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When a chain of superconducting qubits couples to a coplanar resonator in acavity, each of its N qubits (equally-spaced with distance l) experiences adifferent dipole-field coupling strength due to the waveform of the cavityfield. We find that this inhomogeneous coupling leads to a pair of l-dependentladder operators for the angular momentum of the spin chain. Varying the qubitspacing l changes the transition amplitudes between the angular momentumlevels. We derive an exact diagonalization of the general N-qubit Hamiltonianand, through the N=4 case, demonstrate how the l-dependent operators lead to adenser one-excitation spectrum and a probability redistribution of theeigenstates. Moreover, it will be shown that the variation of l between its twolimiting values coincides with the crossover between Frenkel- and Wannier-typeexcitons in the superconducting spin chain.

Excitation spectrum for an inhomogeneously dipole-field-coupled superconducting qubit chain