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-shot noise 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.
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-shot noise 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.
Phase estimation plays a central role in communications, sensing, and information processing. Quantum correlated states, such as squeezed states, enable phase estimation beyond the shot-noise limit, and in principle approach the ultimate quantum limit in precision, when paired with optimal quantum measurements. However, physical realizations of optimal quantum measurements for optical phase estimation with quantum-correlated states are still unknown. Here we address this problem by introducing an adaptive Gaussian measurement strategy for optical phase estimation with squeezed vacuum states that, by construction, approaches the quantum limit in precision. This strategy builds from a comprehensive set of locally optimal POVMs through rotations and homodyne measurements and uses the Adaptive Quantum State Estimation framework for optimizing the adaptive measurement process, which, under certain regularity conditions, guarantees asymptotic optimality for this quantum parameter estimation problem. As a result, the adaptive phase estimation strategy based on locally-optimal homodyne measurements achieves the quantum limit within the phase interval of [ 0 , π / 2 ) . Furthermore, we generalize this strategy by including heterodyne measurements, enabling phase estimation across the full range of phases from [ 0 , π ) , where squeezed vacuum allows for unambiguous phase encoding. Remarkably, for this phase interval, which is the maximum range of phases that can be encoded in squeezed vacuum, this estimation strategy maintains an asymptotic quantum-optimal performance, representing a significant advancement in quantum metrology.
Squeezing currently represents the leading strategy for quantum enhanced precision measurements of a single parameter in a variety of continuous- and discrete-variable settings and technological applications. However, many important physical problems including imaging and field sensing require the simultaneous measurement of multiple unknown parameters. The development of multiparameter quantum metrology is yet hindered by the intrinsic difficulty in finding saturable sensitivity bounds and feasible estimation strategies. Here, we derive the general operational concept of multiparameter squeezing, identifying metrologically useful states and optimal estimation strategies. When applied to spin- or continuous-variable systems, our results generalize widely-used spin- or quadrature-squeezing parameters. Multiparameter squeezing provides a practical and versatile concept that paves the way to the development of quantum-enhanced estimation of multiple phases, gradients, and fields, and for the efficient characterization of multimode quantum states in atomic and optical sensor networks.
We propose a self-guided quantum phase estimation protocol based on a squeezed vacuum state and a stochastic optimization algorithm. Our protocol dynamically aligns the measurement basis with the target phase, eliminating the need for prior knowledge of the squeezing parameter. We demonstrate that this approach successfully tracks the target phase across the entire [0, π) periodicity interval, achieving a measurement precision that surpasses the Standard Quantum Limit and approaches the Heisenberg Limit.
Summary form only given. The use of quantum states of light in optical interferometry improves the precision in the estimation of a phase shift, paving the way for applications in quantum metrology, computation and cryptography. Sub-shot noise phase sensing can for example be achieved by injecting a squeezed vacuum into an interferometer . However, this approach leads to enhanced sensitivity only for small phase shifts. In this work we aim for ab initio sub-shot noise estimation of an unknown phase shift using a pre-determined squeezed probe and an adaptive measurement approach. We experimentally investigate the performances of such protocol under the realistic assumption of thermalization of the probe state. Indeed, adaptive phase estimation schemes with squeezed states and Bayesian processing of homodyne data have been shown to be asymptotically optimal in the pure case, thus approaching the quantum Cramér-Rao bound. In our protocol we take advantage of the enhanced sensitivity of homodyne detection in proximity of the optimal phase which maximizes the homodyne Fisher information. A squeezed thermal probe state (signal) undergoes an unknown phase shift. The first estimation step involves interference on a beam splitter of the signal and a local oscillator followed by homodyne detection. Homodyne data is then processed to compute a rough estimation of the phase through Bayesian inference. The rough estimation is fed back to the local oscillator in order to match the optimal relative phase with the signal. A second estimation step leads to the final estimation of the phase shift. Thermalization of the probe state prevents the attainability of the quantum Cramér-Rao bound. Nevertheless, we show that the studied adaptive scheme still saturates the classical Cramér-Rao bound, showing sub-shot noise behaviour and therefore extracting the maximum available information from homodyne data. In contrast to previous approaches, our scheme is optimized for Gaussian states.
Relevant metrological scenarios involve the simultaneous estimation of multiple parameters. The fundamental ingredient to achieve quantum-enhanced performance is based on the use of appropriately tailored quantum probes. However, reaching the ultimate resolution allowed by physical laws requires nontrivial estimation strategies from both a theoretical and a practical point of view. A crucial tool for this purpose is the application of adaptive learning techniques. Indeed, adaptive strategies provide a flexible approach to obtain optimal parameter-independent performance and optimize convergence to the fundamental bounds with a limited amount of resources. Here, we combine on the same platform quantum-enhanced multiparameter estimation attaining the corresponding quantum limit and adaptive techniques. We demonstrate the simultaneous estimation of three optical phases in a programmable integrated photonic circuit, in the limited-resource regime. The obtained results show the possibility of successfully combining different fundamental methodologies towards transition to quantum sensors applications.
Multiparameter quantum estimation in open optomechanical systems is fundamentally constrained by dissipation, thermal fluctuations, and measurement incompatibility. In this work, we investigate a coupled cavity optomechanical platform in which two mechanical modes interact with driven optical cavities, two mode squeezed vacuum, intracavity degenerate parametric amplification, and coherent optical feedback. Using the continuous variable Gaussian state formalism, we derive the linearized quantum Langevin dynamics and steady state covariance matrix and evaluate the quantum Fisher information matrices associated with simultaneous estimation of the optomechanical coupling strength and cavity dissipation rate. We characterize the precision bounds using the symmetric and right logarithmic derivative formalisms and employ BMI=max{BS,BR} as a comparative figure of merit within the SLD/RLD framework. We find that parametric amplification can substantially reduce the most informative bound, BMI, thereby enhancing multiparameter estimation precision over a broad range of operating conditions. Coherent feedback can further reduce BMI and improve the estimation precision, particularly in the strong feedback regime. Moreover, the simultaneous presence of coherent feedback, parametric amplification, and squeezed light injection provides a combined mechanism for controlling quantum fluctuations and parameter dependent correlations, leading to enhanced and more robust multiparameter estimation. We further analyze the dynamical and steady state regimes to identify the parameter regions in which these quantum resources provide the largest metrological gain. Our results establish coupled cavity optomechanics as a flexible platform for quantum-noise engineering in multiparameter sensing.