Fig. 2

Vacuum Rabi splitting. a The spectral response of device A (155 nm vacuum-gap capacitor) is shown as a function of flux in a single-tone reflection measurement plotted as \(\left| {S_{11}} \right|\) (see Methods and Supplementary Sec. S1). The blue dashed lines indicate the bare (uncoupled) frequency of the fundamental cavity mode, ω ₁, and the transition frequencies of the transmon from the ground state \(\left| g \right\rangle\), to its first and second excited states, \(\left| e \right\rangle\) and \(\left| f \right\rangle\) respectively. The red lines show the hybridized state transitions of the coupled system as fitted from the full spectrum (shown in Supplementary Fig. S2). The green lines indicate the dressed state transitions using the rotating wave approximation (RWA) for the same circuit parameters. Note that the splitting is not symmetric with respect to the point at which qubit and mode frequency cross. This is notably due to the renormalization of the charging energy that comes from considering higher resonator modes.²³ b Line-cut showing the vacuum Rabi splitting of the qubit transition with ω ₁, resulting in a separation of Δ_VRS = 2π × 1.19 GHz, which is about 281 linewidths of separation. c Close up of the spectrum around half a flux quantum (anti-sweet spot), where the qubit frequency is minimal (\(\omega _a \mathbin{\lower.3ex\hbox{$\buildrel<\over\\ {\smash{\scriptstyle \sim}\vphantom{_x}}$}} 2\pi \times 1.1\) GHz from the fitted model). There we observe a discrete transition of the spectral response of the circuit towards the bare cavity, ω ₁, which we attribute to a decoupling of the qubit and resonator due to a thermally populated qubit. Additionally we observe a small avoided crossing with the g ↔ f transition