The Pound-Drever-Hall Method for Superconducting-Qubit Readout

Abstract

Scaling quantum computers to large sizes requires the implementation of many parallel qubit readouts. Here we present an ultrastable superconducting-qubit readout method using the multi-tone self-phase-referenced Pound-Drever-Hall (PDH) technique, originally developed for use with optical cavities. In this work, we benchmark PDH readout of a single transmon qubit, using room-temperature heterodyne detection of all tones to reconstruct the PDH signal. We demonstrate that PDH qubit readout is insensitive to microwave phase drift, displaying $0.73^\circ$ phase stability over 2 hours, and capable of single-shot readout in the presence of phase errors exceeding the phase shift induced by the qubit state. We show that the PDH sideband tones do not cause unwanted measurement-induced state transitions for a transmon qubit, leading to a potential signal enhancement of at least $14$~dB over traditional heterodyne readout.

Figures (4)

• Figure 1: Schematic of drift sensitivities for conventional and PDH readout of transmon qubits. Schematic IQ-plane representation of measurement outcomes corresponding to the ground (blue) and the excited states (orange) of a transmon. (a) For heterodyne readout, the measurement signals are sensitive to both the power (red arrow) and the phase (blue arrow) of the measurement tone. (b) In self phase-referenced PDH readout, the signal is independent of phase drift.
• Figure 2: Long-term phase stability of PDH readout. Average carrier and sideband phases over $1000$ consecutive single shots are monitored over a period of $2$ hours for sideband detunings from $5-30$ MHz. (a) Raw carrier phase $\phi_{0}$, showing roughly $400^\circ$ drift due to a frequency offset between the carrier and the LO. (b) Non-linear residual fluctuations $\tilde{\phi}_0$ after subtracting the linear drift observed in (a). (c) Fluctuations of the PDH differential phase $\phi_{0}$ - $\phi_{-}$ versus sideband detuning. This phase difference displays an RMS phase noise of $0.73^\circ$ due to effective cancellation of common-mode phase noise and drift. Mean phase removed from all data.
• Figure 3: Single-shot phase stability of PDH readout. (a) PDH readout with free running generators for the qubit prepared either in $\vert|g\rangle$ or $\vert|e\rangle$. (a-i) Heterodyne IQ distribution for the carrier. Loss of phase coherence collapses the two states into overlapping IQ distributions. The color map indicates $\vartheta_e$, the fraction of events corresponding to preparation in $\vert|e\rangle$. Transparency is reduced for pixels with less than 10 counts, where $\vartheta_e$ is noisy. (a-ii) PDH readout with the same data, using the optimal combination of real and imaginary parts ($\epsilon_I$ and $\epsilon_Q$), preserves clear state separation, demonstrating robustness against carrier phase errors. Imperfect state preparation results in a small fraction of mislabeled events and overlapping pedestals in the histograms for $\vert|g\rangle$ and $\vert|e\rangle$. See Appendix S3 in the Supplementary Information for details. (b-i) PDH readout with both carrier phase errors and RF timing errors. The latter rotate the real and imaginary parts of the PDH signal, compromising readout fidelity. (b-ii) Scissors-phase readout, defined as $\Sigma = 2\phi_{0} - (\phi_{-}+\phi_{+})$, remains robust against the stated phase and timing errors maintaining state discrimination.
• Figure 4: MIST in PDH readout. MIST due to additional tones during readout is diagnosed using two sequential measurements with a probe tone applied in between (a-inset). Outcomes for each measurement are classified as ground $\vert|g\rangle$, excited $\vert|e\rangle$, or other $\vert|o\rangle$. (a) Conditional probabilities versus probe power and detuning. Rapid increases of $p(o|g)$ and $p(o|e)$ indicate above a critical power MIST and breakdown of QND readout. (b) IQ histograms for sample $2^{\mathrm{nd}}$ measurements from the $0$ MHz detuning data in (a). Decision boundaries for $\vert|g\rangle$ (blue), $\vert|e\rangle$ (orange), and $\vert|o\rangle$ (red) are shown in gold. (c) Semi-log plot of the transition power at which probe-induced MIST becomes dominant. At sufficiently large detunings, no observable probe-induced MIST transition occurs ($\rm{X}$ markers), even with a sideband at $28$ dBc, corresponding to more than $14$ dB of heterodyne gain during PDH error-signal detection.