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

Abstract

The static second hyperpolarizability is derived from the space-fractional Schrödinger 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 $\alpha \ne 1$, where the second hyperpolarizability becomes negative before ultimately damping to zero at the lower fractional limit of $\alpha \to 1/2$.

References in corpus (6)

• Fractional Quantum Mechanics
• Matrix approach to discrete fractional calculus II: partial fractional differential equations
• A new dipole-free sum-over-states expression for the second hyperpolarizability
• Lowest-order relativistic corrections to the fundamental limits of nonlinear-optical coefficients
• Polynomial potentials determined from the energy spectrum and transition dipole moments that give the largest hyperpolarizabilities
• Static hyperpolarizability of space-fractional quantum systems