Efficient simulation framework for modeling collective emission in ensembles of inhomogeneous solid-state emitters

TL;DR

The paper introduces an efficient cumulant-expansion framework to model photon-mediated collective emission in disordered ensembles of solid-state emitters, reducing the computational load from exponential to polynomial and enabling large-scale Monte Carlo sampling over realistic SiV$^{-}$ clusters. By incorporating inhomogeneous broadening and spatial disorder through a dyadic Green’s-function–based interaction model, the approach yields time-domain and frequency-domain predictions that align with experimental conditions. It identifies two robust signatures of collective behavior: (i) a threshold-like superradiant burst that requires enough emitters and high quantum efficiency, mitigated by strong near-field interactions; and (ii) interaction-induced broadening and skewed density of states in PLE spectra with increasing cluster density. The framework is general and applicable to atoms, molecules, and quantum dots, offering a practical tool for interpreting and guiding experiments on collective phenomena in realistic quantum systems.

Abstract

An efficient simulation framework is proposed to model collective emission in disordered ensembles of quantum emitters. Using a cumulant expansion approach, the computational complexity scales polynomially as opposed to exponentially with the number of emitters, enabling Monte Carlo sampling over a large number of realizations. The framework is applied to model negatively charged silicon-vacancy (SiV$^{-}$) centers inside diamond. Incorporating spatial disorder and inhomogeneous broadening, we obtain statistically averaged responses over hundreds of SiV$^{-}$ clusters. These simulations reveal two signatures of collective behavior. First, dynamics of fully inverted clusters show that superradiant emission occurs only with sufficiently large emitter number and high quantum efficiency. Unlike ideal Dicke superradiance, the burst is substantially suppressed by strong near-field dipole-dipole interaction, consistent with existing theoretical predictions. Second, under continuous-wave excitation we compute photoluminescence-excitation spectra, which exhibit interaction-induced broadening in the distribution of resonance peaks. The corresponding density of states also displays a non-zero skewness. Overall, by incorporating realistic inhomogeneities in emitter clusters, our framework is able to predict statistics for disordered ensembles that can be compared to experiments directly. Our approach generalizes to other types of emitters, including atoms, molecules, and quantum dots, thus providing a practical tool for analyzing collective behavior in realistic quantum systems.

Figures (9)

• Figure 1: Schematic illustration of the theoretical approach used in this work. (a) High-level sketch of the considered model. Multiple quantum emitters are randomly positioned within the photonic environment. The emitters are driven by an external laser; the emitted light is collected and analyzed. The photon-mediated dipole-dipole interactions, represented by $J_{ij}$ and $\Gamma_{ij}$, lead to nontrivial collective behavior. (b) When using the full set of Heisenberg equations to simulate $N$ emitters, the complexity scales exponentially. The computational cost can be reduced by truncating the equations of motion via a cumulant expansion approach. (c) The developed framework predicts both time-domain dynamics and spectral responses of emitter ensembles. It captures several important features arising from dipole–dipole interactions, including superradiance, subradiance, and interaction-induced broadening.
• Figure 2: The key properties of the SiV$^{-}$ ensembles modeled in this paper. (a) Schematic illustration of an ensemble of SiV$^{-}$ centers embedded in diamond. The SiV$^{-}$ centers have different orientations, as indicated by the insets. (b) Level structure of a single SiV$^{-}$ center. Although four optical transitions are allowed (marked as A, B, C, D), we only focus on the C transition and model each SiV$^{-}$ center as an ideal two-level system. (c) Experimental linewidth measurements compiled from the literature Rogers2014bSipahigil2014Evans2016Arend2016Sipahigil2016Schroder2017Zhang2018Lang2020Zuber2023. Each filled marker represents the reported mean value, while the horizontal line denotes the corresponding error bar. If no error bar has been reported, an unfilled marker is used.
• Figure 3: Time-resolved collective emission dynamics in SiV$^{-}$ ensembles. (a) Schematic illustration of the TRPL experiment. A pulsed laser is used to excite the SiV$^{-}$ ensemble, and the time-evolution of the collected fluorescence signal is analyzed. In our simulations the depths of SiV$^{-}$ centers are sampled from a Gaussian distribution. (b) Time-evolution of the photon emission rate, averaged over $300$ clusters. Shaded areas denote $\pm1$ standard deviation. Here "w/o RDDI" means that in Equation (\ref{['eq:eff_Hamiltonian']})(\ref{['eq:Lindblad']}), both $J_{ij}$ and $\Gamma_{ij}$ are set to zero for $i\neq j$ pairs. For $N=5$, the curve is almost indistinguishable from the case without dipole-dipole interaction, while for $N=15$ a superradiant burst emerges. (c) Time-evolution of the photon emission rate, with the coherent part of interaction disabled ($J_{ij}=0$ in Equation \ref{['eq:eff_Hamiltonian']}), while the dissipative part remains $\Gamma_{ij}\neq 0$. Results are shown in green. For $N=15$, an apparent superradiant burst appears, with $I_\text{peak}/I_{t=0}\approx 1.173$ for $d=23\pm 7$ nm ($1.086$ for $d=100\pm 22$ nm). Removing $J_{ij}$ eliminates the dephasing induced by strong near-field interaction, thus strengthens the superradiance phenomenon. (d) Maximum photon emission rate $I_\text{peak}$, normalized by $I_{t=0}$, across different parameter settings. The emitter number $N$ and quantum efficiency $\eta_{Q}$ are varied. Threshold behavior is observed for both the "shallow" and "deep" cases: sufficiently large $N$ and $\eta_{Q}$ are required to observe a superradiant peak. The four cases shown in panel (b) are marked by squares.
• Figure 4: Collective behavior obtained from the frequency-domain response of SiV$^{-}$ ensembles. All simulations are assuming an intrinsic quantum efficiency of $\eta_{Q}=60\%$. (a) Simplified schematic of the experimental setup used to measure the PLE spectrum. A tunable laser is used for resonant excitation. A single objective lens is used to focus the incident beam onto the sample, as well as to collect the emission. (b) Flowchart of the simulation framework. We randomly generate $600$ cluster configurations. For each cluster the laser frequency is swept by varying the detune $\Delta$ and calculate the PLE spectrum. Finally, PLE spectra from all clusters are aggregated and analyzed statistically. (c) An example with 5 color centers. The upper panel shows the positions (red dots) and the dipole moments (blue arrows) of all SiV$^{-}$ centers. The unit for $x$, $y$ and $z$ coordinates are all nanometer. The lower panel shows the PLE spectrum (blue) and compares it with the case without dipole-dipole interaction (red). (d) The PLE spectra obtained from Monte Carlo simulation of $600$ clusters. The left panel corresponds to the "shallow" distribution ($23\pm 7$ nm), while the right panel corresponds to the "deep" distribution ($100\pm 22$ nm). Solid lines show the median; shaded bands indicate the $25\%$ to $75\%$ interquartile range. For comparison, blue curves show results when disabling RDDI. (e)(f) Density of single-excitation states in a homogeneous environment. The standard deviation of depth distribution is (e) $\sigma_{d}=7$ nm and (f) $\sigma_{d}=22$ nm, respectively. Solid lines show results obtained for $5, 10, 15$ emitters. The shaded area indicates an ideal Gaussian distribution with a standard deviation of 400 MHz, corresponding to the case without dipole-dipole interactions. The inset shows the skewness of the three distributions. As a reference, the dashed line marks the skewness of the ideal Gaussian, which is close to zero.
• Figure 5: Radiative lifetime $\tau$ of an electric dipole inside diamond, visualized as a funcntion of depth $d$. Two different cases are considered: the blue curve ($\theta_{e} = 0^{\circ}$) corresponds to a dipole that's perpendicular to the interface, while the red curve ($\theta_{e} = 90^{\circ}$) corresponds to a dipole that's parallel to the interface. The lifetime has been normalized using $\tau_{0}$, which stands for the lifetime inside homogeneous environment. The wavelength is chosen as $\lambda_{0} = 637$ nm, corresponding to the zero-phonon line of NV$^{-}$ center inside diamond.