Extracting Dynamical Maps of Non-Markovian Open Quantum Systems

Abstract

The most general description of quantum evolution up to a time $τ$ is a completely positive tracing preserving map known as a dynamical map $\hatΛ(τ)$. Here we consider $\hatΛ(τ)$ 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 $\hatΛ(τ)$ 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 $τ_{\rm m}$. By combining several techniques within a tensor network framework we directly and accurately extract $\hatΛ(τ)$ for a small number of interacting fermionic modes coupled to infinite non-interacting Fermi baths. We employ the Choi-Jamiolkowski isomorphism so that $\hatΛ(τ)$ can be fully reconstructed from a single pure state calculation of the unitary dynamics of the system, bath and their replica auxillary modes up to time $τ$. From $\hatΛ(τ)$ we also compute the time local propagator $\hat{\mathcal{L}}(τ)$. By examining the convergence with $τ$ of the instantaneous fixed points of these objects we establish their respective memory times $τ^Λ_{\rm m}$ and $τ^{\mathcal{L}}_{\rm m}$. Beyond these times, the propagator $\hat{\mathcal{L}}(τ)$ and dynamical map $\hatΛ(τ)$ accurately describe all the subsequent long-time relaxation dynamics up to stationarity. Our numerical examples of interacting spinless Fermi chains and the single impurity Anderson model demonstrate regimes where our approach can offer a significant speedup in determining the stationary state compared to directly simulating the long-time limit.