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arXiv:2606.20281 · Physical Review A

Arrival times of an atomic Bose-Einstein condensate

Pascal Naidon, Lucas Happ, Denis Boiron

cond-mat.quant-gasquant-phSubmitted Jun 18, 2026• Announced Jun 19, 2026• 0 citations
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Abstract

The times of flight of an atomic Bose-Einstein condensate are theoretically investigated in the experimentally unexplored regime corresponding to detection close to the trap of the condensate. In this regime, there is no consensus on how to calculate the distribution of times of arrival onto the detector. For non-interacting particles, distinct theoretical predictions have been made in the past. This work analyses how these predictions are modified for an interacting Bose-Einstein condensate. For this purpose, a time-dependent Gross-Pitaevskii equation is solved analytically and numerically.

Comments
13 pages, 11 figures

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The times of flight of an atomic Bose-Einstein condensate are theoretically investigated in the experimentally unexplored regime corresponding to detection close to the trap of the condensate. In this regime, there is no consensus on how to calculate the distribution of times of arrival onto the detector. For non-interacting particles, distinct theoretical predictions have been made in the past. This work analyses how these predictions are modified for an interacting Bose-Einstein condensate. For this purpose, a time-dependent Gross-Pitaevskii equation is solved analytically and numerically.
We solve the time-dependent Gross-Pitaevskii equation by a variational ansatz to calculate the excitation spectrum of a Bose-Einstein condensate in a trap. The trial wave function is a Gaussian which allows an essentially analytical treatment of the problem. Our results reproduce numerical calculations over the whole range from small to large particle numbers, and agree exactly with the Stringari results in the strong interaction limit. Excellent agreement is obtained with the recent JILA experiment and predictions for the negative scattering length case are also made.
This work shows how a Bose-Einstein condensate of ultracold atoms could be used to address a long-standing question in quantum theory: how much time does it take for a particle to reach a detector? To this end, we propose a realistic experimental setup, whose key idea is not to measure arrival times directly, but the arrival flux on the detector as a function of its position. This novel approach not only solves practical issues with having a detector close to the system, but also results in signals that allow to unambiguously distinguish different theoretical predictions. This proposal raises prospects for resolving the decades-old debate on this fundamental issue.
We present a numerical study of the coupled time-dependent Gross-Pitaevskii equation, which describes the Bose-Einstein condensate of several types of trapped bosons at ultralow temperature with both attractive and repulsive interatomic interactions. The same approach is used to study both stationary and time-evolution problems. We consider up to four types of atoms in the study of stationary problems. We consider the time-evolution problems where the frequencies of the traps or the atomic scattering lengths are suddenly changed in a stable preformed condensate. We also study the effect of periodically varying these frequencies or scattering lengths on a preformed condensate. These changes introduce oscillations in the condensate, which are studied in detail. Good convergence is obtained in all cases studied.
The properties of vortex states in a Bose-Einstein-condensed cloud of atoms are considered at zero temperature. Using both analytical and numerical methods, we solve the time-dependent Gross-Pitaevskii equation for the case when a cloud of atoms containing a vortex is released from a trap. In two dimensions we find the simple result that the time dependence of the cloud radius is given by $(1+{\ensuremath{\omega}}^{2}{t}^{2}{)}^{1/2},$ where $\ensuremath{\omega}$ is the trap frequency. We calculate and compare the expansion of the vortex core and the cloud radius for different numbers of particles and interaction strengths, in both two and three dimensions, and discuss the circumstances under which vortex states may be observed experimentally.
The focusing of a propagating untrapped Bose-Einstein condensate is studied theoretically. We use a scaling solution method comprising a time-dependent scaling function to analytically examine the dynamics of a falling Bose-Einstein condensate in different regimes of propagation including the expansion and compression zones. Our model is based on the Gross--Pitaevskii equation which involves the interparticle interactions between atoms and, consequently, their influence on the focused structures. We investigate the focused profile characteristic factors such as the resolution and peak density for various cases of the focusing optical potential parameters as well as the factors associated with the moving cloud. Our results are compared with numerical solutions of the Gross--Pitaevskii equation.
A variational technique is applied to solve the time-dependent nonlinear Schr\"odinger equation (Gross-Pitaevskii equation) with the goal to model the dynamics of dilute ultracold atom clouds in the Bose-Einstein condensed phase. We derive analytical predictions for the collapse, equilibrium widths, and evolution laws of the condensate parameters and find them to be in very good agreement with our numerical simulations of the nonlinear Schr\"odinger equation. It is found that not only the number of particles, but also both the initial width of the condensate and the effect of different perturbations to the condensate may play a crucial role in the collapse dynamics. The results are applicable when the shape of the condensate is sufficiently simple.