A Hierarchical Approach to Quantum Many-Body Systems in Structured Environments

Abstract: Open quantum systems that feature non-Markovian dynamics are routinely solved using techniques such as the Hierarchical Equations of Motion (HEOM). However, their usage of the entire system density-matrix renders them intractable for many-body systems. Here, we combine the HEOM with the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy to achieve a rigorous description of open many-body systems in contact with structured photonic and phononic baths. We first rationalize that this stacked hierarchy accounts for spin-squeezing and superradiant emission despite its applicability to arbitrarily many emitters. The full potential of BBGKY-HEOM is then illustrated for two relevant applications: (i) the explicit treatment of vibrational modes provides access to resonant enhancements in few-emitter lasing, and (ii) the impact of phononic coupling and charge noise on many-body electronic systems embedded in host materials (e.g. molecules in organic crystals) is as relevant as electronic correlation. Our work establishes an accessible, yet rigorous, route between condensed matter and quantum optics, fostering the growth of a new domain at their interface.