Description
Ultra-intense laser–plasma interactions and plasma-based electron accelerators provide access to regimes where radiation reaction significantly influences particle dynamics. The Landau–Lifshitz (LL) equation provides the standard classical description and admits exact analytical solutions in a plane-wave background, as demonstrated by Di Piazza [1]. In this geometry, the solution can be expressed in terms of integrals over the plane-wave phase, allowing closed-form expressions for the four-velocity and trajectory.
When the quantum nonlinearity parameter $\chi$ reaches the moderately quantum regime, $\chi \sim 0.1$–$1$, the classical LL equation overestimates radiative losses, as quantum recoil suppresses the average emitted power [2, 3]. A fully quantum treatment based on stochastic photon emission captures the discrete nature of radiation but generally requires probabilistic or numerical methods. In large-scale particle-in-cell simulations, quantum effects are therefore commonly incorporated through a deterministic semiclassical correction in which the classical radiation-reaction force is multiplied by a $\chi$-dependent Gaunt factor $g(\chi)$ [2]. While this approach reproduces the quantum-suppressed mean emission rate, it does not describe stochastic broadening effects [4].
Here we show that the Gaunt-modified LL equation remains exactly integrable in a plane-wave background. Using the fact that the quantum parameter $\chi$ depends only on the plane-wave phase, the dynamics can be reduced to a single quadrature determining the four-velocity and trajectory, from which energy and momentum components follow.
[1] Di Piazza A., Lett Math Phys 83, 305-317 (2008)
[2] Niel, F. et al., Phys. Rev. E 97, 043209 (2018)
[3] Di Piazza A., Rev. Mod. Phys. 84, 1177 (2012)
[4] Blackburn, T. G., Phys. Rev. A 109, 022234 (2024)