Extracting dynamical maps of non-Markovian open quantum systems

The most general description of quantum evolution up to a time τ is a completely positive tracing preserving map known as a dynamical mapΛ̂(τ). Here, we consider Λ̂(τ) arising from suddenly coupling a system to one or more thermal baths with a strength that is neither weak nor strong. Given no clear separation of characteristic system/bath time scales, Λ̂(τ) is generically expected to be non-Markovian; however, we do assume the ensuing dynamics has a unique steady state, implying the baths possess a finite memory time τm. By combining several techniques within a tensor network framework, we directly and accurately extract Λ̂(τ) for a small number of interacting fermionic modes coupled to infinite non-interacting Fermi baths. First, we use an orthogonal polynomial mapping and thermofield doubling to arrive at a purified chain representation of the baths whose length directly equates to a time over which the dynamics of the infinite baths is faithfully captured. Second, we employ the Choi-Jamiolkowski isomorphism so that Λ̂(τ) can be fully reconstructed from a single pure state calculation of the unitary dynamics of the system, bath and their replica auxiliary modes up to time τ. From Λ̂(τ), we also compute the time local propagator L̂(τ). By examining the convergence with τ of the instantaneous fixed points of these objects, we establish their respective memory times τmΛ and τmL. Beyond these times, the propagator L̂(τ) and dynamical map Λ̂(τ) accurately describe all the subsequent long-time relaxation dynamics up to stationarity. These timescales form a hierarchy τmL≤τmΛ≤τre, where τre is a characteristic relaxation time of the dynamics. Our numerical examples of interacting spinless Fermi chains and the single impurity Anderson model demonstrate regimes where τre ≫ τm, where our approach can offer a significant speedup in determining the stationary state compared to directly simulating the long-time limit. Our results also show that having access to Λ̂(τ) affords a number of insightful analyses of the open system thus far not commonly exploited.