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

Photon-induced droplet-like bound states in one-dimensional qubit array