Quantum backflow and scattering

Backflow is the phenomenon that the probability current of a quantum particle on the line can flow in the direction opposite to its momentum. In this paper, previous investigations of backflow, pertaining to interaction-free dynamics or purely kinematical aspects, are extended to scattering situations in short-range potentials. It is shown that backflow is a universal quantum effect which exists in any such potential, and is always of bounded spatial extent in a specific sense. The effects of reflection and transmission processes on backflow are investigated, both analytically for general potentials and numerically in various concrete examples.

Figure 1

Lowest eigenvector of the asymptotic current operator. Parameters: $n=2000,p_{\max }=200,x_0=0,\sigma =0.1$. (a) Zero potential. (b) Delta potential, $\lambda =1$.

Figure 2

Backflow bound depending on position. Parameters: $n=1000,p_{\max }=150,\sigma =0.1$. (a) Delta potential, repulsive ($\lambda =1$) and attractive ($\lambda =-1$). (b) Rectangular potential, repulsive ($\lambda =2$) and attractive ($\lambda =-2$). (c) Reflectionless Pöschl-Teller potential, $\mu =1$.

Figure 3

Backflow depending on strength of the potential. Parameters: $n=1000,p_{\max }=150,\sigma =0.1$. (a) Delta potential. (b) Rectangular potential. (c) Pöschl-Teller potential.

Figure 4

Probability distribution and probability current for the maximum backflow vector in configuration space at $t=0$. Dashed vertical lines indicate the position of measurement $x_0$; dotted vertical lines indicate the position of the potential. Parameters: $n=2000,p_{\max }=200,\sigma =0.1$. See the animations in [11] for evolution with time $t$. (a) Zero potential, $x_0=0$. (b) Delta potential, $\lambda =1,x_0=-2$. (c) Delta potential, $\lambda =1,x_0=+2$.