Measuring the decoherence of a quantronium qubit with the cavity bifurcation amplifier

Dispersive readouts for superconducting qubits have the advantage of speed and minimal invasiveness. We have developed such an amplifier, the cavity bifurcation amplifier (CBA), and applied it to the readout of the quantronium qubit. It consists of a Josephson junction embedded in a microwave on-chip resonator. In contrast with the Josephson bifurcation amplifier, which has an on-chip capacitor shunting a junction, the resonator is based on a simple coplanar waveguide imposing a predetermined frequency and whose other rf characteristics such as the quality factor are easily controlled and optimized. Under proper microwave irradiation conditions, the CBA has two metastable states. Which state is adopted by the CBA depends on the state of a quantronium qubit coupled to the CBA’s junction. Due to the megahertz repetition rate and large signal to noise ratio, we can show directly that the coherence is limited by $1∕f$ gate charge noise when biased at the “sweet spot”—a point insensitive to first order gate charge fluctuations. This architecture lends itself to scalable quantum computing using a multiresonator chip with multiplexed readouts.

Figure 1

(Color online) (a) Scanning electron micrograph of a duplicate of the measured device, both of which are fabricated on the same chip simultaneously. (b) Schematic of our sample and its environment. The sample consists of a split Cooper pair box whose loop is interrupted with a large junction (quantronium). The state of the quantronium alters the switching probability $P_{01}$ of the CBA between its two metastable states, which can be inferred from the amplitude and phase change of transmitted microwave pulses through the device.

Figure 2

(Color online) Gate charge and flux modulations of our device. We operated in the weakly nonlinear mode $(P⪡P_b^{\mid 1⟩})$ and monitored the phase of the transmitted signal (grayscale) as we varied the applied gate charge $N_g=C_g{V}_g∕2e$ and flux $\Phi$ $({\Phi }_0=\hslash ∕2e)$. The large ellipsoidal contours can be interpreted as induced transitions between the energy levels of the qubit at multiples of the readout frequency. The green fitted lines are transitions between the 0 and 1 energy levels at the readout frequency of $9.64\mathrm{GHz}$, while the orange fits are for transitions between the 0 and 2 energy levels at twice the readout frequency, $19.28\mathrm{GHz}$.

Figure 3

(Color online) Spectroscopy peak as a function of gate charge $N_g$. Staying at zero flux, we measure $P_{01}$ while sweeping ${\nu }_s$ and stepping $N_g$. The theoretical fit of the resulting sinusoidal-like dependence of the peak with $N_g$ is given by the red dashed line with the fit parameters $E_J=15.02\mathrm{GHz}$ and $E_{CP}=17.00\mathrm{GHz}$. The vertical lines with no $N_g$ dependence are the excitations between qubit energy levels enduced at multiples of the readout frequency, similar to those seen in Fig. 2.

Figure 4

(Color online) (a) Rabi oscillations in the switching probability $P_{01}$, as a function of gate charge modulation $\Delta N_g$ and gate pulse time length $\tau$. $\Delta N_g$ is calculated from the Rabi pulse envelope voltage A reaching the sample through the attentuation in the input lines and is plotted in terms of Cooper pairs, $\Delta N_g=C_{g}A∕2e$. Oscillations in the switching probability $P_{01}$ are seen with both $\Delta N_g$ and $\tau$. (b) Fitted Rabi frequency ${\nu }_{\mathit{\text{Rabi}}}$ vs $\Delta N_g$. As expected from a two-level system, ${\nu }_{\mathit{\text{Rabi}}}$ scales linearly with $\Delta N_g$.

Figure 5

(Color online) (a) 3000 Ramsey oscillations as a function of free evolution time $\Delta t$. Each trace is $2.1\mu \mathrm{s}$ long with $3\mathrm{ns}$ per step. They each take $0.35\mathrm{s}$ to acquire. We can see visually the variation of $T_2$ for the different Ramsey fringes by noticing the variation in contrast in the fringes near $1\mu \mathrm{s}$. (b) Sample data fit. We average five of the acquired Ramsey traces shown in (a) and fit to a decaying sinusoid to extract the $T_2$ which is then plotted in (c). For this particular case, we have a coherence time of $840\mathrm{ns}$ and a Ramsey frequency of $26.9\mathrm{MHz}$. (c) Distribution of $T_2$ for 3000 of the Ramsey traces (600 fits). The black dashed line is the result of a simulation of the free evolution decay of the Ramsey fringes with $1∕f$ noise fluctuations on the gate, $S_q\left(\omega \right)={\alpha }^2∕\mid \omega \mid$. In the simulation, we used ten times more points compared to the data to obtain a smoother curve. (d) Corresponding distribution of Ramsey frequencies at four different flux biasing points. Each distribution has 3000 Ramsey traces. The blue histogram corresponds to the data in (a), (b), and (c). The Ramsey frequency is extracted from the position of the maximum of the power spectral density of each decaying sinusoid. The distributions are lopsided to higher frequencies as would be expected from fluctuations in the gate charge around our operating point at the sweet spot. The dashed line is the expected distribution assuming the same $1∕f$ charge noise as in (c).