Hadamard regularization of open quantum systems coupled to unstructured environments in the Schwinger-Keldysh formalism

The theory of open quantum systems addresses how coupling to external degrees of freedom modifies observables and quantum coherence, a situation central to fundamental condensed-matter research and emerging quantum technologies. Schwinger-Keldysh field theory is a natural framework for both open- and nonequilibrium quantum systems in terms of functional integrals. However, its numerical solution is limited by a cubic scaling with the number of time steps. This is particularly prohibitive for scenarios with widely separated time scales, as is often the case for system and environmental scales. We consider a damped quantum harmonic oscillator as a toy model to study a separation-of-scales ansatz based on Hadamard regularization. A time-stepping algorithm for the Kadanoff-Baym equations on the slow system time-scale is presented that captures both low-temperature non-Markovianity and renormalization effects arising from the much faster environment scale.