FIG. 1. A platform for interacting dressed bound states.—

(a) A 16-site microwave photonic crystal is realized by alternating sections of high and low impedance coplanar waveguide. Two transmon qubits (multilevel, anharmonic energy ladder) are in neighboring unit cells in the middle of the device, centered in the high impedance sections for maximal coupling to the band edge at 7.8 GHz [all values presented in units of (2π) Hz, i.e., ω_BE = 7.8 (2π) GHz]. For this experiment, the passband (band gap) refers to states above (below) the band-edge frequency. Each transmon is individually tunable in frequency via a local flux bias line. (b) Bound-state linewidth, an indirect measure of localization, varies with bare transmon qubit frequency. The wide range over which photon localization can be tuned indicates the feasibility of realizing a chain of strongly interacting bound states. Experimentally measured and simulated linewidths are shown in red and black, respectively. Inset: Overlay of simulated S₂₁ from the transfer matrix method (blue) and measured high-power S₂₁ (black) shows good agreement in bare crystal characteristics. (c) The interaction between bound states will be determined by overlap of their localized photonic envelopes with the qubits. (d) One can couple more qubits to the band edge by adding them to other cells of the photonic crystal. In such a system, the localization-length-dependent interaction of the bound states would preserve the spatial organization of qubits across the crystal, and determine the many-body structure of the interactions. (e) Experimental data and (f) hopping model simulation for S₂₁ vs single-qubit frequency and probe frequency. The bare band edge is at 7.797 GHz. The bright peak in the band gap is the dressed qubit-photon bound state. The bound state always exists within the band gap for qubit frequencies (the other qubit is far detuned and has negligible effect) both above and below the band edge—a clear signature of non-Markovianity.