Ultraintense Laser-Driven Nonthermal Acceleration of Charged Particles in the Near-QED Regime

Here, we have studied the nonthermal acceleration of energetic electrons/protons under the near-QED regime by extending the laser intensity beyond 10 23 W/cm 2 based on a two-dimensional particle-in-cell simulation. The radiation-reaction (RR) effect plays a critical role and brings a quantum stochastic effect to the charged-particle acceleration process. Background electrons in plasma are accelerated in an intense laser eld to several GeVs with strong oscillations and thus radiate γ-ray photons. The emitting γ-photons have a broad energy spectrum with maximal energy up to 3 GeV and result in radiation-reaction trapping of the electrons, forming a relativistic plasma bunch in the plasma channel. The accumulation of electrons and protons produces a charge-separation eld for the acceleration/deceleration of charged particles. The accelerated electrons have a nonthermal spectrum with a power-law index of 1.5 with a laser intensity 10 23 W/cm 2 lower than that in the non-QED regime. As the laser intensity further increases over 10 24 W/cm 2 , the power-law index further drops to 1.2. Moreover, the energy spectrum of accelerated protons has a nonthermal distribution with a power-law index of 0.7, which is much lower than that of electrons in the near-QED regime.


Introduction
The origin and acceleration of cosmics are always unresolved issues 1 . Plasma wake eld collective accelerators, which can provide extremely high acceleration over a short space distance, have been proposed as a potential cosmic-ray acceleration mechanism 2,3 . In astrophysical environments, wake elds can be excited by magnetowaves in relativistic astrophysical out ows 4 or by large amplitude precursor waves upstream of shockwaves 5 . The latter involves a series of processes from shockwave formation to particle acceleration in the wake eld 6 . It is di cult to take a direct observation of astrophysical parameters in extreme celestial events limited by the current technology. The acceleration process of high-energy charged particles based on relativistic laser plasma interactions has been studied, which provides a superhigh acceleration gradient 7 . Kuramitsu et al. reported the simulation of a large-size relativistic laser-induced incoherent wake eld in low-density plasma. The result shows that the nonthermal spectrum of accelerated energetic electrons has a universal power-law index of ~ 2, which is reproduced by subsequent model experiments 8,9 . The relativistic laser used in these works is less than 10 22 W/cm 2 , and the in uence of the radiation process on electron motion is not considered.
The current 10 PW and even future 100 PW laser facilities will provide an intense femtosecond laser pulse with a power density of 10 22-24 W/cm 2 and a focal spot size of a few micrometers 10-13 . Laserplasma interaction employing such an intense laser pulse enters the so-called moderate quantum electrodynamic (or near-QED) regime [14][15][16][17][18] . Electrons in the laser eld are accelerated to GeV in a short time. At the same time, these high-energy electrons oscillating in the laser eld discretely radiate highenergy γ-photons and suffer RR recoil forces [19][20][21] . Radiation events occur randomly or probabilistically, and the energy of a single emitted photon is equivalent to that of an electron, which seriously affects the electron dynamics. Under the in uence of the RR trapping effect, collective electron bunches can also signi cantly alter the plasma eld. Charged particles undergo an entirely different process of acceleration compared with that when the laser intensity is below 10 22 W/cm 2 in the non-QED regime. In the near-QED regime, the stochastic RR effect will participate in the nonthermal acceleration of charged particles.
In this article, we have investigated the acceleration of electrons/protons by taking into account the stochastic QED effect in our simulation based on an ultraintense laser pulse with a peak intensity above 10 23 W/cm 2 . At such a high laser intensity, the radiation loss of electrons is inevitable, and the laser pulse depletes its energy to energetic electrons/protons as well as γ-photons. The nonthermal spectrum of produced electrons is also essentially a power-law distribution function, and its slope varies by approximately 1.5. The radiated -photons also have a broad energy spectrum extending up to 3 GeV when the normalized laser intensity a 0 = 500. Furthermore, the charge separation eld modi ed by the RR effect accelerated the protons to 0.5 GeV and lead to a nonthermal spectrum with a power-law index of 0.7.

Result
Staged power-law spectrum of electrons A relativistic laser pulse propagating in a near-cirtical density plasma (0.1-10 ) induces a plasma channel along the propagation path of the laser pulse instead of a periodic structure wake eld. Both electrons and protons are accelerated inside the plasma channel. There are two simulations (please refer to Methods for speci c parameters) with switching the on/off RR effect to study the accleration process of charged particles in an intense laser eld. As shown in Fig. 1, a clear plasma channel forms in both cases. Electrons are expelled away along the laser axis and transveral direction by strong pondermotive force, and a plasma channel forms after driving the laser pulse. As the RR effect is switched on, a great number of energetic electrons accumulate inside the plasma channel due to the RR trapping effect, as shown in Fig. 1(a). Protons also accumulate at the laser axis due to the transversal electric eld, as plotted in Fig. 1(b), forming a dense quasi-neutral plasma beam 21 . As the laser pulse further propagates, both the electron and proton beam densities greatly increase over the critical density (see Fig. 1(d) and (e)). This trapped plasma bunch has a transverse size of 20 µm and longitudinal length of 50 µm, which is much longer than the laser pulse. The rst half of the electron bunch overlaps with the laser pulse.
They are directly accelerated and modulated by the intense laser eld, resulting in a periodic distribution of electron density. These electrons oscillate in the laser eld, emitting high-energy γ-photons. The production of a high-energy electron beam and γ-photon beam causes serious energy consumption of the local pulse eld and thus leads to a blurry distribution of E y at the back part of the laser pulse, as shown in Fig. 1(f). When the RR is turned off, the plasma channel is almost empty, and there are few electrons and protons inside the channel. Moreover, the laser energy loss is much smaller than that in the case of RR on, as plotted in Fig. 1(f) and (i). It is clear that the RR trapping effect greatly changes the electron density distribution and thus the acceleration process of charged particles bunching inside the plasma channel as well as the -photon beam production when the laser intensity is beyond 10 23 W/cm 2 . The RR effect has a fundamental in uence on the particle acceleration process when the RR force is comparable to the laser pondermotive force and traps electrons at the propagation axis of the laser pulse. Figure 2(a) presents the normalized energy spectrum of energetic electrons in different time steps as the laser intensity a 0 = 500. The maximum energy of electrons quickly increases beyond the GeV level and then stops at 5 GeV, leading to a broad energy spectrum. The energy spectrum has a nonthermal distribution function with a constant power-law slope of ~ 1.5 during the whole acceleration process. The electron spectrum in the RR-off case has a power-law index of ~ 2, which agrees well with that in the non-QED regime 8 . At t = 374 T 0 in Fig. 2(b), the power-law index remains constant over the simulation time, but the cutoff energy of high-energy electrons extends beyond 10 GeV, which is much larger than that in the RR-on case. This con rms that the QED effect leads to a large drop in high-energy electrons because these high-energy electrons e ciently channel energy into the radiated γ-photon beam. The corresponding spectra for γ-photons with energy E γ > 1 MeV are plotted in Fig. 2(c), which is synchrotronlike with maximal energy up to 3 GeV.
A series of simulations have been performed to study the electron acceleration with different laser intensities from a 0 = 50 to a 0 = 1000, where other laser and plasma parameters remain constant (see Methods). When the laser intensity of a 0 = 50 is far from the near-QED regime, the electron spectrum has a power-law distribution with an index of ~ 2, as shown by the black dotted-dashed line in Fig. 3(a), which is similar to the previous result in the non-QED regime 8 . Both the energetic electron number and energy increase by one order of magnitude when the laser intensity increases to a 0 = 100, but the power-law index is still ~ 2. As the laser intensity further increases above 10 23 W/cm 2 (a 0 = 400and 500), the maximum energy of electrons increases to 5 GeV with a sharp knee near the cutoff energy, and the powerlaw index of its spectrum decreases from ~ 2 to ~ 1.5. The QED effect starts to play a signi cant role in the electron acceleration process and stops rapidly increasing the electron cutoff energy since energetic electrons emit a great number of -ray photons. This effect becomes clearer as the laser intensity further increases beyond 10 24 W/cm 2 . Both the electron energy and number have a small increase compared to a 0 = 500, as shown in Fig. 3(c), since a large part of the laser energy is depleted into high-energy γphotons rather than energetic electrons. Thus, the power-law index of the electron spectrum further decreases to ~ 1.2.
The maximum energy evolution of energetic electrons is present for different laser intensities in Fig. 4(a). In the case of a 0 = 50, the maximum electron energy increases rapidly beyond 1 GeV at 100 T 0 and then slowly increases at approximately 3 GeV until the laser pulse eld starts to decrease due to energy loss after 300 T 0 . When the laser intensity increases to a 0 > 500 in the near-QED regime, the maximum energy of accelerated electrons rapidly increases until 150 T 0 and then remains constant, which is completely different from the non-QED regime. Electron energy stops increasing in the near-QED regime since electrons radiate a great number of gamma-ray photons during their acceleration process, which again con rms that QED effects strongly modulate the electron acceleration process. Figure 4(b) shows that the γ maximum energy of electrons increases linearly with instead of and that the QED effect changes this trend.

Nonthermal Electron Accelerating Mechanism
The generation of an acclerated electron bunch with its spectrum of power-law distribution functions in an intense laser eld has a large difference from that in the non-QED regime. These electrons undergo three main different acceleration mechanisms: direct laser acceleration (DLA), nonlinear multiphoton scattering and spatial charge-separation eld acceleration, where the QED effect signi cantly contributes to the laser plasma interaction process. In our simulations, the quantum discrete process in the PIC simulation is realized based on the Monte algorithm, and each radiation event is calculated within an optical depth according to the quantum probability 24 . The most energetic photons are radiated at a certain angle of 0.22 rad along the laser propagation axis. Radiation processes exert transient RR froce in the opposite direction of photon emission to radiated electrons. When the laser intensity exceeds 10 22 W/cm 2 , is comparable to the laser pondermotive force and thus signi cantly participates in the electron aceleration process, bringing random electron trajactories. Each radiation event is accompanied by one multiphoton process, , where and represent laser photons and γ-photons 24,25 . This is a pure quantum phenomenon, and the quantity of is exactly proportional ℏ. Thus, a radiating electron will suddenly change its momentum during γ-photon emission, which brings both random electron direction of motion and kinetic energy. Collective electron bunches that suffer stochastic effects will lead to exotic phenomena, e.g., boarding energy spreading of electrons 27 Fig. 3(b), while the electron spectrum has no sharp end at the cutoff energy in the non-QED regime.
The RR trapping effect leads to both electron and proton accumulation inside the plasma channel as the laser intensity exceeds 10 23 W/cm 2, forming a high-density plasma bunch, as shown in Fig. 1(a, b) and (d, e), and creates a large-amplitude charge-separation eld. The longitudinal electric eld E x at y = 0, marked by the black solid line in Fig. 5(a), has both a positive eld of 100 TV/m located between 230 µm and 255 µm for decelerating electrons (or accelerating protons) and a negative eld of -25 TV/m at x = 220 µm for accelerating electrons (or decelerating protons). One notes that E x lacks the periodic structure of the laser wake eld and is not sensitive to the plasma wavelength because the driving laser pulse has a much larger temporal and spatial scale than the plasma wavelength of under the condition of n e = 0.5 n c . A large number of electrons is accelerated/decelerated by the local charge-separation eld. Their longitudinal momentum p x reaches a peak value of 3000 at approximately 220 µm, as plotted in Fig. 5(b).
Moreover, E x is sensitive to the plasma bunch as well as the laser pulse evolution and thus leads to electrons participating in time-dependent acceleration and deceleration phases. These electrons cannot have continuous energy gain and bring a broad spectrum. However, the maximum momentum of these electrons is much lower than that of electrons accelerated in the DLA regime, which is above 9000 (see Fig. 5(b)).

Proton Acceleration In Near-qed Regime
For driving laser intensities lower than 10 25 W/cm 2 , a proton cannot be accelerated to relativistic energy within a short light period. The principle of proton acceleration in the plasma channel is similar to that of electrons accelerated by a wake eld, and they can be accelerated by a positive longitudinal electric eld in front of the bubble channel 30,31 . The RR trapping effect signi cantly alters the space-charge eld distribution as the laser intensity reaches a 0 = 500 in our simulation. Longitudinal eld E x has a positive value of 100 TV/cm in the front part of the plasma channel, as plotted in Fig. 5 (a), where the proton bunch con ned within the plasma channel can be mainly accelerated. However, the velocities of these protons at the end of the acceleration eld are not high enough to catch up with the laser pulse and then move to the deceleration phase. Moreover, the accelerating space-charge eld E x inside the plasma channel changes over time with the plasma bunch and laser eld, which also brings an unstable acceleration eld. Thus, protons accelerated in the time-dependent charge-separation eld also result in a nonthermal spectrum distribution, as shown in Fig. 6, instead of a quasi-monoenergetic spectrum.
The maximum energy gain for protons in the plasma channel is approximately , which is estimated based on the Coulomb electrostatic potential resulting in radial explosion in the non-QED regime 22 . For laser intensities of a 0 = 50 and 100, the maximal proton energies in our simulation are 23 MeV and 50 MeV, respectively, and agree well with the estimation of . When the laser intensity increases to a 0 = 1000, the maximal proton energy is 5 GeV, which is larger than the estimated by energy gain for protons mainly bene ts from the longitudinal space-charge eld, originally from electron trapping induced by the RR effect. Proton spectra with different laser intensities, as plotted in Fig. 6, show that the cutoff energy of accelerated protons strongly extends with increasing laser intensity. Additionally, the power-law index of the proton spectrum is ~ 1.5in the case of a 0 = 50 and a 0 = 100. However, it decreases to 0.7 with a sudden decrease near the cutoff energy as the intensity increases above 10 23 W/cm 2 . The evolution of the proton energy spectrum with increasing laser intensity is similar to that of electrons. The ultraintense laser pulse drives a strong charge-separation eld due to the RR effect, which accelerates protons to high energy and leads to a step-like power-law proton spectrum.

Discussion
The nonthermal acceleration process of electrons/protons by extending laser intensity into the near-QED regime has been studied in detail based on 2D PIC simulations. The accelerated electron energy spectra exhibit a nonthermal distribution function with a power-law index. The RR effect brings an inherent stochastic process for electron dynamics. γ-photon radiation from energetic electrons induces a step-like spectrum of electrons and prevents the electron spectrum from high-energy expansion beyond 10 GeV. The power-law slope decreases from 2 to 1.2 from to . Protons are mainly accelerated by the space-charge eld modi ed by the RR effect in the plasma channel, and the spectra also show a nonthermal distribution with a slope of 0.7 in the near-QED regime. The nonthermal acceleration of charged particles in the near-QED regime is universal for the next generation of 10PW or even 100PW laser facilities and will help to explore the high-energy cosmic-ray acceleration mechanism in laboratory astrophysics.

Methods
We performed 2D simulations to study the acceleration process of charged particles by using an ultraintense laser pulse based on the Particle-In-Cell code Epoch 32 , where the RR effect was included here. The simulation window of size is divided into cells, and each cell contains 30 macroparticles. A relativistic y-polarized laser pulse with a center wavelength of propagates along the axis. It has Gaussian temporal-spatial pro les with , where its focal size is and the pulse duration is . Peak laser intensities of are used in our simulations, where is the normalized wave potential of the laser. The focus position is set at . Background plasma is located between and , where its density linearly increases from 0 to in the range of and then remains constant at until . Here, is the critical density, with , , and being the mass of electrons, plasma frequency and permittivity of free space, respectively. The effects of electron-position pair production are ignored here due to the quite low quantum e ciency.  Simulation results for switching on/off RR effect. Density distributions of electrons (a) (d) (g), protons (b) (e) (h) and the laser eld Ey (c) (f) (i) are present at time t = 245 T0 and t = 450 T0. The RR effect is turned on at (a)-(f) and turned off at (g)-(i). Ey is normalized to E0 = 1015 V/m. The laser intensity was a0 = 500, and the plasma density was ne = 0.5 nc.