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arXiv:2412.16304 · The European Physical Journal Plus

Quantum-limited estimation of the frequency shift between two interfering photons by time sampling of their quantum beats

Luca Maggio, Danilo Triggiani, Paolo Facchi A, Vincenzo Tamma

Published Oct 7, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract We present a sensing scheme for estimating the frequency difference of two non-entangled photons. The technique consists of time-resolving sampling measurements at the output of a beam splitter. With this protocol, we can achieve the ultimate precision for the estimation of the frequency shift between two non-entangled photons, overcoming the limits in precision and the range of detection of frequency-resolving detectors employed in standard direct measurements of the frequencies. The sensitivity can be increased by increasing the coherence time of the photons. We show that, already with $$\sim 1000$$ ∼ 1000 sampling measurements, the Cramér–Rao bound for the specific input state chosen is saturated independently of the value of the difference in frequency.

DOI
10.1140/epjp/s13360-025-06903-y

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Abstract We present a sensing scheme for estimating the frequency difference of two non-entangled photons. The technique consists of time-resolving sampling measurements at the output of a beam splitter. With this protocol, we can achieve the ultimate precision for the estimation of the frequency shift between two non-entangled photons, overcoming the limits in precision and the range of detection of frequency-resolving detectors employed in standard direct measurements of the frequencies. The sensitivity can be increased by increasing the coherence time of the photons. We show that, already with ∼1000\sim 1000∼1000 ∼ 1000 sampling measurements, the Cramér–Rao bound for the specific input state chosen is saturated independently of the value of the difference in frequency.
We demonstrate the ultimate sensitivity allowed by quantum physics in the estimation of the time delay between two photons by measuring their interference at a beam splitter through frequency-resolving sampling measurements. This sensitivity can be increased quadratically by decreasing the photonic temporal bandwidth even at values smaller than the time delay when standard two-photon interferometers become inoperable and without adapting the path of the reference photon, nor the need of time-resolving detectors with an unfeasible high resolution. Applications can range from the more feasible imaging of nanostructures, including biological samples, and nanomaterial surfaces to quantum enhanced estimation based on frequency-resolved boson sampling in optical networks.
Optical sensing schemes based on two-photon interference offer a powerful platform for precision metrology, but are usually limited by a trade-off between dynamic range and measurement precision. Here, we overcome this limitation by resolving the frequency of two photons impinging on a beam splitter and combining these measurements with a new estimation strategy. This increases the usable dynamic range by up to a factor of 20 compared with conventional frequency nonresolved schemes. We implement the method with independent photon pair sources and characterize its performance in the presence of finite-resolution frequency-resolved detectors, which approach the quantum limit in the lossless regime. Our approach enables scan-free, nanometer resolution depth sensing over millimeter scale distances, with potential applications in biological and nanomaterial imaging.
The Rayleigh criterion is a widely known limit in the resolution of incoherent sources with classical measurements in the spatial domain. Unsurprisingly the estimation of the time delay between two weak incoherent signals is afflicted by an analogue problem. In this work, we show the emergence of two-photon quantum beats in the frequency domain from the interference at a beam splitter of a photon emitted by a reference source and one from the two incoherent weak signals. We demonstrate, based on this phenomena, that with a relatively low number of measurements of the frequencies of the interfering photons either bunching or antibunching at the beam splitter output one can achieve a precision amounting to half of the quantum limit, independently of both the mode structure of the photonic wavepackets and the time delay to be estimated. The feasibility of the technique makes it applicable in astronomy, microscopy, remote clocks synchronization and radar ranging
We present a quantum sensing scheme achieving the ultimate quantum sensitivity in the estimation of the transverse displacement between two photons interfering at a balanced beam splitter, based on transverse-momentum sampling measurements at the output. This scheme can possibly lead to enhanced high-precision nanoscopic techniques, such as superresolved single-molecule localization microscopy with quantum dots, by circumventing the requirements in standard direct imaging of camera resolution at the diffraction limit, and of highly magnifying objectives. Interestingly, we show that our interferometric technique achieves the ultimate spatial precision in nature irrespectively of the overlap of the two displaced photonic wave packets, while its precision is only reduced of a constant factor for photons differing in any nonspatial degrees of freedom. This opens a new research paradigm based on the interface between spatially resolved quantum interference and quantum-enhanced spatial sensitivity.
We present a quantum sensing protocol for three-dimensional estimation of the difference between the momenta of two photons based on spatially resolved interferometric sampling measurements. The protocol attains ultimate quantum precision in the simultaneous estimation of the components of the relative momentum for any values of the parameters already with ∼2000\sim 2000∼2000 sampling measurements and a bias below 1%1\%1%. These results identify 3D spatially resolved two-photon interference as an efficient tool for multi-parameter quantum sensing, with potential applications in high-precision 3D localization, refractometry, and near-field calibration in free-space quantum technologies.