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
A charged particle in a uniform magnetic field undergoes cyclotron motion, constantly accelerating and emitting radiation as it does [1].
Conservation of momentum requires there exist an equal-and-opposite reaction force as the radiation is emitted: what is this force? While a seemingly simple question on its face, the nature of this force -the so-called Radiation Reaction (RR) force -has been a contentious question in physics since its inception. RR solutions have historically been garnered through a variety of different methods [2][3][4][5]. The non-relativistic Abraham-Lorentz (AL) force can be heuristically derived from the Larmor formula, and can be extended to relativistic velocities through a similar argument instead beginning from the Lieńard radiation formula [4]. However, the AL force has well-known theoretical limitations; failing at atomic scales below the Compton wavelength [6]. Furthermore, the AL equation fails to provide an accurate description of the instantaneous particle radiation / energy loss in even as simple a case as cyclotron motion [1]. Fully quantum and relativistic RR forces derived from quantum field theory are beset by unseemly fundamental difficulties such as runaway solutions that predict an exponentially increasing RR force in which the classical point charge retro-causally accelerates prior to the application of the force [1]. In the limit of a weak radiation damping force, the Landau-Lifshitz (LL) radiation damping force avoids these issues but is limited in its applicability [5]. Neither the AL nor the LL models are first principles calculations.
Historically, radiation reaction has been invoked to invalidate the classical picture of an electron gyrating around a nucleus at atomic scales: a classical charged particle in a Kepler orbit would radiate away its kinetic energy in roughly 10 picoseconds and spiral into the nucleus [7]. A quantum description is therefore essential; an electron can be prepared in a coherent state, the quantum analogue of a classical orbit, even at atomic scales. How does such a classical-like coherent quantum state evolve under radiation self-consistently? This is one of the main questions addressed in the present study.
The RR problem belongs to the broader class of selfforce / self-field problems, whereby particles interact with their own self-generated fields [1,8]. The fundamental difficulty of RR forces stems from the fact that an electron is neither a classical point particle nor a classical extended rigid body, cases for which the RR problem has been solved within classical electrodynamics [9][10][11]. Instead, an electron is described by a quantum wave function in spacetime, governed by the Dirac equation, and the dynamics of this wave function are coupled to that of the electromagnetic field (photons) governed by Maxwell’s equations. We note that radiation reaction for an accelerating electron is not well represented as one or a few discrete QED scattering events. It is a cumulative effect arising from a large number of electronphoton interactions. The appropriate framework is the pre-quantized Dirac-Maxwell system, which captures the tree-level dynamics of the underlying QED [12]. What the Dirac-Maxwell equations, viewed as a spacetime PDE system, do not include are loop-level QED corrections. These effects are not the primary concern for the radiation-reaction physics in plasmas considered in the present study.
One strategy to developing a better understanding of RR processes can be garnered by self-consistently evolving the fields of electrons and photons. In the regime of interest, the electrons are nonrelativistic: their typical kinetic energies and potential-energy variations are small compared with their rest energy. In this lowenergy limit the Dirac equation reduces to the Pauli equation, and, when spin-dependent effects are not essential to the phenomena we study, it further reduces to the Schrödinger equation with relative corrections of order (v/c) 2 . Thus, we adopt the Schrödinger-Maxwell (SM) system in the present study as a quantum system that self-consistently evolves the fields of particles and photons and provides an important perspective on the RR processes.
The SM system is growing in importance as high-field physics is becoming increasingly more relevant to active areas of research such as high energy density physics, controlled nuclear fusion, and experimental/laboratory astrophysics [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Particularly, the advent of ultra high-intensity lasers has already began to presage the growing need to understand RR in such systems [25,[28][29][30][31][32][33]. Purely analytical studies in this vein, however, are prohibited by the non-linear nature of the coupled SM system.
We adopt geometric structure-preserving algorithms for the SM system that preserve gauge invariance, symplecticity, and unitarity on the discrete spacetime lattice [34], and implement these algo
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