Dark exciton signatures in the infrared transient absorption of MoS$_2$ monolayer
Dark excitons play a central role in the nonequilibrium dynamics of two-dimensional semiconductors, but remain difficult to characterize. Transient-absorption experiments, with probes tuned in exciton-exciton transitions energy range (exc-tr-abs), can detect excitations from any populated dark excitons, including symmetry-forbidden, spin-forbidden, and finite-momentum ones. In this work, we develop a $GW$+BSE scheme for computing exc-tr-abs spectra from arbitrary populated exciton distributions. These dark excitons are included equally by evaluating exciton-exciton dipoles in a locally smooth gauge, including intra- and inter-band contributions. For monolayer MoS$_2$, the exc-tr-abs signal arises from the $\Gamma$, $K$, $M$, and $Q$ valleys, differing substantially from the $\Gamma$-only interpretation. Exciton-exciton dipoles show similar intensities across these valleys, while their weights are dictated by initial excitonic populations. State- and spin-resolved analyses assign the peaks to $1s \rightarrow 2p$ and $1s \rightarrow 3p$ transitions from both spin-flip and spin-conserving A and B excitons across valleys and momenta.