Detection statistics in a double-double-slit experiment

In this paper, we analyze the statistics of detection data in a general double-double-slit experiment. The two particles are detected at random times which are not equal in general and because we do not have any constraint on the distances of left and right screens from their slits and the ratio they can be detected in completely different timescales. As the detection of the first particle leads to collapse of the wave function, there is no a straightforward and agreed-upon method to study this problem in the orthodox formalism which lacks a clear prediction of these random events and therefore the quantum state afterward. This is not the case in the Bohmian framework which we implement in this paper, and we can predict the system up to the end of experiment. The main result is the joint distribution of detection data including the arrival time and position of the particles on left and right screens. As one of the main consequences, we see that although the joint spatial distribution can be affected by a change to the relative location of screens, the marginals on each side remain intact, compatible with signal locality. At the end, we see how this result is very sensitive to quantum equilibrium conditions.

Figure 1

A schematic representation of the double-double-slit setup. The source emits pairs of entangled particles, each of which passes through a double slit. Two arrays of fast particle detectors are considered, on both sides, recording the detection events. The $X_L, X_R, 2l_y$, and $2l_x$ represent the distance between the left detection screen and left slits, the distance between the right detection screen and right slits, vertical distance between slits, and horizontal distance between slits, respectively.

Figure 2

The effect of collapse on trajectories. The green curves represent the case without considering collapse and the black ones are the modified trajectories due to the collapse effect. One can see that the right black trajectories gradually deviate from green ones as the left screen detects the particle. Here, $X_R=0.5$ m and $X_L=-0.01$ m. The right screen is beyond the box of this figure and the left one is represented by a vertical gray line located at $X=-0.015$ m. Note that the trajectories do not start from exactly identical initial positions on the $x$ axis; they start from various points in the support of the initial wave function. However, since the width of the initial wave packet in the $x$ direction, ${\sigma }_x$, is considered small compared to ${\sigma }_y$, and also because the $x$ and $y$ axes are not plotted in the same scale, the differences between particles' initial positions in the $x$ direction are very small and not observable.

Figure 3

Here we have depicted joint data distributions in $(y_L,y_R)$ space for different choices of $X_L$ where $X_R=0.5$ m for all the cases where $P_Y$ would be the assumed distribution that the data are sampled from. The green ones are for the cases where we do not take the collapse into account and the black ones are the correct distributions with collapse. The right marginal $P_R$ and the left one $P_L$ are also depicted in the top and right of the joint distribution, respectively. The important point here is that the right marginal is independent of whether there is a measurement on the left and, if there is a measurement, it is independent of the location of the left detection screen. Here, for the sake of better visualization, we have only included ${10}^4$ data points in the joint distributions. However, the marginal ones are computed with our ${10}^6$ data points.

Figure 4

Distribution of arrival position on the right screen. The green curve corresponds to the absence of collapse effect whereas black corresponds to the presence of collapse effect. The distributions are computed based on a nonequilibrium initial positions distribution, i.e., a distribution different from $|{\mathrm{\Psi }}_0{|}^2$. The different panels represent the arrival position distributions for different values of $X_L$. In all cases, the right screen surface is located at $X_R=0.5$ m. As we expect the dark and green distribution coincide when $X_L=X_R$ because in this case, there is not much time for the remaining particle to be evolved under the guidance of the collapsed wave function.