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

Abstract

We used Clifford algebra $Cl_{2,0}$ 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 $\alpha \rightarrow 1^{\pm}$, 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 $\hat\psi_0 = e^{\hat\imath\omega_0 t}$, corresponding to the eccentric, deferent, and epicycle in the Copernican construction of planetary orbits.