Title: Real-Time Motion of Open Quantum Systems: Structure of Entanglement, Renormalization Group, and Trajectories

Abstract: In this work we provide a complete description of the lifecycle of entanglement during the real-time motion of open quantum systems. The quantum environment can have arbitrary (e.g. structured) spectral density. The entanglement can be seen constructively as a Lego: its bricks are the modes of the environment. These bricks are connected to each other via operator transforms. The central result is that each infinitesimal time interval one new (incoming) mode of the environment gets coupled (entangled) to the open system, and one new (outgoing) mode gets irreversibly decoupled (disentangled from future). Moreover, each moment of time, only a few relevant modes (3 - 4 in the considered cases) are non-negligibly coupled to the future quantum motion. These relevant mode change (flow, or renormalize) with time. As a result, the temporal entanglement has the structure of a matrix-product operator. This allows us to pose a number of questions and to answer them in this work: what is the intrinsic quantum complexity of a real time motion; does this complexity saturate with time, or grows without bounds; how to do the real-time renormalization group in a justified way; how the classical Brownian stochastic trajectories emerge from the quantum evolution; how to construct the few-mode representations of non-Markovian environments. We provide illustrative simulations of the spin-boson model for various spectral densities of the environment: semicircle, subohmic, Ohmic, and superohmic.