Expansion of a quantum wave packet in a one-dimensional disordered potential in the presence of a uniform bias force

We study numerically the expansion dynamics of an initially confined quantum wave packet in the presence of a disordered potential and a uniform bias force. For white-noise disorder, we find that the wave packet develops asymmetric algebraic tails for any ratio of the force to the disorder strength. The exponent of the algebraic tails decays smoothly with that ratio and no evidence of a critical behavior on the wave density profile is found. Algebraic localization features a series of critical values of the force-to-disorder strength where the $m\mathrm{th}$ position moment of the wave packet diverges. Below the critical value for the $m\mathrm{th}$ moment, we find fair agreement between the asymptotic long-time value of the $m\mathrm{th}$ moment and the predictions of diagrammatic calculations. Above it, we find that the $m\mathrm{th}$ moment grows algebraically in time. For correlated disorder, we find evidence of systematic delocalization, irrespective to the model of disorder. More precisely, we find a two-step dynamics, where both the center-of-mass position and the width of the wave packet show transient localization, similar to the white-noise case, at short time and delocalization at sufficiently long time. This correlation-induced delocalization is interpreted as due to the decrease of the effective de Broglie wavelength, which lowers the effective strength of the disorder in the presence of finite-range correlations.

Figure 1

Scheme of the system. The quantum wave packet has an energy $E$ (dashed blue line). It propagates in a disordered potential $V\left(x\right)$ (solid green line) in the presence of a linear bias potential $-Fx$ (solid red line). The classically forbidden region $x<-{\ell }_0$ is represented by the purple hatched zone.

Figure 2

Time evolution of the density profile of an initially Gaussian quantum wave packet expanding in the presence of white-noise disorder and a uniform bias force. The various panels correspond to different ratios of the force to the disorder: (a) $\alpha =0.2$, (b) $\alpha =0.8$, (c) $\alpha =1$, and (d) $\alpha =2$. The solid color lines correspond to different times indicated on each panel, with the peak propagating towards the right when time increases. The asymptotic probability of transfer (dotted black lines for $\alpha <1$) and the time evolution of the wave packet subjected to the force and in the absence of disorder (dashed color lines) are also shown.

Figure 3

Panels (a) and (b) show the dynamically rescaled density $\tilde{\rho }$ as a function of the dynamically rescaled position $u$ at different times $\tau$ indicated on each panel. They are compared to the algebraic decays of slope $1/u^{{\eta }_{+}}$ for $\alpha <1$ and $1/u$ for $\alpha >1$ (solid black lines). Panel (a) corresponds to $\alpha =0.8$ and panel (b) to $\alpha =2$. In the numerics, we used negative initial velocities ${\kappa }_0<0$. (c) Values of $\eta$ obtained by a linear fit of the dynamically renormalized density in log-log scale on $u\in [0.01,0.1]$, with 10% error bars. The solid blue line shows the analytical prediction ${\eta }_{+}$ for $0<\alpha <{\alpha }_0=1$ [Eq. (11)].

Figure 4

Time evolution of (a) the center-of-mass position and (b) width of the wave packet for various values of the parameter $\alpha$ and, for some curves, two values of $\varepsilon$ (solid lines correspond to $\varepsilon /\alpha =0.1$ and dashed lines to $\varepsilon /\alpha =0.01$). The lower (upper) curves correspond to the smallest (largest) values of $\alpha$ indicated on the panels. Also shown are the analytical prediction in the absence of disorder (black lines), the asymptotic values ${\overline{\xi }}_{\infty }$ and ${\sigma }_{\xi ,\infty }$ (horizontal solid colorful segments), and their maximum values (horizontal dotted black lines).

Figure 5

Values at large time of the first two position moments $\overline{\xi }$ and $\overline{{\xi }^2}$ and the width ${\sigma }_{\xi }$ of the position of the wave packet, as a function of $\alpha$. The lines represent the analytical values at infinite time and the maximum values of ${\overline{\xi }}_{\infty }$ at $\alpha ={\alpha }_1$ and $\overline{{\xi }_{\infty }^2}$ and ${\sigma }_{\xi ,\infty }$ at $\alpha ={\alpha }_2$ are shown as circles. The diamonds, squares, and triangles correspond, respectively, to the values of $\overline{\xi }\left({\tau }_{max}\right)$, $\overline{{\xi }^2}\left({\tau }_{max}\right)$, and ${\sigma }_{\xi }\left({\tau }_{max}\right)$ at the highest time ${\tau }_{max}$ calculated in the numerical simulations.

Figure 6

Power-law exponents of (a) the average position and the (b) the width of the wave packet as a function of the parameter $\alpha$, at different expansion times $\tau$ (color points) and extrapolated to $\tau \to \infty$ (black points). The expected values for $\alpha \to \infty$ are signaled by right-pointing arrows in (a). The error bars for the data at finite time $\tau$ (color points) correspond to the standard deviation due to disorder averaging. The data at infinite time (black points) are extrapolations of the latter as discussed in Appendix pp3, which produces an additional contribution to the error bars.

Figure 7

Ratio of the center-of-mass position to the width of the wave packet as a function of time for various values of the parameter $\alpha$ and, for some curves, two values of $\varepsilon$ (solid lines correspond to $\varepsilon /\alpha =0.1$ and dashed lines to $\varepsilon /\alpha =0.01$). The lower (upper) curves on the right-hand side correspond to the largest (smallest) values of $\alpha$ indicated on the panels.

Figure 8

Evolution of the average position $\overline{\xi }\left(\tau \right)$ of the wave packet (left panel), and its width ${\sigma }_{\xi }\left(\tau \right)$ (right panel) as a function of time $\tau$ in the presence of a bias force. The filled red circles correspond to a correlated disorder with the Gaussian two-point correlation function $\tilde{C}\left(x\right)=\frac{U_{\mathrm{R}}}{\sqrt{2\pi }{\sigma }_{\mathrm{R}}}exp\left(\frac{-x^2}{2{\sigma }_{\mathrm{R}}^2}\right)$, with $\varepsilon /\alpha =0.1$, ${\tilde{\sigma }}_{\mathrm{R}}{\kappa }_0=0.05\alpha$, and for three different values of ${\alpha }_{\mathrm{loc}}\left(0\right)=\alpha exp\left(2{\tilde{\sigma }}_{\mathrm{R}}{\kappa }_0\right)$. Empty light blue squares correspond to white-noise disorder with $\alpha ={\alpha }_{\mathrm{loc}}\left(0\right)$ and $\varepsilon /\alpha =0.1$. Filled dark blue circles correspond to white-noise disorder with $\alpha ={\alpha }_{\mathrm{loc}}\left(0\right)$ and $\varepsilon exp\left(2{\tilde{\sigma }}_{\mathrm{R}}^2{\kappa }_0^2\right)/\alpha =0.1$. The blue horizontal dashed lines correspond to the asymptotic values of the moments for white-noise disorder when they exist.

Figure 9

Power-law exponent of (a) the average position and (b) width of the wave packet as a function of ${\beta }_1^{\infty }$, for various values of the parameter $\alpha$. The analytical result in the absence of disorder is shown as a black line in (a). Long-time linear extrapolations found from linear fits for ${\beta }_1^{\infty }>1.75$ are shown as dashed lines.

Figure 10

Time evolution of the width ${\sigma }_{\xi }$ of the wave packet as a function of time $\tau$ for two values of $\varepsilon$ and four values of the parameter $\alpha$. The solid lines correspond to $\varepsilon =0.05$ and the dashed lines to $\varepsilon =1$. The lower (upper) curves on the right-hand side correspond to the largest (smallest) values of $\alpha$ indicated on the panels. The cases without disorder are plotted in dark black.