  We theoretically demonstrate that the hallmarks of correlation and
fermionization in a one-dimensional exciton-polaritons gas can be observed with
state-of-the-art technology. Our system consists of a chain of excitonic
quantum dots coupled to a photonic waveguide, with a low filling of polaritons.
We analytically identify the Tonks-Girardeau, Tavis-Cummings and mean-field
limits and relate them to different regimes of the excitonic anharmonicity and
photonic bandwidth. Using matrix-product states, we numerically calculate the
ground-state energies, correlation functions and dynamic structure factor of
the system. In particular, the latter has a finite weight in the Lieb-Liniger
hole branch, and the density-density correlator displays Friedel-like
oscillations for realistic parameters, which reveal the onset of fermionization
close to the Tonks-Girardeau regime. Our work encourages future experiments
aimed at observing, for the first time and in spite of the moderate excitonic
anharmonicity, strongly correlated exciton-polariton physics.
