  Many-body localization (MBL) has attracted significant attention due to its
immunity to thermalization, role in logarithmic entanglement entropy growth,
and opportunities to reach exotic quantum orders. However, experimental
realization of MBL in solid-state systems has remained challenging. Here we
report evidence of a possible phonon MBL phase in disordered GaAs/AlAs
superlattices. Through grazing-incidence inelastic X-ray scattering, we observe
a strong deviation of the phonon population from equilibrium in samples doped
with ErAs nanodots at low temperature, signaling a departure from
thermalization. This behavior occurs within finite phonon energy and wavevector
windows, suggesting a localization-thermalization crossover. We support our
observation by proposing a theoretical model for the effective phonon
Hamiltonian in disordered superlattices, and showing that it can be mapped
exactly to a disordered 1D Bose-Hubbard model with a known MBL phase. Our work
provides momentum-resolved experimental evidence of phonon localization,
extending the scope of MBL to disordered solid-state systems.
