Despite the many successes of quantum mechanics since its inception almost a century ago, there are still some significant foundational problems for which we have yet to reach a consensus on solutions. Especially, there exist two interconnected fundamental issues —the measurement problem and the arrival time problem. Various interpretations, alternative theories, and formulations attempt to tackle different aspects of these issues. However, to our knowledge, none of these alternative approaches have been clearly distinguished from each other through strict experiments.

When and where does the wave function collapse? 

This is the essence question that we investigated in our work. We tried to show that there is no agreed-upon answer to this question, even for the double-slit experiment which in the words of Feynman

 The double-slit experiment …has in it the heart of quantum mechanics. In reality, it contains the only mystery ... nobody can give you a deeper explanation of this phenomenon than I have given; that is, a description of it.

The main issue is that, although time can be routinely observed in quantum experiments (see), it is not a quantum observable in the standard scenes— time is a parameter, not a self-adjoint operator. So, there is no agreed-upon way to calculate the spatiotemporal probability distribution of detection events. Note that, the square of the wave function, |ψ(x,t)|², is just a conditional position probability density at a specific time, not joint spatiotemporal distribution. In fact, there are various predictions for the joint spatiotemporal distribution of particle detection events on a screen, which are derived from different formulations and interpretations of the quantum theory. Although the differences are typically small, we show that these predictions can be experimentally distinguished by a proposed unconventional double-slit configuration, which is realizable using present-day single-atom interferometry. 

Different approaches, different predictions, and different technologies

In experiments conducted in the far-field regime, the semi-classical approximation is traditionally used to estimate particle arrival time. However, with the emergence of advanced technologies like gravimetry and ghost imaging via atoms, it has become imperative to make precise predictions about the arrival time in order to achieve better results.

There have been various attempts to introduce a suitable arrival time distribution based on different interpretations and formulations of quantum mechanics (see). The outcomes of these methods differ in principle, and determining which one is more accurate could directly impact technologies.

It is important to note that some conditions may not have specific treatments in all approaches.  For instance, when dealing with entangled particles in the presence of external potential, most approaches are not sufficiently developed, while the Bohmian approach can lead to clear predictions in these situations. On this matter, a recent Bohmian analysis revealed non-local interference in the arrival time of entangled pairs of atoms (see). 

Double-slit experiment as a touchstone for quantum foundations

The purpose of our paper is to make it evident, via numerical simulations, that the famous two-slit experiment could be utilized to distinguish different approaches if we simply use a horizontal screen instead of a vertical one. Using current laser cooling and magneto-optical trapping technologies, this type of experiment can be realized by Bose-Einstein condensates, as a controllable source of coherent matter waves. Moreover, our numerical study shows that the required space-time resolution in particle detection is achievable using present-day fast single-atom detectors.

This is a practical encounter with the measurement problem (see). Our proposed experiment can help us understand the differences between different methods and enhance our knowledge of the foundations of quantum mechanics.