Quantum nondemolition readout using a Josephson bifurcation amplifier

We report an experiment on the determination of the quantum nondemolition (QND) nature of a readout scheme of a quantum electrical circuit. The circuit is a superconducting quantum bit measured by microwave reflectometry using a Josephson bifurcation amplifier. We perform a series of two subsequent measurements, record their values and correlation, and quantify the QND character of this readout.

Figure 1

Quantronium circuit with preparation and readout ports. The qubit consists of two Josephson junctions delimiting an island (black node) and inserted in a superconducting loop. Its eigenstates are tuned using the dc gate voltage $V_{g0}$ and the magnetic flux $\Phi$ through the loop. Resonant microwave pulses $V_g\left(t\right)$ are applied to the gate to manipulate the qubit state. A larger junction and a shunt capacitor $C$ forming an anharmonic oscillator are inserted in the loop for readout. A microwave readout pulse is sent to the system by a microwave generator with internal impedance $R=50\Omega$. The state-dependent inductive behavior of the qubit affects the plasma resonance of the oscillator and modifies the phase $\varphi$ of the microwave readout pulse reflected by the system. In the case of large driving amplitudes, the dynamics of the superconducting phase $\gamma$ across the readout junction can bifurcate between two distinct dynamical states, leading to a jump of $\varphi$.

Figure 2

Experimental setup of the JBA readout. The probing pulses come from the continuous microwave source mixed with a dc pulse $V_M\left(t\right)$ (middle left inset), consisting of a first plateau aimed at inducing the bifurcation or not and of a latching period for measuring the phase $\phi$. The resulting microwave pulses propagate to the microfabricated circuit (bottom left SEM micrograph) along a filtered attenuated line and a directional coupler. The reflected pulse travels through the coupler and to the amplification stage via three cascaded circulators. Then it undergoes a homodyne demodulation; one of the quadratures is recorded with respect to time. The top right inset shows in gray levels thousands of superposed records, with one of them emphasized (shaded line). The observed quadrature either follows the envelope of the readout pulse when no bifurcation occurs (bottom traces, readout outcome $r=l$) or switches upwards in the opposite case (readout outcome $r=h$), corresponding to a phase jump. A threshold (dashed line) is used to count the switching events and deduce a switching probability.

Figure 3

Measurement of the QND fractions of the quantronium-JBA system. Panel (a): the qubit is prepared in state 1 (0) by applying a gate $\pi$ pulse (no $\pi$ pulse). Then two adjacent readout pulses $A$ and $B$ are applied. The two successive output quadrature voltages are averaged during the last $100\mathrm{ns}$ of the latching period of the pulses. Panel (b): bivalued histograms of the quadrature voltages (open symbols, no $\pi$ pulse; solid symbols, $\pi$ pulse). The top and bottom peaks correspond to bifurcation (readout $r=h$) and no bifurcation (readout $r=l$), respectively. A threshold (dashed horizontal line) leads to the determination of the bifurcation probabilities. Panel (c): the eight probabilities of getting two successive responses $(r_A,r_B)$.

Figure 4

Readout outputs and qubit state evolution in a measurement by a single readout pulse. The kets $\mid \psi =0,1⟩$ indicate the qubit state before (left) and after (right) the measurement. The arrows and their associated probabilities correspond to the different possible qubit evolutions. The QND fractions are indicated in bold. The different readout responses for each different scenario are indicated on the right with their probabilities.