Hierarchical equations of motion analog for systems with delay: Application to intercavity photon propagation

Over the last two decades, hierarchical equations of motion (HEOM) Tanimura and Kubo have become equation motion-based tool for numerically exact calculations system-bath problems. The HEOM is today generalized to many cases dissipation transfer processes through an external bath. In spatially extended photonic systems, propagation photons bath leads retardation/delays in coupling quantum emitters. Here, idea behind derivation case photon retardation applied simple example dielectric slabs. derived provide a reliable framework describing may alternative path-integral treatments.