Quantum Limited Source Localization and Pair Superresolution under Finite Emission Bandwidth

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.