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Non-Markovian Dynamical Maps: Numerical Processing of Open Quantum Trajectories

The initial stages of the evolution of an open quantum system encode the key information of its underlying dynamical correlations, which in turn can predict the trajectory at later stages. We propose a general approach based on non-Markovian dynamical maps to extract this information from the initial trajectories and compress it into non-Markovian transfer tensors. Assuming time-translational invariance, the tensors can be used to accurately and efficiently propagate the state of the system to arbitrarily long timescales. The non-Markovian Transfer Tensor Method (TTM) demonstrates the coherent-to-incoherent transition as a function of the strength of quantum dissipation and predicts the non-canonical equilibrium distribution due to the system-bath entanglement. TTM is equivalent to solving the Nakajima-Zwanzig equation, and therefore can be used to reconstruct the dynamical operators (the system Hamiltonian and memory kernel) from quantum trajectories obtained in simulations or experiments. The concept underlying the approach can be generalized to physical observables with the goal of learning and manipulating the trajectories of an open quantum system.