Fig. 3: Enhanced sensing with dual-quadrature measurement.

a, For a given phase angle θ, population measurement in only a single basis, for example Y, can only be inverted within a dynamic range of −π/2 < θ < π/2. By measuring both quadratures X and Y, this dynamic range can be doubled to −π < θ < π, allowing for interrogating larger spreads in phase, such as when measuring for longer times. b, We implement the dual-quadrature readout of Ramsey interrogation by applying local π/2-phase shifts to all the odd sites in the array. c, With single-quadrature readout, the interrogation time is limited due to phase slips, visible by the separation between a decay envelope reconstructed from the single-quadrature phase spread (orange dashed line) and the averaged Ramsey signal (blue and red markers and lines). The equivalent reconstruction with dual-quadrature readout (green dashed line) is accurate up to longer times. d, To perform this reconstruction, we measure the time-resolved probability distributions of the estimated phase relative to the mean from dual-quadrature measurement. As the standard deviation (σ) of the phase distribution grows (inset), the estimated phase begins exceeding the −π/2 < θ < π/2 range for normal spectroscopy (black dashed lines), but is still resolvable via dual-quadrature measurement. Note that the time-dependent contribution from QPN to the standard deviation has been subtracted off in the inset (Methods). e, We estimate the phase-slip probability ϵ for single-quadrature (orange circles) and dual-quadrature (green circles) measurements by fitting a folded Gaussian to the time-resolved estimated phases in d. The fit is folded over at the boundaries of the dynamic range to account for the behaviour of phase slips, as that in a. For the single-quadrature case, we also directly estimate the probability from the underlying data (squares), which is in good agreement with the estimate from the fit. The solid lines are the predicted phase-slip probabilities from the fit in the inset of d. This fit is used to estimate the decay envelopes in c. f, For a given allowable phase-slip probability, the enhanced dynamic range of the dual-quadrature readout improves the maximum possible interrogation time. For our particular phase-growth profile (inset of d), the improvement is a factor of ∼3.43.