Calibrated Quantification of the Dark-Exciton Reservoir via a k-Space-Folding Probe

Technical Summary: Calibrated Quantification of the Dark-Exciton Reservoir via a k-Space-Folding Probe

Problem Statement
Spin-forbidden dark excitons in monolayer transition metal dichalcogenides (TMDs), such as WSe₂, form a dominant "hidden reservoir" that governs exciton dynamics and many-body interactions. However, quantifying the population distribution within this reservoir remains a significant challenge. Standard far-field optical detection is intrinsically inefficient for dark excitons because they predominantly couple to high in-plane wave vectors ($k_{\parallel}$), which fall outside the collection cone of standard objectives with finite numerical aperture (NA). Furthermore, existing methods to access dark excitons (e.g., bright-dark mixing or antenna effects) conflate the actual population with electromagnetic weighting factors, specifically radiative-rate modification and collection efficiency. This entanglement renders the inverse problem of inferring population from detected intensity ill-posed, preventing accurate thermodynamic or many-body inference of the dark reservoir.

Methodology
The authors propose a calibrated far-field quantification approach that decouples radiative-rate modification from collection efficiency. The methodology integrates three core components:

1. Microsphere-Enabled k-Space Folding: A SiO₂ microsphere is placed on a monolayer WSe₂/Au heterostructure. This probe breaks in-plane translational symmetry, redirecting high-$k_{\parallel}$ evanescent emission (including surface plasmon polariton-mediated components) into propagating angles within the objective's collection cone. This "folding" mechanism enables efficient far-field collection of the dark exciton channel.
2. Green-Tensor Quasinormal-Mode (QNM) Calibration: To rigorously separate the detection response into its constituent parts, the authors employ a Green-tensor QNM framework. They identify that the optical response is dominated by two degenerate TM-like QNMs (QNM1 and QNM2) arising from the hybridization of whispering-gallery modes (WGMs) with surface plasmons.
   • Collection Efficiency ($\eta_i$): The framework calculates how the microsphere reshapes the collection efficiency for out-of-plane ($\perp$, dark) and in-plane ($\parallel$, bright) dipoles. The microsphere selectively enhances the out-of-plane efficiency from 8.7% to 43.4% (a $\approx$5$\times$ increase), while the in-plane channel changes modestly.
   • Radiative Rate ($\Gamma_i$): Using the dipole-projected local density of optical states (LDOS), the authors quantify the environment-induced radiative rate enhancement. The out-of-plane channel exhibits a $\approx$40$\times$ larger LDOS than the in-plane channel, compensating for the intrinsically weak radiative decay of dark excitons.
3. Population Retrieval via Spatial Scanning: The detected signal ($I_{det}$) is modeled as a linear superposition of population-weighted response functions ($I_{det} \propto N_{\parallel}R_{\parallel} + N_{\perp}R_{\perp}$). By laterally scanning the excitation spot relative to the microsphere center, the distinct position-dependence of $R_{\parallel}$ and $R_{\perp}$ allows for a two-parameter fit to extract the population ratio $N_{\perp}/N_{\parallel}$. Spectral filtering is used to isolate the dark ($\sim$1.625 eV) and bright ($\sim$1.669 eV) windows, aligned with adjacent microsphere resonances.

Key Results

• Quantified Population Ratio: The calibrated inversion yields a room-temperature dark-to-bright population ratio of $N_D/N_B = 4.3 \pm 1.1$.
• Thermalization Evidence: This ratio is consistent with a near-thermalized exciton manifold under continuous-wave (CW) excitation, aligning with Boltzmann statistics predictions ($N_D/N_B \approx 4.9$ at 300 K) for the known dark-bright splitting in WSe₂.
• Signal Validation: Control experiments using MoSe₂/Au (where the bright state is lower in energy) and WSe₂/SiO₂ (without plasmonic enhancement) show strong correlations between the QNM-coupled signal and bright-exciton-dominated planar PL. In contrast, the WSe₂/Au sample shows weak correlation ($r=0.22$) in the relevant temperature range, confirming that the captured signal originates from the intrinsic dark-exciton reservoir rather than geometric artifacts or disorder-induced bright-exciton projections.
• Efficiency Gains: The microsphere probe successfully transfers the dark-exciton weight into the collection cone, achieving a comparable detection efficiency for both exciton channels, which was previously unattainable with standard far-field optics.

Significance and Claims
The paper claims to establish the first calibrated far-field route to retrieve dark-exciton populations in monolayer WSe₂. By resolving the inverse problem through the decoupling of radiative-rate modification and collection efficiency, the authors provide a quantitative thermodynamic benchmark for the dark reservoir.

• Thermodynamic Anchor: The method establishes a metric for the dark-exciton reservoir under ambient conditions, confirming that the reservoir obeys lattice-temperature statistics despite nonresonant CW driving.
• Benchmark for 2D Phases: This calibrated population metric serves as a critical benchmark for interaction-driven 2D exciton phases. The authors note that in a companion work, this same architecture was used to report room-temperature Berezinskii–Kosterlitz–Thouless (BKT)-type condensation of dark excitons, with the population metrology and coherence diagnostics mutually reinforcing the observation of quasi-thermal equilibrium.
• Generalizability: The approach is presented as a generalizable calibration concept for accessing other elusive out-of-plane dipoles, such as hBN defects and interlayer/moiré excitons, without requiring ultrahigh-Purcell enhancement.