Modified Langevin noise formalism for multiple quantum emitters in dispersive electromagnetic environments out of equilibrium

The control of interactions among quantum emitters through nanophotonic structures offers significant opportunities for quantum technologies. However, a rigorous theoretical description of the interaction of multiple quantum emitters with complex, dispersive dielectric objects remains challenging. Here, we introduce an approach based on the modified Langevin noise formalism that unveils the roles of both the noise polarization currents of the dielectrics and the vacuum fluctuations of the electromagnetic field scattered by the dielectrics. This work extends Refs. \cite{miano_quantum_2025} and \cite{miano_spectral_2025} to the general case of an arbitrary number of emitters. The proposed approach allows us to describe the dynamics of the quantum emitters for arbitrary initial quantum states of the electromagnetic environment, consisting of two independent bosonic reservoirs, a medium-assisted reservoir and a scattering-assisted reservoir, each characterized by its own spectral density matrix. Specifically, we examine situations where both reservoirs are initially in thermal quantum states but have different temperatures. Understanding how these reservoirs shape the dynamics of the emitters is crucial for understanding light-matter interactions in complex electromagnetic environments and for improving intrinsic emitter properties within structured environments.

💡 Research Summary

The manuscript presents a comprehensive theoretical framework for describing the interaction of an arbitrary number of quantum emitters (QEs) with complex, dispersive dielectric structures under non‑equilibrium conditions. Building on the macroscopic quantum electrodynamics (QED) approach, the authors adopt the “modified Langevin noise formalism,” originally introduced by Ciattoni, which treats on equal footing two distinct contributions to the electromagnetic field: (i) the medium‑assisted field generated by the fluctuating polarization currents inside the dielectric, and (ii) the scattering‑assisted field arising from vacuum fluctuations that are scattered by the dielectric body. These two contributions correspond to two independent bosonic reservoirs, termed the medium‑assisted reservoir and the scattering‑assisted reservoir, respectively.

The total Hamiltonian is split into three parts: the bare QE Hamiltonian, the electromagnetic environment Hamiltonian, and the dipole interaction Hamiltonian written in the Power‑Zienau‑Woolley picture. By expanding the electric‑field operator in frequency space, the authors separate the field into the two contributions mentioned above. The medium‑assisted field is expressed through a continuum of bosonic operators (\hat f_\omega(\mathbf r)) weighted by a Green‑function‑derived kernel (G_e(\mathbf r,\mathbf r’;\omega)). The scattering‑assisted field is expressed through a set of free‑space mode operators (\hat g_{\omega,\mathbf n,\nu}) weighted by the scattering modes (E_{\omega,\mathbf n,\nu}(\mathbf r)). Both operator families obey canonical commutation relations and together diagonalize the environment Hamiltonian.

A pivotal result is the integral identity (Eq. 14‑16) that shows the sum of the medium‑assisted and scattering‑assisted contributions reproduces the standard field commutator expressed via the imaginary part of the dyadic Green function (\operatorname{Im}