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arXiv:2004.04152 · Physical Review Research

Entanglement-enhanced estimation of a parameter embedded in multiple phases

Michael R. Grace, Christos N. Gagatsos, Saikat Guha

Published Aug 5, 2021• 15 citations
arXivHTMLPDF

Abstract

Quantum-enhanced sensing promises to improve the performance of sensing tasks using nonclassical probes and measurements that require far fewer scene-modulated photons than the best classical schemes, thereby granting previously inaccessible information about a wide range of physical systems. We propose a generalized distributed sensing framework that uses an entangled quantum probe to estimate a scene-parameter encoded within an array of phases, with a functional dependence on that parameter determined by the physics of the actual system. The receiver uses a laser light source enhanced by quantum-entangled multipartite squeezed-vacuum light to probe the phases and thereby estimate the desired scene-parameter. The entanglement suppresses the collective quantum vacuum noise across the phase array. We report simple analytical expressions for the Cram\'er-Rao bound that depend only on the optical probes and the physical model of the measured system, and we show that our structured receiver asymptotically saturates the quantum Cram\'er-Rao bound in the lossless case. Our approach enables Heisenberg limited precision in estimating a scene-parameter with respect to total probe energy, as well as with respect to the number of modulated phases. Furthermore, we study the impact of uniform loss in our system and examine the behavior of both the quantum and the classical Cram\'er-Rao bounds. We apply our framework to examples as diverse as radio-frequency phased-array directional radar, beam-displacement tracking for atomic-force microscopy, and fiber-based temperature gradiometry.

DOI
10.1103/physrevresearch.3.033114

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Quantum-enhanced sensing promises to improve the performance of sensing tasks using nonclassical probes and measurements that require far fewer scene-modulated photons than the best classical schemes, thereby granting previously inaccessible information about a wide range of physical systems. We propose a generalized distributed sensing framework that uses an entangled quantum probe to estimate a scene-parameter encoded within an array of phases, with a functional dependence on that parameter determined by the physics of the actual system. The receiver uses a laser light source enhanced by quantum-entangled multipartite squeezed-vacuum light to probe the phases and thereby estimate the desired scene-parameter. The entanglement suppresses the collective quantum vacuum noise across the phase array. We report simple analytical expressions for the Cram'er-Rao bound that depend only on the optical probes and the physical model of the measured system, and we show that our structured receiver asymptotically saturates the quantum Cram'er-Rao bound in the lossless case. Our approach enables Heisenberg limited precision in estimating a scene-parameter with respect to total probe energy, as well as with respect to the number of modulated phases. Furthermore, we study the impact of uniform loss in our system and examine the behavior of both the quantum and the classical Cram'er-Rao bounds. We apply our framework to examples as diverse as radio-frequency phased-array directional radar, beam-displacement tracking for atomic-force microscopy, and fiber-based temperature gradiometry.
The quantum Fisher information and quantum correlation parameters are employed to study the application of nonclassical light to the problem of parameter estimation. It is shown that the optimal measurement sensitivity of a quantum state is determined by its intermode correlations (which depends on path entanglement) and intramode correlations (which depends on the photon statistics). In light of these results, we consider the performance of quantum-enhanced optical interferometers. Furthermore, we propose a Heisenberg-limited metrology protocol involving standard elements from passive and active linear optics, for which the quantum Cram\'er-Rao bound is saturated with an intensity measurement. Interestingly, the quantum advantage for this scheme is derived solely from the nonclassical photon statistics of the probe state and does not depend on entanglement. We study the performance of this scheme in the presence of realistic losses and consequently predict a substantial enhancement over the shot-noise limit with current technological capabilities.
Quantum-enhanced interferometers utilize non-classical states of light in order to surpass the limitations imposed by classical physics on phase measurement sensitivity and resolution. We review key scientific developments in this growing field of research and derive fundamental metrology concepts, namely the Cram\'{e}r-Rao bound and the Fisher information, which are used to analyze the metrological performance of quantum states of light. We propose a supersensitive quantum interferometry protocol that can be amplified to the macroscopic realm (i.e. the mean number of photons of the probe state can reach approximately 100,000 photons) using contemporary techniques of spontaneous parametric down conversion. The parity measurement on the output field asymptotically saturates the quantum Cram\'{e}r-Rao bound, which scales like the Heisenberg limit. We uncover the role of photon statistics on phase sensitivity and rule out the necessity of mode entanglement for quantum-enhanced interferometry. Based on these insights we outline a practical metrology technique (independent of mode entanglement) that measures a phase delay of a single-mode anti-squeezing operation. This scheme can also be scaled macroscopically using contemporary techniques and is shown to significantly surpass the shot-noise limit (SNL) in the presence of realistic losses. Finally, the physical resources that enable quantum-enhanced interferometry is studied by analyzing the quantum Fisher information (QFI) using the first quantization and the second quantization formalisms of quantum mechanics. It is shown that increasing the intra-mode correlations of an interferometer (as quantified by the second order Glauber coherence function) can be conducive to attaining a quantum advantage in phase estimation; whereas, introducing mode entanglement can reduce phase sensitivity. Using the first quantization description, we derive a formula for the QFI that shows explicitly how the Heisenberg scaling term depends on particle entanglement.
Distributed quantum sensing exploits entanglement to enhance the estimation of multiple parameters across a network of spatially-separated sensors, achieving sensitivities beyond the classical limit. Potential applications cover a plethora of technologies, from precision navigation to biomedical imaging and environmental monitoring. However, practical implementations are challenged by the complex optimal distribution of entanglement throughout the sensing nodes, which affects scalability and robustness. Here we demonstrate a reconfigurable network of Mach-Zehnder interferometers entangled via a single shared squeezed-vacuum resource. We achieve joint noise suppression of 4.36±0.354.36 \pm 0.354.36±0.35 dB below the standard quantum limit at the phase-uncertainty level of 10−910^{-9}10−9 . Furthermore, after full optimization in the low-intensity regime, we demonstrate a crossover from the standard quantum limit to the Heisenberg limit. The network estimates arbitrary linear combinations of phases, saturates the quantum Cramer-Rao bound in the ideal case, remains robust under realistic photon losses, and scales favorably with the number of sensors.
Distributed quantum sensing uses quantum correlations between multiple sensors to enhance the measurement of unknown parameters beyond the limits of unentangled systems. We describe a sensing scheme that uses continuous-variable multipartite entanglement to enhance distributed sensing of field-quadrature displacement. By dividing a squeezed-vacuum state between multiple homodyne-sensor nodes using a lossless beam-splitter array, we obtain a root-mean-square (rms) estimation error that scales inversely with the number of nodes (Heisenberg scaling), whereas the rms error of a distributed sensor that does not exploit entanglement is inversely proportional to the square root of the number of nodes (standard quantum limit scaling). Our sensor's scaling advantage is destroyed by loss, but it nevertheless retains an rms-error advantage in settings in which there is moderate loss. Our distributed sensing scheme can be used to calibrate continuous-variable quantum key distribution networks, to perform multiple-sensor cold-atom temperature measurements, and to do distributed interferometric phase sensing.
Quantum metrology, such as quantum phase estimation, can surpass classical sensing limits, reaching the Heisenberg‐scaling precision. So far, this kind of metrology has been thought to be only implementable with the quantum systems, which, however, are fragile to environmental noise and hardly contribute to the practical detections. Herein, it is demonstrated both theoretically and experimentally that the parameter encoded by the optical phase can also be estimated at the Heisenberg scaling in classical optics. Inspired by the quantum‐entanglement‐enhanced sensing scheme, the estimation is performed by using classically correlated beams as probes, and obtaining the probes readout after their interaction with the target system. Because the correlated beams considered are spatially separable, a distributed phase estimation scheme is given, which can sense the linear combinations of the phase shifts induced by distinct systems. The results of our experiments show an error reduction up to 3.89 dB below the classical limit when the correlated beam number for probing is 6, approaching the Heisenberg limit. Compared with quantum strategies, the proposal shows a better robustness against the environmental disturbance and keeps their performances even when the correlated beam number is relatively large. Hence, it indicates promising practical applications in the future.
Distributed quantum metrology can improve the sensitivity in estimating arbitrary positive-weight linear combinations beyond the classical limit. It is known that multimode entangled states are crucial for distributed metrology. However, strategies to efficiently engineer multimode entangled states and harness them in distributed quantum metrology remains largely unexplored. In this work, we introduce a concept for distributed quantum sensing utilizing nonlinear optical networks. Unlike linear optical networks, nonlinear optical networks utilize optical parametric amplifiers instead of beam splitters for beam splitting. Here, we employ both simultaneous estimation (global) and separate estimation (local) methods to estimate linear combinations and derive the quantum Cram\'er-Rao bound for the two schemes, respectively. Our findings indicate that the sensitivity of the global strategy surpasses that of the local strategy and reaches the Heisenberg limit. Notably, direct intensity detection saturates the quantum Cram\'er-Rao bound. Furthermore, we characterize loss effects on measurement sensitivity, revealing the reduction in sensitivity with increasing losses and establishing the conditions for surpassing the standard quantum limit. This scheme can be directly applied to applications such as biological imaging and quantum information processing.