Reading out the state inductively and microwave spectroscopy of an interferometer-type charge qubit

We implemented experimentally an interferometer-type charge qubit consisting of a single-Cooper-pair transistor closed by a superconducting loop that is in flip-chip configuration inductively coupled to a radio-frequency tank circuit. The tank permits us to readout the qubit state, acting as an inductance measuring apparatus. By applying continuous microwave power to the quantum device, we observed inductance alterations caused by redistributions of the energy-level populations. From the measured data we extracted the energy gap between ground and upper levels as a function of the transistor quasicharge as well as the Josephson phase across both junctions.

Figure 1

Diagram of the measuring setup, including the micrograph of the Al-based single-Cooper-pair transistor (closed by the superconducting loop inductance $L_q$). On the rhs the tank circuit and some further electronic components are shown (cf. text). The phase $\delta$ is guaranteed to be a good quantum variable for sufficiently large ground capacitance, $C_0⪢C_{\Sigma }$. The angular phase shift $\alpha$ [Eq. (6)] between the tank voltage detected by the amplifier and current signal $I_{\mathrm{rf}}$ is determined by using a lock-in amplifier (not depicted here).

Figure 2

Tank phase shift $\alpha$ vs gate parameter $n_g$ without microwave power (lowest curve) and with microwave power at different excitation frequencies. The data correspond to ${\omega }_{\mathrm{uhf}}∕2\pi =8.9$, 7.5, 6.0 GHz (from top to bottom). The magnetic flux ${\Phi }_e={\Phi }_0∕2$ threading the interferometer loop provides a total phase difference $\delta =\pi$ across the single-Cooper-pair transistor. (For clarity, the upper curves are shifted.)

Figure 3

Energy gap $\Delta E$ between the ground and upper bands of the transistor determined from the experimental data for the case $\delta =\pi$. (Some examples of these data are shown in Fig. 2.) Dots represent the experimental data; the solid line corresponds to the fit within the two-band model (cf. text).

Figure 4

Dependence of the phase shift $\alpha$ on the two parameters $n_g$ and ${\Phi }_e$. The qubit is illuminated by a microwave with the frequency of 8.0 GHz. The periodic circular structure demonstrates the variation of the total interferometer-tank impedance due to transitions from the lower to the upper energy band. Especially, the “crater ridges” correspond to all combinations of the parameters $n_g$ and ${\Phi }_e$ that give the same energy gap between the respective states equal to 8.0 GHz.

Figure 5

Top view of the $\alpha (n_g,{\Phi }_e)$ mountain range of Fig. 4. Here the white ring approximates the ridge of the central crater leading to the fit $E_C=2.2\mathrm{GHz},E_{J1}=70.0\mathrm{GHz}$, and $E_{J2}=74.4\mathrm{GHz}$ within the two-level model.