Directionality emergence and localization in a quantum random Lorentz gas

Abstract

The propagation of a spherical wave through a two-dimensional random Lorentz gas composed of small fixed scatterers is studied. Inspired by the Mott problem (how an initially isotropic quantum wave can give rise to a single particle-like track), we investigate, on a schematic model, whether such a directional behavior can emerge purely from the multiscattering process, without any explicit measurement or decoherence mechanism. Using the Foldy-Lax formalism, we derive the far-field angular behavior of the wavefunction, and introduce a directionality vector to quantify its anisotropy and identify its preferred direction. Numerical simulations reveal the existence of a strongly directional regime within a specific wavenumber range, which emerges from multiscattering with more than $100$ scatterers and which can be related to Anderson localization.

Figures (5)

• Figure 1: Schematic comparison of the asymptotic behavior of a quantum spherical Gamow wave (left) and a classical $\alpha$-particle (right). The very distinct nature of these two propagation modes is the motivation for the study of the angular probability density.
• Figure 2: (a) Evolution of the directionality $w_x$ in a single-scatterer gas as a function of $ks$. The red circles are the zeros of $Y_0$ and the blue circles are the zeros of $J_1$. (b-d) Square modulus of the full wavefunction $|\psi(k, \bm{r} | \bm{x}_0)|^2$ and angular PDF $\mathrm{d} P (\theta) / \mathrm{d} \theta$, respectively for (b) $ks \approx 0.55$ (focalizer regime), (c) $ks \approx 1.75$ and (d) $ks \approx 14.9$ (reflector regimes). The spherical wave source is represented by a star, and the unique scatterer by a black dot. The dotted and dashed lines represent the hyperbolic fringes satisfying \ref{['eq:single_particle_hyperbolic_fringes']}.
• Figure 3: Evolution of $\bm{w}^2$ as a function of $k \varsigma$, for a gas containing $10$, $100$, and $1000$ scatterers (panels (a), (b) and (c) respectively). For each value of $k \varsigma$, $5000$ random configurations were simulated. The average value of $\bm{w}^2$, the interquartile and the full range of the distribution are shown. The blue dotted curves represent the analytical prediction in the ballistic regime from \ref{['eq:ballistic_regime_w']}, while the red dashed curves show the corresponding prediction for $\langle \bm{w}^2 \rangle$ associated with the random model of \ref{['eq:random_model_w']}. The vertical line at $k \ell = 1$ indicates the position of the Ioffe-Regel threshold, around which the strong directionality peak emerges.
• Figure 4: (left) Square modulus of the full wavefunction $|\psi(k, \bm{r} | \bm{0})|^2$ and (right) angular PDF $\mathrm{d} P(\theta) / \mathrm{d} \theta$, for a wave propagating through a Lorentz gas of $N=1000$ scatterers. The gaseous configuration $\{\bm{x}_1, \bm{x}_2, ..., \bm{x}_{1000}\}$ is the same along the three vertical panels. Panel (a): $k \varsigma = 10^{-3}$ and $\bm{w}^2 \approx 0.012$, panel (b): $k \varsigma = 1$ and $\bm{w}^2 \approx 0.83$, panel (c): $k \varsigma = 6$ and $\bm{w}^2 \approx 0.04$. The directionality is very pronounced on panel (b). As $k \varsigma$ increases, the frequency content of $\mathrm{d} P(\theta) / \mathrm{d} \theta$ increases, as it goes from being almost isotropic and dominated by the $0$-th order Fourier term in panel (a), to being highly oscillatory in panel (c). This behavior is explained in \ref{['appsec:fourier_analysis']}.
• Figure 5: Autocorrelation function $C(\tau)$ of the angular PDF $\mathrm{d} P(\theta) / \mathrm{d} \theta$, for a spherical wave propagating through a Lorentz gas of $N=1000$ scatterers. Panel (a): $k \varsigma = 2$, panel (b): $k \varsigma = 50$, panel (c): $k \varsigma = 2000$. In all three panels, the black curve represents the average over respectively $2000$, $800$ and $40$ random configurations of the gas, and the dashed red curve is the corresponding prediction for $C(\tau)$ associated with the random model of \ref{['eq:random_model_correlation_function']}. In panel (c), the dotted blue curve represents the analytical prediction in the ballistic regime from \ref{['eq:correlation_function_delta_ballistic_2d']}.