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arXiv:2101.05849 · npj Quantum Information

Lost photon enhances superresolution

Alexander B. Mikhalychev, Piotr Novik, Ilya Karuseichyk, Dmitry A. Lyakhov, Dominik Ludewig Michels, Dmitri S. Mogilevtsev

Published Aug 9, 2021• 8 citations
arXivHTMLPDF

Abstract

Abstract Quantum imaging can beat classical resolution limits, imposed by the diffraction of light. In particular, it is known that one can reduce the image blurring and increase the achievable resolution by illuminating an object by entangled light and measuring coincidences of photons. If an n-photon entangled state is used and the nth-order correlation function is measured, the point-spread function (PSF) effectively becomes $$\sqrt{n}$$ n times narrower relatively to classical coherent imaging. Quite surprisingly, measuring n-photon correlations is not the best choice if an n-photon entangled state is available. We show that for measuring (n − 1)-photon coincidences (thus, ignoring one of the available photons), PSF can be made even narrower. This observation paves a way for a strong conditional resolution enhancement by registering one of the photons outside the imaging area. We analyze the conditions necessary for the resolution increase and propose a practical scheme, suitable for observation and exploitation of the effect.

DOI
10.1038/s41534-021-00465-4

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Abstract Quantum imaging can beat classical resolution limits, imposed by the diffraction of light. In particular, it is known that one can reduce the image blurring and increase the achievable resolution by illuminating an object by entangled light and measuring coincidences of photons. If an n-photon entangled state is used and the nth-order correlation function is measured, the point-spread function (PSF) effectively becomes n\sqrt{n}n​ n times narrower relatively to classical coherent imaging. Quite surprisingly, measuring n-photon correlations is not the best choice if an n-photon entangled state is available. We show that for measuring (n − 1)-photon coincidences (thus, ignoring one of the available photons), PSF can be made even narrower. This observation paves a way for a strong conditional resolution enhancement by registering one of the photons outside the imaging area. We analyze the conditions necessary for the resolution increase and propose a practical scheme, suitable for observation and exploitation of the effect.
We present a theoretical framework for super-resolution imaging based on quantum coincidence detection. By comparing the classical Airy point spread function with the two-photon wavefunction of entangled photon pairs, we derive that coincidence detection yields an effective narrowing of the point spread function by a factor of two or the square root of two, depending on the maintenance of correlation. This model highlights the role of transverse phase accumulation in the image plane and predicts a quantum resolution limit that is half of the classical diffraction limit when correlation is maintained. The results offer clear insight into recent experimental demonstrations of quantum imaging beyond the classical diffraction limit.
In this paper we study the resolution of images illuminated by sources composed of N+1N+1N+1 photons in which one non-degenerate photon is entangled with NNN degenerate photons. The NNN degenerate photons illuminate an object and are collected by an NNN photon detector. The signal from the NNN photon detector is measured in coincidence with the non-degenerate photon giving rise to a ghost image. We discuss the case of three photons in various configurations and generalize to N+1N+1N+1. Using the Rayleigh criterion, we find that the system may give an improvement in resolution by a factor of NNN compared to using a classical source. For the case that the NNN-photon number detector is a point detector, a coherent image is obtained. If the NNN-photon detector is a bucket detector, the image is incoherent. The visibility of the image in both cases is 1. In the opposite case in which the non-degenerate photon is scattered by the object, then, using an NNN-photon point detector may reduce the Airy disk by a factor of NNN.
In this paper we study the resolution of images illuminated by sources composed of N+1N+1N+1 photons in which one non-degenerate photon is entangled with NNN degenerate photons. The NNN degenerate photons illuminate an object and are collected by an NNN photon detector. The signal from the NNN photon detector is measured in coincidence with the non-degenerate photon giving rise to a ghost image. We discuss the case of three photons in various configurations and generalize to N+1N+1N+1. Using the Rayleigh criterion, we find that the system may give an improvement in resolution by a factor of NNN compared to using a classical source. For the case that the NNN-photon number detector is a point detector, a coherent image is obtained. If the NNN-photon detector is a bucket detector, the image is incoherent. The visibility of the image in both cases is 1. In the opposite case in which the non-degenerate photon is scattered by the object, then, using an NNN-photon point detector may reduce the Airy disk by a factor of NNN.
Quantum imaging emerged quite recently as a breakthrough technology of overcoming the diffraction limit in microscopy and enhancement of optical resolution without the necessity to use hard radiation or perform scanning in the near field [1] , [2] . Both, the quantum imaging itself and the more “classical” techniques inspired by it (for example, super-resolution optical fluctuations imaging – SOFI [3] ), rely on detection and analysis of photon (intensity) correlations. Typically, it is believed that the more correlated illuminating light is used and the higher order of the correlations is measured, the larger super-resolution can be achieved. That conclusion is based on efficient narrowing of the point-spread function, which, however, does not necessarily imply better resolution as the ability to reconstruct smaller features of the investigated object successfully.
Techniques of optical superresolution imaging are vital for uncovering the complex dynamics of biochemistry in cellular environments. However the practical resolution for superresolution imaging is limited by the increased photon budget for superresolution, compared with conventional microscopy. For this reason it is important to determine the optimal methods for analysing all of the incoming information. Most approaches to microscopy use only the wave-like properties of light, but the particle-like nature of light provides extra information that is normally inaccessible and can be used to increase imaging resolution. Here we theoretically study the localisation of quantum emitters using higher-order quantum correlation functions to understand the resolution that is practically achievable for bio-imaging tasks. We show explicit imaging results for varying number of emitters as a function of correlation order to illustrate the necessary tradeoffs between imaging resolution and acquisition time.
Quantum imaging exploits the spatial correlations between photons to image object features with a higher resolution than a corresponding classical light source could achieve. Using a quantum correlated NNN-photon state, the method of optical centroid measurement (OCM) was shown to exhibit a resolution enhancement by improving the classical Rayleigh limit by a factor of 1/N1/N1/N. In this work, the theory of OCM is formulated within the framework of an imaging formalism and is implemented in an exemplary experiment by means of a conventional entangled photon pair source. The expected resolution enhancement of a factor of two is demonstrated. The here presented experiment allows for single-shot operation without scanning or iteration to reproduce the object in the image plane. Thereby, photon detection is performed with a newly developed integrated time-resolving detector array. Multi-photon interference effects responsible for the observed resolution enhancement are discussed and possible alternative implementation possibilities for higher photon number are proposed.