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arXiv:2411.17517 · Journal of Plasma Physics

Variational quantum simulation of the Fokker–Planck equation applied to quantum radiation reaction

Óscar Amaro, Lucas Ivan Iñigo Gamiz, Marija Vranić

Published Aug 1, 2025• 3 citations
arXivHTMLPDF

Abstract

Abstract Near-future experiments with Petawatt class lasers are expected to produce a high flux of gamma-ray photons and electron–positron pairs through strong field quantum electrodynamical processes. Simulations of the expected regime of laser–matter interaction are computationally intensive due to the disparity of the spatial and temporal scales, and because quantum and classical descriptions need to be accounted for simultaneously (classical for collective effects and quantum for nearly instantaneous events of hard photon emission and pair creation). We study the stochastic cooling of an electron beam in a strong, constant, uniform magnetic field, both its particle distribution functions and their energy momenta. We start by obtaining approximate closed-form analytical solutions to the relevant observables. Then, we apply the quantum-hybrid variational quantum imaginary time evolution to the Fokker–Planck equation describing this process and compare it against theory and results from particle-in-cell simulations and classical partial differential equation solvers, showing good agreement. This work will be useful as a first step towards quantum simulation of plasma physics scenarios where diffusion processes are important, particularly in strong electromagnetic fields.

DOI
10.1017/s0022377825100652

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Radiation reaction against a relativistic electron is of critical importance since the experiment to check this "quantumness" becomes possible soon with an extremely high-intensity laser beam. However, there is a fundamental mathematical quest to apply any laser profiles to laser focusing and superposition beyond the Furry picture of its usual method by a plane wave. To give the apparent meaning of q(χ)q(χ)q(χ) the quantumness factor with respect to a radiation process is absent. Thus for resolving the above questions, we propose stochastic quantization of the classical radiation reaction model for any laser field profiles, via the construction of the relativistic Brownian kinematics with the dynamics of a scalar electron and the Maxwell equation with a current by a Brownian quanta. This is the first proposal of the coupling system between a relativistic Brownian quanta and fields in Nelson's stochastic quantization. Therefore, we can derive the radiation field by its Maxwell equation, too. This provides us the fact that q(χ)q(χ)q(χ) produced by QED is regarded as P(Ωτave)\mathscr{P}(\varOmega_τ^{\mathrm{ave}})P(Ωτave​) of an existence probability such that a scalar electron stay on its average trajectory.
In the limit of asymptotically small photon momentum, a Fokker–Planck equation for the center-of-mass Wigner function suffices to describe laser cooling of an atom. We have devised a procedure to compute the light pressure force and the diffusion tensor for an atom with an arbitrary level structure, traveling with an arbitrary velocity in any given three-dimensional light field. The scheme is based on solving a set of differential equations closely related to the density matrix equations of motion encountered in conventional recoilless laser spectroscopy. Computer programs have been written that automatically construct and solve the required equations from broad specifications of the problem, such as energies and angular momenta of the atomic levels. A Langevin equation stochastic simulation has been implemented to solve the Fokker-Planck equation of laser cooling using the computed force and diffusion. Comparisons of optical-molasses experiments and simulations carried out for a model sodium atom with a J = 2→J = 3 transition show qualitative similarities such as sub-Doppler limit temperatures, but the quantitative agreement of theory and experiment is not satisfactory.
In this paper we derive the Fokker-Planck equation that describes the motion of a charged particle, trapped in a static harmonic potential and cooled by radiation pressure forces exerted by a laser beam. This equation is then discussed in both the classical and quantum regimes. The model adopted here is a simple one, in which one-dimensional motion and a two-level atom are assumed, and the time evolution of the system is described in the quasiclassical limit, in which the quantum nature of the photon exchanged between the particle and the field plays no role. We report relevant graphs for the loss of kinetic energy suffered by the particle in the cooling process and estimates of the lowest kinetic temperature that can be reached in these experiments.
The new scheme of stochastic quantization is proposed. This quantization procedure is equivalent to the deformation of an algebra of observables in the manner of deformation quantization with an imaginary deformation parameter (the Planck constant). We apply this method to the models of nonrelativistic and relativistic particles interacting with an electromagnetic field. In the first case we establish the equivalence of such a quantization to the Fokker-Planck equation with a special force. The application of the proposed quantization procedure to the model of a relativistic particle results in a relativistic generalization of the Fokker-Planck equation in the coordinate space, which in the absence of the electromagnetic field reduces to the relativistic diffusion (heat) equation. The stationary probability distribution functions for a stochastically quantized particle diffusing under a barrier and a particle in the potential of a harmonic oscillator are derived.
For quantum computers to become useful tools to physicists, engineers and computational scientists, quantum algorithms for solving nonlinear differential equations need to be developed. Despite recent advances, the quest for a solver that can integrate nonlinear dynamical systems with a quantum advantage, while being realizable on available (or near-term) quantum hardware, is an open challenge. In this paper, we transform a nonlinear dynamical system into a linear system, which we integrate with quantum algorithms. Key to the method is the Fokker–Planck equation, which is a non-normal partial differential equation. Three approaches are proposed: (i) forward-Euler stepping by block encoding; (ii) Schrödingerization and (iii) forward-Euler stepping by a linear combination of unitaries. We emulate the integration of prototypical nonlinear systems with the proposed quantum solvers and compare the output with the benchmark solutions of classical integrators. We find that classical and quantum outputs are in good agreement. This paper opens opportunities for solving nonlinear differential equations with quantum algorithms.