Rabi Oscillations in a Large Josephson-Junction Qubit

We have designed and operated a circuit based on a large-area current-biased Josephson junction whose two lowest energy quantum levels are used to implement a solid-state qubit. The circuit allows measurement of the qubit states with a fidelity of 85% while providing sufficient decoupling from external sources of relaxation and decoherence to allow coherent manipulation of the qubit state, as demonstrated by the observation of Rabi oscillations. This qubit circuit is the basis of a scalable quantum computer.

Figure 1

(a) Model for a current-biased Josephson junction. (b) Cubic potential $U$ showing qubit states and measurement scheme. (c) Circuit diagram for a qubit with high-impedance current bias. Dashed box indicates components fabricated on a superconducting integrated circuit. Microwaves are injected via $C_{\mu \mathrm{w}}$. Component values are $I_0\simeq 21\mu \mathrm{A}$, $C\simeq 6\mathrm{p}\mathrm{F}$, $i_0\simeq 10.5\mu \mathrm{A}$, $C_{\mu \mathrm{w}}\simeq 2\mathrm{f}\mathrm{F}$, $R_{\mu \mathrm{w}}\simeq 50\Omega$, $L\simeq 3.3\mathrm{n}\mathrm{H}$, $M\simeq L/160$, $L_{\varphi }\simeq 0.4\mathrm{n}\mathrm{H}$, $R_{\varphi }\simeq 2\Omega$, and $R_V\simeq 50\Omega$.

Figure 2

Measurement of circuit parameters obtained from a plot of the $I_V$ and $I_{\varphi }$ values at which the circuit switches to the voltage state. Solid arrows indicate trajectory of the swept triangular bias for measurement mode. Dashed arrows indicate optimal biasing for normal operation.

Figure 3

Measured escape rate $\Gamma$ vs junction bias current $I$ with 6.80 GHz microwave irradiation. Lower inset shows detail of resonance for an additional frequency 6.78 GHz. Widths of resonances correspond to 6.1 MHz, giving $Q\simeq 1000$. Upper inset illustrates the procedure for this spectroscopy measurement.

Figure 4

Occupation probability $p_1$ vs microwave power $P_{10}$ at frequency ${\omega }_{10}/2\pi =6.9\mathrm{G}\mathrm{H}\mathrm{z}$, obtained for a measurement pulse of frequency ${\omega }_{21}/2\pi =6.28\mathrm{G}\mathrm{H}\mathrm{z}$, power $P_{21}=5$, and duration 25 ns. Power $P_{10},P_{21}=1(\mathrm{a}\mathrm{.}\mathrm{u}\mathrm{.})$ corresponds to $\sim 1\mathrm{p}\mathrm{W}$ at $R_{\mu \mathrm{w}}$. Theory plotted as a solid curve. Data at low power indicate fidelity of $|0⟩$ state preparation and measurement better than 99%; saturation near 50% gives fidelity of 85% for measurement of the $|1⟩$ state.

Figure 5

Occupation probability $p_1$ of state $|1⟩$ vs time delay $t_{\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{y}}$ between excitation and measurement pulses. Two values of initial-state population are obtained by varying the power $P_{10}$ of the excitation pulse.

Figure 6

Observation of Rabi oscillations from an excitation pulse of 25 ns and frequency ${\omega }_{10}$ followed by a measurement pulse at frequency ${\omega }_{21}$. Plot shows periodic modulation of $p_1$ vs microwave amplitude, as expected from theory. The amplitude of the oscillations are maximized by using an overlap of 7 ns between the pulses.