Angle-resolved stochastic photon emission in the quantum radiation-dominated regime

Signatures of stochastic effects in the radiation of a relativistic electron beam interacting with a counterpropagating superstrong short focused laser pulse are investigated in a quantum regime when the electron’s radiation dominates its dynamics. We consider the electron-laser interaction at near-reflection conditions when pronounced high-energy gamma-ray bursts arise in the backward-emission direction with respect to the initial motion of the electrons. The quantum stochastic nature of the gamma-photon emission is exhibited in the angular distributions of the radiation and explained in an intuitive picture. Although, the visibility of the stochasticity signatures depends on the laser and electron beam parameters, the signatures are of a qualitative nature and robust. The stochasticity, a fundamental quantum property of photon emission, should thus be measurable rather straightforwardly with laser technology available in near future.

represents the angular distribution of the radiated photon number dN R /dΩ, corresponding to Fig. 1 of the paper, which shows the radiated energy dε R /dΩ. We employ the same electron and laser parameters as in Fig. 1 of the paper: the electron beam counterpropagates with a 6-cycle (FWHM) laser pulse with a peak intensity I ≈ 4.9 × 10 23 W/cm 2 (ξ = 600), and the initial polar angle of the electron motion is θ (i) e = 180 • . The electron-beam radius is w e = 1µm, the electron-beam length equals the laser-pulse length, L e = L L = 6λ 0 , the electron density n e ≈ 1.27 × 10 15 cm −3 , and the initial electrons are randomly distributed in the bunch. The initial mean kinetic energy of the electron beam is ε 0 = 180 MeV (γ 0 ≈ 353.25 and χ 0.5), the energy and angular spread of the electron beam are ∆ε/ε 0 = ∆θ = 0.02, and they both have a Gaussian distribution. The distributions of dN R /dΩ including and excluding the stochasticity effects (SE) represent similar structures as those of dε R /dΩ in Fig. 1 of the paper.
For observation of the predicted SE the angular resolution of gamma-ray detection should be less than 10 • , with the precision of the photon number detection less than 10 3 photons, which is within the commonly achieved resolution.

II. APPLICABILITY OF THE CONSTANT CROSSED-FIELD APPROXIMATION FOR THE EMISSION PROBABILITY
To estimate the range of applicability of the constant crossed-field approximation for the emission probability, let us return to the original work of V. I. Ritus, where the radiation probability is calculated in a plane laser field without any approximation, see Ref. 38. In the limit ξ 1 the asymptotic formula for the generalized Bessel function, which describes the emission probability, is derived coinciding with the crossed-field result. The applied approximation is to neglect in the exponent derivative terms higher than the third derivative, see Eq.
Let us analyze the neglected terms in the derivation of the asymptotic formula, which will provide us with an estimate of the order of magnitude of possible deviations from the exact formula. The derivation using the saddle-point integration of the general formula A 0 (s, α, β) ∼ Re 1 π π 0 e f (φ) dφ is given in Appendix B, where α = eap/kp, β = e 2 a 2 /8kp, s is integer, e the Color coded is log10[dNR/dΩ] rad −2 in the polar angle θ and the azimuthal angle φ, with the radiation photon number NR and the emission solid angle Ω. λ0 = 1µm, and the laser focal radius is w0 = 2µm. Other laser and electron beam parameters are given in the text of the article. electron charge, k the four-vector of the laser wave, p a constant four-vector that describes the state of the particle in the field of the wave and coincides with the particle momentum when the field is turned off, and a the amplitudes of the potential. The function in the exponent is expanded near the saddle-point: ∼ s, and higher order derivatives are also ∼ s at s = s 0 . The second derivative term determines the saddle-point integration region (φ − φ 0 ) ∼ 1/s 1/3 from the condition , and therefore, both terms are kept in the exponent, while higher order terms are neglected. The largest one is the forth derivative term f (φ 0 )(φ − φ 0 ) 4 ∼ s(1/s 1/3 ) 4 ∼ s 1/3 . Thus, the neglected term in the exponent is of the order of s 1/3 ∼ 1/ξ, because the effective number of the absorbed laser photons is s ∼ ξ 3 . Then, the neglected terms in the probability is of the order of 1/ξ, which means that at ξ ∼ 10, the accuracy of the synchrotron formula (crossed-field approximation) is 90%, i.e., the relative error is of 10%.

III. ON THE EXPLANATION OF THE EMISSION ANGLE SPREADING
As shown in Figs. 3(a) and 3(b) of the paper, in the case without SE, the single-electron radiation anglē θ e ∼ arctan(mξ/p z ) (0 ≤θ e ≤ π) is well defined at each moment during interaction. The radiation is assumed to be along the electron motion since the radiation-angle spread of order 1/γ is rather small. The electron initially counterpropagating with the laser pulse has a negativē p z , which decreases by an absolute value during the interaction due to radiative energy loss, while the laser field ξ increases in subsequent cycles, and the emission angle decreases. The pointp z = 0 corresponds to the electron reflection, when the emission angle sharply changes from the forward into the backward direction. After reflection FIG. 4. The radiation's angular distribution excluding SE: (d) are corresponding to (a) and (b), respectively, for the case with SE. The initial coordinates of the sample electron are z = x = y = 0, and other electron and laser parameters are the same as in Fig. 1.  7.5 (p z > 0),θ e further decreases because of an increase ofp z when the electron is accelerated along the laser pulse propagation direction.
The angle-resolved radiation intensity and radiation energy are shown in Figs. 3 (e) and 3(f) of the paper, respectively, for the case without SE. In each cycle the smallest θ e corresponds to the peak of the field (the largest ξ). The strong backward radiation arises near the peaks of the cycles aroundη = 8, 9 and 10, respectively, after the reflection point when ξ and χ are both large, creating peaks at θ ≈ 61 • , 28 • and 16 • , respectively, in Fig. 3 (f) of the paper. The radiation atη > 10.5 is weak since ε R ∼ χ is rather small at θ ≈ 0 • . Between adjacent radiation peaks, there is a gap corresponding to the weak-field part of the laser cycle. Therefore, the radiation angular distribution (RAD) reveals the laser-cycle structure when SE are neglected.
The RAD with SE for a single electron is shown in Figs. 3(g) and 3(h) of the paper, which are reproduced via simulating 2.4×10 4 times the electron-radiation process with the same initial conditions. Since with SE the radiation angle in each laser cycle has a very broad spread, the gaps between adjacent radiation peaks are filled out. Consequently, the radiation intensity in this case mostly follows the pulse shape of the laser field producing a single gamma-radiation peak corresponding to the peak of the laser pulse. Note that the discussed qualitative features of the RAD for a sample electron do not depend on the initial position of the electron in the electron beam. The variation of the initial position introduces only a slight modification of a quantitative character. This will be discussed in the next section.
In addition to Fig. 3 of the paper, we show here in Fig. 2 the variations of the invariant laser field parameter ξ and the invariant parameter χ with respect to the laser phaseη = η/(2π) in the case without SE. The laser and electron parameters are the same as in Fig. 3 of the paper. The peak of the laser fields is aroundη ≈ 9, as shown in Fig. 2(a). In addition, the peak of the radiation ∼ χ is aroundη ≈ 7, as shown in Fig. 2(b). Besides, corresponding to Figs. 3(c) and 3(d) of the paper, the variations of the radiated photon energy ε γ and the sample-electron energy ε e with respect to the laser phasē η are illustrated in Fig. 3. The first simulation of the radiation of the sample electron is shown in Figs. 3(a) and 3(c) corresponding to Fig. 3(c) of the paper, and the second simulation with the same laser and electron parameters is shown in Figs. 3(b) and 3(d) corresponding to Fig. 3(d) of the paper. Due to SE the two simulations represent much different results.

IV. THE RADIATION OF ELECTRONS WITH DIFFERENT INITIAL COORDINATES IN THE BEAM
According to Figs. 3(e)-3(h) in the paper, the radiation of a sample electron in other places of the electron beam is investigated. Figures 4 and 5 show the radiation of the sample electron located at z = 0 and z = L e , respectively, of the electron-beam center with x = y = 0, and other parameters are the same as in Fig. 1. In both cases, with SE excluded, the radiation's angular distributions show the structure of the laser cycles, and the main gamma-photon peaks are located in the range of 11 • θ 45 • . Meanwhile, with SE included a single gamma-photon peak is formed in the radiation in the range of 11 • θ 45 • . As the electron moves from the front to the rear of the electron beam the radiation intensity decreases with the interacting laser fields because of the laser defocusing effects (since ε R ∼ χ ∼ ξγ, as ξ decreases ε R decreases as well).
The radiation of the sample electron beyond the electron-beam center with z = L e /2 and x = y = 0.5w e is represented in Fig. 6. Since the electron is out of the electron-beam center, i.e., the propagation axis of the laser pulse, the laser fields experienced by the electron are much smaller than those at the electron-beam center (see Figs. 3(e)-3(h) in the paper), and consequently, the radiation angles of the gamma-photon peaks increase since θ e = θ (i) e − δθ ∼ 180 • − ξ/γ.

V. THE ROLE OF THE ELECTRON BEAM DENSITY
The SE signatures are also obtained with higher electron density. We show this explicitly by re-simulating the results of the paper with the use of a higher density. For instance, in Fig. 2(a) of the paper, the electron beam, with radius w e = λ 0 and length L e = L L = 6λ 0 , includes 2.4×10 4 electrons, and its density is n e ≈ 10 15 cm −3 . Now, we re-simulate this case by employing 4.8×10 4 electrons (the density is 2 times higher). The comparison is illustrated in Fig. 7. As we know, the emitted photon number and intensity is proportional to the electron number, such that to facilitate the comparison the radiation of the high density case is divided by a factor of 2. In the high density case the same SE signatures are observed. Other parameters are the same as in Fig. 1.

VI. COMPARISON BETWEEN THE "SOKOLOV" APPROACH AND THE "LANDAU-LIFSHITZ" APPROACH
We have carried out several simulations using Landau-Lifshits (LL) equation [49] in comparison with the Sokolov equation (SEQ). Generally, the LL equation is derived for the classical regime χ 1. Therefore, firstly we compare the LL and SEQ results in the classical regime of χ 1 and γ 1. An example is shown in Fig. 8, where we compare the radiation's angular distributions simulated via the "Sokolov" approach (blue-solid curves) and with the "Landau-Lifshitz" approach (reddashed curves) in the classical regime. The "Sokolov" approach is the simulation method based on SEQ, which we employ in the manuscript as "no SE" method (excluding the stochasticity effects). In the "Landau-Lifshitz" approach in the applied ultrarelativistic regime, we keep only the leading term in γ in the LL radiation-reaction force, i.e., the third term in the LL radiation-reaction force in Eq. (1) where γ, m, and e is the Lorentz factor, mass and charge of the electron, respectively, and c is the speed of the light in vacuum.
In quantum regime, as χ 1, the quantum recoil effects for the radiation intensity have to be included. The LL equation in its original form is not applicable in the quantum regime, however, heuristically one includes the recoil effects by renormalizing the radiation reaction force by the factor I QED /I C , the ratio of the radiation intensities within QED and classical approaches, which will account for the classical overestimation of the emitted radiation: where, W f i is the radiation probability, r 0 = (k·k ) χ(k·pi) , and E is the electric fields in the electron frame. k, k and p i is the four-vector of the wave vector of the driving laser, the wave vector of the radiation photon, and the momentum of the electron before the radiation, respectively.
In Fig. 9 an example of the radiation's angular distributions in the quantum regime is shown. All employed parameters in Fig. 9 are the same as in Figs. 2(a) and 2(b) of the manuscript.
The radiation's angular distributions demonstrate in Figs. 8 and 9 almost coincidence for SEQ and LL cases. We can conclude that in the applied ultrarelativistic and weakly quantum regime the SEQ and the modified LL equation provide similar results. We prefer SEQ over the modified LL equation in the weakly quantum regime, because the first is derived systematically using energymomentum conservation for the radiating electron in the ultrarelativistic weakly quantum regime, meanwhile the modified LL equation is a heuristic extrapolation of the LL equation out of its applicability region.