Direct Observation of Josephson Capacitance

The effective capacitance has been measured in the split Cooper-pair box (CPB) over its phase-gate bias plane. Our low-frequency reactive measurement scheme allows us to probe purely the capacitive susceptibility due to the CPB band structure. The data are quantitatively explained using parameters determined independently by spectroscopic means. In addition, we show in practice that the method offers an efficient way to do nondemolition readout of the CPB quantum state.

Figure 1

Schematic view of the experiment. The resonant frequency of the $LC$ circuit (made using lumped elements) is tuned by the effective capacitance $C_{\mathrm{eff}}$ of the Cooper-pair box shown in the SEM image. For details, see text.

Figure 2

(a) Phase shift $\Theta$ measured at a probing frequency $803\mathrm{MHz}\sim f_0$, and (b) $\Theta$ calculated using Eq. (1) with the ground band $E_0(\phi ,n_g)$ evaluated numerically with parameters of Table .

Figure 3

(a) Illustration of the microwave spectroscopy used to map the SCPT band gap $\Delta E=E_1-E_0$. The three horizontal planes which intersect the band gap correspond, from bottom to top, to the microwave energy $\hslash {\omega }_{\mathrm{RF}}$ used in (c)–(e), respectively. Whenever $\Delta E=\hslash {\omega }_{\mathrm{RF}}$ (dashed lines), the system experiences resonant absorption; (b) peaks of resonant absorption (arrows) in the measured phase shift at ${\omega }_{\mathrm{RF}}=11\mathrm{GHz}$; (c)–(e) spectroscopy data represented as surfaces in the $\phi$, $n_g$ plane. The resonance conditions shown in (a) are plotted on top of the data; (f) $T_1$ as a function of measurement strength at $\phi =0$, $n_g\sim 1$.

Figure 4

Measured microwave phase shift $\Theta$, (a) vs phase $\phi$ (note different scales in the two panels) and (b) vs gate charge $n_g$ (curves for $\phi =0$ and $\pi /2$ have been shifted vertically for clarity by 20° and 10°, respectively). Solid lines are numerical calculations using Eq. (1) and sample parameters in Table .