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arXiv:1602.07561 · Physical Review A

Quantum-classical boundary for precision optical phase estimation

Patrick M. Birchall, Jeremy L. O’Brien, Jonathan C. F. Matthews, Hugo V. Cable

Published Dec 8, 2017• 17 citations
arXivHTMLPDF

Abstract

Understanding the fundamental limits on the precision to which an optical phase can be estimated is of key interest for many investigative techniques utilized across science and technology. We study the estimation of a fixed optical phase shift due to a sample which has an associated optical loss, and compare phase estimation strategies using classical and nonclassical probe states. These comparisons are based on the attainable (quantum) Fisher information calculated per number of photons absorbed or scattered by the sample throughout the sensing process. We find that for a given number of incident photons upon the unknown phase, nonclassical techniques in principle provide less than a $20%$ reduction in root-mean-square error (RMSE) in comparison with ideal classical techniques in multipass optical setups. Using classical techniques in a different optical setup that we analyze, which incorporates additional stages of interference during the sensing process, the achievable reduction in RMSE afforded by nonclassical techniques falls to only $\ensuremath{\simeq}4%$. We explain how these conclusions change when nonclassical techniques are compared to classical probe states in nonideal multipass optical setups, with additional photon losses due to the measurement apparatus.

DOI
10.1103/physreva.96.062109

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Understanding the fundamental limits on the precision to which an optical phase can be estimated is of key interest for many investigative techniques utilized across science and technology. We study the estimation of a fixed optical phase shift due to a sample which has an associated optical loss, and compare phase estimation strategies using classical and nonclassical probe states. These comparisons are based on the attainable (quantum) Fisher information calculated per number of photons absorbed or scattered by the sample throughout the sensing process. We find that for a given number of incident photons upon the unknown phase, nonclassical techniques in principle provide less than a 2020%20 reduction in root-mean-square error (RMSE) in comparison with ideal classical techniques in multipass optical setups. Using classical techniques in a different optical setup that we analyze, which incorporates additional stages of interference during the sensing process, the achievable reduction in RMSE afforded by nonclassical techniques falls to only \ensuremath≃4\ensuremath{\simeq}4%\ensuremath≃4. We explain how these conclusions change when nonclassical techniques are compared to classical probe states in nonideal multipass optical setups, with additional photon losses due to the measurement apparatus.
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.
Optical absorption measurements characterize a wide variety of systems from atomic gases to in vivo diagnostics of living organisms. Here we study the potential of nonclassical techniques to reduce statistical noise below the shot-noise limit in absorption measurements with concomitant phase shifts imparted by a sample. We consider both cases where there is a known relationship between absorption and a phase shift, and where this relationship is unknown. For each case we derive the fundamental limit and provide a practical strategy to reduce statistical noise. Furthermore, we find an intuitive correspondence between measurements of absorption and of lossy phase shifts, which both show the same analytical form for precision enhancement for bright states. Our results demonstrate that nonclassical techniques can aid real-world tasks with present-day laboratory techniques.
One important problem in quantum optics is to resolve an extremely small change of phase shift. The complementarity between photon number and phase sets an ultimate limit, the so-called Heisenberg limit, on the phase measurement sensitivity. The precise phase estimation has many technological applications, such as optical gyroscopes, gravitational wave detection, quantum imaging and sensing. In this thesis I show that the utilization of the parity measurement in the optical interferometry is actually applicable to a wide range of quantum entangled input states. Comparison of the performance of the various quantum states then can be made within such a unified output measurement scheme. Based on such a universal detection scheme, we present a comparison of the phase sensitivity reduction for various quantum states of light in the presence of photon loss. I also provide a simple condition that could be used to check whether an arbitrary state can achieve the Heisenberg limit independently of the detection scheme. It implies that the fidelity between the two output states with zero phase and minimal detectable phase applied respectively should significantly different from unity as the minimal phase shift scaling as 1/N, whereas the average number of input photons N goes to infinity. Next I give several measures to characterize the which-way information in the interference experiment. We define a new distinguishability associated with the fidelity between two density matrices to measure the which-way information. We demonstrate that the changes of mutual entropy as well as the entanglement of formation of the whole system, i.e. the physical system plus the which-way detector can also be used to describe the which-way information. With such quantities, we show that as the fringe visibility of the interference pattern gets larger, the less which-way information is obtained. Finally, I show that coherent light coupled with photon number resolving detectors can provide a super-resolution much below the Rayleigh diffraction limit, with sensitivity no worse than shot-noise in terms of the detected photon power. This scheme would have applications to laser radar, given the difficulty in making entangled states of light, as well as their susceptibility to atmospheric absorption.
Optical interferometry is a cornerstone technique for precise phase measurements across various fields. In many applications, for example, biological imaging, it often necessitates stringent limits on light intensity to prevent adverse effects on light-sensitive samples, a condition known as dose-limited regimes. Maximizing the precision per dose is therefore crucial. In quantum metrology, quantum correlations enable high precision in phase estimation while adhering to dose constraints. Nevertheless, photon loss, including absorption by a sample, substantially diminishes the benefits of quantum enhancement in interferometry. In this work, we experimentally investigate a dose-efficient approach to quantum phase estimation using sequential strategies in the presence of loss. Performance of sequential strategies with and without control is evaluated through quantum Fisher information (QFI) per dose. Experimental results show that both sequential strategies exceed the classical limit and outperform the parallel strategy using unbalanced N00N states. Notably, the control-enhanced sequential strategy attains superior QFI per dose, approaching the quantum limit. These results highlight the promise of sequential strategy for imaging and sensing in resource-constrained scenarios, marking a significant step toward practical and efficient quantum metrology in lossy environments.
We consider the problem of estimating an unknown but constant carrier phase modulation $\ensuremath{\theta}$ using a general, possibly entangled, nnn-mode optical probe through nnn independent and identical uses of a lossy bosonic channel with additive thermal noise. We find an upper bound to the quantum Fisher information (QFI) of estimating $\ensuremath{\theta}$ as a function of nnn, the mean and variance of the total number of photons NS{N}_{\mathrm{S}}NS​ in the nnn-mode probe, the transmissivity $\ensuremath{\eta}$, and mean thermal photon number per mode n‾B{\overline{n}}_{\mathrm{B}}nB​ of the bosonic channel. Since the inverse of QFI provides a lower bound to the mean-square error (MSE) of an unbiased estimator $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\theta}}$ of $\ensuremath{\theta}$, our upper bound to the QFI provides a lower bound to the MSE. It already has found use in proving fundamental limits of covert sensing and could find other applications requiring bounding the fundamental limits of sensing an unknown parameter embedded in a correlated field.
With the rapid development of quantum technologies in recent years, the need for high-sensitivity measuring techniques has become a key issue. In particular, optical sensors based on quantum states of light have proven to be optimal resources for high-precision interferometry. Nevertheless, their performance may be severely affected by the presence of noise or imperfections. In this work, we derive the quantum Fisher information matrix associated with the simultaneous estimation of an interferometric phase and the indistinguishability characterizing the probe state consisting of an even number of photons. We find the optimal measurement attaining the ultimate precision for both parameters in a single setup and perform an experiment based on a pair of photons with an unknown degree of indistinguishability entering a two-port interferometer.