Fisher information for the momentum parameter from quantum many-particle arrival time measurements

We formulate a quantum arrival time measurement process for a Bosonic many-particle system, with the aim of extracting statistical information on single-particle properties. The arrival time is based on a dynamical multi-particle absorption model in the Fock space, and we consider systems in coherent and incoherent mixtures of N-particle states. We find the resulting probability distributions for arrival time sequences, which we consider as parametric models for the statistical inference of single-particle parameters, and derive a tractable expression for the associated (classical) Fisher information for a general single particle parameter. Subsequently focusing on the concrete case of the momentum parameter of a 1D particle, we consider the idealised limits of a point (Dirac delta) detector and an infinite particle system forming a spatially uniform ‘beam’. We observe that even though no information remains in the spatial distribution, the single-particle momentum is indeed identifiable from the arrival time data, even in the limit of ‘sparse beams’ of vanishing particle density, where we obtain simple analytical form for the Fisher information, which, interestingly, coincides with the one obtained from a hypothetical time-stationary detection model. Our results contribute to the fundamental understanding of temporal measurement data arising from quantum systems consisting of freely evolving particles.