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

TL;DR

This paper models the hydrogen electron's orbit under circularly polarized light using Clifford algebra, revealing complex nonlinear dynamics and epicyclical orbits analogous to Copernican astronomy, with specific resonant behaviors.

Contribution

It introduces a novel Clifford algebra-based approach to analyze electron orbits under circularly polarized light, deriving nonlinear differential equations and identifying epicyclical orbit structures.

Findings

Electron orbits are sums of five Fourier components with specific frequencies.

Resonant light frequencies cause divergent orbits approximating Keplerian ellipses.

Non-resonant frequencies produce nondivergent, periodic orbits.

Abstract

In this paper, we use Clifford algebra $Cl_{2,0}$ to find the 2D orbit of 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 derive the complex nonlinear differential equation for the perturbation which is similar to but different from the Lorentz oscillator equation: (1) the acceleration terms are similar, (2) the damping term coefficient is not real but imaginary due to Coriolis force, (3) the term similar to spring force is not positive but negative, (3) there is a complex conjugate of the perturbation term which has no Lorentz analog but which makes the equation nonlinear, and (4) the angular frequency of the forcing term is not $\omega$ but $\omega - \omega_0$. By imposing that the position and velocity of the electron are continuous at time $t = 0$, we show that the orbit of the electron is a sum of five exponential Fourier terms with frequencies 0, $\omega_0$, $2\omega_0$, $(2\omega_0 - \omega)$, and $\omega$, which correspond to the eccentric, deferent, and three epicycles in Copernican astronomy. We show that at the three resonant light frequencies $0$, $\omega_0$, and $2\omega_0$, the electron's orbit is divergent, but approximates a Keplerian ellipse. At other light frequencies, the orbits are nondivergent with periods that are integer multiples of $\pi/\omega_0$ depending on the frequency ratio $\omega/\omega_0$. And as $\omega/\omega_0\rightarrow \pm\infty$, the orbit approaches the electron's unperturbed circular orbit.