The second hyperpolarizability of systems described by the space-fractional Schrodinger equation

TL;DR

This paper derives the static second hyperpolarizability for systems described by the space-fractional Schrödinger equation, revealing how fractional parameters dampen optical responses and alter system linearity.

Contribution

It introduces a novel derivation of hyperpolarizability within the space-fractional quantum framework and analyzes how fractional parameters influence optical properties.

Findings

The oscillator strength decreases with lower fractional parameter $\alpha$.

Maximum hyperpolarizability is reduced due to fractional effects.

The quantum harmonic oscillator becomes non-linear for $\alpha eq 1$.

Abstract

The static second hyperpolarizability is derived from the space-fractional Schr\"{o}dinger equation in the particle-centric view. The Thomas-Reiche-Kuhn sum rule matrix elements and the three-level ansatz determines the maximum second hyperpolarizability for a space-fractional quantum system. The total oscillator strength is shown to decrease as the space-fractional parameter $\alpha$ decreases, which reduces the optical response of a quantum system in the presence of an external field. This damped response is caused by the wavefunction dependent position and momentum commutation relation. Although the maximum response is damped, we show that the one-dimensional quantum harmonic oscillator is no longer a linear system for $\alpha \neq 1$, where the second hyperpolarizability becomes negative before ultimately damping to zero at the lower fractional limit of $\alpha \rightarrow 1/2$.