  When a chain of superconducting qubits couples to a coplanar resonator in a
cavity, each of its N qubits (equally-spaced with distance l) experiences a
different dipole-field coupling strength due to the waveform of the cavity
field. We find that this inhomogeneous coupling leads to a pair of l-dependent
ladder operators for the angular momentum of the spin chain. Varying the qubit
spacing l changes the transition amplitudes between the angular momentum
levels. We derive an exact diagonalization of the general N-qubit Hamiltonian
and, through the N=4 case, demonstrate how the l-dependent operators lead to a
denser one-excitation spectrum and a probability redistribution of the
eigenstates. Moreover, it will be shown that the variation of l between its two
limiting values coincides with the crossover between Frenkel- and Wannier-type
excitons in the superconducting spin chain.
