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arXiv:1603.02375 · Physical Review A

Physical resources for optical phase estimation

Jaspreet Sahota, Nicolás Quesada, Daniel F. V. James

Published Sep 9, 2016• 18 citations
arXivHTMLPDF

Abstract

We study the role of quantum entanglement (particle entanglement and mode entanglement) in optical phase estimation by employing the mode description and the particle description of bosonic probe states. To determine the physical resources responsible for Heisenberg scaling (attainable only in the noiseless case), we restrict our analysis to noiseless quantum-limited optical phase estimation. The quantum Fisher information (QFI) is expressed as a function of the first- and second-order optical coherence functions. The resulting form of the QFI elucidates the deriving metrological resources for quantum phase estimation: field intensity and photon detection correlations. Our analysis confirms that mode entanglement is not required for quantum-enhanced interferometry, whereas particle entanglement is a necessary requirement. Furthermore, the derived forms of the QFI equations are summations of two terms: the classical (shot-noise-limit scaling) term and the quantum (Heisenberg scaling) term. This allows us to clearly identify the physical resources responsible for quantum enhancement in optical phase estimation.

DOI
10.1103/physreva.94.033817

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We study the role of quantum entanglement (particle entanglement and mode entanglement) in optical phase estimation by employing the mode description and the particle description of bosonic probe states. To determine the physical resources responsible for Heisenberg scaling (attainable only in the noiseless case), we restrict our analysis to noiseless quantum-limited optical phase estimation. The quantum Fisher information (QFI) is expressed as a function of the first- and second-order optical coherence functions. The resulting form of the QFI elucidates the deriving metrological resources for quantum phase estimation: field intensity and photon detection correlations. Our analysis confirms that mode entanglement is not required for quantum-enhanced interferometry, whereas particle entanglement is a necessary requirement. Furthermore, the derived forms of the QFI equations are summations of two terms: the classical (shot-noise-limit scaling) term and the quantum (Heisenberg scaling) term. This allows us to clearly identify the physical resources responsible for quantum enhancement in optical phase estimation.
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.
Quantum mode parameter estimation determines parameters governing the shape of electromagnetic modes occupied by a quantum state of radiation. Canonical examples, time delays and frequency shifts, underpin radar, lidar, and optical clocks. A comprehensive framework recently established that broad families of quantum states can attain the Heisenberg limit, surpassing any classical strategy. This raises a fundamental question: among all quantum-enhanced strategies, which is truly optimal? Answering this requires identifying physically meaningful resources governing each estimation task, so quantum states can be compared on equal footing. We show these resources are connected to the eigenmode basis of the generator of the relevant mode transformation. For time-shift estimation, whose generator is diagonal in the frequency domain, the pertinent resources are the mean frequency and bandwidth; analogous quantities emerge for other transformations. Our framework unifies two historically separate perspectives: the particle-number aspect and the mode-structure of quantum light, providing a coherent picture of quantum-enhanced sensing with multimode radiation. Within this unified framework, we derive a tight upper bound on the quantum Fisher information for multimode Gaussian states, expressed in terms of these natural resources, and analytically identify the optimal Gaussian states saturating it. These optimal states take a particularly transparent form in the generator eigenbasis, a structural simplicity reflecting the deep connection between the geometry of the mode transformation and the architecture of the optimal probe. We further demonstrate that multimode homodyne detection constitutes the optimal measurement, achieving this bound and completing the end-to-end characterization of optimal quantum metrology strategies for mode parameter estimation.
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.
Bosonic systems, particularly in quantum optics and atomic physics, are leading platforms for achieving quantum enhanced precision in parameter estimation. By exploiting properties such as mode and particle entanglement, it is possible to attain precisions that surpass the shot noise limit with respect to key resources like probe number or energy. Yet the mechanisms by which these bosonic resources enable quantum enhancement remain unclear. Consequently, the design of optimal probes and evolutions often relies on case by case analyses, where continuous and discrete variable regimes are treated separately and their connection is still unclear. We develop a comprehensive framework for quantum metrology that unifies all known precision enhancement mechanisms based on bosonic systems. Our approach employs a superselection rule compliant representation of the electromagnetic field that explicitly includes the phase reference, thereby enforcing total particle number conservation and bridging the discrete and continuous limits of quantum optics and symmetric massive systems. Within this unified formalism, of which established results emerge as special cases, we identify the distinct roles of mode and particle entanglement for quantum enhanced precision. The framework further provides general measurement optimization strategies for arbitrary multimode entangled probe states and naturally incorporates noise and non-unitary dynamics, ensuring applicability to realistic experimental conditions.
The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of NNN resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with NNN better than when resources would be used independently. Consequently, the optimal use of all resources involves accumulating them in a single execution of the experiment. Unfortunately, that implies that the most common theoretical tool used to analyze metrological protocols - quantum Fisher information (QFI) - does not allow for a reliable description of this problem, as it becomes operationally meaningful only with multiple repetitions of the experiment. In this thesis, using the formalism of Bayesian estimation and the minimax estimator, I derive asymptotically saturable bounds on the precision of the estimation for the case of noiseless unitary evolution. For the case where the number of resources NNN is strictly constrained, I show that the final measurement uncertainty is πππ times larger than would be implied by a naive use of QFI. I also analyze the case where a constraint is imposed only on the average amount of resources, the exact value of which may fluctuate (in which case QFI does not provide any universal bound for precision). In both cases, I study the asymptotic saturability and the rate of convergence of these bounds. In the following part, I analyze the problem of the Heisenberg limit when multiple parameters are measured simultaneously on the same physical system. In particular, I investigate the existence of a gain from measuring all parameters simultaneously compared to distributing the same amount of resources to measure them independently. I focus on two examples - the measurement of multiple phase shifts in a multi-arm interferometer and the measurement of three magnetic field components.