Superpositions of coherent states determined by Gauss sums

TL;DR

This paper introduces a new family of quantum superposition states of the Schrödinger cat type, characterized by coefficients derived from quadratic Gauss sums, and explores their mathematical properties and connections to fractional Fourier transforms.

Contribution

It defines a novel class of quantum states using Gauss sums and links them to eigenfunctions of transformed lowering operators, extending known coherent states.

Findings

States include the Yurke-Stoler coherent state as a special case

States are eigenfunctions of transformed lowering operators

Provides a mathematical framework connecting Gauss sums and quantum states

Abstract

We describe a family of quantum states of the Schr\"odinger cat type as superpositions of the harmonic oscillator coherent states with coefficients defined by the quadratic Gauss sums. These states emerge as eigenfunctions of the lowering operators obtained after canonical transformations of the Heisenberg-Weyl algebra associated with the ordinary and fractional Fourier transformation. The first member of this family is given by the well known Yurke-Stoler coherent state.