  Lattice polarons, quasiparticles arising from the interaction between an
impurity and its surrounding bosonic environment confined to a lattice system,
have emerged as a platform for generating complex few-body states, probing
many-body phenomena, and addressing long-standing problems in physics. In this
study, we employ a variational ansatz to investigate the quasiparticle and
spectral properties of an impurity coupled to a condensate gas of hard-core
bosons in a two-dimensional optical lattice. Our findings demonstrate that the
polaron features can be tuned by adjusting the filling factor of the bath,
revealing intriguing polaron characteristics in the strongly interacting
regime. These results offer valuable insights for lattice polaron experiments
with ultracold gases and can serve as a guide for new experiments in emergent
quantum devices, such as moir\'e materials, where optical excitations can be
described in terms of hard-core bosons.
