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Physical Review A

Comment on “Quantum time-of-flight distribution for cold trapped atoms”

José Viana‐Gomes, Michael Scott Belsley, Denis Boiron

Published Feb 21, 2008• 4 citations
Publisher

Abstract

Ali et al. [Phys. Rev. A 75, 042110 (2007)] have claimed that empirically testable differences exist between the quantum and semiclassical time of flight distribution for cold trapped atoms. We believe these claims to be misleading. For the particular case of noninteracting bosonic particles released from a harmonic trapping potential, we show that the quantum and semiclassical calculations for the time-of-flight distribution are equivalent, provided one is not too close to the quantum degeneracy point.

DOI
10.1103/physreva.77.026101

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Reply to “Comment on ‘Quantum time-of-flight distribution for cold trapped atoms’ ”

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In their comment Gomes et al. [Phys. Rev. A 77, 026101 (2008)] have questioned the possibility of empirically testable differences existing between the semiclassical time of flight distribution for cold trapped atoms and a quantum distribution discussed by us recently [Ali et al., Phys. Rev. A 75, 042110 (2007).]. We argue that their criticism is based on a semiclassical treatment having restricted applicability for a particular trapping potential. Their claim does not preclude, in general, the possibility of differences between the semiclassical calculations and fully quantum results for the arrival time distribution of freely falling atoms.

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Ali et al. [Phys. Rev. A 75, 042110 (2007)] have claimed that empirically testable differences exist between the quantum and semiclassical time of flight distribution for cold trapped atoms. We believe these claims to be misleading. For the particular case of noninteracting bosonic particles released from a harmonic trapping potential, we show that the quantum and semiclassical calculations for the time-of-flight distribution are equivalent, provided one is not too close to the quantum degeneracy point.
In their comment Gomes et al. [Phys. Rev. A 77, 026101 (2008)] have questioned the possibility of empirically testable differences existing between the semiclassical time of flight distribution for cold trapped atoms and a quantum distribution discussed by us recently [Ali et al., Phys. Rev. A 75, 042110 (2007).]. We argue that their criticism is based on a semiclassical treatment having restricted applicability for a particular trapping potential. Their claim does not preclude, in general, the possibility of differences between the semiclassical calculations and fully quantum results for the arrival time distribution of freely falling atoms.
The time-of-flight distribution for a cloud of cold atoms falling freely under gravity is considered. We generalize the probability current density approach to calculate the quantum arrival time distribution for the mixed state describing the Maxwell-Boltzmann distribution of velocities for the falling atoms. We find an empirically testable difference between the time-of-flight distribution calculated using the quantum probability current and that obtained from a purely classical treatment which is usually employed in analyzing time-of-flight measurements. The classical time-of-flight distribution matches with the quantum distribution in the large mass and high temperature limits.
It is shown that a class of exponentially decaying time-of-arrival probability distributions, suggested by W\l{}odarz [Phys. Rev. A 65, 044103 (2002)], Marchewka and Schuss [Phys. Lett. A 240, 177 (1998), Phys. Rev. A 63, 032108 (2001); Phys. Rev. A 65, 042112 (2002)], and Jurman and Nikoli\ifmmode \acute{c}\else \'{c}\fi{} [Phys. Lett. A 396, 127247 (2021)], and a semiclassical distribution implicit in time-of-flight momentum measurements do not show the expected behavior for a Gaussian wave train. This casts doubts on the physical adequacy of these arrival-time proposals. In contrast, the quantum flux distribution (a special case of the Bohmian arrival-time distribution) displays the expected behavior.
We point out that a certain kind of combined classical translational and spin dynamics -- claimed in [M. Pletyukhov, et al. Phys. Rev. Lett. 89 (2002) 116601] to arise from the Pauli equation in the semiclassical limit ℏ→0\hbar\to0ℏ→0 for fixed spin -- only shows up if one simultaneously considers the high spin limit S→∞S\to\inftyS→∞.
Motivated by a recent prediction [Commun. Phys. 6, 195 (2023).] that time-of-flight experiments with ultracold atoms could test different interpretations of quantum mechanics, this work investigates the arrival times predicted by the stochastic interpretation, whereby quantum particles follow definite but nondeterministic and nondifferentiable trajectories. The distribution of arrival times is obtained from a Fokker-Planck equation and confirmed by direct simulation of trajectories. It is found to be, in general, different from the distribution predicted by the Bohmian interpretation, in which quantum particles follow definite deterministic and differentiable trajectories. This result suggests that trajectory-based interpretations of quantum mechanics could be experimentally discriminated.
We study the expansion of an interacting atomic system at zero temperature, following its release from an isotropic three-dimensional harmonic trap and calculate the time dependence of its density and momentum distribution, with special focus on the behavior of the contact parameter. We consider different quantum systems, including the unitary Fermi gas of infinite scattering length, the weakly interacting Bose gas, and two interacting particles with highly asymmetric mass imbalance. In all cases analytic results can be obtained, which show that the initial value of the contact, fixing the 1/k41/{k}^{4}1/k4 tail of the momentum distribution, disappears for large expansion times. Our results raise the problem of understanding the recent experiment of R. Chang et al. [Phys. Rev. Lett. 117, 235303 (2016)] carried out on a weakly interacting Bose gas of metastable 4He^{4}\mathrm{He}4He atoms, where a 1/r41/{r}^{4}1/r4 tail in the density distribution was observed after a large expansion time, implying the existence of the 1/k41/{k}^{4}1/k4 tail in the asymptotic momentum distribution.