QED cascades induced by circularly polarized laser fields
N.V. Elkina, A.M. Fedotov, I.Yu. Kostyukov, M.V. Legkov, N.B. Narozhny, E.N. Nerush, H. Ruhl
aa r X i v : . [ h e p - ph ] O c t QED cascades induced by circularly polarized laser fields
N. V. Elkina, A. M. Fedotov, I. Yu. Kostyukov, M. V. Legkov, N. B. Narozhny, E. N. Nerush, and H. Ruhl Ludwig-Maximilians Universit¨at M¨unchen, 80539, Germany National Research Nuclear University MEPhI, Moscow, 115409, Russia Institute of Applied Physics, Russian Academy of Sciences, 603950, Nizhny Novgorod, Russia
The results of Monte-Carlo simulations of electron-positron-photon cascades initiated by slowelectrons in circularly polarized fields of ultra-high strength are presented and discussed. Ourresults confirm previous qualitative estimations [A. M. Fedotov, et al., PRL 105, 080402 (2010)] ofthe formation of cascades. This sort of cascades has revealed the new property of the restorationof energy and dynamical quantum parameter due to the acceleration of electrons and positrons bythe field and may become a dominating feature of laser-matter interactions at ultra-high intensities.Our approach incorporates radiation friction acting on individual electrons and positrons.
PACS numbers: 41.75.Jv, 42.50.Ct, 52.27.Ep, 52.50.DgKeywords: strong laser field, pair creation, QED cascades, electron-positron plasma, Monte-Carlo simulations
I. INTRODUCTION
The dramatic progress in laser technology has enableda novel area of studies exploring laser-matter interac-tions at ultra-high intensity [1]. The intensity level of2 × W/cm has recently been achieved [2] and twoprojects [3, 4] aiming at intensity levels up to 10 W/cm have been supported and are under way. Furthermore,several original proposals have been suggested e.g. [5–7], which reach even higher intensities with almost thepresent level of technology. One of the key phenomenaof laser-matter interactions, that probably dominates atultra-high intensities of our interest, is the occurrenceof QED cascades [4, 8–10]. These cascades (also calledavalanches, or showers) are caused by successive events ofhard photon emissions and electron-positron pair photo-production by hard photons. As predicted in [10] basedon qualitative estimations, the cascades may arise as soonas the laser field strength exceeds the threshold value of E ∗ = αE S , where α = e / ~ c ≈ /
137 is the fine struc-ture constant and E S = m c /e ~ = 1 . × V/cm isthe characteristic QED field. Such a field strength corre-sponds to an intensity of ∼ W/cm .Previously QED cascades have been observed andstudied as a part of Extensive Air Showers (EAS) in thecontext of the passage of ultra-high energy particles, thatoriginate from Cosmic Rays, through the atmosphere[11–13]. However, similar processes can be observed inthe external electromagnetic field as well. In this case,Bremsstrahlung is replaced by the non-linear Comptonscattering and Bethe-Heitler process is replaced by thenon-linear Breit-Wheeler process. The latter processesare well studied both theoretically [14–19] and in laserexperiments [20] and are probably of great importancefor astrophysics (see, e.g., [21]).An important novel distinctive feature of the cascadesin the ultra-strong laser field, compared to situations everstudied previously, is that the laser field is not only ableto be a target for ultra-relativistic electrons and hardphotons, but can also accelerate the charged particles to ultra-relativistic energies.As a result, the cascades can be produced even byinitially slow electrons or positrons, if they were some-how injected into the strong field region. Moreover, themean energy of the particles is no longer decreasing inthe course of the cascade development due to its redis-tribution among the permanently growing number of thecreated particles, but rather is restoring at the expenseof the energy extracted from the laser field. This mustlead to a vast increase of the cascade yield, as comparedto the cascades in media or in strong magnetic fields. Inthis case the cascade multiplicity would be restricted ei-ther by the duration of stay of the particles in the focalregion of the laser field, or even, under more extreme con-ditions, by the total energy stored in the laser field. Inthe latter case the focused laser pulses would be depletedby cascade production.As it will be explained in more details below, therestoration mechanism works if the particles can be ac-celerated transversely to the field. It was conjectured[8, 10] on the basis of qualitative analysis for the modelof a uniformly rotating electric fields, that this may beindeed the case.In the EAS theory, the 1D approximation is often usedbecause spreading in transverse direction is inessential forultrarelativistic particles and has no significance for thatproblem. Besides, the cascade equations can be solved inthis case analytically within the ultra-relativistic approx-imation by means of the Mellin transform [12, 13]. Theresults of such analytic theory are in good agreement withboth experiments [11, 12] and direct Monte-Carlo simu-lations [22]. The attempts to treat the cascades in strongmagnetic fields on similar grounds are also known [23].However, though the 1 D approximation remains valid,the cascade equations can not be simplified via the Mellintransform unless some further approximation is made.According to Monte-Carlo simulations [24], the resultinganalytical approach works much worse here than in thecase of 1D approximation for EAS. In our case of cas-cades arising in a laser field, the structure of the cascadeequations (see Appendix A) is the same as for the mag-netic field, but it is impossible to incorporate restorationmechanism within the 1D approximation in momentumspace. This means that our problem is essentially two-or three dimensional.In this work we report on the first results of the Monte-Carlo simulations of cascades produced by initially slowelectrons in a uniformly rotating homogeneous electricfield. Such a field can be obtained practically at theantinodes of a standing electromagnetic wave. The choiceof the field model is uniquely specified by the existence ofreasonable qualitative estimations for scaling of the basiccascade characteristics for this particular case [10]. Ourgoal was to prove explicitly the existence of the restora-tion mechanism and to test the estimations [10] by directnumerical simulations.The paper is organized as follows. In Sec. II, which canbe considered as a technical introduction, we review andcollect the known information on the elementary quan-tum processes: single photon emission by electrons andpair creation by hard photons in strong fields of arbitraryconfiguration. Though this information is not completelynew, it is of essential importance for our presentation andis spread among the literature on the subject. After that,in Sec. III we present the reasoning in favor of the en-ergy restoration mechanism for cascades in electromag-netic fields. In Sec. IV we formulate the assumptions ofour model, present the details of our Monte-Carlo rou-tine and discuss the results obtained by numerical sim-ulations. These results are compared to the known es-timations. Summary and discussion is given in Sec. V.Finally, in Appendix A, we discuss the cascade equa-tions for our problem and explicitly demonstrate that,contrary to the recent doubts [25], the approach we usetakes proper account of radiation friction by ultrarela-tivistic electrons. II. QUANTUM PROCESSES WITHHIGH-ENERGY PARTICLES IN A STRONGELECTROMAGNETIC FIELD
General properties of radiation of ultrarelativistic par-ticles are well known [26]. Due to the relativistic aberra-tion effect the momenta of the products of any decay ofan ultrarelativistic particle are directed within the nar-row angle ∆ θ ∼ γ − with its momentum, where γ = p p /mc ) . Thus, radiation of a charged ultrarela-tivistic particle is visible at a point of observation onlyfor a short period of time τ during which its momentumturns through the angle of the order ∆ θ . The momen-tum turning angle can be estimated by ∆ θ ∼ eF ⊥ τ /mcγ ,where F ⊥ denotes the characteristic value of the trans-verse component of the field. Thus, τ ∼ mc/eF ⊥ . Thecharacteristic frequency Ω of classical radiation can bemost simply estimated in the proper reference frame ofthe particle, by transition to which the duration τ trans-forms into τ ′ = τ /γ . Thus, Ω ′ ∼ τ ′− ∼ eF ⊥ γ/mc . Inthe laboratory frame, due to the Doppler up-shift, we would thus have Ω ∼ ( eF ⊥ /mc ) γ . Such a scaling with γ is typical for congeneric problems and arises e.g. in thetheory of synchrotron radiation.The parameter χ = ~ Ω / ( γmc ) = F ⊥ γ/E S [27], beingthe ratio of the classically estimated mean energy of anemitted photon to the energy of the radiating particle,determines whether the process of radiation is controlledby classical electrodynamics or QED. Namely, if χ ≪ χ & χ is Lorentz and gauge invariant and is preciselydefined as χ = e ~ / ( m c ) p − ( F µν p ν ) , where F µν is thestrength tensor of the field and p µ is the 4-momentumof a particle. In what follows, we assume that F ≪ E S .On the other hand, we assume that the field is of rela-tivistic strength in the sense that the dimensionless fieldamplitude a = e p − A µ A µ / ( mc ) ≫
1, where A µ is the4-vector of the field potential. The latter means, in par-ticular, that it varies on the scale that exceeds τ and thuscan be considered constant with respect to the decay pro-cesses.Two different theoretical approaches have been devel-oped in order to study photon emission by ultrarelativis-tic ( γ ≫
1) charged particles in electromagnetic fieldsof ultrarelativistic ( a ≫ χ &
1) but still subcritical( F ≪ E S ) intensities. Nikishov and Ritus (NR) have cal-culated the appropriate quantum amplitudes in terms ofVolkov solutions in a constant crossed field ( E − H = 0and E · H = 0) of arbitrary strength [15, 27]. Their re-sults can be applied directly to our problem because, asthey have pointed out especially, under the abovemen-tioned conditions any field looks locally as constant andcrossed, the latter in the sense that both field invariants( E − H ) /E S and E · H /E S are much less than χ .The other approach by Baier and Katkov (BK) [19, 27] isbased on the observation that the motion of a particle inbetween two acts of photon emission (which correspondsto free lines of Feynman diagrams) can be consideredclassically if F ≪ E S , so that all the relevant quantumcorrections are reduced to quantum recoil and, possibly,to field-spin interactions. In a sense, the BK approachis equivalent to the replacement of the aforementionedVolkov solutions by the localized wave packets movingalong the classically prescribed trajectories. Both ap-proaches are based essentially on the same approxima-tions and provide the same results for the energy spectraof emitted photons.The energy distribution of the probability rate for pho-ton emission by ultrarelativistic electrons in an electro-magnetic field is given by [15, 19, 27] dW rad ( ε γ ) dε γ = − αm c ~ ε e ∞ Z x Ai ( ξ ) dξ + (cid:18) x + χ γ √ x (cid:19) Ai ′ ( x ) , (1)where x = ( χ γ /χ e χ ′ e ) / , Ai( x ) = (1 /π ) R ∞ cos( ξ / ξx ) dξ is the Airy function, ε γ and ε e are the energies ofthe emitted photon and the initial electron, respectively. χ e , χ ′ e = χ e − χ γ and χ γ (0 < χ γ < χ e ) are the di-mensionless quantum parameters for the electron beforeand after emission, and for the emitted photon, respec-tively. In terms of the field strengths, this parameter isrepresented as χ = e ~ m c s(cid:18) ε E c + p × H (cid:19) − ( p · E ) , (2)where ε and p are the energy and the momentum of thecorresponding particle.Note that expression (1) suffers from the infrared sin-gularity at ε γ →
0. However, in our constant field ap-proximation this singularity dW rad ( ε γ ) /dε γ = O ( ε − / γ )is weaker than the usual O ( ε − γ ) scaling of the infraredbehavior of perturbative QED [27, 29]. In particular, thetotal radiation probability rate is infrared convergent inour approximation. The infrared sector, however, is notimportant for the parameters considered in our paper,because most of the emitted radiation are found to havemuch larger frequencies than the frequency of the drivingfield.The energy distribution of the probability rate for di-rect pair creation by hard photons ( ε γ ≫ mc ) is givenby [15, 19, 27] dW cr ( ε e ) dε e = αm c ~ ε γ ∞ Z x Ai ( ξ ) dξ + (cid:18) x − χ γ √ x (cid:19) Ai ′ ( x ) , (3)where the indices “ γ ” and “ e ” this time refer to the initialphoton and to the created electron, respectively. Forthe created positron, we have χ ′ e = χ γ − χ e (0 < χ γ <χ e ). Formula (3) is completely symmetric with respectto electron and positron remaining unchanged under thereplacement χ e ↔ χ ′ e . Similarity between formulas (1)and (3) is explained by the fact that these two processesare related by the cross-symmetry [27].The total probability rates for both processes W rad = ε e Z dW rad ( ε γ ) dε γ dε γ , W cr = ε γ Z dW rad ( ε e ) dε e dε e , (4)cannot be written in terms of known special functions andshould be obtained by numerical integrations. However,they allow simple asymptotic expressions in the limits ofsmall and large χ e , χ γ , respectively. Namely, we have W rad ≈ . αm c ~ ε e χ e , χ e ≪ , (5a) FIG. 1: The sign of ˙ χ e ( t ) along the particle trajectory at t = t in different zones of the p x p y plane. The shaded zonescorrespond to acceleration (increase of χ e ) of positrons anddeceleration (decrease of χ e ) of electrons. The non-shadedzones correspond to the vice-versa situation. W rad ≈ . αm c ~ ε e χ / e , χ e ≫ , (5b)and W cr ≈ . αm c ~ ε γ χ γ e − / χ γ , χ γ ≪ , (6a) W cr ≈ . αm c ~ ε γ χ / γ , χ γ ≫ . (6b)Eqs. (5a), (6a) in these formulas describe the quasiclas-sical regime. For small values of the quantum parameter χ γ , the probability rate for pair photoproduction W cr issuppressed exponentially, in accordance with the essen-tially quantum nature of this process. At the same time, W rad remains O ( ~ − ), thus providing a finite classicallimit for the mean radiated intensity I rad . As for thelimit of large χ , both rates (5b) and (6b) differ only bya numerical factor of the order of unity.Given the energy and the momentum of the electronbefore emission, Eq. (1) determines the probability dis-tribution for the energy ε γ of the emitted photon. Underour assumption γ e ≫
1, the momentum of this photon isgiven by p γ = ( ε γ /p e ) p e . The energy and the momen-tum of the electron after emission should be determinedfrom the conservation laws. In the electromagnetic back-ground, they are of the form p µe + q µ = p ′ eµ + p µγ , where q µ is the four-momentum extracted from the field. Theexact value of this q µ essentially depends on the globalstructure of the field. This is because the whole space-time contributes to the integrals in the QED matrix ele-ment that yield delta-functions expressing the conserva-tion laws. For example, in a crossed constant field withthe Poynting vector directed along the z -axis the con-served quantities are ε/c − p z , p x and p y . In a constantelectric field, the canonical momentum is conserved. Ina constant magnetic field, directed along the z axis and (cid:0) x, c/ (cid:1)(cid:2) y , c / (cid:3) Initial positionNon-perturbed trajectory e (cid:4) e + e (cid:5) e + e (cid:6) e + e (cid:7) e + e (cid:8) e + e (cid:9) e + e (cid:10) e + e (cid:11) e + e (cid:12) e + e (cid:13) e + e (cid:14) e + e (cid:15) e + e (cid:16) e + FIG. 2: Spatial picture of the formation of the cascade ini-tiated by a positron in the homogeneous uniformly rotat-ing electric field (obtained by a Monte-Carlo simulation with a = 2 × and ~ ω = 1eV). Legend: Trajectories of elec-trons and positrons are shown as black and gray curves, re-spectively. The hard photons which have created pairs duringthe simulation time are shown as the dashed lines. The tra-jectory of the primary positron ignoring any QED processesis plotted as the thick light gray curve. with symmetric gauge, the conserved quantities are thenumber of the Landau level, the angular momentum, and p z . Nevertheless, for ultrarelativistic particles there is ac-tually no difference between these possibilities, and eitherof them can be adopted with the accuracy of our approx-imation. The reason is that q . eF τ . mc ≪ p e , p ′ e , p γ .In particular, we can assume p ′ e = p e − p γ . The same ar-gument can be applied to pair creation by hard photonsas well.In addition to one-photon emission and direct pairphotoproduction reviewed above, there exist more com-plicated higher-order processes, such as e.g. the two-photon emission e − → e − γγ or the trident process e − → e − e − e + . Their specific feature is that the interme-diate particle is off the mass shell, i.e. is virtual. How-ever, in strong field situations of our interest the two-stepprocesses dominate [8, 9, 20]. For this reason, we do notconsider higher-order processes in the sequel. III. BASIC ESTIMATIONS FOR CASCADEPRODUCTION IN A ROTATING ELECTRICFIELD
Since there is no difference whether an electron or apositron initiates a cascade, we assume in this section that our cascade is initiated by a positron ( e > E ( t ) = { E cos ωt, E sin ωt } . (7)The equation of motion˙ p ( t ) = e E ( t ) , (8)with the initial condition p ( t ) = p , can be easily solved: p x ( t ) = p x + mca (sin ωt − sin ωt ) ,p y ( t ) = p y − mca (cos ωt − cos ωt ) . (9)Here, a = eE /mωc is the dimensionless field amplitude.Let us assume first that the positron is at rest ( p = 0)initially ( t = 0). Equations (7) and (9) show that theenergy and the quantum parameter χ of the positron forthis case depend on time as ε e ( t ) = mc r a sin ωt , (10a) χ e ( t ) = e ~ E m c r a sin ωt . (10b)Both quantities are increasing initially. They are os-cillating with the period 2 π/ω of the rotation of thefield. The amplitudes of these oscillations, ε m ≈ mc a , χ m ≈ a ( E /E S ) = 2( ~ ω/mc ) a are proportional to a and a , respectively, and are quite large under ourbasic assumptions. For example, χ m approaches unityalready at a ∼ a c = 500 for an optical rotation fre-quency of ~ ω = 1eV. This corresponds to the fieldstrength E ∼ − E S ∼ V/cm and the intensity10 W/cm . Since χ m ∼ χ γ ∼ χ e ∼
1, which, in turn,can create an electron-positron pair. However, at such in-tensities a new generation of pairs is typically producedon the time scale π/ω , and the whole pair generation pro-cess may be rather sensitive to peculiarities of the fieldmodel. As we discuss below, stable cascade formation isexpected at higher intensity levels.The formulas (10a) and (10b) become especially simplefor stronger fields a ≫ a c , because in this case the value χ e ∼ t acc of the rotation period. Namely, we have ε e ( t ) ≈ eE ct, ωa ≪ t ≪ ω , (11a) χ e ( t ) ≈ (cid:18) E E S (cid:19) mc ω ~ t , ω √ a ≪ t ≪ ω . (11b)Eq. (11a) is easy to understand, because initially thepositron is accelerating almost along the field. In or-der to understand Eq. (11b) better, let us note that in t/t rad -2 -1 N ee [ (cid:17) > (cid:18) (cid:19) ] our codealternative event generatorbenchmark FIG. 3: Comparison of the cascade profile obtained with ourcode [black thin line], with a code applying an alternativeevent generator [circles] and from previous independent sim-ulations [thick gray line, see Fig. 5 in [24]]. Depicted is thenumber of pairs with energy ε > − ε versus the elapsedtime. The simulation parameters are ε = 100GeV and E /E S = 0 . the case p (0) = 0, according to Eqs. (7) and (9), themomentum of the positron constitutes exactly the angle ωt/ x -axis. Intuitively, this is because due to its inertia theparticle does not follow the rotation of the field precisely.As a consequence, the transverse component of the fieldwith respect to momentum of the particle increases as E ⊥ = E sin( ωt/ ≈ E ωt/
2. Since χ e ( t ) ≈ E ⊥ γ/E S ,we arrive immediately at Eq. (11b). Qualitatively, thesame growth of the energy and the parameter χ withtime has been observed for generic field configurations[10].As it follows from Eq. (11b), the quantum parameter χ e becomes of the order of unity over the period of time t acc , t acc ∼ ~ αmc µ r mc ~ ω . (12)Here we have introduced a new dimensionless field inten-sity parameter µ = E/E ∗ , E ∗ = αE S ≈ E S / µ is related to the commonly accepted parameter a by µ = ( ~ ω/αmc ) a . According to Ref. [10], the cascadescan be caused by initially slow particles if µ & χ e is shared between the positronand the emitted photon [35], χ e ≈ χ γ + χ ′ e . If χ e & χ γ and χ ′ e are less than χ e but are of compa-rable value χ ′ e ∼ χ γ . χ e . Although propagation of theresulting hard photon is not affected by the field, nev-ertheless its χ γ continues to increase after emission justdue to rotation of the field.In order to understand better what must happen af-ter the first hard photon emission, let us come back to Eq. (9) and consider the general initial condition. It iseasy to see that in the case H = 0 the sign of the deriva-tive of the quantity (2) is determined completely by theexpression − e ( p · E )( p · ˙ E ). The zones in the p -plane inFig. 1, where ˙ χ e > t = t are shaded. In the shaded areas χ e can be expected toincrease for some time. The time t can be identifiedwith the creation time of a new pair.Since the momentum of a primary positron is confinedto the shaded region to the right, and the new secondaryparticles are created with momenta parallel to the mo-mentum of parental particle in our approximation, wesee that momenta of the secondary particles also lie inthe shaded sector. Thus, the newly created positrons areaccelerating with ˙ χ ( t ) >
0, while the newly created elec-trons are initially decelerating ( ˙ χ < µ ≫
1. The idea is that, due to similarity of therates (5b) and (6b), and also because the variation of theangles between the momenta of all particles (positrons,electrons, and photons) and the field is determined bythe same temporal scale ω − , there is actually no needto distinguish between all three sorts of particles. So, anorder of magnitude estimation can be provided withinthe model of a simple doubling chain process.Let us denote by t e the typical lifetime for electronsand positrons with respect to hard photon emission. Thesame quantity up to an order of magnitude defines thelifetime of photons with respect to pair creation. Thelifetime t e , together with the typical energy and the valueof quantum parameters of the particles, as well as withthe angle between their momenta and the field, can beestimated as [10] t e ∼ ~ αmc µ / r mc ~ ω , (13a) ε ∼ mc µ / r mc ~ ω , χ ∼ µ / , (13b) θ ∼ ωt e ∼ αµ / r ~ ωmc . (13c)Under the condition µ ≫
1, as is assumed here, we have θ ≪ χ ≫
1. The latter inequality approves thechoice of the asymptotic expressions (5b) and (6b). In ad-dition, we have the following hierarchy of the time scales t acc ≪ t e , which assures that exactly hard photons with χ γ & t, (cid:20)(cid:21) (cid:22) e without QED effects Run 1Run 2 Run 3 t, (cid:23)(cid:24) N e + e (cid:25) Run 1Run 2 Run 3
FIG. 4: Left plot: Temporal evolution of the quantum dynamical parameter χ e of the primary electron for three independentMonte-Carlo simulations. The thick gray curve corresponds to the analytical solution Eq. (10b) for χ e ( t ) in the absence ofany QED processes. The three other curves (Run1, Run2, and Run 3) are the results of the three independent Monte-Carlosimulations with parameters a = 2 × and ~ ω = 1 eV. Right plot: The total number of electrons and positrons N e + e − vs.time for the same independent simulations. must grow exponentially, N ( t ) ∼ e Γ t , Γ ∼ t e ∼ αµ / r mc ω ~ . (14)In the next section, we are checking the estimations (13)and (14) by direct Monte-Carlo simulations. IV. DESCRIPTION OF MONTE-CARLOAPPROACH AND NUMERICAL RESULTS
In our simulations we are using a Monte-Carlo ap-proach for the integration of the cascade equations [seethe Eqs. (A1), (A2)]. We trace the motion of the elec-trons and positrons in between the photon emissions clas-sically, whereas for hard photons we exploit the ray trac-ing approximation in between their emission and con-version into pairs. Even though there exists the exactanalytical solution (9) for equations of motion (8) forpositrons and electrons, we are integrating Eq. (8) nu-merically for each of the particles. This is done in orderto incorporate the probabilistic events of photon emissionand pair creation in the routines as described below, aswell as for the purpose of future generalization to morerealistic field configurations.Our numerical algorithm works as follows. At eachtime step t i < t < t i +∆ t we are calculating the momentaof all the particles created at preceding time steps by p i +1 = p i + q i E i +1 / ∆ t , where E i +1 / = E ( t i + ∆ t/ q i = + e, − e, p i and E i = E ( t i ), we attach the value χ i at time t i using Eq. (2) to each electron and positron and computethe total probability rate W rad (see Eq. (4)). In order toisolate the infrared singularity, we set the lower limit ofintegration to ε min . For each electron and positron, weassume that it emits a photon between t i and t i +1 if r 1) is a uniformly distributedrandom number. If the above inequality is fulfilled, thenthe energy ε γ of the emitted photon is obtained as theroot of the sampling equation1 W rad ε γ Z ε min dW rad ( ε γ ) dε γ dε γ = r ′ , (15)where r ′ (0 < r ′ < 1) is an independent random number.The time step ∆ t , which remains fixed in the course ofcomputation, must be chosen such that ∆ t ≪ W − rad , W − cr holds. The direction of propagation of the newly emittedphoton is parallel to the momentum p i of the parentalelectron or positron, whose momentum after emission wefind from the conservation law as discussed in Sec. II. Forpair creation, the event generator works similarly, apartfrom the fact that there is no need for the regularizationparameter ε min .Within the constant crossed field approximation ap-plied here we assume that ε γ ≫ mc . However, the pho-tons with energies ε γ . mc are not able to create pairsin a subcritical field, for which χ γ ≪ ε min to mc for both W rad andthe sampling Eq. (15).As a benchmark for our code we have simulated thedevelopment of a cascade initiated by a high-energy( ε = 2 × mc ) initial electron in a constant homo-geneous transverse field with E = 0 . E S . Our results t, (cid:26)(cid:27) < (cid:28) e > Run 1Run 2Run 3 t, (cid:29)(cid:30) (cid:31) e , ! , " m c e e + $ FIG. 5: Left plot: The dynamical quantum parameter h χ e i for the electrons averaged over the cascade vs. time for the samesimulations as in Fig. 4. Right plot: Evolution of the mean energy of the electrons, positrons, and photons averaged over thecascade in a typical simulation run ( a = 5 × and ~ ω = 1eV). are averaged over 10 simulation runs. In this particularsimulation, the curvature of trajectories of electrons andpositrons has been neglected, so that the results of oursimulations can be directly compared with previous sim-ulations of cascades produced by high-energy electronsin a magnetic field [24]. Comparison of cascade profilesobtained in both simulations is given in Fig. 3 by thesolid red and dashed green lines, respectively. The figurerepresents the number of pairs with an energy exceed-ing 0 . 1% of the energy of the primary electron versus theelapsed time. In our notation the reference characteristicradiation time t rad as adopted in Ref. [24] in our notationis t rad = 3 . × ( γ in /αχ / in ) × ( ~ /mc ) = 5 . × W − rad,in ,where the subscript “in” refers to the initial data for pri-mary electron. We see that our results are in reasonableagreement with the paper [24].We have also implemented and tested a different eventgenerator, which provides significant speed up due to theabsence of numerical integrations. The idea is to exploitsome explicit algebraic fits for the energy spectrum (1),and to exchange the order of testing the occurrence ofphoton emission and of sampling its energy. In this al-ternative version of the algorithm, within each time stepone first samples the possible energy of an emitted pho-ton just as an uniformly distributed random quantity, ε γ = ε e r ′ in the above notation. After that, photon emis-sion is assumed to take place if r < [ dW rad ( ε γ ) /dε γ ] ε e ∆ t .In this case, the time step must satisfy the condition∆ t < [ ε e dW rad ( ε ∗ γ ) /dε γ ] − for all appearing electronsand positrons, where ε ∗ γ is the photon energy that cor-responds to the maximum of the emission spectrum (1).The same scheme can be applied to the simulation of pairphotoproduction as well. Note that in this case there isno need to introduce the energy cutoff ε min , althoughthis may serve as a useful trick if one wants to restrictthe number of soft photons that are traced by the code.The test of the modified event generator is included by a dashed blue line in Fig. 3. This test demonstrates thatboth versions of the event generator are in fact equiva-lent.The results of our simulations are collected in the fig-ures 2, 4-7. Fig. 2 represents a typical spatial pictureof the formation and development of a cascade initiatedby a positron. The electrons and positrons are deflectedby the field in opposite directions, whereas the directionsof propagation of photons are distributed randomly, ascould be expected. For the rest of the paper we assumefor all simulations in an uniformly rotating field that at t = 0 we have a single electron at rest ( p e = 0) andno photons and positrons. The typical evolution of thequantum dynamical parameter χ e of the primary electronis illustrated with the left panel of Fig. 4. Before the emis-sion of a first photon, the electron is gaining energy andits parameter χ e is growing as the square of time in ac-cordance with Eq. (11b). After the first photon emission,which for our parameters happens typically on the timescale t e smaller than ω − , the curves become stochasticand consist of smooth sections with typical growth of χ e due to acceleration by the field. These sections are sep-arated by sudden breakdowns resulting from recoils dueto successive photon emissions. Since these recoils arerandom, the three curves in the figure corresponding toindependent simulation runs deviate at later times. Af-ter the transient period which typically lasts for severallifetimes t e , the momentum losses due to quantum recoilsare coming into equilibrium on the average with the trendof acceleration by the field. After that, function χ e ( t ) foran individual electron describes a stationary stochasticprocess.As it was predicted in Ref. [10], the development of acascade results in exponential growth with time of thetotal numbers of secondary hard photons and electron-positron pairs. This is illustrated with the right panel inFig. 4. The plot N e − e + ( t ) is given by a random stairway,with each stair corresponding to creation of a single pair.The successive stairs are well separated initially, whenthe total number of pairs remains small. At later timewith the number of pairs growing rapidly the stair-likestructure of the lines in the plot becomes invisible andstraight lines are obtained. Although these straight linesfor independent simulation runs are typically different,mostly because emission of the first photon starts ran-domly from one simulation run to another, neverthelesstheir gradients are varying weakly in different runs andcan be used to determine the growth rate Γ in Eq. (14).For example, the growth rates extracted from the curves1-3 at Fig. 4 are Γ = 4 . , . 84 and 4 . 90, respectively.We have studied the averages of the quantities χ e , ε e and θ over the cascade. For example, temporal evolutionof the mean value h χ e ( t ) i = 1 N e − ( t ) N e − ( t ) X i =1 χ e i ( t ) , (16)where N e − ( t ) is the instant number of present electronsand χ e i ( t ) is the instant value of the quantum dynami-cal parameter for the i -th electron, as depicted in the leftplot of Fig. 5. One can see that at later times the ran-dom fluctuations are smoothed out and the quantity (16)stabilizes acquiring some definite constant value which isindependent of the simulation run. The same behaviorwas observed for h ε e i and h θ i , which are defined in amanner similar to Eq. (16). The typical evolution of theaveraged energy of all the components of the cascade isrepresented in the right plot of Fig. 5. At later times,the mean energies of electrons and positrons coincide asis expected from symmetry consideration, whereas themean photon energy typically remains smaller. At thesame time, the energy spectrum of created electrons andpositrons is wider than the photon spectrum (see Fig. 6).Both features are explained naturally by the fact that inour setup the hard photons ( χ γ & 1) are quickly con-verted into pairs which survive, whereas soft photons( χ γ . 1) are stable with respect to pair photoproductionand hence are accumulated. In the high energy region,all the spectra are likely to show exponential decay.One of our main tasks was the investigation of the va-lidity of estimations (13) and (14) which were suggestedpreviously in Ref. [10] and are of crucial importance. Inparticular, Eq. (14) was serving as an argument for theestimation of the maximal value of the intensity attain-able with focused laser fields. In order to test these es-timations we have performed parametric studies of thestabilized values of the quantities h χ e i , h ε e i , h θ i and ofthe increment Γ. These results are presented in Fig. 7.The left plot demonstrates the dependence of the ratiosof the quantities obtained by simulations to their estima-tions (13) on µ for fixed rotation frequency ~ ω = 1eV.It is clear from the figure that for large values of µ eachratio acquires some definite limit of the order of unity.According to our results the formulas (13) are valid upto some numerical coefficients of the order of unity, whichvary no more than twice in the whole range µ > 1. The % e , &’ , 10 ( MeV d N / d ) , M e V * + e , FIG. 6: The energy spectra for different components of thecascade at t = 1 . × ω − for a = 5 × and ~ ω = 1 eV. results of simulations for Γ are compared with Eq. (14)for two different rotation frequencies ~ ω = 0 . ~ ω = 1eV on the right panel of Fig. 7. One can see thatfor large values of µ the estimation (14) is justified withgood accuracy even without any correction factor. For µ . 30 formula (14) overestimates Γ but not more thanby half of an order of magnitude. This may be neverthe-less crucial for the estimation of the total cascade yielddue to its exponential dependence on Γ. For the par-ticular value µ ≈ 10, which was exploited in Ref. [10],formula (14) overestimates Γ by approximately a factorof 1 . 5. This, however, can be compensated in principleby simultaneous underestimation of the escape time inRef. [10].In order to apply the results of our simulations to es-timate the cascade yield by a realistic focused laser field,we assume that the appearing electrons and positronsare pushed away as a whole from the focus by the pon-deromotive potential in radial direction with almost thespeed of light. Assuming the Gaussian profile of the fo-cused beam we can write µ ( t ) = µ e − c t /w , where µ is the value of the parameter µ at the center of the focusand w is the focal radius. Then the total number ofpairs produced by the cascade can be estimated to theln( N e + e − ) ∼ ∞ Z Γ( µ ( t )) dt = Γ( µ ) ∞ Z e − c t / w dt. The remaining integral defines the effective time of escapefrom the focus and equals √ π ( w /c ), i.e. is √ π ≈ . µ ≈ 10 that we have observed in our sim-ulations. Thus, we hope that the quantitative predictionsin Ref. [10] must remain unaffected by our corrections.One can see that we are currently neglecting the com-plicating details in our code such as, e.g. the elastic colli-sions, the Compton scattering, and the annihilation pro-cesses. Such phenomena must become important only at - . / / est / est / est / simulation =1 eVestimation =1 eVsimulation ;< =0.66 eVestimation => =0.66 eV FIG. 7: Left plot: Parametric studies of the mean energy ε e , the mean dynamical quantum parameter χ e , and the mean angle θ e between the momentum and the field for electrons and positrons. The ratios of the simulation results to the approximations(13b), (13c) are plotted vs. the parameter µ for ~ ω = 1eV. Right plot: Parametric study of the increment Γ as a function ofthe dimensionless field strength µ for two rotation frequencies ~ ω = 1 eV and ~ ω = 0 . 66 eV. The approximation (14) is shownby the dashed lines. later times, when the plasma is dense enough. Thoughsuccessive collisions and annihilations of the electron andpositron from the same created pair may be important[30], we are currently ignoring these effects for simplicity[36]. We ignore the higher-order processes, such as two-photon creation and the trident processes as well (see theremark at the end of Sec. II). All these assumptions arenatural and commonly accepted in the present cascadetheory [23], even though they may be revised in futurestudies.Due to limitations of computer power, we cur-rently stop our simulations after the creation of around N e − e + . pairs. This was shown to be enough toestimate the growth rate Γ, as well as to average thecharacteristics of a cascade over the ensemble of pairswith reasonable accuracy. In the time interval of simula-tion these pairs occupy a volume of the order d , where d ∼ c/ω ∼ µ m. This corresponds to the pair density n e + e − ∼ cm − . Typical values of the γ -factor forelectrons and positrons are γ ∼ ÷ , see Fig. 6. Thiscorresponds to their energies ε e = γmc ∼ . ÷ T ∼ ε e /k ∼ K, we can es-timate the Debye screening radius r D ∼ p kT /e n e + e − ∼ ≫ d . The relativistic plasma frequency Ω pe ∼ c/r D ∼ sec − remains about five orders of magni-tude smaller than the optical frequency. For these rea-sons we have completely neglected Coulomb interactionbetween electrons and positrons and all the accompany-ing collective plasma effects in the present simulations.However, the density of pairs is growing exponentially, n e + e − ( t ) ∝ e Γ t , and hence r D ∝ e − Γ t/ and Ω pe ∝ e Γ t/ .After a relatively short period of time . π/ω , whenthe number of pairs becomes macroscopic ( ∼ ), thequantities r D and Ω pe attain the values d and ω , respec-tively, and the collective plasma effects may come intoplay. Within our approximation of a homogeneous field, the total number of created pairs would be restricted bythe screening of the external field by the self-field of aris-ing plasma.Let us note, that despite some doubts in the literature[25], the radiation friction is taken into account properlyin our version of the algorithm by the recoils happen-ing at the times of photon emission (see, e.g., Ref. [31]and the Appendix A in our paper). Hence, there is noneed to include an additional radiation friction force inthe equations of motion (8) for electrons and positrons,otherwise this would cause double counting. Moreover,our approach transfers the concept of classical radiationfriction into the quantum domain in a correct fashion. Itcan be asked how the classical continuously acting radi-ation friction can be recovered from the sudden jumpsof momentum similar to those in Fig. 4. In fact thishappens on the average with respect to the ensemble ofMonte-Carlo realizations, since the moments of succes-sive photon emissions are distributed randomly. At latertimes, when the number of created pairs becomes large,the cascade forms a representative ensemble itself andthere is in principle no need for taking the average overindependent realizations. V. SUMMARY AND DISCUSSION In this paper we have presented the first results of nu-merical simulations of the formation and development ofelectron-positron-photon cascades by initially slow elec-trons in a uniformly rotating homogeneous electric field.In such a situation the cascades reveal a principally newfeature, i.e. the restoration of the energy and the dy-namical quantum parameter due to the acceleration ofelectrons and positrons by the field. This feature maybe of crucial importance for the whole physics of laser-0matter interactions in the strong field domain, as it wasdemonstrated in Ref. [10]. We have explicitly identifiedthis restoration mechanism in the course of our simula-tions. Also, our simulations clearly confirm the qualita-tive analysis of Ref. [10], including the basic scalings (13)and the estimation (14) for the cascade yield. So, theycan be used to fix the remaining numerical correction fac-tors in Eqs. (13) and (14), which turn out to be of theorder of unity.The numerical approach that we adopt is based onMonte-Carlo simulations of the cascade equations. Wehave shown explicitly that contrary to some recentdoubts in literature [25] such an approach incorporatesradiation friction acting on individual electrons andpositrons and, moreover, is doing this in a manner whichis consistent with intense field QED.The codes designed for our task can be readily adoptedfor simulating cascades in the laser fields with more re-alistic configurations, such as tightly focused Gaussianbeams and pulses. This is required in order to makemore definite predictions on the impact of cascade pro-duction for possible future experiments, as well as forfurther corrections of the maximal value of intensity thatcan be attained with optical lasers [10]. Simulation ofcascades in a focused laser field will be presented in aseparate publication. However, let us make several briefcomments about cascades in focused laser fields, possibleexperimental scenarios, and some yet unresolved techni-cal problems that may require further studies.The restoration mechanism arises due to the curva-ture of the trajectories of the charged particles across thefield and may be sensitive to its polarization. Althoughwe expect that this mechanism must work for genericfield configurations (e.g., for generic tightly focused laserfields), there may exist several particular configurationsfor which the restoration mechanism does not work. Forexample, in an arbitrary constant electromagnetic field ora circularly polarized propagating plane electromagneticwave the dynamical quantum parameter χ e is conservedexactly in the course of motion. In the case of a genericpropagating plane wave the amplitude of oscillations ofthe parameter χ e for an initially slow electron does notexceed E /E S , i.e., always remains smaller than unity.Another example is a linearly polarized oscillating elec-tric field [25], since in this case the initially slow parti-cles are accelerated strictly along the field and hence thegrowth of the transverse component of the field is absent.In some intermediate cases, e.g. for elliptical polarizationor a weakly focused Gaussian beam, restoration of χ e must exist but may be less effective than in the case of cir-cular polarization. However, in all the cases at least theusual cascades would be caused by external high-energyelectrons or hard photons passing through the high-fieldregion transverse to the field. In this case, the cascadeyield remains microscopic and would be determined byboth the initial energy of an external energetic particleand the laser field strength. In order to initiate a cascade in a tightly focused laserfield, it is required to inject a primary particle into thecenter of a focal region. This task may be not triv-ial because the focal region is surrounded by a pon-deromotive potential wall of the characteristic height U ∼ mc p a ≈ mc a . The external high-energyelectrons will be most likely deflected rather than pen-etrate inside. The most direct and elegant scenario isbased on the exploration of pairs that are created spon-taneously from vacuum by the laser field itself [10], sincethey are appearing exactly at the center of the focus asrequired. However, this possibility implies high intensi-ties & W/cm . Another possible resolution wouldbe the initiation of cascades by energetic γ -quanta. Inour opinion, the final conclusion whether or not cascades with macroscopic yield can arise in generic real exper-iments exploring laser-matter interaction at intensitieslower than ∼ W/cm requires further studies. Wenote that for cascades that arise in the course of the in-teraction of high-intensity laser radiation with materialtargets it may be necessary to take the impact of ordinarycascades in matter into account as well [33].If the cascade yield attains macroscopic values( N e − e + ∼ ), the self-field of the electron-positronplasma becomes comparable to the guiding field. In thisregime screening of the external field and its absorptionby the electron-positron plasma self-field will restrict fur-ther pair production. Such a regime can be simulatedby combining our codes with the Particle-in-Cell (PIC)method [34]. We hope to address this problem in one ofour next publications. Acknowledgments We are grateful to G. Korn for his permanent interestin this work and useful advise. We are also grateful toS.V. Bulanov, J. Kirk, A. Bell and I. Sokolov for fruit-ful disputes that gave us several useful ideas and to E.Vilenius and S. Rykovanov for reading this manuscriptand suggesting a number of stylistic corrections. Thiswork was supported by the grant DFG RU 633/1-1, theCluster-of-Excellence ’Munich-Centre for Advanced Pho-tonics’ (MAP), the Dynasty foundation, Russian Fundfor Basic Research, the Ministry of Science and Educa-tion of the Russian Federation and the Russian FederalProgram “Scientific and scientific-pedagogical personnelof innovative Russia”. Appendix A: Cascade equations and radiationreaction The cascade equations for a uniformly rotating homo-geneous electric field1 ∂f ± ( p e , t ) ∂t ± e E ( t ) · ∂f ± ( p e , t ) ∂ p e = Z w rad ( p e + p γ → p γ ) f ± ( p e + p γ , t ) d p γ − f ± ( p e , t ) Z w rad ( p e → p γ ) d p γ + Z w cr ( p γ → p e ) f γ ( p γ , t ) d p γ , (A1) ∂f γ ( p γ , t ) ∂t = Z w rad ( p e → p γ )[ f + ( p e , t ) + f − ( p e , t )] d p e − f γ ( p γ , t ) Z w cr ( p γ → p e ) d p e . (A2)differ from the standard equations of EAS [12, 13] only bythe addition of the second term to the LHS of Eq. (A1),which takes electron and positron acceleration into ac-count. Here, f ± and f γ are the distribution functionsfor positrons, electrons and photons, respectively. In ourapproximation of photon and pair emission in strictly for-ward direction the differential probability rates are of theform w rad ( p e → p γ ) = Z dλ dW rad dε γ (cid:12)(cid:12)(cid:12)(cid:12) ε γ = λε e δ ( p γ − λ p e ) ,w cr ( p γ → p e ) = Z dλ dW cr dε e (cid:12)(cid:12)(cid:12)(cid:12) ε e = λε γ δ ( p e − λ p γ ) , (A3)so that the integrals standing on the RHS of Eqs. (A1),(A2) are essentially one-fold. Nevertheless, in the maincase of interest, the direction of the field E ( t ) varies intime, so that the problem does not reduce to 1D. Notethat the scalings (13) can be obtained from (A1) and(A2) in the limit of large χ via dimensional analysis.It is interesting to note that in the view of the approx-imation (A3) it follows from the Eqs. (A1), (A2) that ddt (cid:26)Z p e ( f + + f − ) d p e + Z p γ f γ d p γ (cid:27) = e E ( t ) Z ( f + − f − ) d p e , (A4) ddt (cid:26)Z ε e ( f + + f − ) d p e + Z ε γ f γ d p γ (cid:27) = e E ( t ) Z p e ε e ( f + − f − ) d p e . (A5)These identities ensure that in our approximation themomentum and the energy are extracted from the fieldonly during the acceleration of electrons and positrons,i.e., both energy and momentum of the electron-positron-photon plasma are conserved during photon emission andphotons to pairs conversion.The first two terms on the RHS of the Eq. (A1) de-scribe the influence of photon emission on electrons andpositrons. Let us now demonstrate that classical radi-ation reaction is taken into account properly by theseterms.For this aim let us assume in what follows that theelectrons are slow in the sense that distributions f ± are restricted to such momenta for which χ e ± ≪ N ± = R f ± d p e are conserved.The relation between the variables χ γ and x in Eq. (1)can be expressed in the alternative form χ γ = x / χ e x / χ e . (A6)It is clear from this formula by taking into account thatthe spectrum (1) of emitted photons is effectively con-centrated in the range x . 1, that χ γ ≈ x / χ e . χ e ≪ χ e . (A7)As a consequence, p γ ≪ p e in the remaining integrals onthe RHS of Eq. (A1). Thus, the expansion w rad ( p e + p γ → p γ ) f ± ( p e + p γ ) − w rad ( p e → p γ ) f ± ( p e ) ≈ p γ · ∂∂ p e [ w rad ( p e → p γ ) f ± ( p e )] , (A8)is valid.Consider the average momenta P ± ( t ) =(1 /N ± ) R p e f ± ( p e , t ) d p e . By multiplying Eq. (A1),truncated as described above, by p e and integrating itover p e by parts we have˙ P ± ( t ) = ± e E ( t ) + h R i ± ( t ) , (A9)where R ( p e ) = − Z p γ w rad ( p e → p γ ) d p γ (A10)is the mean rate of momentum losses of electrons andpositrons due to photon emission, and h R i ± ( t ) = 1 N ± Z R ( p e ) f ± ( p e , t )are its averages over the momentum distributions ofpositrons and electrons, respectively.2Taking into account that p γ ≈ x / χ e p e and ε γ =( ε e /χ e ) χ γ and using Eqs. (A3) and (1), we are comingto R ( p e ) = − ε e p e χ e Z x / dW rad dε γ dχ γ . (A11)From this point, let us pass to the integration over thevariable x . In the approximation χ e ≪ χ γ = x / χ e . Contribution to the integral in Eq. (A11) comesfrom the range x ∼ 1. 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