Coherent Backscattering of Bose-Einstein Condensates in Two-Dimensional Disorder Potentials

We study quantum transport of an interacting Bose-Einstein condensate in a two-dimensional disorder potential. In the limit of a vanishing atom-atom interaction, a sharp cone in the angle-resolved density of the scattered matter wave is observed, arising from constructive interference between amplitudes propagating along reversed scattering paths. Weak interaction transforms this coherent backscattering peak into a pronounced dip, indicating destructive instead of constructive interference. We reproduce this result, obtained from the numerical integration of the Gross-Pitaevskii equation, by a diagrammatic theory of weak localization in the presence of nonlinearity.

Figure 1

Scattering geometry and stationary scattering state associated with a randomly generated disorder potential. The left-hand side displays $V(x,y)$ in a gray-scale plot and shows the spatial variation of the nonlinearity $g(x)$. The upper right panel shows the density of the corresponding scattering state that is populated through the numerical integration of the inhomogeneous Gross-Pitaevskii equation (1). The lower right panel shows the decay of the coherent mode $|⟨\psi ⟩{|}^2$ and the density $⟨|\psi {|}^2⟩$ with $x$, averaged over $y$ for $\sim {10}^3$ randomly generated disorder configurations. Parameters: $kL=40$, $kW=120$, $k\sigma =0.5$, $V_0=0.614\mu$, $g=0.005$, $j_{\mathrm{in}}=\hslash k^3/m$, with $k\equiv \sqrt{2m\mu }/\hslash$.

Figure 2

Scattering mean free path ${\ell }_s$ (left panel) and transport mean free path ${\ell }_{\mathrm{tr}}$ (right panel) in the disorder potential for $k\sigma =0.5$ in the absence of nonlinearity. In the left panel, the numerically computed lengths are compared with the Born approximation (2) (solid line) and in the right panel with the Boltzmann mean free path (3) (${\ell }_B$, dashed line) and the expression (4) that takes into account weak localization corrections (${\ell }_{\mathrm{tr}}$, solid line). We find ${\ell }_{\mathrm{tr}}\simeq {\ell }_s$ for $V_0<\mu$.

Figure 3

Angle-resolved current density of backscattered atoms in the absence and presence of nonlinearity, obtained from the average over $\sim {10}^3$ disorder configurations (parameters as in Fig. 1; the error bars denote the statistical standard deviation). The coherent backscattering cone for $g=0$ (black line) is transformed into a pronounced dip for intermediate nonlinearities ($g=0.02$, bold red line), and turns into a smooth peak structure at larger values of $g$ ($g=0.06$, dashed blue line). The inset shows the angle-resolved current for the case of a tilted incident beam where the source term in Eq. (1) populates the transverse eigenmode ${\chi }_{n=6}(y)$ defined by the angle ${\theta }_6\simeq 0.32$. In contrast to the smooth peak, the cone and dip structures are indeed found at the angle that corresponds to retroreflection of the incident beam, which confirms that they both arise due to interference between reflected paths.

Figure 4

Backscattered current at $\theta =0$ as a function of the nonlinearity $g$ (parameters as in Fig. 1), obtained from the numerical simulation (symbols) and from the diagrammatic theory, Eqs. (5, 6, 7) (solid line). The horizontal dashed line indicates the diffuse background intensity $j_L(0)$. Negative cone heights $j_C(0)<0$ leading to a dip in the angle-resolved current density $j(\theta )$ appear for $g>0.01$.