Hydrogen Atom as a Nonlinear Oscillator Under Circularly Polarized Light: Epicyclical Electron Orbits

We used Clifford algebra Cl2,0 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 ω, 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 ω0, 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, ω0, 2ω0, (2ω0−ω), and ω, whose coefficients depend on the light-to-atom frequency ratio ω/ω0 and light-to-atom force magnitude ratio A/r0. We showed that the electron orbits are approximately Keplerian at light-to-atom frequency ratio α=ω/ω0={0,1,2}, with the orbits becoming discontinuous and divergent as α→1±, but continuous and non-divergent at α={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 ψ^0=eı^ω0t, corresponding to the eccentric, deferent, and epicycle in the Copernican construction of planetary orbits.