Figure 1

Circuit diagram and sample images. (a) Circuit diagram of the device. (b) Optical image of the device. The meandering trace in the center is a half-wavelength CPW resonator made of 50-nm-thick Nb. (c) Interdigitated capacitance between the microwave feedline and the resonator, which is designed to be 15 fF. The gap of the capacitance is 2 $\mu$m wide. (d) Magnified image of the qubit part. The qubit made of Al is located in the gap between the center conductor and the ground plane of the CPW resonator. One of the islands containing the smaller junction is made larger (a white rectangle in the center) so that it has large enough capacitance $C_{\mathrm{c}}$ to the center conductor of the resonator. (e) Scanning electron micrograph of the three-junction flux qubit.Reuse & Permissions

Figure 2

Spectroscopy of the coupled system. (a) Phase of the reflection coefficient $\Gamma$ as a function of the flux bias and the probe frequency. The vacuum Rabi splittings are clearly observed at $f=0.488$ and 0.512. (b) $\left|\Gamma \right|$ as a function of the flux bias and the qubit excitation frequency. $\left|\Gamma \right|$ in the panel is normalized by $\left|\Gamma \right|$ measured at ${\omega }_{\mathrm{r}}$. Annotations on the energy levels denote the corresponding excitation from the ground state; ${|i\rangle }_{\mathrm{q}}{|j\rangle }_{\mathrm{r}}$ indicates that the qubit and the resonator are in the $i$th and $j$th states, respectively. The dashed curves represent the fit by Eq. (1). The same data as in (a) is shown in the blue box for comparison. There is no data in the regions filled with the meshes.Reuse & Permissions

Figure 3

Dispersive shift and Rabi oscillations observed at $f=0.5$. (a) Amplitude of the normalized reflection coefficient ${\Gamma }^{\prime }$ as a function of ${\omega }_{\mathrm{p}}$. The blue curve is taken when the qubit is in the $|0\rangle$ state, and the red curve is when the qubit is in the $|1\rangle$ state which is prepared by a $\pi$ pulse 3 ns long. The shift of the dip frequency represents the dispersive shift. The residual dip at ${\omega }_{\mathrm{p}}/2\pi =10.656$ GHz for the qubit in the $|1\rangle$ state can be explained by the energy relaxation of the qubit. (b) High contrast Rabi oscillations. $|{\Gamma }^{\prime }|$ is plotted as a function of the control pulse length. The blue dots are the data and the red curve is a fit by a sinusoidal function with exponential decay. The inset depicts the applied pulse sequence. After the control pulse with a length of $\Delta t$ is applied, we wait for $t_{\mathrm{d}}$ before acquiring the readout signal. The length of the readout pulse was 400 ns, and the IF frequency was 50 MHz. The signal was sampled at 1 GS/s, and a 100-ns-long time trace was used to extract the amplitude and the phase.Reuse & Permissions

Figure 4

Flux bias dependence of $|g_{ij}|$, ${\chi }_{ij}$, and 2$|{\chi }_{\mathrm{eff}}|$. (a) Calculated matrix elements of the coupling Hamiltonian $H_{\mathrm{c}}$ between the two states $|i\rangle$ and $|j\rangle$, i.e., $|g_{ij}|=|\langle i|H_{\mathrm{c}}|j\rangle |$ ($i=0,1$, and $0\le j\le 3$). (b) Calculated ${\chi }_{ij}$ and ${\chi }_{\mathrm{eff}}$. The three largest components are plotted here. The ${\chi }_{\mathrm{eff}}$ includes the terms up to $j=4$ in Eq. (5). (c) 2$|{\chi }_{\mathrm{eff}}|$ obtained from the measurement (blue dots), estimated from the energy band calculation (dashed red line), and calculated from Eq. (5) (solid green line). Experimental ${\chi }_{\mathrm{eff}}$ was obtained by Lorentzian fit to the measured spectra, and has errors of a few MHz.Reuse & Permissions