Huygens-Fresnel-Kirchhoff construction for quantum propagators with application to diffraction in space and time

We address the phenomenon of diffraction of nonrelativistic matter waves on openings in absorbing screens. To this end, we expand the full quantum propagator, connecting two points on the opposite sides of the screen, in terms of the free particle propagator and spatiotemporal properties of the opening. Our construction, based on the Huygens-Fresnel principle, describes the quantum phenomena of diffraction in space and diffraction in time, as well as the interplay between the two. We illustrate the method by calculating diffraction patterns for localized wave packets passing through various time-dependent openings in one and two spatial dimensions.

Figure 1

Probability density ${\left|\Psi (x,t)\right|}^2$, calculated in accordance with Eq. (37) for various characteristic functions $\chi \left(t_1\right)$, as a function of position $x$. The initial wave packet is characterized by its position $Q=0$, average momentum $P=200$, and dispersion $\sigma =0.1$. The position of the shutter is given by $d=8$, and the propagation time is $t=0.05$. The shutter switching time $t_0=0.04$ and the time window $\epsilon =5\times {10}^{-4}$. All quantities are given in atomic units, $m=\hslash =1$.

Figure 2

Diffraction in time on a curved shutter. Part (a) of the figure illustrates the setup. The initial wave packet is characterized by its average position $\mathbf{Q}=(0,0)$, average momentum $\mathbf{P}=(200,0)$, and dispersion ${\sigma }_1={\sigma }_2=0.1$. The shutter is positioned along the curve $s(x_1,y_1)=x_1-d-\alpha y_1^2=0$ with $d=8$, and its time dependence is specified by $\chi \left(t_1\right)=\Theta (1-|t_1-t_0|/\epsilon )$ with $t_0=0.04$ and $\epsilon =5\times {10}^{-4}$. The propagation time is $t=0.05$. The (light-blue) rectangle, covering $8.9<x<11.1$ and $-1.6<y<1.6$, outlines the spatial region, in which the probability density is computed. Parts (b), (c), and (d) of the figure show ${\ln \left|\Psi (x,y;t)\right|}^2$ for $\alpha =-0.3$, 0, and $0.3$ respectively. All quantities are given in the atomic units, $m=\hslash =1$.

Figure 3

Diffraction on a flat absorbing screen with a time-dependent aperture. Part (a) of the figure illustrates the setup. Parameters of the initial wave packet are the same as in Fig. 2: $\mathbf{Q}=(0,0)$, $\mathbf{P}=(200,0)$, and ${\sigma }_1={\sigma }_2=0.1$. The propagation time is $t=0.05$. The plane of the absorbing screen is defined by $s(x_1,y_1)=x_1-d$ with $d=8$. The (light-blue) rectangle, covering $8.9<x<11.1$ and $-1.6<y<1.6$, outlines the spatial region, in which the probability density is computed. Parts (b), (c), and (d) of the figure show ${\ln \left|\Psi (x,y;t)\right|}^2$ for $\chi ({\mathbf{q}}_1;t_1)$ equal, respectively, to $\Theta (1-|y_1|/\delta )$, $\Theta (1-|y_1|/\delta )\Theta (1-|t_1-t_0|/\epsilon )$, and $\Theta (1-y_1^2/{\delta }^2-{(t_1-t_0)}^2/{\epsilon }^2)$ with $\delta =0.05$, $t_0=0.04$, and $\epsilon =5\times {10}^{-4}$. All quantities are given in atomic units, $m=\hslash =1$.