  The localization properties of one-dimensional degenerate Fermi gases with
cavity-assisted non-local quasiperiodic interactions are numerically studied.
Although the cavity-induced interaction is typically nonlocal, it is proved
that the eigenstate thermalization hypothesis (ETH) is still applicable in our
system depending on the system parameters. We also find the segment of the
spectrum corresponding to infinite effective temperature varies for different
system parameters, which indicates the spectral range employed in the spectral
statistical analysis should be varied accordingly. The features of many-body
localization (MBL) are numerically identified by analyzing the spectral
statistics and the entanglement entropy using exact diagonalization. These
features are further confirmed by our time evolution results. In addition, the
number of cavity photons are found stable over long time dynamics in the MBL
phase. Such a feature can not only be utilized to nondestructively diagnose the
MBL phase by monitoring the number of leaking photons from the cavity, but
leveraged for constructing a device to produce a stable number of photons.
