Superpositions of coherent states determined by Gauss sums

We describe a family of quantum states of the Schrödinger 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 transformations. The first member of this family is given by the well known Yurke–Stoler coherent state.