Quantum-Limited Time-Frequency Multiparameter Estimation through Mode-Selective Photon Measurement

In optics, when attempting to find the distance separations s between two incoherent emitters, it is standard practice to take a full image and extract the single parameter s from it. In such scenarios, the smallest precisely measurable separation is limited by the point-spread function on the image plane, with a vanishingly small amount of informa­tion available for smaller and smaller separations especially when photon shot noise is the dominant noise source (as in, for example, astronomical observations). This limit can be formalised by the Fisher information, which quantifies the amount of information accessible per photon detection and is directly associated with Cramér-Rao lower bound. For direct intensity imaging, the Fisher information drops to zero for object separations smaller than the spread of the optical field, sometimes referred to as the Rayleigh's curse, which limits the usefulness of photon counting for metrology.