Solution to the Landau–Lifshitz equation in laser and electrostatic fields

Laser intensity of ${10}^{23}\text{ W}/{\text{cm}}^2$ has been available recently, at which radiation reaction begins to significantly influence electron motion within the laser field. In vacuum, the motion is typically described by solutions to the Landau-Lifshitz equation considering a laser field. In plasma, which is the most common scenario in laser-matter interactions, electron motion is governed by the laser field as well as static electric and magnetic fields through plasma response. Here, we theoretically investigate electron motion in the radiation reaction regime, considering the presence of both laser and electrostatic fields. We solve the Landau-Lifshitz equation using the harmonic balance method. The solution is benchmarked against single-particle simulation and the differences are less than 5% when the QED parameter ${\chi }_e<1$. We find that there is a significant difference in electron motion when considering both the laser field and the electrostatic field simultaneously, compared to when considering each field separately. When the electrostatic field and the laser propagation direction are the same, the electron motion resembles the one described by the Lorentz equation in the plane-wave case with a stable dephasing rate controlled by the field strengths. This provides a way to control the electron energy spread via the radiation reaction. While the electrostatic field and the laser propagation direction are opposite, the electron motion approaches the one described by the Lorentz equation with a lowered dephasing rate. Our results contribute to investigating the radiation reaction effect in laser-plasma interactions where the electrostatic fields generated via plasma response cannot be ignored.