Multiparameter quantum-enhanced adaptive metrology with squeezed light

Abstract

Squeezed light enables quantum-enhanced phase estimation, with crucial applications in both fundamental physics and emerging technologies. To fully exploit the advantage provided by this approach, estimation protocols must remain optimal across the entire parameter range and resilient to instabilities in the probe state. In this context, strategies that rely on pre-calibrated squeezing levels are vulnerable to degradation over time and become sub-optimal when experimental conditions fluctuate. Here, we develop an adaptive multiparameter estimation strategy for ab-initio phase estimation, achieving sub-standard quantum limit precision in the full periodicity interval $[0,π)$, without relying on prior knowledge of the squeezing parameter. Our approach employs real-time feedback to jointly estimate both the optical phase and the squeezing level, ensuring robustness against experimental drifts and calibration errors. This self-calibrating scheme establishes a reliable quantum-enhanced sensing framework, opening new routes for practical scenarios and scalable distributed sensor networks using squeezed light.

Figures (4)

• Figure 1: Sketch of the experimental setup and of the implemented protocol.(a) Squeezed-vacuum probes are generated by pumping an optical parametric amplifier (OPA) with a single-mode continuous-wave laser at $532$ nm. To achieve high levels of squeezing, the pump and LO beams are both spatially filtered in identical mode-cleaner cavities to maximize the overlap between the squeezed field and the LO at the homodyne BS. The relative phase between the squeezed state and the LO is stabilized with a phase-locked loop (PLL). The LO phase $\theta$ is finely tuned by means of a piezoelectric stage, while an FPGA-based feedback system allows dynamic selection and locking of the measurement quadrature. The phase $\phi$ can be changed with a liquid crystal device in the path of the squeezed beam. (b) At each step of the protocol, the homodyne data collected are used to reconstruct the multiparameter posterior distribution. From this posterior, the estimate is updated and the optimal measurement angle $\theta$ is computed. A feedback $\Delta$, derived from the current estimates, is then applied to adjust the LO phase for the next measurement. Repeating this cycle progressively refines the posterior until, at the end of the protocol, the estimates converge to the true values of phase $\phi_t$ and of the squeezing parameter $r_t$.
• Figure 2: Single- and multiparameter variances of the phase estimation. We report the variances of the adaptive phase estimation (radial axis) as a function of the phase $\phi$ (angular axis), spanning the entire range $[0, \pi)$. The experimental results $\mathrm{Var}[\phi]$ obtained in the single- [panel (a)] and multiparameter [panel (b)] adaptive protocols are reported as green and blue dots, respectively. The corresponding error bars represent the standard deviations over 5 (10) repetitions of the experiment, in the single-parameter (multiparameter) case. In both panels, we also report the corresponding classical bounds i.e., the QCRB for a coherent probe with the same average number of photons, as a blue line for the former and a green line for the latter case, while the QCRB is represented by a red line in both cases. For the multiparameter protocol we also report the actual CRB (orange dashed line). The experimental single-parameter protocol employs a total of $M=5000$ homodyne measurements, among which $M_{\mathrm{R}}=200$ are used to obtain the rough estimate. Instead, in the multiparameter estimation, the total number of measurements amounts to $M=20000$, while the rough estimation needs $M_{\mathrm{R}} = 1200$ of them.
• Figure 3: Adaptive phase estimation single- and multi-parameter protocols. In panel (a), we report the evolution of the posterior distribution $p(\phi|X)$ as a function of the number of homodyne data measured along the adaptive single-parameter protocol in the range $M\in[100,25000]$. During the rough estimation, highlighted by a yellow background, we observe that measuring different quadrature angles ($\theta=\{0,\pi/4\}$ and $\theta=\{0,\pi/4,\pi/2\}$ in the single- and multiparameter estimations, respectively) allows us to disambiguate the estimate between the ranges $\phi \in [0,\pi/2]$ and $\phi \in [\pi/2, \pi)$. At the end of this stage, the current estimation $\hat{\phi}$ is employed to shift the LO phase to its optimal value $\theta = \hat{\phi} - \phi_{\mathrm{opt}}$. The remaining measurements (purple background) are taken within the optimal homodyne configuration, which is further updated two times before reaching the final estimation, extrapolated from the posterior distribution reported in panel (b). This inset shows how the precision of the phase estimation (blue shaded area), tightly approaches the squeezed QCRB (red line), while surpassing the classical bound (black dotted line). The changes of the LO angle, along the adaptive protocol, are illustrated in panel (c). It shows that the quantity $\theta - \phi$ is close to its optimal value, depending on the chosen protocol. In the single parameter estimation (in blue), it concides with the value corresponding to the maximum of the FI [in panel (d)], while in the multiparameter approach (green), we aim at canceling the off-diagonal element of the FI matrix $F_{\phi r}$ [shown in panel (d)]. In panel (e), the normalized variance $\widetilde{\mathrm{Var}}[\phi] \equiv \mathrm{Var}[\phi]\cdot F_{Q}^{\mathrm{sq,eff}}$ is reported as a function of $M$, thus having a common ultimate precision bound (solid red line) for both the single- and multiparameter variances, illustrated as blue and green lines, respectively. This panel also shows how the disambiguation and the LO feedback positively affect the precision of both the estimation protocols, although the multiparameter one exhibits a slower convergence to values below the classical bound (black dotted line). Panels (f) and (g) provide zoomed-in views of the marked area, demonstrating convergence to the bound for the single-parameter strategy and for the multi-parameter strategy, respectively. The latter converges to the CRB computed for the two-parameter estimate that does not coincide with the QCRB.
• Figure 4: Robustness of the multiparameter adaptive estimation protocol to different squeezing strength and fully black-box approach. In (a) the green squares represents the estimated squeezing level $\hat{r}$, within the multiparameter adaptive protocol, compared to the actual values $r$. As an example, we illustrate the experimentally reconstructed posterior distributions $p(\phi, r|X)$, corresponding to the lowest (top-left inset) and the highest (bottom right inset) squeezing levels. (b) Scaling of the experimental phase variances compared to the relative bounds: the QCRB for squeezed vacuum probes (red), the CRB (orange), and the QCRB for coherent states (green). The data lie all below the coherent bounds, computed without considering losses. Panel (c), reports the results obtained with a fully black-box approach, in which the efficiency $\eta$ is also estimated. The normalized variances $\widetilde{\mathrm{Var}}[\phi]$ (purple dots) are compared with the classical lossless (dash-dotted purple line) bound. In all the panels, the reported errors correspond to standard deviations over 10 repetitions of the experiments.