Metamaterial resonant structures made from arrays of superconducting lumped circuit elements can exhibit microwave mode spectra with left-handed dispersion, resulting in a high density of modes in the same frequency range where superconducting qubits are typically operated, as well as a bandgap at lower frequencies that extends down to dc. Using this novel regime for multi-mode circuit quantum electrodynamics, we have performed a series of measurements of such a superconducting metamaterial resonator coupled to a flux-tunable transmon qubit. Through microwave measurements of the metamaterial, we have observed the coupling of the qubit to each of the modes that it passes through. Using a separate readout resonator, we have probed the qubit dispersively and characterized the qubit energy relaxation as a function of frequency, which is strongly affected by the Purcell effect in the presence of the dense mode spectrum. Additionally, we have investigated the ac Stark shift of the qubit as the photon number in the various metamaterial modes is varied. The ability to tailor the dense mode spectrum through the choice of circuit parameters and manipulate the photonic state of the metamaterial through interactions with qubits makes this a promising platform for analog quantum simulation and quantum memories.
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