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

Abstract: Spin-forbidden dark excitons in monolayer transition metal dichalcogenides constitute a dominant hidden reservoir that governs exciton dynamics and many-body interactions. Yet determining the population distribution within this reservoir remains challenging because detected brightness conflates radiative-rate modification with collection efficiency, obscuring the link between intensity and population. Here we make this inverse problem well posed by calibrating the position- and orientation-resolved detection response. Combining microsphere-enabled k-space folding with Green-tensor quasinormal-mode calibration, we decouple radiative-rate modification from collection efficiency. We extract a room-temperature dark-to-bright population ratio ND/NB = 4.3, consistent with a near-thermalized manifold under continuous-wave excitation. This calibrated population metric provides a quantitative thermodynamic benchmark for the dark reservoir and interaction-driven 2D exciton phases.