Coherent Forward Scattering Peak Induced by Anderson Localization

Numerical simulations show that, at the onset of Anderson localization, the momentum distribution of a coherent wave packet launched inside a random potential exhibits, in the forward direction, a novel interference peak that complements the coherent backscattering peak. An explanation of this phenomenon in terms of maximally crossed diagrams predicts that the signal emerges around the localization time and grows on the scale of the Heisenberg time associated with the localization volume. Together, coherent back and forward scattering provide evidence for the occurrence of Anderson localization.

Figure 1

Ensemble-averaged momentum distribution $\overline{n}(k_x,k_y,t)$ of a simulated matter wave launched with initial momentum ${\mathit{k}}_0=(1.5/\zeta ,0)$ inside a 2D random potential with correlation length $\zeta$. The time unit is ${\tau }_{\zeta }=m{\zeta }^2/\hslash$. (a) $t=10{\tau }_{\zeta }$: multiple scattering generates a coherent backscattering (CBS) peak at $-{\mathit{k}}_0$. (b) $t=90{\tau }_{\zeta }$: a coherent forward scattering (CFS) peak has emerged at $+{\mathit{k}}_0$. (c) $t=950{\tau }_{\zeta }$: the CBS and CFS twin peaks in the Anderson localization regime.

Figure 2

Contrast of CBS (red circles) and CFS peak (blue squares) as a function of time, as obtained by fitting the data of Fig. 1 with the theoretical shapes, Eqs. (2, 4), and a running average over a time window of $10{\tau }_{\zeta }$.

Figure 3

Constructive interference of amplitudes along maximally crossed trajectories, leading to peaks at (a) $-{\mathit{k}}_0$ for CBS and (b) $+{\mathit{k}}_0$ for CFS, respectively.

Figure 4

Ladder (a) and maximally crossed (b) multiple scattering Feynman diagrams, giving rise to the diffusive background and the CBS signal, Eq. (2). Diagram (c) is responsible for the CFS peak, Eq. (5). Diagram (d) is exactly zero.

Figure 5

Early-time CFS contrast for disorder strengths $V=5E_{\zeta }$ (red circles), $V=4E_{\zeta }$ (green squares), $V=3E_{\zeta }$ (blue triangles), and $V=2E_{\zeta }$ (purple diamonds). According to our theory, the CFS begins to rise when localization sets in. The inset shows ${\tau }_{\mathrm{loc}}$, estimated by the earliest CFS appearance, as a function of $V$.