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Scientific Reports

Retrodiction of single emitter photon emission rates from nanoscale light emitters in random, dynamic systems

Richard S. K. Lane, Niclas Westerberg, Luna Hu, Eleanor Stride, Daniele Faccio

Published Aug 25, 2026• 0 citations
Publisher

Abstract

Abstract Photon emission in biological and chemical systems often consists of large numbers of moving nanoscale emitters with a high spatial density. Estimating the flux of a single emitter within the ensemble is challenging and can be beyond the limit of current imaging techniques. We introduce a generalised quantum retrodiction technique that estimates the single emitter flux from measurements in complex, dynamic systems. We first experimentally validate this using a solution of fluorescent molecules, and then apply it to a cloud of sonoluminescent bubbles in which the single bubble flux is otherwise unattainable. Our approach provides a quantum-informed experimental determination of single emitter photon emission rates in nanophotonic and photoluminescent systems.

DOI
10.1038/s41598-026-67530-w

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Abstract Photon emission in biological and chemical systems often consists of large numbers of moving nanoscale emitters with a high spatial density. Estimating the flux of a single emitter within the ensemble is challenging and can be beyond the limit of current imaging techniques. We introduce a generalised quantum retrodiction technique that estimates the single emitter flux from measurements in complex, dynamic systems. We first experimentally validate this using a solution of fluorescent molecules, and then apply it to a cloud of sonoluminescent bubbles in which the single bubble flux is otherwise unattainable. Our approach provides a quantum-informed experimental determination of single emitter photon emission rates in nanophotonic and photoluminescent systems.
The fundamental understanding of biological pathways requires minimally invasive nanoscopic optical resolution imaging. Many approaches to high-resolution imaging rely on localization of single emitters, such as fluorescent molecules or quantum dots. Additionally, the exact determination of the number of such emitters in an imaging volume is essential for a number of applications; however, in standard intensity-based microscopy it is not possible to determine the number of individual emitters within a diffraction limited spot without initial knowledge of system parameters. Here we explore how quantum measurements of the emitted photons using photon number resolving detectors can be used to address this challenging task. In the proposed new approach, the problem of counting emitters reduces to the task of determining differences between the emitted photon distribution and the Poisson limit. We show that quantum measurements of the number of photons emitted from an ensemble of emitters enable the determination of both the number of emitters and the probability of emission. This method can be applied for any type of single-photon emitters. The scaling laws of this new approach are presented by the Cramer-Rao Lower Bounds, and this technique has great potential in quantum optical imaging with nanoscopic resolution.
Photons emitted from a single emitter are promising candidates for an efficient single-photon source, which is essential for many novel nano- and quantum-optical applications. As a photon source having a larger flux and high reliability is highly desirable. In this paper, we quantitatively studied the dependencies of the numbers of photons emitted and collected from single terrylene molecules on different experimental conditions, the refractive index of the substrate and the numerical aperture of the collection lens. Also, back-focal plane images are used to visualize the emission pattern of a single emitter.
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The use of techniques for analyzing the fluorescence photon statistics of a single molecule for modeling single-emitter dynamics is demonstrated. The photon distribution function measured in the fluorescence of a single tetra-tert-butylterrylene molecule embedded in polyisobutylene is used to devise a theoretical model for single emitters with complicated fluorescence photon statistics. Our analysis was carried out with the theoretical approach developed by Osad'ko and co-workers [J. Chem. Phys. 130, 064904 (2009); J. Phys. Chem. C 114, 10349 (2010)] for photon distribution functions. Although the experimental data were obtained at cryogenic temperature where narrow zero-phonon lines are present, the method is based on a purely statistical approach and does not require spectrally resolved data. It can also be applied to the analysis of broad fluorescence bands as measured at room temperature. Therefore, the method has prospects for revealing the quantum dynamics of single biological objects and other single quantum emitters in ambient conditions.
The lack of reliable quantitative methods in fluorescence microscopy prevents the detailed investigation of basic molecular processes. Single-molecule fluorescence spectroscopy offers several techniques to determine the stoichiometry of fluorescent probes. Among them, an approach using the single-photon emission of an individual quantum system, also known as photon antibunching, yields access to both the emitter number and their molecular brightness. Previous work demonstrated the feasibility of the method to estimate up to five dyes without further physical characterization. In this work, I investigated and improved the reliability and robustness of this method. I developed different fluorescent probes with a defined label stoichiometry to explore the performance of our approach \nin experiments. I could demonstrate, in combination with simulations, that it can provide reliable estimates of up to 18 fluorescent emitters at a time resolution of approximately 200 ms. Furthermore, I inspected the dependency of the estimation bias and precision on the laser intensity, the analysis time and the number of expected molecules in detail. My results designate the scope for future stoichiometry determinations. In initial experiments, I analyzed the labeling distributions of different fluorescent probes. In addition, I could combine our counting approach with super-resolution STED-microscopy and identify critical parameters for possible applications