Extracting Dynamical Maps of Non-Markovian Open Quantum Systems

The most general description of quantum evolution up to a time $\tau$ is a completely positive tracing preserving map known as a dynamical map $\hat{\Lambda}(\tau)$. Here we consider $\hat{\Lambda}(\tau)$ 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 $\hat{\Lambda}(\tau)$ 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 $\tau_{\rm m}$. By combining several techniques within a tensor network framework we directly and accurately extract $\hat{\Lambda}(\tau)$ for a small number of interacting fermionic modes coupled to infinite non-interacting Fermi baths. We employ the Choi-Jamiolkowski isomorphism so that $\hat{\Lambda}(\tau)$ 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 $\tau$. From $\hat{\Lambda}(\tau)$ we also compute the time local propagator $\hat{\mathcal{L}}(\tau)$. By examining the convergence with $\tau$ of the instantaneous fixed points of these objects we establish their respective memory times $\tau^{\Lambda}_{\rm m}$ and $\tau^{\mathcal{L}}_{\rm m}$. Beyond these times, the propagator $\hat{\mathcal{L}}(\tau)$ and dynamical map $\hat{\Lambda}(\tau)$ 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.