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Laser Physics Letters

On precision in HOM interferometer delay time measurements and estimation of photon loss probability in an interferometer

Dmitry Kuts, Alexander A Koksharov, А. Г. Воронцов, V. L. Ushakov, S. P. Kulik

Published Dec 1, 2025• 1 citation
Publisher

Abstract

Abstract An analysis of the precision of temporal delay estimation in a Hong–Ou–Mandel (HOM) interferometer is carried out via the error propagation formula and the classical Fisher information. The study finds delay times characterized by a minimum precision, where precision is defined as the standard deviation of the estimated delay, and those for which this precision is a minimal fraction of the delay time itself. The temporal delay is measured in units set by the location of the inflection point on the HOM coincidence dip, with the corresponding delay time in the considered models depending solely on the biphoton spectrum width. The paper also discusses the estimation of photon loss probability in an interferometer, which is based on the temporal delay estimate that possesses minimum variance.

DOI
10.1088/1612-202x/ae245b

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Journal of the Optical Society of America B•Nov 7, 2024•1 citation
Abstract An analysis of the precision of temporal delay estimation in a Hong–Ou–Mandel (HOM) interferometer is carried out via the error propagation formula and the classical Fisher information. The study finds delay times characterized by a minimum precision, where precision is defined as the standard deviation of the estimated delay, and those for which this precision is a minimal fraction of the delay time itself. The temporal delay is measured in units set by the location of the inflection point on the HOM coincidence dip, with the corresponding delay time in the considered models depending solely on the biphoton spectrum width. The paper also discusses the estimation of photon loss probability in an interferometer, which is based on the temporal delay estimate that possesses minimum variance.
The paper deals with the measurement error of the delay time between the arrivals at the beam splitter of two photons traveling through different channels in an optical system based on a HongOu-Mandel (HOM) interferometer. We consider the source of two-photon light to be the radiation of spontaneous parametric scattering (SPS) of light of the second type. The photon delay in one of the HOM interferometer channels can be induced by a change in the optical path length due to various factors: mechanical displacement of interferometer parts, changes in the refractive index of the medium caused by temperature variations, electric fields, etc. In the scheme, the outputs of the beam splitter are connected to single-photon detectors, which send signals to a coincidence counter upon photon detection. The coincidence counter data determine the delay time. The authors determine the absolute error in estimating the delay time in terms of the biphoton spectral width, the visibility, calculated from the maximum and minimum values of the coincidence counter signal, and the estimated delay time value itself. The derived expression facilitates finding the estimated delay time value. The study reveals that for a fixed biphoton field width, the delay time corresponding to the minimum absolute error value approaches the inflection point of the coincidence-time curve with decreasing visibility. This should be taken into account when designing measuring devices based on the Hong-Ou-Mandel interferometer.
In this work we study the application of two-photon correlated quantum states for high-precision time estimation. To achieve this, photon pairs were fed into a Hong-Ou-Mandel polarization interferometer to generate N00N entangled states with N=2N = 2N=2. Two- and four-detector schemes were considered, and the achievable time estimation precision of each was studied, with particular emphasis on the system's Fisher information in each case. Our results reveal that using a four-detector scheme, enabling the measurement of both coincidence and photon-bunching terms, achieves the maximum precision given by the quantum Fisher information, without the need of characterizing system losses. We also show that correlations between repetitions arise due to systematic errors, limiting the precision enhancement that can be obtained by increasing the number of repetitions. Taking these correlations into account, we show that the four-detector scheme outperforms the two-detector scheme in the precision of time-delay estimation. With the experimentally determined calibration uncertainties, the four-detector scheme improves the dispersion by a factor of 2 for the repetitions performed, reaching up to a factor of 4 as both schemes approach their systematic floors.
The paper addresses the issue of selecting the optimal operating point for a sensor based on a Hong–Ou–Mandel (HOM) interferometer with collinear degenerate type-II phase-matching. Experimental data on coincidence counting rates from single-photon detectors are compared to a model constructed using an approximation where the spectral amplitude of the biphoton field is modeled by a double-Gaussian function. Errors in determining the delay time in the HOM interferometer are estimated based on the Cramér–Rao bound. The delay times corresponding to the minimum absolute and minimum relative errors are calculated for the interferometer model. The authors have found that these delay times occur in the regions where the discrepancy between the model and the experimentally measured coincidence counting rate is approximately 2–3 %. Therefore, when selecting the operating point of the sensor, the deviation between the modeled and experimentally observed coincidence count function should be additionally taken into account.
Quantum interference forces two indistinguishable input photons to depart a beamsplitter in the same (of two possible) spatial output modes; this is known as the Hong-Ou-Mandel (HOM) effect. It provides a quantitative way of measuring the distinguishability between two photons and also has been utilised to determine their relative temporal delays [1, 2]. HOM interferometry offers numerous advantages over its classical counterpart, most importantly it lacks any dependence on the phase of the photons and therefore can potentially provide reliable measurements in environments where the phase is unstable. To date there have been very few investigations on the precision that is achievable with HOM interferometry. Experiments with collinear geometries have yielded sensitivities as low as 0.1fs however these approaches are limited to measurements of birefringence and other polarisation dependent effects [3, 4], as the precision derives from the inherently stable shared path of the photon pairs. Typical non-shared path HOM interferometers exhibit resolution of roughly 1–3 fs (0.3–1 μm).
Hong-Ou-Mandel (HOM) interferometry enables delay estimation at the quantum precision limit but is traditionally constrained to path differences within the coherence time of the interfering photons. Here, we demonstrate single-measurement path-delay sensing at the measurement Cramer-Rao bound using spectrally resolved HOM interference, thereby removing the conventional dynamic-range limitation imposed by the photon coherence window, with no scanning required for calibration. By extracting delay information from the spectral interference fringes of spectrally entangled photon pairs, we retain near-optimal sensitivity over an operational range exceeding the photon coherence time by over two orders of magnitude. Using one million detected photon pairs, we achieve a time-delay precision of 20 attosecond (6 nm), while real-time operation (at 1 Hz) yields 330 attosecond (100 nm) precision. Because the estimator relies on fringe periodicity rather than absolute coincidence rates, the method is intrinsically robust to photon losses and variations in interference visibility, eliminating the need for recalibration. As a practical demonstration, we measure the thickness of a 300 um transmissive target with nanometer-scale precision. These results mark a significant step towards deploying quantum-limited measurements in real-world sensing applications using HOM interferometry.
We theoretically extend the original Hong–Ou–Mandel (HOM) interferometer to three parameters by concatenating 50:50 beam splitters with three independent and adjustable time delays. The coincidence probability of such an interferometer is obtained based on the linear transformation of the beam splitter matrix. We present a comprehensive analysis of the interference characteristics of the interferometer at different time delays with various types of frequency-correlated resources and input states. It is found that by properly setting the time delays, one can obtain typical interferograms that include multiple sub-interferograms associated with one- and two-parameter HOM interferometers, revealing richer and more complicated two-photon interference phenomena. In particular, additional enhancements of the interference signal at a specific delay can be achieved, and the symmetry of the interferograms on both sides can also be changed. This work provides comprehensive insights into the interference characteristics of the three-parameter HOM interferometer and enables the simultaneous optimal estimation of multiple time-delay parameters within a single interferometer, which holds potential for applications in multiparameter estimation and quantum metrology.