Abstract
High brightness gamma rays can be generated by colliding an ultraintense laser pulse with a high energy electron beam. This collision phenomenon also represents a powerful approach to explore new physics in the exotic strong field Quantum ElectroDynamics (QED) regime. Here we show that in the crosscollision geometry, there exists a barrier induced by the classical radiationreaction force that prohibits electrons of arbitrarily high energies to pass. However, such classical barrier vanishes in the QED picture, where electrons can be well reflected (transmitted) in the regimes forbidden by classical theory. This effect can be measured in the upcoming 10–100 PW laser facilities for laser intensities at 2 × 10^{23} W cm^{−2} and electron energies of ~10^{2} MeV. The results are capable of identifying the boundaries between classical and QED approaches in the strong field regime and confirming the various models describing this fundamental process.
Introduction
Understanding the electron dynamics in relativistic laser fields has been of core interest in strongfield physics and inspired numerous key applications such as fast ignition fusion, acceleration of charged particles, and producing bright X/gammaray sources. These advances become strong motivations for developing the 10–100PW highpower laser systems^{1,2,3,4,5,6}. Light intensity is likely to approach 10^{23−24} W cm^{−2} in the foreseeable future and promote light–matter interaction to the radiationdominated regime^{7} or even the quantum electrodynamics (QED) regime^{8,9,10}. In these regimes, an interest in the unique electron dynamics at extreme laser fields arises, in which electrons are accelerated and radiate photons of considerable energies, such that the recoil force is not negligible. This phenomenon is usually referred to as radiation reaction (RR).
Theoretical attempts were made to account for classical RR, such as the Lorentz–Abraham–Dirac equation^{11} and the Landau–Lifshitz (LL) equation^{12}, both of which were derived from the assumption of continuous classical radiation. The latter is widely accepted because it resolves the nonphysical runaway solution^{13}. Classical treatment is successful in describing accumulative RR effects. In the QED regime, stochastic radiation of highenergy photons^{14,15} no longer allows one to treat RR as a continuous effect and the quantum effects become effective. While photon emission and the RR force could lead to profound effects in light–matter interaction, such as electron cooling^{16,17}, energy redistribution^{18}, and anomalous trapping of electrons^{19,20,21}, identifying RR in the QED regime has been challenging due to the insufficient peak laser intensities. A direct approach is to headon collide a high intensity laser with an energetic electron beam. The laser field can be boosted by a factor of ~γ (electron gamma factor) in the electron rest frame, so that the QED parameter χ = eℏF⋅p/m^{3}c^{4} can reach unity^{22}, where F^{μν} is the electromagnetic tensor and p_{μ} is the electron fourmomentum. Considerations based on this scenario have been made to observe classical RR^{7,16,23,24} and quantum effects^{25,26,27,28,29}, by identifying the signature from either the radiated gammaphotons^{17,27} or the electron dynamics^{26,28}. For the latter in particular, a quantum “quenching” effect is revealed in the headon colliding geometry for fewcycle laser pulses^{26}, by which some electrons can radiate zero energy and go through the laser field freely.
In this article, we show that new quantum features in electron dynamics arise in the crosscollision geometry at extreme laser intensities. We noticed that classical RR can form a distinctive barrier that blocks electrons with energy below the barrier and allows electrons with energy beyond to pass. However, in the regime where the classical RR barrier allows for full transmission (reflection), we found that an electron can be reflected (transmitted) due to the quantum nature of its dynamics. Intuitively, an electron transmits through the laser field by emitting a smaller amount of energy than it does classically, as suggested in the quantum “quenching” picture^{26}. However, in the quantum reflection (transmission) mechanism, we see that a considerable portion of electrons are reflected (transmitted) by the laser field, even when they radiate lower energy (more energy) than they do classically. The anomalous behavior can be observed by a proposed experiment that measures the reflected electron signal.
Results
Reflection beyond the classical RR barrier
We examine the electron dynamics by perpendicularly colliding a focused laser beam to a highenergy electron bunch. The tightly focused laser^{30} is polarized in the xdirection and propagates in the zdirection with a profile of \({\mathbf{E}} = {\mathbf{E}}_0(x,y,z,w_0){\mathrm{cos}}^2(\psi /2N)\). Here w_{0} is the radius of beam waist, ψ (ψ < Nπ) the phase term, and N the pulse length in wavelength that focuses at the origin at t = 0, respectively. We start with the simplest case where monoenergetic electrons are injected in the y = 0 plane. A more realistic electron bunch will be discussed later. The reflection ratio of electrons is evaluated in a large parametric region.
We compare the results between the classical approach and the QED calculation in Fig. 1, where a = 300 and γ_{0} = 250,400 (2 × 10^{23} W cm^{−2}, 128 MeV, and 204 MeV) are considered. Here λ = 800 nm, w_{0} = 2λ, pulse length is 20λ, and a = Eeλ/2πmc^{2} (E is the laser electric field) is the Lorentzinvariant field strength and γ_{0}mc^{2} is the electron initial energy. The case of pure Lorentz force (LF) is also included for full comparison. In the parameter range of χ < 1, the electron trajectories are greatly diverged. When the RR effect is excluded, all electrons transmit freely through the laser beam, with minor perturbation in the particle trajectories, as shown in Fig. 1a, b. However, a distinctive behavior arises when the classical RR (LL equation is employed) is turned on. At a fixed laser intensity of 2 × 10^{23} W cm^{−2}, for γ_{0} = 250, one sees complete reflection of electrons in the colliding vicinity (Fig. 1c). When we increase the electron energy to γ_{0} = 400, we find that the picture flips where all electrons transmit through the laser beam (Fig. 1d). The drastically different dynamic behavior by slightly varying the electron energy indicates the presence of a clear threshold between the two sets of parameters, i.e., γ_{0} = 250 and γ_{0} = 400. These features vanish when we switch to the QED description (see the “Methods” section). In the classical reflection (γ_{0} = 250, below threshold) or transmission (γ_{0} = 400, above threshold) regime, we see that 5 out of 100 of test electrons transmit through the laser field in the former and 4 out of 100 get reflected in the latter, as illustrated in Fig. 1e, f, respectively.
This disparity is universal in a large parameter range. We quantize the electron reflectance ratio in the (a, γ_{0}) domain in Fig. 1g–i. When RR is turned off, the only barrier for electrons to overcome is the laser ponderomotive potential. Thus, an electron passes through the laser field freely while its initial energy dominates over the ponderomotive potential. This threshold is perfectly fitted by the criterion γ_{0}~a^{31,32} in Fig. 1g. The RR effect imposes another barrier for electrons. When the laser amplitude increases, the lowest energy for penetration (reddashed line in Fig. 1h), namely, the barrier, grows higher than that in the noRR case (whitedashed line in Fig. 1g). Therefore, one sees a clear and sharp threshold that defines reversed features in the electron reflectance, as shown in Fig. 1c, d.
However, the classical RR barrier smears when viewed in the QED perspective. In the parameter region beyond the barrier where LL allows for total transmission, more than 1% of the electrons still get reflected in QED, as shown by the nonzero reflectance above the reddashed line in Fig. 1i. Moreover, electrons can tunnel through the beam below the barrier due to quantum behavior, which significantly lowers the reflectance as compared with the LL case.
QED effects: anomalous reflection
The QED effect can be best understood by looking at one electron injected at one fixed position for multiple times. The lowest penetration energy for classical LF and LL (whitesolid lines) is definite as shown in Fig. 2a. These two curves coincide with the manyparticle modeling in Fig. 1g, i. In the QED picture, we repeat the colliding process for 10^{4} times at each set of (γ_{0}, a), where the reflectance probability is defined by \(\frac{{{\mathrm{reflected}}}}{{{\mathrm{total}}}}\). The reproduced probability map in Fig. 2a shows three distinctive regimes. Below the LF boundary is the forbidden zone where the initial electron energy is too small to overcome the ponderomotive barrier. The LL boundary indicates the definite electron dynamics in the classical picture and deviates the transmission and reflection regions shown in Fig. 1a, d. However, in QED, electron dynamics are stochastic and the reflectance is smeared near the LL boundary. The quantum transmission region lies between the LL and the LF boundaries, which becomes more prominent as χ increases. Beyond the LL boundary is the abovethreshold zone, where electron energy dominates the classical barrier. The nonzero reflectance above the LL boundary in QED indicates the quantum reflection that is forbidden by the classical LL equation. It should be noticed that the LL boundary can be modified by quantum correction^{33}. However, these quantum effects can always happen due to the stochastic behavior.
We further count the radiated energy for each electron in Fig. 2b for a = 300, γ_{0} = 400 during t < 0, as simulation starts at t = −t_{0} and the electron is initialized at x_{0} = ct_{0}, z_{0} = 0 outside the laser. The LL equation gives a definite singlevalued radiated energy at each interaction time. In contrast, energy loss in the QED modeling deviates from the LL equation in several ways. First, QEDMC exhibits a broadened distribution ranging from 0 to 200 MeV due to random radiation, meaning that the electron could lose all the initial energy or radiate nothing. Second, the averaged radiated energy is higher for reflected electrons and lower for transmitted electrons, which is similar to the quantum quenching effect^{26}.
The most interesting feature in Fig. 2b is that some electrons may radiate more (less) than classical calculation, but still transmit (get reflected) as shown by the colored area. These anomalously transmitted/reflected electrons are presented at each colliding phase t = −1.2 T, −1 T, and 0, as recorded in Fig. 2b. The mechanism is further confirmed in Fig. 2c, d by tracking the energy loss of each electron trajectory during collision. These results indicate a new QED feature that has not been revealed previously. Possible electron trajectories are determined by the LL equation or QED; the former contains only one solution for a given condition, while the latter includes various solutions due to the stochastic process, which allows for diverged patterns of radiation and electron trajectories. Therefore, an electron does not necessarily radiate more energy to be reflected or less to transmit. This classically forbidden phenomenon is a most profound reflection of the stochastic quantum laws.
QED effects: angular feature
We consider at a = 350 a more realistic electron bunch of spatial size Δy = Δz = 100λ at the full width at half maximum (FWHM) of the superGaussian profile, Δx = 4λ at FWHM of Gaussian profile with E = 102 MeV, 5% energy spread, 10 mrad angular divergence^{34,35}, and peak density of ~10^{15} cm^{−3}, respectively. We assume electrons located further away from the bunch center to have larger divergence angles with respect to the propagation axis. The relatively large transverse size of the electron bunch can significantly lower down the difficulty of overlapping the bunch and the laser pulse in the crosscollision geometry. The collective plasma field can be neglected due to the very low electron density, such that test particle modeling is sufficient. Otherwise particleincell (PIC) simulations are required.
The results are presented in Fig. 3a, b. The laser pulse drills through the bunch and scatters electrons to large angles. By focusing on the scattered electrons, the measurement is free from the disturbance of background electrons. For longdistance propagation at the order of meter, it is very important to identify the angular distribution after collision. In Fig. 3c, we see clear periodic structures in the LL case, where θ is the polar angle to −x and ϕ the azimuthal angle to z; when we switch to the QED case, such structures vanish in Fig. 3d. In the classical case, there is a specific correlation between the scattering angle and the injected phase, as shown in Fig. 3e. Thus, the oscillation along the scattering angle θ (black horizontal bars) represents the structure of the laser field. However, in the QED case, the scattering angle is no longer singlevalued for a certain injecting phase. This is induced by the stochastic effect that allows electrons to enter phases forbidden by the classical model. The broadened scattering angle in QED thus smears the oscillating structure presented in the classical case.
Experimental consideration
One can leverage the new feature to experimentally probe the QED nature at extreme intensities. In this work, we focus on the quantumreflected electrons as the reflected electron signal is free of the background signal from the abundant transmitted electrons. These effects can be captured experimentally by distributing electron number detectors along θ and ϕ angles and recording the scattered electrons, as shown in Fig. 4a. The records, accumulated from ϕ = −15° to 15° (sections between the dashed lines in Fig. 3c, d) as a function of θ (50 sets of detectors along θ ranging from π/2 to π) are shown in Fig. 4b. Each detector corresponds to an accepting area of ~6 × 6 cm for 2 m propagation distance. Clear oscillations (spikes and valleys) are reproduced in the detectors along θ in the classical case. This feature is absent in the QED picture, providing an explicit evidence of the QED process. We integrate electron numbers in π/2 < θ < π and obtain the total reflectance at different field strengths in Fig. 4c. The reflection curves show a sharp jump at a = 220 for LL, corresponding to the classical RR barrier. For QED, due to the mechanisms revealed in Fig. 2a, one observes a smoothed transition rather than an abrupt change. We thus conclude that the measurement of electron reflectance along with the angular structure will provide a clear proof of quantum RR.
Discussion
The crosscollision geometry can minimize the electron oscillation times when compared with the headon collisions, because the current laser systems can focus the spot size to 1 ~ 2λ, but the laser pulse length is usually at about 10λ. To achieve the regime where the QED feature is active, singlecycle or even subcycle laser pulses are required in the headon collision setup, as shown in ref. ^{26}, which would be very challenging for the stateoftheart techniques. Crosscollision, on the other hand, is less demanding on the pulse profile to create quantum reflection events. For experimental consideration, the reflected electrons get separated from the electron bunch; i.e., the signal and background are not coaxial, which will be favored for making highcontrast measurements. The critical spatial overlapping/aiming issue in the headon collision scenario is also mitigated in the crosscollision geometry, where the electron beam can in principle be extended to hundreds of micrometers in diameter, such that the laser pulse can cross it at a picosecond timescale.
Breit–Wheeler pairproduction^{36} in the crosscollision geometry is investigated in a large parameter range with the PIC code SMILEI^{37}. We use the same laser and electron bunch configuration as that in Fig. 3 but in a twodimensional form. The cell size in the PIC simulations is 0.04λ × 0.04λ with 10 particles per cell. The results are presented in Fig. 5, where the pairproduction is strongly suppressed for χ ~ γ ⋅ a ≪ 1. The fraction of the pairproduced electrons is also small when compared with all the reflected electrons. Therefore, we conclude that pairproduction does not affect the phenomenon of quantum reflection in the considered parameter space.
All the results are calculated with our singleparticle model. Here, we compare our results of angular distribution in Fig. 3 with PIC simulation, where the laser and electron bunch configurations in the PIC simulation are the same with the singleparticle modeling. The cell size in the PIC simulation is 0.04λ × 0.08λ × 0.08λ with 1 particle per cell, where the first dimension is the laser propagation direction. Comparison of the results is shown in Fig. 6. Although the singleparticle modeling ignores the interaction between the electrons, the consistency with each other shows fidelity of our model.
In conclusion, we found that perpendicular laserelectron collision provides a unique approach to distinguish the classical and quantum dynamics in the strongfield regime. By calculating the reflectance of injected electrons, we revealed that the classical RRbarrier threshold is not active when viewed in the QED picture. This classical barrier grows to infinity and can block electrons of arbitrarily high energy when the field strength reaches a critical value. In the perspective of QED, electrons may get reflected even when the electron energy goes beyond the classical barrier and transmit for electron energy below the barrier.
We notice that recent experimental efforts have been made to collide laserwakefield accelerated electrons, with counterpropagating highintensity laser pulses to produce X/gammarays via Compton scattering^{38,39,40} and to create RR events^{41,42}. In the former, RR was not active due to relatively low laser intensities. In the experiment by Cole et al.^{41}, the collision probability was limited by the techniques of time synchronization and spatial overlap. Thus, successful events were necessarily identified through extensive theoretical modeling and comparison with experimental results^{41}. In the experiments by Poder et al.^{42}, a more precise model beyond constant crossfield approximation is required to account for the results. Wistisen et al.^{43} measure the photon spectrum in a positroncrystal collision but did not find a valid theory that fully accounts for the results. We believe that the new QED features of electron dynamics revealed here can be explicitly probed by the measurement proposed in Fig. 4, where the double measurement of the angular structure and the reflectance is selfconsistent in identifying the QED effect.
Methods
The classical RR barrier
In a classical picture, we solve the equation of motion dp/dt = F_{L} + F_{RR} numerically for each electron and track their trajectories. Here, F_{L} is the Lorentz force and F_{RR} is the RR force taking the dominating term of LL equation^{12}. In the classical framework, the electron dynamics is deterministic in the sense that it always experiences the same amount of damping if the injection position is constant. We consider an electron initially located at the upper edge of the laser field, as seen in Fig. 7a. The trajectory is presented in the ψ − x space, where ψ = ωt − kz. One notices that before the trajectory turns staggered, there is a depletion zone where it loses the kinetic energy rapidly along a straight line in the space–time domain. This process, namely rapid exhaustion, is the key making electron possible to be reflected by the laser field. For electrons with energy far beyond ponderomotive potential, they must lower down their momenta to a certain level through the rapid exhaustion phase in RR to be reflected by the laser ponderomotive potential.
We simplify the geometry to onedimensional (1D), since the electron moves in the polarization plane and maintains a straight trajectory. To give an explicit formula, the laser is approximated to a plane wave
with beam waist w_{0}. Since RR is dominant in the depletion zone, we only consider RR force
where we only take the dominant term proportional to γ^{2}. We have the rate of energy loss from classical radiation dE/dt = F_{RR} ⋅ v, where v_{x} ≈ −c. Along the straight trajectory, if one has t = (w_{0} − x)/c then dE/dt only depends on x. Then electron energy evolves as
where \(\beta _x^2 \approx 1\). Then we have
Finally we obtain the evolution of γ(x) along x
where \({\Bbb I}_{w_0}(x)\) is an integral relevant to the laser profile. It weakly depends on initial z_{0}, thus, we take z_{0} = 0 for convenience. From Fig. 7, we see that the depletion zone ends at about w_{0}/4 away from the laser axis, thus, we evaluate the integration \({\Bbb I}_{w_0}(x)\) from x = w_{0} to w_{0}/4. The lowest energy for an electron to penetrate ponderomotive potential is approximated by demanding γ(w_{0}/4) ~ k ⋅ a, from which we can determine the lowest energy (γ_{0}mc^{2}) to penetrate the laser beam considering a strong RR
where k can be determined by the boundary (whitedashed line) in Fig. 1f. Equation (1) depicts the boundary of the classical LL transmittance map. For comparison, we numerically calculate the LL boundary of transmittance in Fig. 2a and compare it with Eq. (1) in Fig. 7b. One sees excellent agreement between the two. For comparison, the w_{0} of the cos^{2} profile is scaled to fit the Gaussian profile of the laser in our simulation \(\left( {w_{0,\mathrm{cos}^2} = 1.71w_{0,\mathrm{Gaussian}}} \right)\). Equation (1) reveals the unique LL threshold by setting the denominator to null \(1/ka = a^2{\Bbb I}_{w_0}(w_0/4)\), suggesting an infinite value for the initial electron energy. The critical field strength then scales as
where \({\Bbb I}_{w_0}(x)\) is basically linear to w_{0}. Equation (2) reveals the important fact that the LL barrier threshold does not depend on the initial momentum of the colliding electrons. It is only a function of the laser profile. We notice that the scaling of Eq. (2) is consistent with the criteria by which RR dominates over LF^{44,45} and radiation trapping happens^{20,21}. Particularly, we see that from Eq. (2) that although relevant, the barrier dependence on the laser beam size is relatively weak. Therefore, it is favored by tight focusing of finite laser peak powers, as \(a\sim w_0^{  2}\) while \(a_{{\mathrm{cr}},w_0}\sim w_0^{  1/3}\).
The QED radiation algorithm
Quantum radiation is treated stochastically, while the motion between the emission events is classical. One can do so because the de Broglie wavelength of an ultrarelativistic electron is much smaller than the optical laser wavelength and the photon formation length mc^{2}/eE~10^{−3}λ is much smaller than the gradient of the field^{46} in the region we discuss. The emission rate in the QED regime is given in refs. ^{22,47}. We use the synchrotron radiation configuration^{48,49} based on the solution of a “dressed” electron in external fields; the photon emission is modeled by the synchrotron spectrum^{50}
where \(y = \frac{2}{3}\chi ^{  1}\frac{\delta }{{1  \delta }}\) and δ = F ⋅ ℏk/F ⋅ p≈ℏω/γmc^{2} represents the photon energy normalized by the electron energy; ℏk_{μ} is fourmomentum of the photon. Photon emission is triggered by a modified event generator^{51} that resolves the cutoff issue in the lowenergy region of the radiation model. In the QEDMC algorithm, emission probability and photon energy are determined by two independent random numbers r_{1} and r_{2}. If r_{2} < P(r_{1}), a photon of ℏω = r_{2}γmc^{2} is emitted, where P(r) is the probability density function constructed from F(χ, δ).
Data availability
Simulation data and figures are accessible from Geng upon reasonable requests.
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Acknowledgements
This work is supported by the Ministry of Science and Technology of the People’s Republic of China (2018YFA0404803 and 2016YFA0401102), the National Science Foundation of China (No. 11875307), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB16010000), and the Recruitment Program for Young Professionals.
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L.L.J. and X.S.G. proposed the research. X.S.G. did the numerical modeling and simulations under supervision by L.L.J. and B.F.S.; X.S.G. and L.L.J. prepared the paper, with suggestion from B.F.S., B.F., Z.G., Q.Y., L.G.Z. and Z.Z.X.
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Geng, X.S., Ji, L.L., Shen, B.F. et al. Quantum reflection above the classical radiationreaction barrier in the quantum electrodynamics regime. Commun Phys 2, 66 (2019). https://doi.org/10.1038/s4200501901642
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