  We investigated the dynamics of Bose-Einstein condensates (BECs) under an
external periodic potential. We consider two such systems, the first being made
of exciton-polaritons in a nanoribbon of transition metal dichalcogenides
(TMDCs), such as MoSe$_2$, embedded in a microcavity with a special curvature,
which serves as the source of the external potential. The second, made of bare
excitons in a nanoribbon of twisted TMDC bilayer, which naturally creates a
periodic Moir\'e potential that can be controlled by the angle of twist. We
proved that such systems exhibit a Time Crystal (TC) phase. This was
demonstrated by the fact that the calculated BEC spatial density profile shows
a non-trivial two-point correlator that oscillates in time. These BECs density
profiles were calculated by solving the quantum Lindblad master equations for
the density matrix within the mean-field approximation. We then go beyond the
usual mean-field approach, by adding a stochastic term to the master equation,
which corresponds to quantum corrections, and we show that the TC phase is
still present.
