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Applied Optics

Bounds of parameter estimation for interference signals

Chengshuai Li, Yizheng Zhu

Published Aug 11, 2017• 3 citations
Publisher

Abstract

Parameter estimation, especially frequency estimation, from noisy observations of interference is essential in optical interferometric sensing and metrology. The Cramer-Rao bound (CRB) of such estimation determines measurement sensitivity limit. Unlike the well-studied complex sinusoids in communication theory, an optical interference signal is distinctly different in its model parameters and noise statistics. The connection between these parameters and their estimation bounds has not been well understood. Here we propose a complete, realistic multiparameter interference model corrupted by a combination of shot noise, dark noise, and readout noise. We derive the Fisher information matrix and the CRBs for all model parameters, including intensity, visibility, optical path length (frequency), and initial phase. We show that the CRBs of frequency and phase are coupled but not affected by the knowledge of intensity and visibility. Knowing the initial phase offers significant sensitivity advantage, which is verified by both theoretical derivations and numerical simulations. In addition to the complete model, a shot noise-limited case is studied, permitting the calculation of the CRBs directly from measured data.

DOI
10.1364/ao.56.006867

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Parameter estimation, especially frequency estimation, from noisy observations of interference is essential in optical interferometric sensing and metrology. The Cramer-Rao bound (CRB) of such estimation determines measurement sensitivity limit. Unlike the well-studied complex sinusoids in communication theory, an optical interference signal is distinctly different in its model parameters and noise statistics. The connection between these parameters and their estimation bounds has not been well understood. Here we propose a complete, realistic multiparameter interference model corrupted by a combination of shot noise, dark noise, and readout noise. We derive the Fisher information matrix and the CRBs for all model parameters, including intensity, visibility, optical path length (frequency), and initial phase. We show that the CRBs of frequency and phase are coupled but not affected by the knowledge of intensity and visibility. Knowing the initial phase offers significant sensitivity advantage, which is verified by both theoretical derivations and numerical simulations. In addition to the complete model, a shot noise-limited case is studied, permitting the calculation of the CRBs directly from measured data.
Interference frequency estimation is essential in spectral-domain interferometric sensing and imaging, and its performance determines system sensitivity. To date, an objective and practical criterion is still absent for the proper evaluation of fundamental sensitivity limit in a given system. Here, we report the derivation of Cramer-Rao bound (CRB) for unbiased estimators of spectral interference frequency, which imposes a theoretical limit on measurement sensitivity. Results based on a shot noise-limited model are presented. A more complete model, including dark and read noises, is also discussed and its approximate CRB is obtained. Poisson statistics were used in both cases. Asymptotic behaviors and simplified forms of CRB are studied for fringe visibility approaching 0 or 1. Further, we show that the current Fourier transform-based estimation algorithm achieves CRB only for low visibility, but is inferior by as much as √2 times for high visibility. This performance gap may potentially permit a sensitivity gain if better algorithms can be devised. Finally, we introduce a practical means to accurately estimate CRB from experimentally acquired spectral data. The results are verified in both simulation and experiments. Although spectral interference is discussed here, the same derivation can be applied to spatial or temporal interference as well, provided they can be similarly modelled.
Precision measurement has been an important research area in sensing and metrology. In classical physics, the Fisher information determines the maximum extractable information from statistically unknown signals, based on a joint probability density function of independently and identically distributed random variables. The Cramer-Rao lower bound (CRLB) indicates the minimum error of the Fisher information, generally known as the shot-noise limit. On the other hand, coherence has pushed the resolution limit further overcoming the diffraction limit using many-wave interference strictly confined to the first-order intensity correlation. However, practical implementation is limited by the lithographic constraints in, e.g., optical gratings. Recently, a coherence technique of superresolution has been introduced to overcome the diffraction limit in phase sensitivity using higher-order intensity correlations of a phase-controlled output field from an interferometer. Here, the superresolution is adopted for precision metrology in an optical spectrometer, whose enhanced frequency resolution is linearly proportional to the intensity-product order, overcoming CRLB. Unlike quantum sensing using entangled photons, this technique is purely classical and offers robust performance against environmental noises, benefiting from the interferometer's scanning mode for fringe counting.
Bounding the optimal precision in parameter estimation tasks is of central importance for technological applications. In the regime of a small number of measurements, or low signal-to-noise ratios, the meaning of common frequentist bounds, such as the Cramér-Rao bound (CRB), becomes questionable. Here, we discuss a different (Barankin-like) approach to derive the conditions for the existence of optimal estimators and calculate their variance, for any number of measurement repetitions. We elaborate on connections to the shot-noise limit and the Kitaev phase estimation algorithm and derive an extended Cramér-Rao bound that is compatible with a finite variance in situations where the Barankin bound (BaB) is undefined. Finally, we show an exemplary numerical confrontation between frequentist and Bayesian approaches to parameter estimation. Published by the American Physical Society 2024
Due to the potential of quantum advantage to surpass the standard quantum limit (SQL), nonlinear interferometers have garnered significant attention from researchers in the field of precision measurement. However, many practical applications require multiparameter estimation. In this work, we discuss the precision limit of multi-parameter estimation of pure Gaussian states based on nonlinear interferometers, and derive the Holevo Cramér−Rao bound (HCRB) for the case where both modes undergo displacement estimation. Furthermore, we compare our analytical results with the quantum Cramér−Rao bound based on the symmetric logarithmic derivative (SLD-CRB), and with the result of the dual homodyne measurement. Through numerical analysis, we find that the HCRB equals the result of the dual homodyne measurement, whereas SLD-CRB is not saturable at small squeezed parameters. Therefore, this indicates that the HCRB is tight. Additionally, we provide intuitive analysis and visual representation of our numerical results in phase space.
Due to the potential of quantum advantage to surpass the standard quantum limit (SQL), the nonlinear interferometers have garnered significant attention from researchers in the field of precision measurement. However, many practical applications require multi-parameter estimation. In this work, we discuss the precision limit of multi-parameter estimation of pure Gaussian states based on nonlinear interferometers, and derive the Holevo Cramér-Rao Bound (HCRB) for the case where both modes undergo displacement estimation. Furthermore, we compare our analytical results with the quantum Cramér-Rao Bound based on the symmetric logarithmic derivative (SLD-CRB), and with the result of the dual homodyne measurement. Through numerical analysis, we find that the HCRB equals the result of the dual homodyne measurement, whereas SLD-CRB is not saturable at small squeezed parameters. Therefore, this indicates that the HCRB is tight. Additionally, we provide intuitive analysis and visual representation of our numerical results in phase space.
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.