Rényi and Tsallis information entropies for the Darboux III quantum nonlinear oscillator

TL;DR

This work presents a comprehensive information-theoretic analysis of the one-dimensional Darboux III quantum nonlinear oscillator, focusing on Rényi and Tsallis entropies. It provides exact analytical expressions for entropic moments and the corresponding entropies in position space, while treating momentum-space quantities numerically due to the absence of a closed-form Fourier transform. The study reveals a rich interplay between the nonlinearity parameter $\lambda$ and the entropy index $\alpha$, including regimes where entropies exceed those of the harmonic oscillator and strong effects in highly excited states. The results recover the harmonic oscillator limits as $\lambda\to0$ and satisfy entropy-based uncertainty relations, offering new insights into quantum information measures for position-dependent mass systems and nonlinear oscillators.

Abstract

The Darboux III oscillator is an exactly solvable $N$-dimensional nonlinear oscillator defined on a radially symmetric space with non-constant negative curvature. Its one-dimensional version can be seen as a position dependent mass system whose mass function $μ= (1 + λx^2)$ depends on the nonlinearity parameter $λ$, such that in the limit $λ\to 0$ the harmonic oscillator is recovered. In this paper, a detailed study of the entropic moments and of the Rényi and Tsallis information entropies for the quantum version of the one-dimensional Darboux III oscillator is presented. In particular, analytical expressions for the aforementioned quantities in position space are obtained. Since the Fourier transform of the Darboux III wave functions does not admit a closed form expression, a numerical analysis of these quantities has been performed. Throughout the paper the interplay between the entropy parameter $α$ and the nonlinearity parameter $λ$ is analysed, and known results for the Shannon entropy of the Darboux III and for the Rényi and Tsallis entropies of the harmonic oscillator are recovered in the limits $α\to 1$ and $λ\to 0$, respectively. Finally, motivated by the strong non-linear effects arising when large values of $λ$ and/or highly excited states are considered, an approximation to the probability density function valid in those regimes is presented. From it, an analytical approximation to the probability density in momentum space can be obtained, and some of the previously observed effects arising from the interplay between $α$ and $λ$ can be explained.

Figures (14)

• Figure 1: Energy levels $E_n^\lambda$ (A) and effective frequency $\Omega_n^\lambda$ (B) vs $n$ for $\omega = 1$ and different values of $\lambda$ (indicated within the plot). The energy increases with the quantum number $n$ but decreases with $\lambda$. The frequency decreases with both. NUmerical data are given in Tables \ref{['table: energy 1D']}, \ref{['table: Omega 1D']}.
• Figure 2: Density function in position space for several values of $\lambda$ and $n=0$(A), $n=1$(B), $n=3$(C) and $n=6$(D).
• Figure 3: Density function in momentum space for several values of $\lambda$ and $n=0$(A), $n=1$(B), $n=3$(C) and $n=6$(D).
• Figure 4: Effect of the parameter $\alpha$: entropy in position space vs $n$ for several $\alpha$ values given within panel (A). (A)&(B) Rényi entropy $\mathcal{R}_\rho^{(\alpha,n,\lambda)}$, (C)&(D) Tsallis entropy $\mathcal{T}_\rho^{(\alpha,n,\lambda)}$. (A)&(C) Harmonic oscillator $(\lambda=0)$, (B)&(D) Darboux III oscillator $(\lambda=0.4)$. Numerical data in Tables \ref{['p2']} (A), \ref{['p3']} (B), \ref{['p4']} (C), \ref{['p5']} (D).
• Figure 5: Rényi $\mathcal{R}_\rho^{(\alpha,n,\lambda)}$ (\ref{['eq: entropia final renyi subs N']}) (A) and Tsallis $\mathcal{T}_\rho^{(\alpha,n,\lambda)}$ (\ref{['eq: Tsallis 1D darboux']}) (B) entropies in position space vs $\lambda$ (up to $\lambda=20$) for $\alpha=2$, $\omega=1$, from $n=0$ to $n=9$ (C).