Quantum chaos and the double-slit experiment

We report on the numerical simulation of the double-slit experiment, where the initial wave packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behavior.

Figure 1

The geometry of the numerical double-slit experiment. All scales are in proper proportions. The two slits are placed at a distance $s$ on the lower side of the billiard

Figure 2

(Color) The total intensity after the double-slit experiment as a function of the position on the screen. $I\left(x\right)$ is obtained as the perpendicular component of the probability current, integrated in time. The red full curve indicates the case of regular billiard, while the blue dotted curve indicates the case of chaotic one. The green dashed curve indicates the averaged intensity over two 1-slit experiments, with either the regular or chaotic billiard (with results being practically the same, see Fig. 3).

Figure 3

(Color) The two pairs of curves represent the intensities on the screen for the two 1-slit experiments (with either one of the two slits closed). The red full curves indicate the case of the regular billiard while the blue dotted ones indicate the case of chaotic billiard.

Figure 4

(Color) Typical snapshots of the wave function (plotted is the probability density) for the two cases: (a) for the regular billiard at $t=0.325$ and (b) for the chaotic billiard at $t=0.275$ (both cases correspond to about half the Heisenberg time). The probability density is normalized separately in both parts of each plot, namely, the probability density, in absolute units, in the radiating region is typically less than 1% of the probability density in the billiard domain. The screen, its center, and the positions of the slits are indicated with thin black lines. Please note that the color code on the top of the figure is proportional to the square root of probability density.