Monolayer two-dimensional transition metal dichalcogenides (2D TMDs) represent a class of atomically thin semiconductors with unique optical properties. Similar to graphene, but with a three-layer (staggered) honeycomb lattice, TMDs host direct-gap transitions at their $\pm$K valleys that exhibit circular-dichroism due to their finite Berry curvature. The reduced dimensionality of materials in this system, combined with large effective carrier masses, leads to enhanced Coulomb interaction and extremely tightly bound excitons ($E_{\rm{B}} \approx 150-300:\rm{meV})$. Here, we seek to exploit the unusually tight binding of the excitons to probe two different types of higher energy exciton species in TMDs.

First, we experimentally probe the magneto-optical properties of 2$s$ Rydberg exciton species in WSe$2$. The magnetic response of excitons gives information on their spin and valley configurations, nuanced carrier interactions, and insight into the underlying band structure. Recently, there have been several reports of 2$s$/3$s$ charged excitons in TMDs, but very little is still known about their response to external magnetic fields. Using photoluminescence excitation spectroscopy, we verify the 2$s$ charged exciton and report for the first time its response to an applied magnetic field. We benchmark this response against the neutral exciton and find that both the 2$s$ neutral and charged excitons exhibit similar behavior with $g$-factors of g${\rm{X_0^{2s}}}$=-5.20$\pm0.11$ $ \mu_{\rm{B}}$ and g${\rm{X-^{2s}}}$=-4.98$\pm0.11$ $ \mu_{\rm{B}}$, respectively.

Second, via theoretical calculations, we investigate the exciton spectrum generated in 2D semiconductors under illumination by twisted light. Twisted light carries orbital angular momentum (OAM) which can act as an additional tunable degree of freedom in the system. We demonstrate that twisted light does not have the ability to modify the exciton spectrum and induce dipole-forbidden excitons, in contrast to atoms. This result stems from the fact that the additional OAM is transferred preferentially to the center-of-mass (COM) of the exciton, without modifying the relative coordinate which would allow dipole-forbidden, higher energy excitons to form.