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

TL;DR

This work generalizes the modified Langevin noise formalism to multiple quantum emitters embedded in dispersive dielectric environments, introducing two independent bosonic reservoirs (medium-assisted and scattering-assisted) with potentially different thermal states. By employing emitter-centered bright modes, the authors derive a reduced, tractable Hamiltonian and define matrix-valued spectral densities ${\mathcal J}^{(M)}(\omega)$ and ${\mathcal J}^{(S)}(\omega)$ that encode emitter–environment couplings, mutual interactions, and energy exchange. A surrogate vacuum environment with an effective spectral density ${\mathcal J}^{\text{eff}}(\omega;\beta_M,\beta_S)$ reproduces the exact reduced dynamics under product initial states, enabling non-Markovian analysis via standard methods. Application to two emitters near a Drude sphere and a rod–disk nanostructure reveals entanglement decay, revivals, and generation driven by the detailed structure of the spectral density matrices, highlighting routes to optimize quantum correlations in structured nanophotonic environments. The framework provides a rigorous, scalable foundation for designing emitter interactions in realistic environments with dispersion and losses, with potential extensions to molecular polaritons and larger ensembles.

Abstract

The control of interactions among quantum emitters through nanophotonic structures offers significant potential for quantum technologies. However, a rigorous theoretical description of the interaction of multiple quantum emitters with complex dispersive dielectric objects remains highly 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 extends Refs. \cite{miano_quantum_2025}, \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. Understanding how these reservoirs shape emitter dynamics is crucial to understanding light-matter interactions in complex electromagnetic environments and to enhancing intrinsic emitter properties in structured environments.

Figures (13)

• Figure 1: Two equal quantum emitters with dipole moment $\mu$ interact with a Drude sphere with radius $a$, plasma frequency $\omega_p$ and damping rate $\nu$. Normalized spectral densities $\mathcal{J}^{(M)}_{ij}/\mathcal{J}_0$ and $\mathcal{J}^{(S)}_{ij}/\mathcal{J}_0$, for $i,j=1,2$, as a function of the normalized frequency $\omega/\omega_p$ with $k_p \, a=1$, $\nu/\omega_p = 0.01$ where $k_p=\omega_p/c$. The two emitters are positioned as in the inset, at the same distance $d$ from the sphere center, $d/a=1.75$. The characteristic spectral density $\mathcal{J}_0$ is given by $\mathcal{J}_0 = \frac{1}{6 \pi^2} \frac{k_p}{\hbar\, \varepsilon_0 } \left({\mu k_p}\right)^2$. The spectral density matrices are Hermitian.
• Figure 2: Two equal quantum emitters with dipole moment $\mu$ interact interact with a nanostructure composed of two rods and a disk. The first rod has length $3a$, width $2a$, and height $a$; the second rod has length $4a$, width $2 a$, and height $a$; the disk has radius $a$ and height $a$. All three particles are made of a Drude material with plasma frequency $\omega_p$ and damping rate $\nu$ such as $\nu/\omega_p = 0.01$. The two quantum emitters are placed at the midpoint of the gaps between adjacent particles. Normalized spectral densities $\mathcal{J}^{(S)}_{ij}/\mathcal{J}_0$ and $\mathcal{J}^{(M)}_{ij}/\mathcal{J}_0$ versus $\omega/\omega_p$ for $i,j=1,2$ and $k_p \, a=1$ where $k_p=\omega_p/c$. The characteristic spectral density $\mathcal{J}_0$ is given by $\mathcal{J}_0 = \frac{1}{6 \pi^2} \frac{k_p}{\hbar\, \varepsilon_0 } \left({\mu k_p}\right)^2$. The spectral density matrices are Hermitian.
• Figure 3: Spherical particle, see Fig. \ref{['fig:Sphere']}. Relaxation dynamics of populations of quantum emitters: $p_a^e(t)$ is the population of the quantum emitter $a$, for $a=1,2$, in the excited state $|e\rangle$. The initial composite state has been chosen as $\hat{\rho}_{12}(0)=\ketbra{\Psi^{-}}$, while $\mu=10^{-3}\omega_{p}$ and $\beta_{S}=\beta_{M}=1000$. solid blue (red) curves denote the case $\Omega_1=0.54\omega_p,\Omega_2=\omega_p$, while cyan (orange) are computed for $\Omega_1=\Omega_2=0.54\omega_p$.
• Figure 4: Spherical particle, see Fig. \ref{['fig:Sphere']}. Relaxation dynamics of populations of the quantum emitters: $p_a^e(t)$ is the population of the quantum emitter $a$, for $a=1,2$, in the excited state $|e\rangle$. The initial composite state has been chosen as $\rho_{12}(0)=\ketbra{\Psi^{-}}$, while $\mu=10^{-3}\omega_{p}$ and $\Omega_1=\Omega_2=0.54\omega_p$. Solid blue (red) curves are computed for $\beta_{S}=\beta_{M}=1000$, and dashed curves denote the case $\beta_{M}=1,\beta_{S}=1000$.
• Figure 5: Spherical particle, see Fig. \ref{['fig:Sphere']}. Time evolution of negativity $\mathcal{N}(\hat{\rho}_{12})$ with $\mu=10^{-3}\omega_{p}$, corresponding to the same initial state as in Figs. \ref{['fig:popAB1']} and \ref{['fig:popAB2']}. Solid blue(red) lines denote the cases $\Omega_1=0.54\omega_p,\Omega_2=\omega_p$ and $\Omega_1=\Omega_2=0.54\omega_p$ with equal inverse temperatures $\beta_{S}=\beta_{M}=1000$, respectively, while dashed curves denote the same frequency choices for $\beta_{M}=1$ and $\beta_{S}=1000$.