On the deformed oscillator and the deformed derivative associated with the Tsallis q-exponential

TL;DR

This paper explores the mathematical structure of a deformed oscillator linked to the Tsallis q-exponential, revealing its eigenfunctions, coherent states, and energy spectrum variations with the parameter q.

Contribution

It introduces a novel deformed oscillator associated with the Tsallis q-exponential, connecting it to deformed derivatives, coherent states, and energy spectra.

Findings

Tsallis q-exponential functions are eigenfunctions of a deformed derivative.

Coherent states of the deformed oscillator are given by Tsallis q-exponentials.

Energy levels vary with q, from boson-like to two-level systems.

Abstract

The Tsallis $q$-exponential function $e_q(x) = (1+(1-q)x)^{\frac{1}{1-q}}$ is found to be associated with the deformed oscillator defined by the relations $\left[N,a^\dagger\right] = a^\dagger$, $[N,a] = -a$, and $\left[a,a^\dagger\right] = \phi_T(N+1)-\phi_T(N)$, with $\phi_T(N) = N/(1+(q-1)(N-1))$. In a Bargmann-like representation of this deformed oscillator the annihilation operator $a$ corresponds to a deformed derivative with the Tsallis $q$-exponential functions as its eigenfunctions, and the Tsallis $q$-exponential functions become the coherent states of the deformed oscillator. When $q = 2$ these deformed oscillator coherent states correspond to states known variously as phase coherent states, harmonious states, or pseudothermal states. Further, when $q = 1$ this deformed oscillator is a canonical boson oscillator, when $1 < q < 2$ its ground state energy is same as for a boson and the excited energy levels lie in a band of finite width, and when $q \longrightarrow 2$ it becomes a two-level system with a nondegenerate ground state and an infinitely degenerate excited state.