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 the 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.
Strong one-dimensional (1D) lattices usually lead to unconnected two-dimensional gases. The long-range character of the dipole-dipole interactions leads to a scenario where nonoverlapping gases at different sites may interact significantly. We show that the excitations of nonoverlapping condensates in 1D optical lattices acquire a bandlike character, being collectively shared by different sites. In particular, the hybridization of the modes significantly enhances the rotonization of the excitations and may induce roton instability. We discuss the observability of this effect in ongoing experiments.