  Quasiparticles and their interactions are a key part of our understanding of
quantum many-body systems. Quantum simulation experiments with cold atoms have
in recent years advanced our understanding of isolated quasiparticles, but so
far they have provided limited information regarding their interactions and
possible bound states. Here, we show how exploring mobile impurities immersed
in a Bose-Einstein condensate (BEC) in a two-dimensional lattice can address
this problem. First, the spectral properties of individual impurities are
examined, and in addition to the attractive and repulsive polarons known from
continuum gases, we identify a new kind of quasiparticle stable for repulsive
boson-impurity interactions. The spatial properties of polarons are calculated
showing that there is an increased density of bosons at the site of the
impurity both for repulsive and attractive interactions. We then derive an
effective Schr\"odinger equation describing two polarons interacting via the
exchange of density oscillations in the BEC, which takes into account strong
impurity-boson two-body correlations. Using this, we show that the attractive
nature of the effective interaction between two polarons combined with the
two-dimensionality of the lattice leads to the formation of bound states --
i.e. bipolarons. The wave functions of the bipolarons are examined showing that
the ground state is symmetric under particle exchange and therefore relevant
for bosonic impurities, whereas the first excited state is doubly degenerate
and odd under particle exchange making it relevant for fermionic impurities.
Our results show that quantum gas microscopy in optical lattices is a promising
platform to explore the spatial properties of polarons as well as to finally
observe the elusive bipolarons.
