Double-slit experiment revisited

TL;DR

The paper argues that a complete quantum description of the double-slit experiment requires the joint distribution $ ho(x,y,t_f)$ of detection position and time, beyond the standard fringe pattern. It adopts Bohmian mechanics to define a first-passage time distribution for detections, connecting it to the flux $j_ot(oldsymbol{r},t_f)$ and illustrating the approach with numerical simulations of an initial Gaussian wave packet passing through a double slit, including a dynamic variant where a slit closes during flight. The comparison with the Kurtsiefer–Pfau–Mlynek experiment shows that observed ToF features arise mainly from velocity spreading, not quantum longitudinal spreading, and the authors propose ToF-resolved double-slit and dynamic double-slit experiments with improved state control to test quantum time distributions. Overall, the work provides a concrete framework for arrival-time questions in quantum mechanics and outlines practical experimental paths to test quantum ToF concepts in the DSE.

Abstract

The double-slit experiment is one of the quintessential quantum experiments. However, it tends to be overlooked that a theoretical account of this experiment requires the specification of the joint position and time distribution of detection at the screen, whose position marginal yields the famous interference pattern. The difficulty then arises what this distribution should be. While there exists a variety of proposals for a quantum mechanical time observable, there is no consensus about the right choice. Here, we consider Bohmian mechanics, which allows for a natural and practical approach to this problem. We simulate this distribution in the case of an initial Gaussian wave packet passing through a double-slit potential. We also consider a more challenging setup in which one of the slits is shut during flight. To experimentally probe the quantum nature of the time distribution, a sufficient longitudinal spread of the initial wave packet is required, which has not been achieved so far. Without sufficient spread, the temporal aspect of the distribution can be treated classically. We illustrate this for the case of the double-slit experiment with helium atoms by Kurtsiefer et al. [Nature 386, 150 (1997)], which reports the joint position and time distribution.

Figures (6)

• Figure 1: Figures from the KPM experiment Pfaudetector (reproduced with permission). (a) Double-slit interference pattern observed on a screen after many single-atom detection events. (b) Joint distribution of the ToF $t_{\!f}$ and the $x$-screen-coordinate of the detection events. Impact positions of atoms arriving in the $t_{\!f}$ range indicated by the dashed lines in (b) were accumulated to produce (a). A fraction of very fast atoms emanating from the source cast a shadow of the slits at the bottom.
• Figure 2: (a) Schematic illustration of the double-slit potential \ref{['eq:pot']} for ${f_1=f_2=1}$. (b) A collection of $5\times10^5$ Bohmian trajectories in the $xz$-plane for the DSE with initial conditions (black dots) sampled randomly from the $|\psi(\vb{r},0)|^2$-distribution. Only the trajectories which pass through the slits (i.e. those which are not back-scattered by the double-slit potential) are shown. Each curve tracks a particle that eventually strikes the detection plane ${z=d}$. The following potential barrier and wave packet parameters were assumed: ${V_0=10^3}$, ${\sigma_b=0.125}$, ${\sigma_s=0.25}$, ${k_x=x_0=0}$, ${k_z=15}$, ${z_0=2}$, and ${\sigma_x=3,\sigma_z=0.25}$ in units where ${\hbar=m=\Delta=1}$. A number of selected trajectories (in white) display the characteristic non-Newtonian meandering of the trajectories.
• Figure 3: Numerical data generated for ${V_0=10^3}$, ${\sigma_b=0.125}$, ${\sigma_s=0.5}$, ${k_x=0}$, ${k_z=15}$, ${z_0=3}$, ${\sigma_x=3}$, ${\sigma_z=0.25}$, with ${d=0.9}$ (top), ${d=10}$ (middle), and ${d=15}$ (bottom), taking ${\hbar=m=\Delta=1}$. (The slits were closed after the passage of the wave packet to avoid reflection from the boundary of the grid, see \ref{['eq:dynamicT']}; $t_c=0.5$, $\gamma=4$.)
• Figure 4: A few snapshots of the evolving $|\psi|^2$-density, using the same parameters as in Fig. \ref{['fig:triple_hist']}. Note the closing of the slits (see \ref{['eq:dynamicT']}; $t_c=0.75$, $\gamma=4$) to prevent the reflected wave from influencing the arrivals.
• Figure 5: Scatter plot of ${\approx10^5}$ Bohmian trajectories arriving at ${d=10}$ over time for the dynamic DSE, where the slit centred at ${x=-\Delta}$ (${=-1}$ in our units) is closed in flight around time ${t_c=0.25}$: (a) Fast switching ${\gamma=100}$, (b) Slow switching ${\gamma=20}$. Data points for trajectories passing through the closing (open) slit are rendered in brown (cyan). The potential barrier and initial wave packet parameters are same as for Fig. \ref{['fig:triple_hist']}.