Quasiprobability for the arrival-time problem with links to backflow and the Leggett-Garg inequalities

Abstract

The arrival time problem for the free particle in one dimension may be formulated as the problem of determining a joint probability for the particle being found on opposite sides of the $x$-axis at two different times. We explore this problem using a two-time quasi-probability linear in the projection operators, a natural counterpart of the corresponding classical problem. We show that it can be measured either indirectly, by measuring its moments in different experiments, or directly, in a single experiment using a pair of sequential measurements in which the first measurement is weak (or more generally, ambiguous). We argue that when positive, it corresponds to a measurement-independent arrival time probability. For small time intervals it coincides approximately with the time-averaged current, in agreement with semiclassical expectations. The quasi-probability can be negative and we exhibit a number of situations in which this is the case. We interpret these situations as the presence of `quantumness', in which the arrival time probability is not properly defined in a measurement-independent manner. Backflow states, in which the current flows in the direction opposite to the momentum, are shown to provide an interesting class of examples such situations. We also show that the quasi-probability is closely linked to a set of two-time Leggett-Garg inequalities, which test for macroscopic realism.