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In this work we develop EPOCH: Equilibrium Propagator by Orthogonalpolynomial CHain, a computationally efficient method to calculate thetime-dependent equilibrium Green's functions, including the anomalous Green'sfunctions of superconductors, to capture the time-evolution in largeinhomogeneous systems. The EPOCH method generalizes the Chebyshev wave-packetpropagation method from quantum chemistry and efficiently incorporates theFermi-Dirac statistics that is needed for equilibrium quantum condensed mattersystems. The computational cost of EPOCH scales only linearly in the systemdegrees of freedom, generating an extremely efficient algorithm also for verylarge systems. We demonstrate the power of the EPOCH method by calculating thetime-evolution of an excitation near a superconductor-normal metal interface intwo and three dimensions, capturing transmission as well as normal and Andreevreflections.

Efficient numerical method for evaluating normal and anomalous time-domain equilibrium Green's functions in inhomogeneous systems