Single-photon sub-Rayleigh precision measurements of a pair of incoherent sources of unequal intensity

Interferometric methods have been recently investigated to achieve sub-Rayleigh imaging and precision measurements of faint incoherent sources up to the ultimate quantum limit. Here we consider single-photon imaging of two point-like emitters of unequal intensity. This is motivated by the fact that pairs of natural emitters typically have unequal brightness, for example, binary star systems and exoplanets. We address the problem of estimating the transverse separation d or the relative intensity \epsilon$\epsilon$. Our theoretical analysis shows that the associated statistical errors are qualitatively different from the case of equal intensity. We employ multi-plane light conversion technology to implement Hermite–Gaussian (HG) spatial-mode demultiplexing (SPADE), and demonstrate sub-Rayleigh measurement of two emitters with a Gaussian point-spread function. The experimental errors are comparable with the theoretical bounds. The latter are benchmarked against direct imaging, yielding an \epsilon ^{-1/2}$\epsilon ^{-1/2}$ improvement in the signal-to-noise ratio, which may be significant when the primary source is much brighter than the secondary one, for example, as for imaging of exoplanets.