Quantum dynamics of an ion in an RF quadrupole trap with weak octupolar anharmonicity
We investigate the quantum dynamics of a single ion confined in a radio-frequency (RF) Paul trap in the presence of weak octupole-induced anharmonicity, as can occur in realistic ion-trap potential. The explicitly time-dependent quadrupole potential is treated in a reference-oscillator Fock basis, allowing the effects of the RF drive and the quartic correction to be resolved directly at the level of motional-state couplings. While the quadratic RF interaction generates $Δn=\pm2$ transitions and Gaussian squeezing, the quartic contribution opens additional $Δn=\pm4$ coupling pathways. Starting from the reference-oscillator vacuum, we show that the octupole interaction broadens the even-parity Fock-state distribution and enhances the phonon-number fluctuations more strongly than the mean occupation. At the same time, the RF-induced sub-vacuum quadrature squeezing remains robust, although its magnitude and orientation are modified by the quartic interaction. The number statistics and quadrature fluctuations therefore respond differently to the same nonlinear perturbation, reflecting the distinct roles of Fock-space redistribution and phase coherence. The resulting dynamics provide a quantum-mechanical characterization of weak octupole corrections beyond the ideal quadrupole approximation and clarify how realistic anharmonicities modify nonclassical trapped-ion motion. Importantly, the motional nonclassicality remains robust against the weak octupole perturbation, as evidenced by persistent sub-vacuum quadrature squeezing and Klyshko-inequality violation. These results characterize how realistic anharmonic corrections reshape trapped-ion motion beyond the ideal quadrupole approximation.