Circuit QED with a Nonlinear Resonator: ac-Stark Shift and Dephasing

We have performed spectroscopic measurements of a superconducting qubit dispersively coupled to a nonlinear resonator driven by a pump microwave field. Measurements of the qubit frequency shift provide a sensitive probe of the intracavity field, yielding a precise characterization of the resonator nonlinearity. The qubit linewidth has a complex dependence on the pump frequency and amplitude, which is correlated with the gain of the nonlinear resonator operated as a small-signal amplifier. The corresponding dephasing rate is found to be close to the quantum limit in the low-gain limit of the amplifier.

Figure 1

(a) Experimental setup: A transmon qubit is strongly coupled to a coplanar resonator made nonlinear with a Josephson junction. The sample is cooled to 20 mK and is driven through an attenuator $A$ by three microwave sources. Source $p$ is used to establish a pump field in the resonator, source $q$ for qubit spectroscopy, and source $r$ as a JBA readout: Its signal at ${\omega }_r$ is reflected from the resonator and routed through circulators to a cryogenic amplifier, a demodulator, and a digitizer, which yields the JBA switching probability $p_s$ and thus the probability of the qubit excited state $e$. (b) Frequency dependence of the maximum current $I$ in the resonator calculated for the sample parameters and for reduced powers $P_p/P_c=0.1$, 0.635, 1, 3.5, and 16.35 (bottom to top). (c) Stability diagram of the resonator in the $\Omega \mathrm{\text{−}}{P}_p$ plane. The solid lines indicate the highest parametric amplifier gain below ${\Omega }_c$ and the power at which the resonator bifurcates from the low- ($L$) to the high- ($H$) amplitude state, respectively. The hatched area is the bistability region. Vertical dotted lines correspond the two data sets of Figs. 2, 3.

Figure 2

2D plots of the switching probability $p_s({\omega }_q,P_p)$ for ${\Omega }_g/{\Omega }_c=3.1$ and 0.7 (left and right panels). A few qubit lines are shown in overlay for negligible field amplitude in the resonator (top left), near the switching point at $P_p=1.0\mathrm{dBm}$ for ${\Omega }_g/{\Omega }_c=3.1$, and at $P_p=-12.4$ and $-8.6\mathrm{dBm}$ for ${\Omega }_g/{\Omega }_c=0.7$. The horizontal dashed line indicates the qubit frequency at zero cavity field. Lorentzian fits of the qubit lines (see the example at $-8.6\mathrm{dBm}$) yield the ac-Stark shift $\Delta {\omega }_q$ and the FWHM linewidth $w$. Inset: Microwave pulse sequence used.

Figure 3

(a) Experimental (open symbols) average photon number $\overline{n}$ and corresponding fits (lines—see the text) as a function of $P_p$ for the same data sets ${\Omega }_g/{\Omega }_c=3.1$ (circles) and ${\Omega }_g/{\Omega }_c=0.7$ (triangles) as in Fig. 2. Experimental points are obtained by converting the measured $\Delta {\omega }_q$ according to the ac-Stark shift model of the inset (solid line—see the text; the dashed line shows the linear approximation). (b) Qubit dephasing rate ${\Gamma }_2$ measured for the same data sets (open symbols) and calculated either by numerical integration of the system master equation (solid symbols) or by using Eq. (2) (solid lines). The horizontal dashed-dotted line indicates the intrinsic dephasing at zero field. The measured parametric power gain (dashed line) is also shown for comparison for ${\Omega }_g/{\Omega }_c=0.7$. (c) Complex cavity field amplitude ${\alpha }_g$ (solid and dashed lines) calculated from Eq. (1) for increasing pump powers and for the same data sets. Both ${\alpha }_g$ (squares) and ${\alpha }_e$ (triangles) are also shown at six points labeled A–F. Their separation (segments) is to be compared to the uncertainty disk of a coherent state [open circles or disks shown at point C and at points A–F in (b)].