Spatial statistics of single-quantum detection

In a single-particle detection experiment, a wavefront impinges on a detector but observers only see a point response. The extent of the wavefront becomes evident only in statistical accumulation of many independent detections, with probability given by the Born rule. Drawing on concepts from quantum optics, we analyze a simple model to reverse-engineer how this behavior can come about in terms of wave mechanics alone without a measurement axiom. The model detector consists of many molecules, each of which can be resonantly excited by the incoming particle and then emit a detection signature (e.g., localized flash of light). Different molecules have different resonant energies because local conditions (proximity of other molecules, Doppler shifts, etc.) vary. The detector is thus a quasi-continuum, and the incoming particle preferentially excites the molecule that it matches most closely in energy. (In actuality, molecules can be so numerous that many could closely match the incoming particle in energy; but in that case only one will be the first, and there will be nothing left for the others by the time the first match resonates and then emits. The model does not explicitly take into account the temporal advance of the particle wave packet through the detector medium.) The excited molecule can emit a detection signature, but that process competes with fluctuation-driven dephasing. We estimate the probability that a given molecule is resonantly excited, and we also estimate the probability that a detection signature is produced before being overwhelmed by fluctuations in the detector medium. The product of these two probabilities is proportional to the absolute-square of the incoming wavefunction at the molecule in question, i.e. the Born rule. We discuss ways to probe these mechanisms experimentally, and check the model with numbers from a neutron interference experiment.