Hierarchical equations of motion model for phonon and photon correlations

The hierarchical equations of motion (HEOM), constructed through Feynman and Vernon path integrals, are used to model the dynamics of quantum systems within a statistical ensemble representing the condensed phase. We show that second-order, two-time correlation functions for phonons and photons emitted from a vibronic molecule in a thermal bath produce signatures relating to exchange with the environment and discuss how these arise from the hierarchical equations of motion. Additionally, we discuss the physical justification for the fundamental steps involved in deriving the HEOM and highlight the profound impact on the system memory effects that changes in the spectral density, or bilinear system-environment coupling, can have.