Self-Consistent Dynamics of Electron Radiation Reaction via Structure-Preserving Geometric Algorithms for Coupled Schrödinger-Maxwell Systems

TL;DR

The paper develops structure-preserving geometric algorithms for the Schrödinger-Maxwell system to study self-consistent radiation reaction (RR) in an electron under a uniform magnetic field. By implementing these on SPHINX, the authors preserve gauge invariance, symplecticity, and unitarity while evolving the quantum wavefunction and electromagnetic fields on a lattice. Analyses of coherent Landau-state dynamics reveal rapid decoherence and strong radiation when RR is included, while Landau levels can renormalize into stationary dressed eigenstates under proper boundary conditions. The findings provide a computational window into RR physics relevant to extreme-field plasmas, astrophysics, and next-generation laser experiments, highlighting the limits of classical RR models and the importance of fully self-consistent quantum-field dynamics.

Abstract

Classically, a charged particle in a magnetic field emits radiation, losing momentum and experiencing the Abraham-Lorentz (AL) / Landau-Lifshitz (LL) radiation reaction (RR) force. However, at atomic scales and outside the range of their applicability, the AL/LL equations fail and RR destroys the coherent state of an electron-undermining the very concept of a RR force. This process can be described by the coupled Schrödinger-Maxwell (SM) system under appropriate limits, but the system's nonlinear complexity has long limited purely analytical studies. We present geometric structure-preserving algorithms for the SM system that preserve gauge invariance, symplecticity, and unitarity on the discrete space-time lattice, which are implemented in our Structure-Preserving scHrodINger maXwell (SPHINX) code. By constructing coherent states from the Landau levels, SPHINX simulates the fully-coupled nonlinear dynamics of an electron coherent state, the energy partition evolution, and decoherence/relaxation of the electron wave packet in time due to RR. These simulations indicate that, in an external magnetic field, an electron prepared in an atomic-scale coherent state can radiate strongly, rapidly losing coherence and dispersing into a decoherent wave packet. Additionally, we also present the fully-coupled nonlinear evolution of the non-degenerate ground- and first-excited Landau levels themselves to understand how the coupled SM system modifies the well-known ideal (i.e., Schrödinger-only) dynamics of the Landau Levels. With appropriate boundary conditions, simulations show that the Landau levels are renormalized into stationary dressed eigenstates with constant electromagnetic and kinetic energies. This opens a new computational window into RR physics and advances modeling of extreme-field phenomena in fusion plasmas, astrophysics, and next-generation laser experiments

Figures (11)

• Figure 1: Uncoupled coherent state evolution. Here, $T_{c} = 2 \pi / \omega_c$ is the cyclotron period and figures (a) - (e) depict the evolution of the coherent state over a full cyclotron period respectively
• Figure 2: Energy partition evolution of static-field cyclotron simulation. Here, $T_{c} = 2 \pi / \omega_c$ is the cyclotron period, $H_{\text{Total}} = H_{qm} + H_{em}$ is the total energy of the system, $H_0$ is the initial total energy of the system, $P = \int dV |\psi|^2$, and $P_0$ is the initial value of $P$. Figures (a) - (d) depict evolution of $H_{em}, H_{qm}, H_{\text{Total}}, P$, while figures (e) - (h) respectively depict the error in these values.
• Figure 3: Dynamic evolution of $|\Psi|^2$ and $\delta B_{z}$ over 6 cyclotron periods ($T_{c} = 2 \pi /\omega_c$) for the fully coupled $\beta = 10$ simulation. The left hand column presents the evolution of $|\Psi|^2$ for the coherent state $\Psi$ over the time periods $t /T_{c} = 0,~1,~2.4,~3.53,~5.08, ~\text{and } 6$ in plots (a), (c), (e), (g), (i), and (k) respectively. Similarly, the right hand column presents the evolution of the z-component of the non-uniform/perturbed magnetic field $\delta \textbf{B}$ over the same respective times in plots (b), (d), (f), (h), (j), and (l). In both sets of figures, the contours of $|\Psi|^2$ and a circle of radius $\rho_{\ell}$ (the Larmor radius) are overlaid atop the figures to act as visual aid
• Figure 4: Evolution of the energy partition of the SM system for the fully coupled $\beta = 10$ simulation over 6 ideal cyclotron periods ($T_{c} = 2 \pi / \omega_c$). Here, $H_{\text{Total}} = H_{qm} + H_{em}$ is the total energy of the system, $H_0$ is the initial total energy of the system, $P \equiv \int dV |\psi|^2$, and $P_0$ is the initial value of $P$. Plots (a)-(c) depict the conservation of the electromagnetic subsystem energy ($H_{em}$), quantum subsystem energy ($H_{qm}$), and $H_{\text{Total}}$ relative to $H_{0}$ respectively, and plot (d) depicts the conservation of $P$ relative to $P_0$. Plots (g) and (h) depicts the error in $H_{\text{Total}}$ and $P$ respectively. Plot (e) and (f) depict how the energy of the partition of each subsystem system evolves relative to the time-dependent subsystem energies $H_{em}(t)$ and $H_{qm}(t)$ respectively
• Figure 5: Dynamic evolution of $|\Psi|^2$ and $\delta B_{z}$ over 3 cyclotron periods ($T_{c} = 2 \pi /\omega_c$) for the fully coupled $\beta = 5$ simulation. The left hand column presents the evolution of $|\Psi|^2$ for the coherent state $\Psi$ over the time periods $t /T_{c} = 0,~1,~1.5,~1.9,~2.22, ~\text{and } 3$ in plots (a), (c), (e), (g), (i), and (k) respectively. Similarly, the right hand column presents the evolution of the z-component of the non-uniform/perturbed magnetic field $\delta \textbf{B}$ over the same respective times in plots (b), (d), (f), (h), (j), and (l). In both sets of figures, the contours of $|\Psi|^2$ and a circle of radius $\rho_{\ell}$ (the Larmor radius) are overlaid atop the figures to act as visual aid