ac Stark Shift and Dephasing of a Superconducting Qubit Strongly Coupled to a Cavity Field

We have performed spectroscopy of a superconducting charge qubit coupled nonresonantly to a single mode of an on-chip resonator. The strong coupling induces a large ac Stark shift in the energy levels of both the qubit and the resonator. The dispersive shift of the resonator frequency is used to nondestructively determine the qubit state. Photon shot noise in the measurement field induces qubit level fluctuations leading to dephasing which is characteristic for the measurement backaction. A crossover in line shape with measurement power is observed and theoretically explained. For weak measurement a long intrinsic dephasing time of $T_2>200\mathrm{n}\mathrm{s}$ of the qubit is found.

Figure 1

(a) Simplified circuit diagram of measurement setup. The phase $\varphi$ and amplitude $T$ of a microwave at ${\omega }_{\mathrm{r}\mathrm{f}}$ transmitted through the resonator, amplified, and mixed down to an intermediate frequency ${\omega }_{\mathrm{I}\mathrm{F}}={\omega }_{\mathrm{r}\mathrm{f}}-{\omega }_{\mathrm{L}\mathrm{O}}$ using a local oscillator at ${\omega }_{\mathrm{L}\mathrm{O}}$ is measured. An additional spectroscopy microwave at ${\omega }_{\mathrm{s}}$ is applied to the input port of the resonator. (b) Ground $|\downarrow ⟩$ and excited $|\uparrow ⟩$ state energy levels of CPB vs gate charge $n_g$. (c) Calculated phase shift $\varphi$ in ground and excited states vs $n_g$ for ${\Delta }_{\mathrm{a},\mathrm{r}}/2\pi =100\mathrm{M}\mathrm{H}\mathrm{z}$.

Figure 2

(a) Probe microwave phase shift $\varphi$ vs gate charge $n_{\mathrm{g}}$ at spectroscopy frequency ${\nu }_{\mathrm{s}}=6.125\mathrm{G}\mathrm{H}\mathrm{z}$ (lower panel), $6.15\mathrm{G}\mathrm{H}\mathrm{z}$ (middle panel), and $6.2\mathrm{G}\mathrm{H}\mathrm{z}$ (upper panel). (b) Density plot of $\varphi$ vs $n_{\mathrm{g}}$ and ${\nu }_{\mathrm{s}}$; white (black) corresponds to large (small) phase shift. Horizontal arrows indicate line cuts shown in (a); vertical arrows indicate line cuts shown in Fig. 4a. Measurements in (a) and (b) were performed populating the resonator with $n\sim 25$ photons on average.

Figure 3

(a) Measured qubit linewidth $\delta {\nu }_{\mathrm{H}\mathrm{W}\mathrm{H}\mathrm{M}}$ vs input spectroscopy power $P_s$ (solid circles) with fit (solid line). Probe beam power $P_{\mathrm{r}\mathrm{f}}$ is adjusted such that $n<1$. (b) Measured peak depth $\varphi$ and excited state population probability $P_{\uparrow }$ on resonance ${\Delta }_{\mathrm{s},\mathrm{a}}=0$ vs $P_{\mathrm{s}}$ (solid circles) with fit (solid line).

Figure 4

Measured spectroscopic lines (wide lines with noise) at (a) intraresonator photon number $n\approx 1$ ($P_{\mathrm{r}\mathrm{f}}=-30\mathrm{d}\mathrm{B}\mathrm{m}$) with fit to Lorentzian line shape (solid line) and at (b) $n\approx 20$ ($P_{\mathrm{r}\mathrm{f}}=-16\mathrm{d}\mathrm{B}\mathrm{m}$) with fit to Gaussian line shape (solid line). Dashed lines are best fits to (a) Gaussian or (b) Lorentzian line shapes, respectively. The qubit transition frequency ${\nu }_{\mathrm{a}}$ at low $P_{\mathrm{r}\mathrm{f}}$, the half width half maximum $\delta {\nu }_{\mathrm{H}\mathrm{W}\mathrm{H}\mathrm{M}}$, and the ac Stark shift ${\nu }_{\mathrm{a}\mathrm{c}}$ of the lines are indicated.

Figure 5

(a) Measured qubit level separation ${\tilde{\nu }}_{\mathrm{a}}$ and fit (solid line) vs input microwave probe power $P_{\mathrm{r}\mathrm{f}}$. The ac Stark shift ${\nu }_{\mathrm{a}\mathrm{c}}$ and the intraresonator photon number $n$ extracted from the fit are also indicated. (b) Measurement broadened qubit linewidth $\delta {\nu }_{\mathrm{H}\mathrm{W}\mathrm{H}\mathrm{M}}$ vs $n$. Error bars are reflecting estimated systematic uncertainties in the extracted linewidth. The corresponding total dephasing time $T_{\phi }=1/2\pi \delta {\nu }_{\mathrm{H}\mathrm{W}\mathrm{H}\mathrm{M}}$ is also indicated. The solid line is obtained from Eq. (4) with a spectroscopy power broadened $T_2^{\prime }\approx 80\mathrm{n}\mathrm{s}$.