Multi-mode ultra-strong coupling (I): spectroscopic experiments using a vacuum-gap transmon circuit architecture

In circuit QED, multi-mode extensions of the quantum Rabi model suffer from divergence problems. Here, we spectroscopically study multi-mode ultra-strong coupling using a transmon circuit architecture, which provides no clear guidelines on how many modes play a role in the dynamics of the system. As our transmon qubit, we employ a suspended island above the voltage anti-node of a $\lambda/4$ coplanar microwave resonator, thereby realising a circuit where 88\% of the qubit capacitance is formed by a vacuum-gap capacitor with the center conductor of the resonator. We measure vacuum Rabi splitting over multiple modes up to 2 GHz, reaching coupling ratios of $g/\omega=0.18$, well within the ultra-strong coupling regime. We observe a qubit-mediated mode coupling, measurable up to the fifth mode at 38 GHz. Using a novel analytical quantum circuit model of this architecture, which includes all modes without introducing divergencies, we are able to fit the full spectrum and extract a vacuum fluctuations induced Bloch-Siegert shift of up to 62 MHz. This circuit architecture expands the versatility of the transmon technology platform and opens many possibilities in multi-mode physics in the ultra-strong coupling regime.