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

• URL: http://arxiv.org/abs/2306.02141v2
• Date: Tue, 23 Jan 2024 12:30:26 GMT
• Title: Quantum delay in the time of arrival of free-falling atoms
• Authors: Mathieu Beau and Lionel Martellini
• Abstract summary: We show that 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.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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