Optimal quantum superresolution for full distance between incoherent optical sources in two dimensions

TL;DR

The authors address resolving the full two-dimensional distance $r$ between two incoherent point sources beyond the classical Rayleigh limit. They formulate a four-parameter quantum estimation problem for $g=(\bar X,\bar Y,r,\alpha)^T$ and derive the quantum Fisher information matrix, proving that the distance information remains finite as $r\to0$ and that the bound is saturable with a real PSF. They show that the optimal relative azimuth $\alpha$ depends on the PSF characteristics, enabling significant precision enhancement by aligning the sources along principal axes; for Gaussian PSFs they provide closed-form expressions and identify minor-axis alignment as optimal in general non-circularly symmetric cases. These results offer practical strategies for quantum-enhanced 2D superresolution and extend Rayleigh-limit breaking techniques to full distance estimation in realistic, multidimensional imaging systems.

Abstract

The Rayleigh criterion has long served as a fundamental limit for the resolution of optical imaging. Recent advances in multiparameter quantum metrology have led to quantum superresolution that can break this limit and achieve nonvanishing precision in estimating the separation between a pair of closely located incoherent point sources. For two-dimensional optical systems, the quantum superresolution has been studied for the Cartesian components of separation between two incoherent point sources. However, the precision limit of estimating the full distance between two point sources remains unknown so far. In this paper, we investigate the estimation precision of the full distance between two incoherent point sources with arbitrary intensities in a two-dimensional imaging system. Through the multiparameter quantum estimation theory, we obtain the ultimate estimation precision for the distance and show that it remains nonzero when the distance approaches zero, which surpasses the Rayleigh criterion. We further show the dependence of the estimation precision on the relative orientation between the two point sources, which leads to a novel scheme that can enhance the precision by aligning the sources along proper directions if the point-spread functions are not circularly symmetric, and the enhancement is determined by the extent to which the point-spread functions deviate from circular symmetry. Finally, the results are illustrated by incoherent sources with Gaussian point-spread functions.

Figures (3)

• Figure 1: Two point optical sources located at $(X_{1},Y_{1})$ and $(X_{2},Y_{2})$. The distance between the two sources is $r$, and the relative orientation between the two point sources characterized by the azimuth $\alpha$, i.e., the angle between the displacement of the two sources and a given reference axis which is the $x$ axis in the figure.
• Figure 2: The probability distribution $p(x,y)$ for detecting a single photon from two point sources with identical Gaussian point-spread functions on a two-dimensional image plane with respect to the different azimuth $\alpha$. The plots show how the spatial overlap $\delta$ between the two point-spread functions and the probability distribution of photon detection varies with the azimuth $\alpha$.
• Figure 3: Density plot of the distance precision $\hr$ for two Gaussian point-spread functions with respect to the azimuth $\alpha$ and and the ratio $\epsilon$ of the photon number difference to the total photon number as $r\rightarrow0$. $\sigma_{1}=1$, $\sigma_{2}=1.4$ and $\beta=0.4$. It can be observed that when the azimuth $\alpha$ is fixed, the estimation precision of the distance $r$ will decrease as the ratio $\epsilon$ increases and the optimal estimation precision can be always reached when the photon numbers of the two sources are the same, i.e., $\epsilon=0$, showing the negative impact of photon number difference on the superresolution precision. On the other hand, when $\epsilon$ is fixed, the estimation precision of the distance varies with the azimuth $\alpha$, and the maximum precision of the distance $r$ can be obtained when $\alpha$ is approximately $-\pi/7$, which corresponds to the direction along the minor axis of the Gaussian point-spread function, implying the precision of estimating the distance can be enhanced by optimizing the relative orientation between the two point sources.