Towards strong multi-mode coupling between a transmon and a metamaterial resonator

A high density of modes can be produced using metamaterial resonant structures made from arrays of lumped circuit elements, to which a flux-tunable transmon qubit can be coupled. For such a system, we have measured the coupling strength of the qubit to multiple modes by tuning the flux and observing the splitting in the transmission of each mode. In these initial measurements, the coupling strengths were larger than the individual mode linewidths, but did not exceed the inter-mode spacing. We discuss approaches to decrease the mode spacing in this system and simulate the spectrum numerically. In addition, we discuss techniques for increasing the coupling strength between these modes and the transmon qubit. For appropriate parameters, we show that it will be to possible to reach the regime where the qubit can be coupled to multiple modes simultaneously, which will have applications in analog quantum simulations and multi-mode cQED.