FIG. 3. Interacting bound states.—

Interaction between bound states is characterized by the avoided crossing (observed in S₂₁ measurement) that arises while tuning one qubit (y axis) through resonance with the other (fixed). (a) An avoided crossing of 240 MHz is observed when the fixed qubit is at 7.73 GHz. The two points where transmission amplitude of a bound state dims are understood as the bound-state peak being resonant with the qubit frequency. (a), inset—Hopping model simulation of the one-excitation manifold is consistent with experimental observation. The lamb shift in the hopping model originates from next-nearest-neighbor interaction between coupled cavities. (b),(c),(d) Tunable bound-state interaction strength is illustrated in example bound-state avoided level crossings for a fixed qubit whose bare frequency is circa 6.125, 6.75, and 7.625 GHz. As qubits are detuned further from the band edge, bound states are more tightly localized, reducing overlap and thus reducing interaction. (e),(f) Transmission when the qubits are on resonance across a range of qubit frequencies in the experiment and the simulation, respectively. The uneven linewidths of the two bound states when they occur at the same frequency suggest they are symmetric (higher-frequency bound state) and antisymmetric (lower-frequency bound state) states (see main text). (g) Bound-state avoided crossing and qubit population (from simulation) as a function of average bound-state frequency. A steady reduction in interaction strength occurs with increasing detuning from the band edge (moving deeper into the band gap) due to increasing localization of the bound states. Hopping model simulation (black) captures this detuning-dependent behavior observed in experiment (red). Near the band edge, both bound states (blue and cyan) have a significant photonic contribution.