Hydrogen atom as a nonlinear oscillator under circularly polarized light: epicyclical electron orbits

Abstract

We used Clifford algebra $C{l}_{2,0}$ to find the 2D orbit of a hydrogen electron under a Coulomb force and a perturbing circularly polarized electric field of light at angular frequency~$\omega$, which is turned on at time $t=0$ via a unit step switch. Using a coordinate system co-rotating with the electron's unperturbed circular orbit at angular frequency ${\omega }_0$, we derived a complex differential equation that is similar to but different from that of the Lorentz oscillator equation for light--atom interaction. We solved the homogeneous and particular differential equation and showed that the position of the electron is a linear combination of five exponential Fourier terms or orbital wave functions with frequencies $0$, ${\omega }_0$, $2{\omega }_0$, $(2{\omega }_0-\omega )$, and $\omega$, whose coefficients depend on the light-to-atom frequency ratio $\omega /{\omega }_0$ and light-to-atom force magnitude ratio $A/r_0$. We showed that the electron orbits are approximately Keplerian at light-to-atom frequency ratio $\alpha =\omega /{\omega }_0=\{0,1,2\}$, with the orbits becoming discontinuous and divergent as $\alpha \to 1^{\pm }$, but continuous and non-divergent at $\alpha =\{0,2\}$. These Keplerian orbits are approximated by the sum of the zeroth, first, and second harmonics of the electron's unperturbed orbital wave function ${\hat{\psi }}_0=e^{\hat{ı}{\omega }_{0}t}$, corresponding to the eccentric, deferent, and epicycle in the Copernican construction of planetary orbits.

The full revised paper has 42 pages, 184 equations, 9 figures, and 2 tables. The sign error in the 2024 ArXiv version was corrected, leading to new results and conclusions