Bragg Scattering from a Random Potential

A potential for propagation of a wave in two dimensions is constructed from a random superposition of plane waves around all propagation angles. Surprisingly, despite the lack of periodic structure, sharp Bragg diffraction of the wave is observed, analogous to a powder diffraction pattern. The scattering is partially resonant, so Fermi’s golden rule does not apply. This phenomenon would be experimentally observable by sending an atomic beam into a chaotic cavity populated by a single mode laser.

Figure 1

Bragg scattering of a wave packet in a random (Berry) potential. The potential is drawn in gray scale on the background. An initial wave packet on the top launched downward into the potential with an average momentum $\hslash \overrightarrow{k}$ is depicted in green and magenta scale. The wave function at a later time is shown in red and blue scale. The inset shows the probability density of the wave at a later time in the reciprocal space in dark blue scale.

Figure 2

Dependence of the wave evolution on $\hslash$. The real part of the wave function is plotted in red and blue scale and an inset in each panel shows the probability density distribution in the reciprocal space. For small $\hslash$, i.e., $k\gg q$ (small wavelength and classical limit), the wave dynamics is particlelike, diffusive, showing branched flow [16]. This is consistent with a very small Bragg angle which leads to repeated almost-forward scattering. For intermediate $\hslash$, i.e., $k\sim q$ the Bragg angle is not small, giving less classical-like diffractive behavior. For large $\hslash$, i.e., $k\ll q$ the wave is in a transparency regime: the wavelength is large enough that the small scale fluctuations of the potential are averaged to zero. Or equivalently, in reciprocal space, there are no intersections between the energy contour and the nonzero Fourier components of the potential. Therefore, there is effectively no scattering, and the potential is transparent.

Figure 3

The real part of the first-order wave function ${\psi }^{(1)}(\overrightarrow{r},t)$ from a wave packet initially launched downward, showing the interference of the instantaneous scattered amplitudes from different times.

Figure 4

Population of the scattered waves as a function of time. Blue and red curves correspond to simulation and first order time-dependent perturbation theory results, respectively. The population growth is not linear, i.e., Fermi’s golden rule is not valid. Also, the population growth stops occasionally, showing that the pulses (scattered waves) come out sporadically as shown in Figs. 1 and 2. The two results coincide at early times when single scattering dominates. The discrepancy later is due to multiple scattering, absent in the first order time-dependent perturbation theory.

Figure 5

Bragg scattering in a Berry potential: a special case of the 90° scattering angle. The real part of the wave function was plotted in red and blue scale. The inset shows the momentum space probability distribution of the wave. This is a snapshot, after a long time propagation of a downward launched wave packet. It is not diffusing in angle beyond the 90° turns, in spite of the random nature of the potential. A periodic boundary condition is used.