Design of quantum backflow in the complex plane

TL;DR

This paper introduces a method to design quantum wave functions exhibiting backflow, using rational complex functions and Padé approximations, enabling control over negative probability currents in quantum systems.

Contribution

It presents a novel approach to engineer quantum backflow wave functions through complex analysis and rational functions, expanding the toolkit for quantum control and analysis.

Findings

Successfully designed wave functions with controlled backflow behavior

Demonstrated approximation of desired backflow profiles using Padé-type methods

Provided a mathematical framework linking complex functions to quantum probability currents

Abstract

A way is presented to design quantum wave functions that exhibit backflow, namely negative probability current despite having a strictly positive spectrum of momentum. These wave functions are derived from rational complex functions which are analytic in the upper half-plane and have zeros in the lower half-plane through which the backflowing behavior is controlled. In analogy, backflowing periodic wave functions are derived from rational complex functions which are analytic in the interior and have appropriately placed zeros or poles in the exterior of the unit circle. The concept is combined with a Pad\'e-type procedure to design wave functions of this type that approximate a desired profile along the interval of backflow.