Quantum delay in the time of arrival of free-falling atoms

Using standard results from statistics, we show that for Gaussian quantum systems the distribution of a time measurement at a fixed position can be directly inferred from the distribution of a position measurement at a fixed time as given by the Born rule. In an application to a quantum particle of mass $m$ falling in a uniform gravitational field $g$, we use this approach to obtain an exact explicit expression for the probability density of the time-of-arrival (TOA). In the long time-of-flight approximation, we predict that the average positive relative shift with respect to the classical TOA in case of a zero initial mean velocity is asymptotically given by $\delta = \frac{q^2}{2}$ when the factor $q\equiv \frac{\hbar}{2m\sigma \sqrt{2gx}} \ll 1$ (semi-classical regime), and by $\delta = \sqrt{\frac{2}{\pi}}q$ when $q\gg 1$ (quantum regime), where $\sigma$ is the width of the initial Gaussian wavepacket and $x$ is the mean distance to the detector. We also discuss experimental conditions under which these predictions can be tested.