Sequential Dispersive Measurement of a Superconducting Qubit

We present a superconducting device that realizes the sequential measurement of a transmon qubit. The device disables common limitations of dispersive readout such as Purcell effect or transients in the cavity mode by turning on and off the coupling to the measurement channel on demand. The qubit measurement begins by loading a readout resonator that is coupled to the qubit. After an optimal interaction time with negligible loss, a microwave pump releases the content of the readout mode by upconversion into a measurement line in a characteristic time as low as 10 ns, which is 400 times shorter than the lifetime of the readout resonator. A direct measurement of the released field quadratures demonstrates a readout fidelity of 97.5% in a total measurement time of 220 ns. The Wigner tomography of the readout mode allows us to characterize the non-Gaussian nature of the readout mode and its dynamics.

Figure 1

(a) The readout resonator (dark gray) is first prepared in a coherent state to probe a transmon qubit. Owing to the dispersive interaction between them, the resonator and the qubit evolve unitarily during a time $t_{\mathrm{int}}$. The readout mode is then upconverted and released into an output line (via purple valve). The complex amplitude $\beta$ of the released mode is encoded in the recorded output voltage $V(t)$. Both the Wigner function $W(\alpha )$ of the readout mode and the probability distribution $P(\beta )$ can be measured. (b) Scheme of the device. The readout mode is a $\lambda /2$ coplanar waveguide resonator (dark gray) that is capacitively coupled to a transmon qubit (green). The readout resonator is coupled to another $\lambda /2$ resonator (buffer in orange) by a Josephson ring modulator (JRM in purple). The readout resonator is driven by $r_{\mathrm{in}}$, the transmon (green) by $q_{\mathrm{in}}$, the buffer resonator (orange) outputs in $b_{\mathrm{out}}$, and the JRM is pumped at amplitude $p_{\mathrm{in}}$ by the buffer input. (c) Pulse sequence of the qubit measurement (top) and the released field measurement scheme (bottom).

Figure 2

Top row: measured Wigner function $W(\alpha )$ of the readout mode after various interaction times $t_{\mathrm{int}}$ for a qubit initialized in $|e⟩$ or $|g⟩$ and initial coherent state amplitude ${\alpha }_0=5.8$. A global rotation of the quadrature phase space was done numerically [24]. Bottom row: corresponding histograms of the measured complex amplitude $\beta$ of the output field (see text for definition) after it has been released into the transmission line following the interaction time $t_{\mathrm{int}}$ for ${10}^5$ realizations of the experiment.

Figure 3

(a) Raw traces of the output voltage $V(t)$ for two realizations of the experiment with the qubit prepared either in $|g⟩$ or in $|e⟩$ and with an initial displacement ${\alpha }_0=5.8$ and interaction time $t_{\mathrm{int}}=100\mathrm{ns}$. (b) Average traces over ${10}^5$ realizations. (c) Real and imaginary part of the weight function $w(t)$ from which the complex amplitude of the released mode is obtained.

Figure 4

(a) Scalar product of the measured distributions $P_{g,e}(\beta )$ of released mode amplitudes when the qubit is prepared in $|g⟩$ or $|e⟩$ as a function of interaction time $t_{\mathrm{int}}$, and for various values of initial displacement ${\alpha }_0$. (b) Same scalar product as a function of ${\alpha }_0$ and $t_{\mathrm{int}}$. Colored dashed lines indicate cuts corresponding to (a).