Extracting Dynamical Maps of Non-Markovian Open Quantum Systems

Abstract

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

v2, 12 pages, 10 figures