Fig. 4

Qubit mediated mode–mode interactions. a Two-tone spectroscopy measurement showing the flux dependence of the third mode ω ₃ as obtained and shown in Fig. 3. Due to the strong hybridization over multiple modes, we observe that the qubit-mediated mode–mode coupling is sufficient to observe the effect of driving ω ₃ by monitoring the response of the fundamental mode ω ₁. The red dashed line indicates the dressed state of ω ₃ and the blue dashed line indicates the bare cavity. The green dashed line indicates the predicted line using the RWA, showing the effect of removing the counter-rotating terms, from which we obtain a Bloch–Siegert shift of 45 MHz. b Using the same measurement technique, we show a trace of the normalized reflection coefficient of a weak probe tone positioned at the slope of the resonance of the fundamental mode as a function of a higher frequency drive tone. Here we observe clearly the harmonics of the cavity, including the fourth-mode (ω ₄) and fifth-mode (ω ₅). The response for the same drive tone power clearly decreases for higher modes as these are further detuned. The data traces measuring ω ₅ is 39.5 dB higher in drive power than the other traces, but making power comparisons is impractical as our microwave measurement setup uses components specified up to 18 GHz. The leftmost peak corresponds to the onset of a frequency region where the system is driven to its linear regime³² due to the strong drive powers necessary to acquire this data. c Three panels showing a close up of the resonances of ω ₃, ω ₄, and ω ₅, following the harmonics of the fundamental frequency of ω ₁ = 2π × 4.268 GHz