  We consider an array of $N_e$ non-interacting qubits or emitters that are
coupled to a one-dimensional cavity array with tunneling energy $J$ and
non-linearity of strength $U$. The number of cavities is assumed to be larger
than the number of qubits. Working in the two-excitation manifold, we focus on
the bandgap regime where the energy of two excited qubits is off-resonant with
the two-photon bound state band. A two-step adiabatic elimination of the
photonic degrees of freedom gives rise to a one-dimensional spin Hamiltonian
with effective interactions; specifically, the Hamiltonian features constrained
single-qubit hopping and pair hopping interactions not only between nearest
neighbors but also between next-to-nearest and next-to-next-to-nearest spins.
For a regularly arranged qubit array, we identify parameter combinations for
which the system supports novel droplet-like bound states whose characteristics
depend critically on the pair hopping. The droplet-like states can be probed
dynamically. The bound states identified in our work for off-resonance
conditions are distinct from localized hybridized states that emerge for
on-resonance conditions.
