  Engineering quantum phases with spontaneously broken symmetries is a major
goal of research in different fields. Trapped ultracold Rydberg-excited atoms
in optical lattices are a promising platform for realizing quantum phases with
broken lattice translational symmetry since they are interacting over distances
larger than the lattice constant. Although numerous theoretical works on
trapped Rydberg-excited gases have predicted such phases, in particular density
wave or supersolid phases, their experimental observation proves to be
difficult due to challenges such as scattering processes and the limited
experimentally achievable coupling strength. Most of these previous studies
have focused on isotropically interacting gases dressed with Rydberg
$s$-states, while the effect of anisotropic interactions due to Rydberg-excited
$p$-states in trapped quantum gases remains much less investigated.
Additionally, it was shown that the excitation scheme used to excite Rydberg
$p$-states possesses advantages regarding achievable coupling strengths and
limitation of scattering processes compared to its $s$-state counterpart, which
makes the investigation of Rydberg $p$-state dressed quantum gases even more
interesting. In the present work, we study the extended, two-component
Bose-Hubbard model, realized with a bosonic quantum gas with Rydberg-excited
$p$-states trapped in an optical lattice, within Gutzwiller mean-field theory.
We compute the ground state phase diagram and investigate its different
regimes. By comparison to the phase diagram of the isotropic case, we find the
anisotropic interaction to be more advantageous for the observation of
supersolid phases.
