Figure 5. Calibration of engineered noise.

(a) Resonator transmission spectra measured with the qubit prepared in the ground and excited states. In contrast to typical transmon qubits with ω_q<ω_r, an excited-state C-shunt flux qubit shifts the resonator to higher frequencies because of interactions with higher-level qubit transitions. (b) Engineered thermal-photon noise source. A coherent tone near the resonator frequency is mixed with white-noise of nominally equal power from two independent arbitrary waveform generators (AWGs) applied to the in-phase (I) and quadrature (Q) ports of the I/Q mixer. The AWG noise bandwidth (80 MHz) is much greater than the cavity linewidth, creating effectively a thermal-photon noise source with power P_add. (c) Qubit spectral line shape (dots) and lorentzian fits (solid lines) for various added noise powers P_add. The equivalent photon population added to the resonator is derived from d. The blue trace corresponds to no added noise from the source in b. (d) Stark-shifted qubit frequency versus applied noise power (dots). Coloured dots correspond to traces in c. Combining the linear fit (solid line) with the first-order dependence of Stark shift on photon population (equation (4)) yields the power-per-added-photon in the low-power limit. Inset: wider range of applied noise powers; dashed box indicates the range in the main panel. At large photon populations the frequency shift becomes nonlinear, following equation 43 in ref. 40 (solid line). (e) Spin-echo decay (dots) with exponential fit (solid lines) for several values of added photons. Inset: spin-echo pulse sequence. (f) Spin-echo pure-dephasing rate (echo decay rate without the T₁ contribution) plotted versus injected photon population (dots). The linear fit (solid line) has slope , in agreement with the value of 2.5 × 10⁶ s⁻¹ calculated from equation (5). The intercept indicates a residual photon population in the resonator.