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Experimental Advances in Phase Estimation with Photonic Quantum States

Laura T. Knoll, Agustina G. Magnoni, M. A. Larotonda

Published Jul 1, 2025• 2 citations
Publisher

Abstract

Photonic quantum metrology has emerged as a leading platform for quantum-enhanced precision measurements. By taking advantage of quantum resources such as entanglement, quantum metrology enables parameter estimation with sensitivities surpassing classical limits. In this review, we describe the basic tools and recent experimental progress in the determination of an optical phase with a precision that may exceed the shot-noise limit, enabled by the use of nonclassical states of light. We review the state of the art and discuss the challenges and trends in the field.

DOI
10.3390/e27070712

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Quantum metrology exploits quantum correlations to perform measurements with precision higher than can be achieved with classical approaches [1]. Photonic approaches promise transformative advances in the family of interferometric phase measurement techniques, a vital toolset used to precisely determine quantities including distance, velocity, acceleration and materials properties. Without quantum enhancement, the precision limit in optical phase sensing (i.e. the minimum uncertainty) is the shot noise limit (SNL): 1/sqrt(N) where N is the number of resources (e.g. photons) used. Entangled photons promise sensitivity surpassing the shot noise limit achievable with classical probes. The maximally phase-sensitive state is the NOON state [2], a path-entangled state of definite photon number N. Despite theoretical proposals stretching back decades [3], no measurement using such photonic (i.e. definite photon number) states has unconditionally surpassed the shot noise limit: by contrast, all such demonstrations employed postselection to discount photon loss in the source, interferometer or detectors. Here, we use the state of art single photon generation and detection technology to respectively make and measure a two-photon NOON state, and use it to perform unconditional phase sensing beyond the shot noise limit — that is, without artificially correcting for loss or any other source of imperfection [4]. We performed a two-photon NOON state polarisation interferometry measurement on a birefringent test phase. We use photons generated from a high-heralding-efficiency, high purity source of telecom-wavelength photon pairs [5], and we employ high efficiency superconducting photon detectors. Unlike previous experiments, our apparatus does not require postselection to achieve phase uncertainty below that achievable in an ideal, lossless classical interferometer. Our results show a clear violation (for a range of phases) of the stringent SNL Fisher-information bound, F_{SNL} = 2.09635, that takes into account the information in unrecorded trials arising from loss and higher order terms — making our demonstration unconditional. We also performed a direct phase sensing measurement and observed phase uncertainties more than 10 standard deviations below the SNL [4]. Our results enable quantum-enhanced phase measurements at low photon flux and open the door to the next generation of optical quantum metrology advances. References [1] V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011). [2] J. P. Dowling, “Quantum optical metrology–the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008). [3] R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kołodynski, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345–435 (2015). [4] S. Slussarenko, M. M.Weston, H. M. Chrzanowski, L. K. Shalm, V. B. Verma, S.W. Nam, and G. J. Pryde, “Unconditional violation of the shot noise limit in photonic quantum metrology,” Nat. Photon. 11, 700-703 (2017). [5] M. M. Weston, H. M. Chrzanowski, S. Wollmann, A. Boston, J. Ho, L. K. Shalm, V. B. Verma, M. S. Allman, S. W. Nam, R. B. Patel, S. Slussarenko, and G. J. Pryde, “Efficient and pure femtosecond-pulse length source of polarization-entangled photons,” Opt. Express 24, 10,869–10,879 (2016).

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Photonic quantum metrology has emerged as a leading platform for quantum-enhanced precision measurements. By taking advantage of quantum resources such as entanglement, quantum metrology enables parameter estimation with sensitivities surpassing classical limits. In this review, we describe the basic tools and recent experimental progress in the determination of an optical phase with a precision that may exceed the shot-noise limit, enabled by the use of nonclassical states of light. We review the state of the art and discuss the challenges and trends in the field.
Quantum metrology exploits quantum correlations to perform measurements with precision higher than can be achieved with classical approaches [1]. Photonic approaches promise transformative advances in the family of interferometric phase measurement techniques, a vital toolset used to precisely determine quantities including distance, velocity, acceleration and materials properties. Without quantum enhancement, the precision limit in optical phase sensing (i.e. the minimum uncertainty) is the shot noise limit (SNL): 1/sqrt(N) where N is the number of resources (e.g. photons) used. Entangled photons promise sensitivity surpassing the shot noise limit achievable with classical probes. The maximally phase-sensitive state is the NOON state [2], a path-entangled state of definite photon number N. Despite theoretical proposals stretching back decades [3], no measurement using such photonic (i.e. definite photon number) states has unconditionally surpassed the shot noise limit: by contrast, all such demonstrations employed postselection to discount photon loss in the source, interferometer or detectors. Here, we use the state of art single photon generation and detection technology to respectively make and measure a two-photon NOON state, and use it to perform unconditional phase sensing beyond the shot noise limit — that is, without artificially correcting for loss or any other source of imperfection [4]. We performed a two-photon NOON state polarisation interferometry measurement on a birefringent test phase. We use photons generated from a high-heralding-efficiency, high purity source of telecom-wavelength photon pairs [5], and we employ high efficiency superconducting photon detectors. Unlike previous experiments, our apparatus does not require postselection to achieve phase uncertainty below that achievable in an ideal, lossless classical interferometer. Our results show a clear violation (for a range of phases) of the stringent SNL Fisher-information bound, F_{SNL} = 2.09635, that takes into account the information in unrecorded trials arising from loss and higher order terms — making our demonstration unconditional. We also performed a direct phase sensing measurement and observed phase uncertainties more than 10 standard deviations below the SNL [4]. Our results enable quantum-enhanced phase measurements at low photon flux and open the door to the next generation of optical quantum metrology advances. References [1] V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011). [2] J. P. Dowling, “Quantum optical metrology–the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008). [3] R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kołodynski, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345–435 (2015). [4] S. Slussarenko, M. M.Weston, H. M. Chrzanowski, L. K. Shalm, V. B. Verma, S.W. Nam, and G. J. Pryde, “Unconditional violation of the shot noise limit in photonic quantum metrology,” Nat. Photon. 11, 700-703 (2017). [5] M. M. Weston, H. M. Chrzanowski, S. Wollmann, A. Boston, J. Ho, L. K. Shalm, V. B. Verma, M. S. Allman, S. W. Nam, R. B. Patel, S. Slussarenko, and G. J. Pryde, “Efficient and pure femtosecond-pulse length source of polarization-entangled photons,” Opt. Express 24, 10,869–10,879 (2016).
Quantum entanglement offers powerful opportunities for enhancing measurement sensitivity beyond classical limits, with optical atomic clocks serving as a leading platform for such advances. This chapter introduces the principles of entanglement-enhanced quantum metrology and explores their applications to timekeeping. We review the theoretical framework of quantum phase estimation, comparing frequentist and Bayesian approaches, and discuss paradigmatic entangled states such as spin-squeezed and GHZ states. Particular emphasis is placed on the challenges posed by decoherence, which constrain the practical advantages that can be realized in large-scale devices. The discussion then turns to frequency estimation in atomic clocks, highlighting how experimental constraints shape the translation of abstract quantum limits into real performance gains. Finally, we outline emerging directions of contemporary quantum metrology. Together, these developments underscore the increasingly close interplay between quantum information processing and precision metrology.
Quantum metrology is one of the most promising applications of quantum technologies. The aim of this research field is the estimation of unknown parameters exploiting quantum resources, whose application can lead to enhanced performances with respect to classical strategies. Several physical quantum systems can be employed to develop quantum sensors, and photonic systems represent ideal probes for a large number of metrological tasks. Here, the authors review the basic concepts behind quantum metrology and then focus on the application of photonic technology for this task, with particular attention to phase estimation. The authors describe the current state of the art in the field in terms of platforms and quantum resources. Furthermore, the authors present the research area of multiparameter quantum metrology, where multiple parameters have to be estimated at the same time. The authors conclude by discussing the current experimental and theoretical challenges and the open questions toward implementation of photonic quantum sensors with quantum-enhanced performances in the presence of noise.
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.
Quantum metrology holds the promise of improving the measurement precision beyond the limit of classical approaches. To achieve such enhancement in performance requires the development of quantum estimation theories as well as novel experimental techniques. In this article, we provide a brief review of some recent results in the field of quantum metrology. We emphasize that the unambiguous demonstration of the quantum-enhanced precision needs a careful analysis of the resources involved. In particular, the implementation of quantum metrology in practice requires us to take into account the experimental imperfections included, for example, particle loss and dephasing noise. For a detailed introduction to the experimental demonstrations of quantum metrology, we refer the reader to another article ‘Quantum metrology’ in the same issue.