynamics, and on the precursors of non-Markovianity, by comparing and contrasting when system and ancilla are DV qubits with when they are CV Gaussian states.…”

“…A significant limitation of the approach is that it calls for evaluation of features of both the system and the total environment, which is generally too demanding a task, and as such investigations along these lines have rarely been considered [26][27][28]. However, it has been shown that when the environment can be described as a collection of individual subsystems, often only a small subset of the total environmental degrees of freedom plays a significant role in driving the evolution [29,30]. By relying on the Bures distance and considering collision models of open quantum system dynamics [31][32][33][34], where the environment is modeled by an ensemble of individual ancillae with which the system sequentially interacts, we are able to introduce and study a hierarchy of precursors of non-Markovianity for systems in both discrete and continuous variable (CV) settings.…”