  A single nanotube synthesized from a transition metal dichalcogenide (TMDC)
exhibits strong exciton resonances and, in addition, can support optical
whispering gallery modes. This combination is promising for observing
exciton-polaritons without an external cavity. However, traditional
energy-momentum-resolved detection methods are unsuitable for this tiny object.
Instead, we propose to use split optical modes in a twisted nanotube with the
flattened cross-section, where a gradually decreasing gap between the opposite
walls leads to a change in mode energy, similar to the effect of the barrier
width on the eigenenergies in the double-well potential. Using
micro-reflectance spectroscopy, we investigated the rich pattern of polariton
branches in single MoS$_2$ tubes with both variable and constant gaps. Observed
Rabi splitting in the 40 - 60 meV range is comparable to that for a MoS$_2$
monolayer in a microcavity. Our results, based on the polariton dispersion
measurements and polariton dynamics analysis, present a single TMDC nanotube as
a perfect polaritonic structure for nanophotonics.
