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arXiv:2502.21295 · Journal of Physics Conference Series

Quantum arrival times in free fall

Mathieu Beau, Timothey Szczepanski, Rafael Martellini, Lionel Martellini

Published Jun 1, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract The probability distribution of a time measurement Tx at position x can be inferred from the probability distribution of a position measurement Xt at time t as given by the Born rule [1, 2]. In an application to free-fall, this finding has been used to predict the existence of a mass-dependent positive relative shift with respect to the classical time-of-arrival in the long time-of-flight regime for dropped quantum particles [1]. The present paper extends these results in two important directions. We first show that for a Gaussian quantum particle of mass m dropped in a uniform gravitational field g, the uncertainties about time and position measurements are related by the relation Δ T x Δ X t ≥ ℏ 2 m g This novel form of uncertainty relation suggests that choosing the initial state so as to obtain a lower uncertainty in the measured position leads to a higher uncertainty in the measured arrival time. Secondly, we examine the case of a free-falling particle starting from a non-Gaussian initial superposed state, for which we predict the presence of gravitationally induced interferences and oscillations in the mean time-of-arrival as a function of the detector’s position that can be interpreted as the signature of a Zitterbewegung-like effect.

DOI
10.1088/1742-6596/3017/1/012022

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Abstract The probability distribution of a time measurement Tx at position x can be inferred from the probability distribution of a position measurement Xt at time t as given by the Born rule [1, 2]. In an application to free-fall, this finding has been used to predict the existence of a mass-dependent positive relative shift with respect to the classical time-of-arrival in the long time-of-flight regime for dropped quantum particles [1]. The present paper extends these results in two important directions. We first show that for a Gaussian quantum particle of mass m dropped in a uniform gravitational field g, the uncertainties about time and position measurements are related by the relation Δ T x Δ X t ≥ ℏ 2 m g This novel form of uncertainty relation suggests that choosing the initial state so as to obtain a lower uncertainty in the measured position leads to a higher uncertainty in the measured arrival time. Secondly, we examine the case of a free-falling particle starting from a non-Gaussian initial superposed state, for which we predict the presence of gravitationally induced interferences and oscillations in the mean time-of-arrival as a function of the detector’s position that can be interpreted as the signature of a Zitterbewegung-like effect.
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 mmm falling in a uniform gravitational field ggg, 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 $\ensuremath{\delta}=\frac{{q}^{2}}{2}$ when the factor $q\ensuremath{\equiv}\frac{\ensuremath{\hbar}}{2m\ensuremath{\sigma}\sqrt{2gx}}\ensuremath{\ll}1$ (semiclassical regime) and by $\ensuremath{\delta}=\sqrt{{\frac{2}{\ensuremath{\pi}}}^{\phantom{\ensuremath{\int}}}}q$ when $q\ensuremath{\gg}1$ (quantum regime), where $\ensuremath{\sigma}$ is the width of the initial Gaussian wave packet and xxx is the mean distance to the detector. We also discuss experimental conditions under which these predictions can be tested.
We study the time-of-arrival (TOA) distribution of a Gaussian wave packet in the semiclassical regime for a general one-dimensional quadratic Hamiltonian, namely a forced harmonic oscillator with time-dependent frequency. Expanding the Gaussian probability density as a series in derivatives of the Dirac delta and using the standard rules for the composition of distributions with the classical trajectory, we obtain compact closed-form expressions for the leading semiclassical mean arrival time and its standard deviation. The mean acquires a quantum shift of order σ2σ^2σ2 set by the classical velocity and acceleration and by the time derivative of the Ermakov width, while the spread reduces to the simple form ΔTx≃σ(tx)/vxΔ\mathcal{T}_x\simeqσ(t_x)/v_xΔTx​≃σ(tx​)/vx​. The latter implies a time--position uncertainty relation that depends only on the fundamental solutions of the classical equation of motion. We apply the framework to free fall and to two models of a time-dependent opening trap under gravity, and we validate every expression against exact numerical integration of the TOA distribution. These results extend earlier free-fall predictions to realistic settings in which the trapping potential cannot be neglected.
The weak equivalence principle of gravity is examined at the quantum level in two ways.First, the position detection probabilities of particles described by a non-Gaussian wave packet projected upwards against gravity around the classical turning point and also around the point of initial projection are calculated.These probabilities exhibit mass dependence at both these points, thereby reflecting the quantum violation of the weak equivalence principle.Second, the mean arrival time of freely falling particles is calculated using the quantum probability current, which also turns out to be mass dependent.Such a mass dependence is shown to be enhanced by increasing the non-Gaussianity parameter of the wave packet, thus signifying a stronger violation of the weak equivalence principle through a greater departure from Gaussianity of the initial wave packet.The mass dependence of both the position detection probabilities and the mean arrival time vanishes in the limit of large mass.Thus, compatibility between the weak equivalence principle and quantum mechanics is recovered in the macroscopic limit of the latter.A selection of Bohm trajectories is exhibited to illustrate these features in the free fall case.
We reconsider Baute et al.'s free crossing states [Phys. Rev. A 61, 022118 (2000)] and show that if we require a generalization in the interacting case that goes in complete parallel with the free-particle case, then this generalized crossing state cannot be arbitrary but is determined by the null space of the particle's quantum time-of-arrival operator. Nonetheless, the free crossing states appear as the leading term in the asymptotic expansion of our generalized crossing state in the limit of large momentum. We then examine the quantum time-of-arrival problem of a spinless particle in a uniform gravitational field. Mass-dependent time-of-arrival probability distributions emerge, signifying quantum departures from the weak equivalence principle. However, in the classical limit of large mass and vanishing uncertainty in position, the mass dependence of the quantum time-of-arrival distribution becomes exponentially small and the mean quantum time of arrival reduces to the classical time of arrival.
Using standard results from statistics, we show that for any continuous quantum system (Gaussian or otherwise) and any observable $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}$ (position or otherwise), the distribution ${\ensuremath{\pi}}_{a}(t)$ of time measurement at a fixed state aaa can be inferred from the distribution ${\ensuremath{\rho}}_{t}(a)$ of a state measurement at a fixed time ttt via the transformation ${\ensuremath{\pi}}_{a}(t)\ensuremath{\propto}|\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}{\ensuremath{\int}}_{\ensuremath{-}\ensuremath{\infty}}^{a}{\ensuremath{\rho}}_{t}(u)du|$. This finding suggests that the answer to the long-lasting time-of-arrival problem is in fact secretly hidden within the Born rule and therefore does not require the introduction of a time operator or a commitment to a specific (e.g., Bohmian) ontology. The generality and versatility of the result are illustrated by applications to the time of arrival at a given location for a free particle in a superposed state and to the time required to reach a given velocity for a free-falling quantum particle. Our approach also offers a potentially promising new avenue toward the design of an experimental protocol for the yet-to-be-performed observation of the phenomenon of quantum backflow.
Violation of the gravitational weak equivalence principle in quantum mechanics (WEQ) has been earlier studied using Gaussian wave packets in free fall.Here we study the effect of quantum statistics on the -Gaussian as well as non rrival time distribution of quantum particles computed through the probability current density.We show that a symmetrization or asymmetrization of the wave function impacts the arrival time distribution of wavepackets.The . the mass of the particles, and varies according to the statistics mean arrival time is dependent on