Charged particle motion and radiation in strong electromagnetic fields

The dynamics of charged particles in electromagnetic fields is an essential component of understanding the most extreme environments in our Universe. In electromagnetic fields of sufficient magnitude, radiation emission dominates the particle motion and effects of quantum electrodynamics (QED) in strong fields are crucial, which triggers electron-positron pair cascades and counterintuitive particle-trapping phenomena. As a result of recent progress in laser technology, high-power lasers provide a platform to create and probe such fields in the laboratory. With new large-scale laser facilities on the horizon and the prospect of investigating these hitherto unexplored regimes, this review explores the basic physical processes of radiation reaction and QED in strong fields, how they are treated theoretically and in simulation, the new collective dynamics they unlock, recent experimental progress and plans, and possible applications for high-flux particle and radiation sources.

Figure 1

Cube of theories. Its three axes correspond to relativistic ($c$), quantum ($\hslash$), and high-intensity effects ($E_{\mathrm{cr}}$) and its vertices correspond to the theories: $(0,0,0)$ is classical mechanics; $(c,0,0)$ is special relativity; $(0,\hslash ,0)$ is quantum mechanics; $(c,\hslash ,0)$ is quantum field theory; $(c,0,E_{\mathrm{cr}})$ is classical electrodynamics; $(0,\hslash ,E_{\mathrm{cr}})$ is atomic, molecular, and optical physics; and $(c,\hslash ,E_{\mathrm{cr}})$ is high-intensity particle physics. Here $E_{\mathrm{cr}}=m^2{c}^3/e\hslash$ is the critical field of QED, where $m$ is the electron mass and $e$ is the elementary charge. Adapted from [84].

Figure 2

Laser facilities on the map of pulse energy and pulse duration. Data were taken from [404], [127], and individual facilities; see Table 2. For illustration, we show the nominal peak intensity levels estimated, assuming ideal $f/2$ focusing of a linearly polarized pulse with Gaussian spatial and temporal profiles and a nominal wavelength of $\lambda =1\mu \mathrm{m}$. The physics in access is denoted by color and the following characteristic intensity scales (see Sec. 1c): $3.5\times {10}^{16}\mathrm{W}/{\mathrm{cm}}^2$ [atomic field corresponding to a field strength of 1 a.u.; see 314], $1.37\times {10}^{18}\mathrm{W}/{\mathrm{cm}}^2$ (relativistic electron corresponding to $a_0=1$), $8\times {10}^{22}\mathrm{W}/{\mathrm{cm}}^2$ [radiation-dominated dynamics corresponding to $L_d=\lambda$, assuming that $\gamma =a_0$; see Eq. (13)], $3.5\times {10}^{23}\mathrm{W}/{\mathrm{cm}}^2$ [avalanche-type cascades corresponding to $L_p=\lambda$, assuming that $\gamma =a_0$; see Eq. (16)], and $4.65\times {10}^{29}\mathrm{W}/{\mathrm{cm}}^2$ (corresponding to the Schwinger field strength). Inset: effective intensity boost that one would gain by splitting the laser power among the specified number of beams and colliding them so that the electric field is summed up coherently; see the Appendix of [212]. The boost is shown via the corresponding relative shift of location on the map for the cases using $f/2$ (left side) and $f/1$ (right side) focusing, and the dotted line shows the ultimate boost given by the dipole wave; see Sec. 5e3.

Figure 3

Interaction regimes in the space of the normalized field amplitude $a_0$ and the quantum nonlinearity parameter $\chi$. We identify the importance of different processes by comparing key space scales with a characteristic laser wavelength $\lambda =1\mu \mathrm{m}$. The decreasing depletion length $L_d$ [Eq. (13)] denotes the transition from weak to strong radiation losses ($L_d>{10}^3\lambda$ in white, $L_d<\lambda$ in dark gray). A quantization length $L_q$ [Eq. (14)] comparable to $\lambda$ indicates that the discrete nature of radiation emission is important ($L_q>\lambda$, checkered area). As the pair-creation length $L_p$ [Eq. (16)] becomes smaller than $\lambda$, electron-positron pair creation by photons in strong fields becomes prolific ($L_p<\lambda$ in light blue). The field amplitude necessary for the onset of vacuum pair creation by the Schwinger mechanism, shown in violet, assumes a nonzero $\mathcal{F}=a_0(\hslash {\omega }_0/m{c}^2)$. Key limits on theoretical treatments are as follows: the field cannot be assumed to be constant if the formation length $L_f<\lambda$ [Eq. (15)] (yellow line), perturbation theory in SFQED is expected to break down when $\alpha {\chi }^{2/3}\gtrsim 1$ (blue line), and a general electromagnetic field cannot be approximated as crossed if ${\chi }^2<\mathcal{F},\mathcal{G}$. We also outline the prospects for laser experiments in interactions with a stationary plasma target ($\gamma \lesssim a_0$, red dashed line) or ultrarelativistic electron beams (green lines). Numbered green points indicate experiments that have probed the strong-field regime (in order of publication): (1) [75] and [97], (2) [575], (3) [119], and (4) [425].

Figure 4

Interaction regimes as a function of the amplitude $a_0$ and the wavelength $\lambda$ of the electromagnetic wave driving an electron, using the same key as in Fig. 3. The importance of RR effects is parametrized by $L_d$ [Eq. (13)] and $L_q$ [Eq. (14)], which are both wavelength dependent.

Figure 5

The Lorentz factor $\gamma$ of an electron in a rotating electric field of normalized amplitude $a_0$, as predicted by the LAD and Landau-Lifshitz equations, Eqs. (18) and (19), respectively: linear (left panel) and log scaled (right panel). The vertical lines indicate, from left to right, onset of classical radiation-reaction effects, Eq. (73) (red line); a quantum parameter of unity, Eq. (74) (orange line); and the QED-critical field strength, Eq. (1) (black line). Here ${ϵ}_{\mathrm{rad}}=1.47\times {10}^{-8}$, which is equivalent to a wavelength of $0.8\mu \mathrm{m}$. Adapted from [89].

Figure 6

Radiation at unphysically large frequencies is predicted by classical theory if ${\chi }_e\gtrsim 1$: classical power spectra (dashed lines) and quantum-corrected power spectra (solid lines) at ${\chi }_e=0.1$ (blue lines, left), 1 (green lines, center), and 10 (orange lines, right), using Eqs. (22) and (23), respectively.

Figure 7

Nonperturbativity in strong-field quantum electrodynamics arises in two ways: from the coupling between the charge and the background (external) field (lower set of wavy lines, in red) and the coupling between the charge and the radiation (quantized) field (upper set of wavy lines, in orange). For $a_0>1$, the former interaction must be taken into account exactly, i.e., to all orders in $a_0$, as indicated by the double fermion lines in a diagrammatic representation. For fields with a sufficient magnitude or duration, higher-order contributions to the coupling with the radiation field can dominate lower-order terms.

Figure 8

Evolution of a Volkov wave packet (in a time $t$ and a coordinate $z$) in a laser pulse with a normalized amplitude $a_0=2$ and duration $\mathrm{\Delta }\varphi =20$ (color or gray scale density indicates the wave function). The wave packet begins with a Gaussian distribution of light-front momentum (mean $mc$ and standard deviation $0.05mc$) and zero transverse momentum. The centroid closely follows the equivalent classical trajectory, which is indicated as a solid black line. The laser pulse propagates between the two dashed lines. From [477].

Figure 9

Feynman diagrams of the Compton ($e\to e\gamma$) and Breit-Wheeler ($\gamma \to ee$) processes. Double fermion lines indicate that the process occurs in an external field.

Figure 10

Classically, the photon formation length $L_f$ can be related to the emission angle ${\theta }_{\gamma }$ and the instantaneous radius of curvature $r_c$ of the electron trajectory. From [61].

Figure 11

One-loop mass operator, polarization operator, and vertex correction displayed from left to right.

Figure 12

Dressed loop expansion of the polarization operator $P$ (top row) and mass operator $M$ (bottom row). According to the Ritus-Narozhny conjecture, these diagrams represent the dominant contribution at $n$ loops, and $\alpha {\chi }^{2/3}$ is the true expansion parameter of SFQED in the regime $\chi \gg 1$. From [581].

Figure 13

Examples of higher-order SFQED processes. Trident pair creation, double nonlinear Compton scattering, phototrident pair creation, and photon splitting are displayed from left to right.

Figure 14

$G({\chi }_e)$ is the factor by which quantum effects reduce the radiation power from its classically predicted value: the full expression (blue solid line) and limiting values at small and large ${\chi }_e$ (black dashed line on the left and red dashed line on the right, respectively) from Eqs. (62) and (63), respectively.

Figure 15

Overview of the ways that spin influences, and is influenced by, particle degrees of freedom (top and right blue panels) and radiation (left yellow panel). Individual couplings are denoted by arrows, with those introduced in Sec. 2d given in black. Adapted from [526].

Figure 16

Spin-resolved photon-emission rates $d{N}^{s,s^{\prime }}/d\tau$, normalized to the spin-averaged rate $dN/d\tau$, for electrons orbiting in a rotating electric field of normalized amplitude $a_0$. Here $s$ and $s^{\prime }$ are the projections of the electron initial and final spin on the axis of rotation. From [131].

Figure 17

Schematic representation of the energy spectrum of the electromagnetic field in a high-intensity laser-matter interaction. The frequencies shown (from left to right) are characteristic of the target (${\omega }_t=c/L$, with target size $L$), the laser (${\omega }_L$), the plasma density (${\omega }_p$), the upper limit of coherent emission processes (${\omega }_{\mathrm{coh}}$), and incoherent synchrotron emission (${\omega }_c$). From [213].

Figure 18

Particle merging in a simulation of a laser-driven QED cascade indicating energy conservation and growth in the number of particles over the course of the simulation. Adapted from [551].

Figure 19

Principal experimental schemes aimed at the study of nonlinear QED. Upper panel: laser–electron-beam interactions (all-optical setup). Lower panel: colliding laser pulses. Adapted from [86].

Figure 20

Radiation-free direction ${\mathbf{n}}_{\mathrm{RFD}}^{+}$, as a function of the end point of the electric-field vector $\mathbf{E}$, given a fixed magnetic-field vector $\mathbf{B}$ (blue arrow). Arrows denote the projection of ${\mathbf{n}}_{\mathrm{RFD}}^{+}$ onto the plane spanned by $\mathbf{E}$ and $\mathbf{B}$, and the color scale shows the projection on $\mathbf{B}\times \mathbf{E}$: values are negative (positive) in the upper (lower) half. From [215].

Figure 21

Radiation-reaction trapping in simulations of laser-wakefield acceleration. Shown are the densities of (a),(d) the electrons, (b),(e) the protons, and (g) the $\gamma$ photons, as well as (c),(f) the transverse component of the electric field obtained using 3D PIC simulations with (right column) and without (left column) radiation reaction. From [274].

Figure 22

Electron number density as a function of the wave amplitude (here called $a=a_0$) in a linearly polarized, standing electromagnetic wave. As $a$ increases, the ponderomotive, normal, and anomalous radiative trapping regimes are accessed. The curves in the lower region indicate the ponderomotive potential (thick gray line), as well as the electric (thin red line) and magnetic (dot-dashed blue line) field amplitudes, in arbitrary units. From [212].

Figure 23

Explanation of ART through the net migration of electrons toward electric-field maxima. The particle trajectory (black curve, axis to the left), its gamma factor (red dashed curve, axis to the right), and asymptotic trajectories (thin green curves) are shown together with the regions of electric-field (red) and magnetic-field (blue) dominance (see details in the text). From [212].

Figure 24

Poincaré sections showing particle coordinates in the space of position $x$ and momentum $p_x$ at time intervals equal to the period of the driving field. (a) $a_0=617$, ${ϵ}_{\mathrm{rad}}=1.2\times {10}^{-8}$. (b) $a_0=778$, ${ϵ}_{\mathrm{rad}}=6\times {10}^{-9}$. (c) $a_0=1996$, ${ϵ}_{\mathrm{rad}}=1.2\times {10}^{-9}$ From [92].

Figure 25

Electron trajectories in the standing wave formed by two colliding circularly (upper frame) and linearly (lower frame) polarized laser pulses propagating along the $x$ axis with $I=1.37\times {10}^{24}\mathrm{W}/{\mathrm{cm}}^2$, $\lambda =1\mu \mathrm{m}$, duration 33 fs, and focal spot $3\mu \mathrm{m}$. Modeled with semiclassical radiation reaction [Eq. (64)]. From [179].

Figure 26

Electron trajectories in the standing wave formed by two colliding circularly (upper frame) and linearly (lower frame) polarized laser pulses propagating along the $x$ axis with $I=1.11\times {10}^{24}\mathrm{W}/{\mathrm{cm}}^2$, $\lambda =1\mu \mathrm{m}$, duration 30 fs, and focal spot $3\mu \mathrm{m}$. Modeled with quantum stochastic radiation reaction. From [276].

Figure 27

Feedback between SFQED processes and classical relativistic plasma dynamics in a QED plasma.

Figure 28

Experimental configurations envisaged for (top panel) laser-electron and (bottom panel) laser-$\gamma$ collisions at LUXE. From [3].

Figure 29

All-optical realization of a laser–electron-beam collision experiment. One laser pulse is focused using a short-focal-length optic with a hole in it to allow for counterpropagation of an electron beam that is accelerated by a laser wakefield in a gas jet. Electrons, and the radiation they emit, are transmitted through this hole before being diagnosed. The collision is timed so that it occurs close to the edge of the gas jet, where the electron beam is smallest, to maximize overlap with the laser pulse. From [119].

Figure 30

Experimental evidence of radiation reaction in the correlation between the energy of the electron beam after the collision and the critical energy of the $\gamma$-ray spectrum as measured in four successful collisions (points), which are compared to simulations of the interaction that include various models of radiation reaction and fluctuations in the electron energy and laser intensity. Colored (shaded) regions from left to right: classical RR (blue), stochastic RR (orange), and no RR (green). From [119].

Figure 31

Experimental evidence of radiation reaction and quantum corrections: electron energy spectra as measured at best overlap, without the scattering laser (black line) and with it (red line), and from simulations without (a) radiation reaction, (b) classical (Landau-Lifshitz) radiation reaction, (c) a quantum-corrected Landau-Lifshitz model, and (d) stochastic radiation reaction (PIC and single-particle codes in green and blue, respectively), all including experimental uncertainties in the initial electron spectrum, the magnetic spectrometer, and the laser intensity. Adapted from [425].

Figure 32

Interaction regimes in the collision of an electron beam of energy ${ϵ}_0$ with a dipole wave generated by MCLPs with total power $P$. Number of high-energy ($\hslash \omega >{ϵ}_0/2$) photons per incident electron, red-orange color scale, solid contours; number of electron-positron pairs per incident electron, blue-green color scale, dashed contours; final electron-beam energy, percent of initial value, gray color scale, dotted contours. From [361].

Figure 33

Brightness of radiation emitted in high-intensity laser interactions with electron beams (empty circles) or plasma targets (filled circles), as well as in nonlaser strong-field environments, as (1 and 2) measured in recent experiments and (3–10) predicted using simulations.

Figure 34

Efficiency of $\gamma$-ray generation in plasmas driven by single (filled circles), dual (double circles), and multiple (diamonds) lasers as predicted by simulations.

Figure 35

Number of positrons produced in high-intensity laser-plasma interactions. For laser–electron-beam interactions (empty circles), the energy of the electron beam is noted in brackets. Points marked with asterisks indicate experimental results from LWFA electron-beam interactions with high-$Z$ foils; in these cases the laser power is not indicated.

Figure 36

Dedicated SFQED facility housing multiple petawatt-class lasers. The facility can operate in several modes, including (i) an $e^{+}{e}^{-}$ collider, with all lasers used to drive the staged acceleration of electron and positron beams; (ii) a laser–electron-beam collider where half of the lasers drive the staged acceleration of the electron beam and the remaining half provides the high-field region via the multiple colliding pulses configuration; and (iii) all the laser pulses are brought to the interaction point to generate the highest intensity possible. From [592].