Fisher information 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. Subsequently focusing on the concrete case of the momentum parameter of a 1D particle, we consider the idealized 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.