A hierarchical equations of motion (HEOM) analog for systems with delay: illustrated on inter-cavity photon propagation

TL;DR

This paper extends the hierarchical equations of motion (HEOM) framework to include photon delay effects in spatially extended photonic systems, demonstrated on a two-slab example, offering a new computational approach for such problems.

Contribution

The authors generalize HEOM to account for photon retardation in extended systems, providing a new, reliable framework as an alternative to path integral methods.

Findings

Derived equations effectively model photon retardation in dielectric slabs.

The framework offers a numerically reliable alternative to path integral approaches.

Application to two dielectric slabs demonstrates the method's utility.

Abstract

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