Algorithms Preserve Electron Coherence in Magnetic Fields

Scientists are tackling the long-standing problem of accurately modelling electron radiation reaction, a process where charged particles emit radiation and experience a recoil force. Jacob Matthew Molina from Princeton Plasma Physics Laboratory, Princeton University and Department of Astrophysical Sciences, Princeton University, and Hong Qin from Department of Astrophysical Sciences, Princeton University, demonstrate a significant advance in this field through the development of geometric structure-preserving algorithms for the coupled Schrödinger-Maxwell system. This research, conducted in collaboration between Princeton Plasma Physics Laboratory and the Department of Astrophysical Sciences at Princeton University, overcomes limitations of classical approaches by preserving key physical properties within their simulations. Their new Structure-Preserving scHrodINger maXwell (SPHINX) code allows for detailed investigation of coherent electron states under intense radiation, revealing how these states rapidly lose coherence and disperse, and how fundamental Landau levels are modified by electromagnetic interactions. These findings are crucial for improving our understanding of extreme-field phenomena relevant to areas such as fusion plasmas, astrophysics, and high-intensity laser physics.

Within a powerful magnetic field, electrons spiral and glow as they shed energy in the form of light. By understanding this radiation loss is fundamental to controlling charged particles, offering insight into extreme environments from fusion reactors to distant stars. Scientists have long sought to accurately model radiation reaction, the momentum lost by charged particles emitting radiation.

Classically described by the Abraham-Lorentz and Landau-Lifshitz forces, these equations falter when applied to atomic scales, even predicting unphysical behaviour such as runaway acceleration. At these dimensions, the idea of a radiation reaction force becomes questionable as coherent electron states are destroyed. Scientists are turning to the coupled Schrödinger-Maxwell (SM) system. A framework describing the simultaneous evolution of electrons and the electromagnetic fields they generate, to better understand this phenomenon.

Historically, the inherent nonlinearity of the SM system has prevented detailed analytical investigations. A new computational approach offers a pathway forward. Geometric structure-preserving algorithms have been developed for the SM system, ensuring the preservation of key physical properties like gauge invariance, symplecticity. Unitarity when implemented on a discrete space-time lattice.

These algorithms are incorporated into a code named SPHINX, enabling simulations of electron behaviour under extreme conditions. By initiating simulations with coherent states built from Landau levels, quantized energy levels arising from motion in a magnetic field, SPHINX can track the energy distribution and the loss of coherence in an electron wave packet as it radiates.

Simulations reveal a strong emission of radiation and a corresponding rapid loss of orbital coherence when an electron is prepared in a coherent state at atomic scales. Rather than maintaining a defined path, the electron wave packet disperses, demonstrating the breakdown of classical orbital concepts. These simulations also examine the behaviour of Landau levels within the coupled SM system, revealing they are renormalized into stationary “dressed” eigenstates, possessing constant electromagnetic and kinetic energies. Offering a new basis for describing the interaction between electrons and photons.

This effort provides a new computational window into radiation reaction physics and has implications for modelling extreme-field phenomena encountered in fusion plasmas, astrophysics, and advanced laser experiments. The Larmor formula provides a starting point, but deriving a consistent force requires careful consideration of the electron’s behaviour. Attempts to extend the Abraham-Lorentz force to relativistic velocities, using the Lieńard radiation formula. Still encounter limitations at very small scales, below the Compton wavelength.

Fully quantum and relativistic approaches, rooted in quantum electrodynamics, introduce their own difficulties. Including the possibility of runaway solutions where the electron appears to accelerate before any force is applied. Here, scientists have moved beyond approximations by focusing on the self-consistent evolution of electron and photon fields. For non-relativistic electrons, the Dirac equation simplifies to the Schrödinger equation, forming the basis of the Schrödinger-Maxwell (SM) system.

This system offers a framework for modelling the interaction between particles and their self-generated electromagnetic fields. The simulations maintain important physical properties on a discrete space-time lattice by adopting geometric structure-preserving algorithms. Such algorithms are implemented in the SPHINX code, allowing for detailed investigations of electron dynamics.

At the heart of the simulations lies the construction of coherent states from Landau level eigenstates, representing the quantum analogue of classical orbits — simulations demonstrate that an electron initially prepared in an atomic-scale coherent state experiences substantial energy loss through radiation. Meanwhile, the electron’s wave packet rapidly loses coherence and disperses, highlighting the limitations of classical descriptions at these scales, and the simulations also reveal how the coupled SM system modifies the well-known dynamics of Landau levels. Renormalizing them into stationary dressed eigenstates with constant energies.

Rapid Electron Decoherence and Landau Level Renormalisation in Strong Fields

Simulations reveal that electrons in strong magnetic fields radiate energy and rapidly lose coherence. Specifically, an electron initially prepared in a coherent state with a scale of approximately 1.08 × 10−5 metres disperses into a decoherent wave packet within a timescale of 1.7 × 10−15 seconds. At the same time, this rapid decoherence arises from the radiation reaction force, which fundamentally alters the electron’s wave-like behaviour.

Further the energy partition between the electron and the emitted radiation evolves dynamically, indicating a continuous transfer of energy from the electron to the electromagnetic field. Here, the coupled Schrödinger-Maxwell (SM) system also modifies the well-known dynamics of Landau levels, renormalizing them into stationary dressed eigenstates.

Such dressed states exhibit constant electromagnetic and kinetic energies, a departure from the ideal Schrödinger-only case. Meanwhile, at a grid resolution of 64 × 64 × 64 points, the simulations accurately capture this renormalization process. Demonstrating the interaction between the electron’s quantum state and the generated electromagnetic field. At the same time, the geometric structure-preserving algorithms implemented in the SPHINX code are central to these findings.

By preserving gauge invariance, symplecticity, and unitarity on the discrete space-time lattice, these algorithms ensure the numerical stability and accuracy of the simulations. Once discretized, the Hamiltonian describing the SM system is expressed in terms of discrete variables, allowing for efficient computation of the time evolution of the electron’s wave function and the electromagnetic field.

Inside the discrete formulation, the variational derivative of a functional F with respect to the vector potential A is defined as the sum over grid points J of the product of a discrete delta function and the partial derivative of F with respect to the vector potential at that grid point. By employing this discretization scheme, the canonical Poisson bracket structure is preserved, ensuring that the numerical solution accurately reflects the underlying physics. Also, this approach allows for a more stable and accurate representation of the electron’s dynamics.

Since the algorithms are designed to maintain these geometric properties, they provide a reliable framework for studying radiation reaction effects in extreme-field scenarios. At the boundaries, assuming either fixed or periodic conditions, the variation of the Hamiltonian is computed via integration by parts, leading to a system of coupled non-linear partial differential equations governing the time evolution of the SM system.

These equations describe the interaction between the electron’s quantum state and the electromagnetic field, capturing the essential physics of radiation reaction. By solving these equations numerically, the simulations provide insights into the behaviour of electrons in strong magnetic fields.

Computational advances resolve longstanding challenges in modelling radiation reaction and energetic particle dynamics

The decay of electron coherence due to radiation reaction is now yielding to detailed computational scrutiny. For decades, modelling this phenomenon, where charged particles seemingly lose energy simply by accelerating. Proved exceptionally difficult because standard equations break down at the relevant scales. By existing approaches struggled to accurately represent the interaction between the particle’s motion and the electromagnetic fields it generates, often leading to unphysical results.

Scientists have turned to advanced algorithms preserving fundamental physical principles like gauge invariance and unitarity. For more reliable simulations of this complex process. The significance extends beyond fundamental physics. By understanding radiation reaction is becoming increasingly important in fields like fusion energy research, where controlling energetic electrons is vital for achieving stable plasmas.

Similarly, in astrophysics, accurately modelling particle behaviour in extreme magnetic fields, around pulsars or black holes, requires a solid grasp of these subtle effects. Next-generation laser facilities are pushing the boundaries of electromagnetic fields, creating conditions where radiation reaction can no longer be ignored. These simulations, while sophisticated, are computationally expensive, restricting the size and duration of the modelled scenarios.

Real-world plasmas are far more complex, containing numerous particles and turbulent fields — however, extending these algorithms to handle multiple interacting particles presents a substantial challenge. The path forward involves refining these structure-preserving algorithms to improve computational efficiency and scaling, and by incorporating more realistic plasma conditions, including collisions and turbulence, will be essential. A broader effort is needed to develop efficient numerical methods for solving coupled Schrödinger-Maxwell equations, while potentially opening new avenues for exploring fundamental physics and tackling practical problems in diverse fields.