Detector models for the quantum time of arrival

In this chapter we have discussed a model to measure the time of arrival of an atom. The basic idea is that upon entering a laser-illuminated region the atom will start emitting photons, and the first-photon detection can be taken as a measure of the arrival time of the atom in that region. We have shown the explicit connection between this atom-laser model and approaches where the detector is modeled in a simplified manner by a local complex potential, with the arrival probability distribution given by the decrease rate of the norm of the quantum wave function. By using deconvolution techniques, we can measure, according to this model, the quantum mechanical flux in a limit. The quantum mechanical flux is in many cases a good approximation to Kijowski?s distribution, an ideal reference distribution for the arrival time of a particle. By applying operator normalization we can get also Kijowski?s distribution directly from the fluorescence in some limits, which provides an operational interpretation of Kijowski?s distribution. The quantum mechanical flux and Kijowski?s distribution are examples of local densities associated with a single classical local density. Other examples are the quantum kinetic energy densities that could also be measured in some limits. Summarizing, we have shown that several local densities of an atomic wavepacket (quantum mechanical flux, Kijowski?s time-of-arrival distributions, kinetic energy densities) can be measured for different limits of the laser shape or intensity and by means of different manipulations of the fluorescence signal or the initial state. One could in this manner observe quantum dynamical effects that have not been realized experimentally so far, for example, the backflow effect (negative fluxes for positive?momentum wavepackets), and quite generally distinguish quantum from classical dynamics and arrival times. Of course actual measurements will generally approximate these limits and operations only to some degree and this will determine the quantity which is really measured. We emphasize though that it is only through extreme operations and idealized limits that we get access to the properties of the bare, freely moving system. There are still many open questions. For example, the atom-laser model has been examined until now on a single-particle level. An interesting task is to extend the model and examine many-particle effects. Another route of research is to apply the atom-laser model to other time quantities.