Quantum Limited Source Localization and Pair Superresolution under Finite Emission Bandwidth

TL;DR

This paper investigates the fundamental limits of localizing and resolving pairs of quantum sources with finite emission bandwidths, revealing how bandwidth affects localization precision and superresolution capabilities through quantum Fisher information analysis.

Contribution

It introduces a quantum Fisher information framework using prolate spheroidal wave functions to quantify bandwidth effects on source localization and pair superresolution.

Findings

Localization precision degrades with increasing emission bandwidth.

Quantum Fisher information quantifies the minimum achievable estimation variance.

Bandwidth influences the effective dimensionality of the localization problem.

Abstract

Optically localizing a single quasi-monochromatic source to sub-diffractive precisions entails, in the photon-counting limit, a minimum photon cost that scales as the squared ratio of the width, $w$, of the optical system's point-spread function (PSF) and the sought localization precision, $d$, i.e., as $\alpha(w/d)^2$. For sources with a finite emission-frequency spectrum, while the inverse quadratic scaling is expected to remain unchanged, the coefficient $\alpha$ must increase due to a degrading fidelity of localization as the imaging bandwidth increases and PSF undergoes a frequency-dependent widening. We specifically address how rapidly $\alpha$ must increase with increasing width of a flat-top spectral profile of emission of a point source being localized in two dimensions by an imager with a clear circular aperture by calculating quantum Fisher information (QFI), whose inverse yields the lowest possible unbiased-estimation variance of source-localization error. The novel use of prolate spheroidal wave functions as a basis for obtaining a solution of the eigenvalue problem of the single-photon density operator needed for the QFI calculation helps us develop the notion of an effective dimensionality of the continuous-state problem in terms of the associated space-bandwidth parameter. We subsequently extend our considerations of QFI to treat the finite-bandwidth pair superresolution problem in two dimensions, obtaining similar results. We also consider generalizations to emission power spectra of arbitrary profiles.