Theory and application of Fermi pseudo-potential in one dimension

TL;DR

This paper develops a one-dimensional theory of Fermi pseudo-potentials, revealing new interaction terms and applying them to model quantum memory in two-channel scattering scenarios.

Contribution

It introduces a comprehensive one-dimensional Fermi pseudo-potential framework, including the odd and even components, and demonstrates its application to quantum memory modeling.

Findings

The general one-point interaction includes delta and two Fermi pseudo-potentials.

The even Fermi pseudo-potential is a novel and significant addition.

Application to quantum memory shows potential for quantum computing implementations.

Abstract

The theory of interaction at one point is developed for the one-dimensional Schrodinger equation. In analog with the three-dimensional case, the resulting interaction is referred to as the Fermi pseudo-potential. The dominant feature of this one-dimensional problem comes from the fact that the real line becomes disconnected when one point is removed. The general interaction at one point is found to be the sum of three terms, the well-known delta-function potential and two Fermi pseudo-potentials, one odd under space reflection and the other even. The odd one gives the proper interpretation for the delta'(x) potential, while the even one is unexpected and more interesting. Among the many applications of these Fermi pseudo-potentials, the simplest one is described. It consists of a superposition of the delta-function potential and the even pseudo-potential applied to two-channel scattering. This simplest application leads to a model of the quantum memory, an essential component of any quantum computer.