arXiv:2112.07613 · Theoretical and Mathematical Physics

Superpositions of coherent states determined by Gauss sums

Vyacheslav Pavlovich Spiridonov

Published Sep 1, 2022• 1 citation

Abstract

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.

DOI: 10.1134/s0040577922090069

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arXiv:2112.07613 · Theoretical and Mathematical Physics

Superpositions of coherent states determined by Gauss sums

Vyacheslav Pavlovich Spiridonov

Published Sep 1, 2022• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.1134/s0040577922090069

Related papers

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{\mathit{q}}^{2}

${\mathit{A}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$A=1. Coherent states for this and other reflectionless potentials whose discrete spectra consist of N geometric series are analyzed. In the harmonic oscillator limit, the surviving part of these states takes the form of orthonormal superpositions of N canonical coherent states \ensuremath{\Vert}${\mathrm{\ensuremath{\epsilon}}}^{\mathit{k}}$\ensuremath{\alpha}〉, k=0,1,...,N-1, where \ensuremath{\epsilon} is a primitive Nth root of unity, ${\mathrm{\ensuremath{\epsilon}}}^{\mathit{N}}$=1. A class of q-coherent states related to the bilateral q-hypergeometric series and Ramanujan type integrals is described. It includes an unusual set of coherent states of the free nonrelativistic particle, which is interpreted as a q-algebraic system without a discrete spectrum. A special degenerate form of the symmetry algebras of self-similar potentials is found to provide a natural q analog of the Floquet theory. Some properties of the factorization method, which is used throughout the paper, are discussed from the differential Galois theory point of view.

Overcomplete families of states of the type of Barut-Girardello coherent states (BG CS) are constructed for noncompact algebras

u(p,q)

and

sp(N,C)

in quadratic bosonic representation. The

sp(N,C)

BG CS are obtained in the form of multimode ordinary Schrödinger cat states. A set of such macroscopic superpositions is pointed out which is overcomplete in the whole

N

mode Hilbert space (while the associated

sp(N,C)

representation is reducible). The multimode squared amplitude Schrödinger cat states are introduced as macroscopic superpositions of the obtained

sp(N,C)

BG CS.}

We construct two-parameters family of nonlinear coherent states by replacing the factorial in coefficients

z^n/\sqrt{n!}

of the canonical coherent states by a specific generalized factorial

x_n^{γ,σ}!

where parameters

γ

and

σ

satisfy some conditions for which the normalization condition and the resolution of identity are verified. The obtained family is a generalization of the Barut-Girardello coherent states and those of the philophase states. In the particular case of parameters

γ

and

σ

, we attache these states to the pseudo-harmonic oscillator depending on two parameters

α,β> 0

. The obtained nonlinear coherent states are superposition of eigenstates of this oscillator. The associated Bargmann-type transform is defined and we derive some results.