Dynamically assisted Schwinger effect beyond the spatially-uniform-field approximation
DDynamically assisted Schwinger effect beyondthe spatially-uniform-field approximation
I. A. Aleksandrov , , G. Plunien , and V. M. Shabaev Department of Physics, St. Petersburg State University,7/9 Universitetskaya Naberezhnaya, Saint Petersburg 199034, Russia NRC “Kurchatov Institute” — ITEP, Moscow 117218, Russia Institut f¨ur Theoretische Physik, Technische Universit¨at Dresden,Mommsenstrasse 13, Dresden D-01062, Germany
Abstract
We investigate the phenomenon of electron-positron pair production from vacuum in the presence of a strongelectric field superimposed by a weak but fast varying pulse which substantially increases the total particle yield.We employ a nonperturbative numerical technique and perform the calculations beyond the spatially-uniform-field approximation, i.e. dipole approximation, taking into account the coordinate dependence of the fast com-ponent. The analysis of the main characteristics of the pair-production process (momentum spectra of particlesand total amount of pairs) reveals a number of important features which are absent within the previously usedapproximation. In particular, the structure of the momentum distribution is modified both qualitatively and quan-titatively, and the total number of pairs created as well as the enhancement factor due to dynamical assistancebecome significantly smaller. a r X i v : . [ h e p - ph ] M a y . INTRODUCTION The process of the vacuum decay accompanied by the production of electron-positron pairs in thepresence of strong external fields was predicted decades ago [1–3] and still remains a very intriguingphenomenon. From the theoretical viewpoint, the interest in this effect is due to the nonperturbativenature of the pair-production process taking place in strong quasistatic backgrounds. In order to probethe quantum vacuum in this regime, i.e. to study the Schwinger mechanism, one has to employ non-perturbative evaluation methods instead of using perturbation theory, which is not applicable in thisstrong-coupling domain. The essential point is that the Schwinger effect has never been observed ex-perimentally as the required field strength is extremely large. In the case of a static and spatially uniformelectric field, the characteristic critical field strength is E c = m c / ( | e | (cid:126) ) ≈ . × V/cm whichis - orders of magnitude larger than the peak electric field strength reached in modern laser pulses.Nevertheless, the laser technologies develop very rapidly, so one may expect the Schwinger mechanismto become experimentally accessible in the not too distant future. To theoretically support these studies,it is necessary to find the most promising scenarios that can be implemented in experiments.One of the possible schemes was proposed a decade ago in Ref. [4]. The configuration involvestwo laser pulses of different intensity and frequency. While the first pulse is strong and slowly varying,the second one is weak and fast. Let E ( ε ) and Ω ( ω ) be the peak strength and frequency of the strong(weak) pulse. If one introduces the Keldysh parameters γ E = mc Ω / | eE | and γ ε = mcω/ | eε | [5], theyshould satisfy γ E (cid:28) and γ ε (cid:29) . This means that the strong pulse alone acts in the nonperturbative(Schwinger) regime whereas the individual weak pulse can be treated in the framework of perturbationtheory. It turns out that the combination of these two pulses can lead to a dramatic enhancement of theparticle yield. This phenomenon was first studied in Ref. [4], where the external field was representedas a sum of two spatially uniform Sauter pulses without a subcycle structure (see also Refs. [6–9]).The carrier of the laser pulses was taken into account in a number of subsequent studies [10–15].Nevertheless, a systematic analysis of the pair-production process beyond the spatially-uniform-fieldapproximation (we will also call it the dipole approximation) still has not been conducted.In fact, the previously used dipole approximation (DA) can hardly be justified due to the pres-ence of the fast pulse. The usual ansatz approximating the monochromatic external electric field bya uniform background is justified by the requirement that the laser wavelength λ be much larger thanthe characteristic length scale of the pair-production process (cid:96) = 2 mc / | eE | . This is equivalent to thecondition γ (cid:28) which is not satisfied in the case of a weak and fast pulse since γ ε (cid:29) . One may2xpect that in the presence of both the strong and the weak components, the relevant parameter is the“combined” Keldysh parameter γ c = mcω/ | eE | , but as was demonstrated in a number of studies (see,e.g., Refs. [4, 6, 9, 11]), the efficient dynamical assistance is likely to occur only when γ c (cid:38) . Thissuggests that the spatial variations of the weak fast pulse should be taken into account, which is themain goal of the present investigation.In this study we consider a combination of a uniform time-dependent strong field and a standingwave containing rapid oscillations in space and time. Both pulses have a finite duration. We examinethe key aspects of the dynamically assisted Schwinger mechanism both within the dipole approximationand beyond it (bDA). According to the results of Ref. [9], the particle yield is exponentially suppressed,and the corresponding exponent does not change when one goes beyond the uniform-external-fieldapproximation. Nevertheless, in this study we carry out numerical calculations which provide the exactvalues of the number density of particles created, while the worldline instanton approach employed inRef. [9] allows one only to estimate the total particle yield. Besides, we take into account the temporaldependence of the strong pulse and examine various characteristics of the pair-production process. Inparticular, we analyze the momentum spectra of particles created and the integrated number density.The corresponding calculations are performed by means of a nonperturbative numerical technique.It turns out that taking into consideration the spatial dependence of the weak pulse uncovers a fewsignificant features in the momentum spectra which do not appear within the dipole approximation.Furthermore, the enhancement due to the dynamical assistance as well as the total particle yield alsonotably alters.After completion of the present investigation we noticed the very recent study [16], where it wasdemonstrated that the spatial dependence of the external field plays a crucial role in the context ofthe Breit-Wheeler process, where a combination of two fast-varying laser pulses is considered. It wasshown that one can hardly approximate the resulting field of two pulses with large γ by a spatiallyuniform background. In Ref. [17] this conclusion was drawn regarding a combination of two pulseswith γ ∼ . In the present study, we demonstrate that the same applies to the case of the dynamicallyassisted Schwinger effect.In Sec. II we describe the field configuration to be studied and introduce an approximate enhance-ment factor which is used to identify the values of the field parameters in the dynamical assistanceregime. A similar analysis is carried out beyond the dipole approximation, which reveals a number ofnew important features. In Secs. III and IV, we turn to the study of the momentum distribution of par-3icles produced within the dipole approximation and beyond it, respectively. In Sec. V we examine thetotal number of e + e − pairs and thus provide the exact quantitative comparison of the two approaches.Finally, in Sec. VI we discuss the main findings of the study and the future prospects. Relativistic units( (cid:126) = 1 , c = 1 ) are employed throughout the paper. II. APPROXIMATE ENHANCEMENT FACTOR
The external electromagnetic field is described by the following vector potential: A x ( t, z ) = F ( t ) (cid:18) E Ω sin Ω t + εω sin ωt cos k z z (cid:19) , A y = A z = 0 , (1)where k z = ω and F ( t ) is a smooth envelope function ( ≤ F ( t ) ≤ ). This external background canbe formed by two pairs of counterpropagating laser pulses with a large number of carrier cycles. Theenvelope F ( t ) is chosen in the following form: F ( t ) = sin (cid:2) ( πN − Ω | t | ) (cid:3) if π ( N − / Ω ≤ | t | < πN/ Ω , if | t | < π ( N − / Ω , otherwise . (2)Accordingly, the field (1) contains N cycles of the slow laser pulse including switching on and switch-ing off parts of half a cycle each and a flat plateau of N − cycles. The fast pulse governed by thesecond term in Eq. (1) contains ( ω/ Ω) N cycles. In what follows, we choose N = 10 , which guaran-tees that both pulses contain a large number of cycles, and therefore the external background can beapproximated by a sum of two standing waves. Since γ E (cid:28) , the strong pulse can be considered as aspatially uniform time-dependent field according to the first term in Eq. (1). We also choose E = 0 . E c , Ω = 0 . m , and γ ε = 10 . and vary ω . This leads to γ E = 0 . and γ c = 5 ( ω/m ) .Within the dipole approximation, the spatial dependence of the second term in Eq. (1) is neglectedby replacing cos k z with . This dependence can be partially taken into account by averaging the resultsobtained in the dipole approximation for the amplitude ε ( z ) = ε cos k z being considered at variouspositions z ∈ [0 , π/ Ω] . This approach will be referred to as the local dipole approximation.The method employed in this study is based on the well-known Furry picture formalism incor-porating vacuum instability [18]. The external field is assumed to act only within the time interval t in < t < t out . One can demonstrate that the number density of particles produced can be directly4xtracted from the two specific sets of solutions of the Dirac equation. The in ( out ) solutions are de-termined by their asymptotic behavior in the region t < t in ( t > t out ). After propagating a given out solution backwards in time, we decompose it in terms of the in solutions and obtain the number den-sity of the particles created in the corresponding out state. Since the external field (1) is periodic (andmonochromatic) in space at each time instant t , and it does not depend on x and y , a given momentum p z along the z axis can be changed only by an integer number of ω , while components p x and p y areconserved. This allows one to propagate only a discrete set of Fourier components for each one-particlesolution. This approach was described in detail in Refs. [19, 20]. As a result, our computations providethe number density of electrons (positrons) produced per unit volume: n ( p ) = (2 π ) V d N p ,s d p , (3)where p is the momentum of the particle and s = ± determines its spin state. Due to the symmetryof the external field, the spectra of particles produced are invariant under the reflection p → − p andindependent of s .The local number density n ( p ) considered at a given point p cannot yield a reliable quantitativemeasure of the dynamical assistance. In this perspective, the total number of pairs, i.e. the function n ( p ) integrated over p , seems to be the most suitable parameter. However, its evaluation becomes verytime consuming beyond the dipole approximation. For this reason, we study in more detail the numberdensity integrated over p y at p x = p z = 0 : n y = + ∞ (cid:90) n (0 , p y , p y . (4)The y direction is chosen since the magnetic field, which appears beyond the dipole approximation,is directed along the y axis and does not affect much the p y distribution computed for the spatiallyhomogeneous configuration. This was confirmed by studying an individual pulse as a uniform back-ground and a standing wave, respectively. It turns out that the momentum spectrum in the transversaldirection (either y or z ) in the former case is more similar to the spectrum along the y direction inthe latter case (this fact was also indicated in Ref. [17]). Moreover, the integral (4) converges fasterthan the analogous p x and p z integrals. We use the parameter n y as a guide for searching for the do-main of the dynamical assistance and then study the effect in more detail by calculating the density n ( p ) and the total number of particles created. We also introduce an approximate enhancement factor K = n y ( I + II ) / [ n y ( I ) + n y ( II )] where n y ( I ) and n y ( II ) denote the value of n y in the case of the indi-5idual strong and individual weak pulse, respectively, and n y ( I+II ) is associated with the combinationof the both pulses.Let us first discuss the results obtained within the spatially-uniform-field approximation. In Fig. 1we present the values of n y as a function of the fast-pulse frequency ω for the case of the individualpulses (I and II) and the combined pulses (I+II). Obviously, the particle yield provided by the strongslow pulse alone (horizontal line) does not depend on ω . On the other hand, the function n y ( II )( ω ) exhibits a quite nontrivial behavior. Its plot contains a set of large leaps. Each of them correspondsto the appearance of the next n -photon channel, and its position can be determined from the condition m ∗ = nω , where m ∗ is the effective laser-dressed electron mass. In the presence of a weak field( γ ε (cid:29) ), one has m ∗ ≈ m , so the leaps in Fig. 1 appear at ω/m = 2 , / , / ... The even leaps donot take place here. As was demonstrated in many numerical studies [10, 21–23], the dependence of n ( p = 0) on ω has a resonant structure which consists of sharp peaks at ω = 2 m ∗ /n for odd valuesof n , while the even- n resonances are forbidden. This can be understood if one notes that the angularmomentum of the e + e − pair equals zero for p = 0 , and thus its charge-conjugation parity is − . Sincethe C parity of the photon is also − , the pair can be generated only by absorbing an odd numberof photons. It turns out that this selection rule remains valid even if the transverse component of theparticle momentum differs from zero, i.e. p y and p z can be arbitrary, provided p x = 0 [10, 21, 23]. -12 -10 -8 -6 -4 -2
1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 n y ω / mn = 3 n = 1 n = 5 I+IIIII
Figure 1. The number density integrated over p y according to Eq. (4) as a function of the fast-pulse frequency ω in the case of the individual pulses (I and II) and in the presence of both pulses (I+II). When both pulses are present (line “I+II” in Fig. 1), the pair-production yield becomes substan-6ially larger. In Fig. 2 the approximate enhancement coefficient K is depicted versus ω . One observesthat the enhancement can reach several orders of magnitude, but for smaller values of ω , it is also quitesmall. Furthermore, in order to preserve the nonperturbative character of the pair-production process,one should also make sure that n y ( I ) (cid:29) n y ( II ) which holds true only in the region ω (cid:46) . m . Thismeans that the domain of the dynamically assisted Schwinger mechanism is . m (cid:46) ω (cid:46) . m . K ω / m Figure 2. The approximate enhancement factor defined by K = n y ( I + II ) / [ n y ( I ) + n y ( II )] as a function of thefast-pulse frequency ω . A special emphasis should be placed on the fact that the more physical characteristic of the pair-production process is the total number of pairs, unlike the rough estimate n y . One should at least verifythe findings of such an analysis by the complete calculations of the particle yield. This is especiallyimportant for the quantitative comparison of various field configurations and various computational ap-proaches. Besides, the oscillatory behavior of n y ( I+II ) (and accordingly K ) proves to be a nonphysicalartifact which does not show up in the total number of particles created. In Sec. V we will address thesepoints in more detail.In Fig. 3 we present the results obtained within the local dipole approximation. Although theyquantitatively differ from the dipole-approximation results for the case of the second pulse alone (II),the qualitative behavior as well as the results for the combined pulses remain almost the same. Theanalysis of the momentum distribution of particles produced also brings us to the conclusion that thelocal dipole approximation does not provide any significant findings besides those established in theusual dipole approximation. 7 -12 -10 -8 -6 -4 -2
1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 n y ω / mn = 3 n = 1 n = 5 I+II (DA)III (DA)I+II (LDA)II (LDA)
Figure 3. The values of n y as a function of ω calculated within the dipole approximation (DA) and the localdipole approximation (LDA) for the three configurations: I, II, and I+II. In Fig. 4 we display the dependences calculated beyond the dipole approximation, i.e. using theexpression (1). First, we observe that the dipole approximation considerably overestimates the particleyield, especially in the large- ω region. It is no surprise since the Keldysh parameter γ c increases withincreasing ω while the dipole approximation appears to be better justified for smaller γ c . Second,one observes a different multiphoton structure in the case of the individual weak pulse (II). Since thephotons now possess not only energy, but also momentum along the z axis (the projection equals + ω or − ω ), the “resonance” condition has a different form. Let q and p be the initial and final -momentaof a certain electronic state, respectively. The conservation laws read p = q + n + k + + n − k − , (5)where k ± = ( ω, , , ± ω ) t and n ± are integer numbers. Taking into account p x = p z = 0 and therelations p = q = m (again, the effective mass in a weak field approximately equals the electronmass), one obtains n + n − ω = p ( n + + n − ) , (6)which means that the particle yield should considerably increase with increasing ω at the points ω/m =( n + + n − ) / (2 n + n − ) . One can assume here that n + ≥ n − . The relation derived now allows one to ex-plain the structure of the graph II (bDA) depicted in Fig. 4. The numbers in the graph denote thecorresponding values of n + and n − . A quite similar analysis was performed in Ref. [22] in order to8 -12 -10 -8 -6 -4 -2
1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 n y ω / mn = 3 n = 1 n = 5 I+II (DA)III (DA)I+II (bDA)II (bDA)
Figure 4. The values of n y as a function of ω calculated within the dipole approximation (DA) and beyond it(bDA) for the three configurations: I, II, and I+II. explain the positions of the multiphoton resonances in the scenario involving two counterpropagatinghigh-intensity laser pulses. In Fig. 4 one notices that beyond the dipole approximation in the vicinity ω ≈ m no resonances occur. Another distinctive feature of the n y dependence consists in the presenceof the – (or – ) resonance which corresponds to an even total number of photons absorbed. Thisdemonstrates that the previously discussed selection rule can be violated beyond the dipole approxima-tion. We observe that the different dynamics taking place beyond the spatially-uniform-field approxi-mation leads to the substantially different patterns (this aspect will also be emphasized in the followingtwo sections). Besides, the more accurate results indicate that the enhancement due to the dynamicalassistance is, in fact, weaker. The latter point will also be discussed in Sec. V. In the next two sections,we study the momentum distribution of particles created for the specific choice of ω (and accordingly γ c ). III. MOMENTUM DISTRIBUTION WITHIN THE DIPOLE APPROXIMATION
In this section, we examine the momentum spectra of particles produced within the spatially-uniform-field approximation. The major part of the results is presented for ω = 0 . m .9 . Transversal direction As was pointed out above, within the dipole approximation, all of the directions in the y − z plane,i.e. perpendicular to the electric field, are equivalent. Without loss of generality, we set p z = p x = 0 and vary p y . In Fig. 5 we present the momentum distribution of particles created as a function of p y forthe three configurations: I, II, and I+II. The so-called shell structure revealed here was accounted for inRefs. [12, 13]. The peaks in Figs. 5(a) and 5(c) have the positions that satisfy E (0 , p y ,
0) = n Ω with E ( p ) being the effective energy in the external field: E ( p ) = 12 π π (cid:90) d x (cid:113) m + (cid:2) p x + γ − E sin x + γ − ε sin( ωx/ Ω) (cid:3) + p y + p z , (7)where the term with γ − ε should be omitted in the case I. The peaks in Figs. 5(a) and 5(c) correspond to -8 -6 -4 -2
0 0.2 0.4 0.6 0.8 1 1.2 n ( p ) p y / m I
651 653 655 657 (a) -14 -12 -10 -8
0 0.2 0.4 0.6 0.8 1 1.2 n ( p ) p y / m II (b) -8 -6 -4 -2
0 0.2 0.4 0.6 0.8 1 1.2 n ( p ) p y / m I+II
651 653 655 657 659 661663 (c)
Figure 5. The momentum distribution of particles created as a function of their transversal momentum p y ( p x = p z = 0 ) for the three external field configurations (I, II, and I+II) and ω = 0 . m . n = 651 , ... In the case II of the weak external background, the effective energy can be estimated as E ( p ) ≈ (cid:112) m + p , and the peak in Fig. 5(b) is related to the condition (cid:112) m + p y = nω with n = 5 ( p y ≈ . m ). Note that the number of photons is always odd in accordance with the selection rulediscussed in the previous section.The appearance of the fast pulse leads to lifting the momentum distribution evaluated in thecase of the slow pulse alone (I). Note that the presence of the weak pulse almost does not affect theexpression (7) since γ ε (cid:29) . Accordingly, the lifting effect is not accompanied by any shift of thepeaks.However, the e + e − pair can be now produced by absorbing n photons of the strong pulse and ˜ n photons of the weak one. Supposing that n corresponds to a certain peak in Fig. 5(a), in the presenceof both pulses, the combination of ˜ n photons of the weak pulse and n − ( ω/ Ω)˜ n photons of the strong10ulse corresponds to the same resonance. Since in our case ω/ Ω = 25 , the total number of photonsis n − n , and thus the additional photons of the weak field do not change its parity. This explainswhy the even resonances do not appear in Fig. 5(b). Nevertheless, this might as well not be the case. InFig. 6 we display the I+II spectrum for ω = 0 . m . Since ω/ Ω = 30 is now even, the even peaks now -8 -6 -4 -2
0 0.2 0.4 0.6 0.8 1 1.2 n ( p ) p y / m I+II Figure 6. The momentum distribution of particles created as a function of their transversal momentum p y ( p x = p z = 0 ) for the combination of the two pulses (I+II) and ω = 0 . m . The subscripts indicate the number ofphotons absorbed from the weak pulse. take place, although they do not appear in Fig. 5(a). The numbers in Fig. 6 denote the values of n (largenumbers) and ˜ n (subscripts). For each resonance, ˜ n can be increased by an arbitrary even number k ,provided n is decreased by k . B. Longitudinal direction
In Fig. 7 we display the spectrum of particles produced with p y = p z = 0 and various values of p x . By means of a similar analysis in terms of resonance conditions, one identifies in Figs. 7(a) and7(c) the peaks with n = 650 , ... The even peaks are now not forbidden. In Fig. 7(b) we observenow three sharp peaks which correspond to n = 5 , , and . As p x tends to the value of E ( p x , , almost reaches , which explains the rapid rise of the distribution function. However, at the very point p x = 0 the pair-production probability is again very low. This indicates that the even- n processes arenot permitted if the longitudinal momentum vanishes.Next we will investigate how the patterns discussed above change when one goes beyond thedipole approximation. 11 -8 -6 -4 -2
0 0.2 0.4 0.6 0.8 1 1.2 n ( p ) p x / m I (a)
651 652 653 654 -14 -12 -10 -8
0 0.5 1 1.5 2 n ( p ) p x / m II (b) -8 -6 -4 -2
0 0.2 0.4 0.6 0.8 1 1.2 n ( p ) p x / m I+II (c)
651 652 653 654
Figure 7. The momentum distribution of particles created as a function of their longitudinal momentum p x ( p y = p z = 0 ) for the three external field configurations (I, II, and I+II) and ω = 0 . m . IV. MOMENTUM DISTRIBUTION BEYOND THE DIPOLE APPROXIMATION
The field configuration (1) now consists of both the electric field along the x axis and the magneticfield along the y axis, so the cylindrical symmetry is not present now. In this section we analyze thespectra in the three spatial directions. A. Magnetic field direction y We now set p x = p z = 0 . The p y spectra contain again a set of pronounced peaks (see Fig. 8).However, their positions differ from those found in the dipole approximation. In order to describe -14 -12 -10 -8
0 0.2 0.4 0.6 0.8 1 1.2 n ( p ) p y / m II (a) -8 -6 -4 -2
0 0.2 0.4 0.6 0.8 1 n ( p ) p y / m I+II (b)
M M M M MA A A A A
DAbDA
Figure 8. The momentum distribution of particles created as a function of p y ( p x = p z = 0 ) for the fieldconfigurations II and I+II and ω = 0 . m . The solid lines represent the results obtained beyond the dipoleapproximation (bDA). In panel (b) the spectrum is compared with the DA results. this difference in the case of the individual weak pulse [Fig. 8(a)], we turn again to the conservation12aw (5). This expression can now be used to determine the resonance position p y for given n + , n − , and ω . Taking into account p = q = m , one obtains p y /m = (cid:115) (2 n + n − ) ( n + + n − ) (cid:18) ωm (cid:19) − . (8)This expression predicts a resonant peak at p y ≈ . m (resonance – or – ) which is clearly seenin Fig. 8(a). The other resonances are considerably suppressed as they appear in higher orders ofperturbation theory. The resonance – would correspond to p y = 0 , but it does not show up inFig. 8(a) since it has an even sum n + + n − . The analysis of the momentum distributions beyond thedipole approximation reveals that the processes with even photon numbers are suppressed only in thecase of the p y spectra.In the presence of the two pulses, the spectrum possesses a more complicated structure. Besidesthe peaks predicted within the dipole approximation, there exist also additional peaks in between. Theycan be accounted for by means of the conservation laws, which in this case take the following form: p = q + n + k + + n − k − + nk , (9)where k = (Ω , , , t is the -momentum of the strong pulse photon. Then we set p x = p z = 0 anduse the relations p = E ( p ) and q = E ( q ) . The resonance condition reads: E (0 , p y ,
0) + E (0 , p y , ( n − − n + ) ω ) = ( n + + n − ) ω + n Ω . (10)In order to evaluate the effective energy E ( p ) , we employ again the expression (7) even though we gobeyond the dipole approximation. The reason for this is that the second pulse contribution (the termwith γ − ε ) is always very small, so it does not need to be modified. Using Eqs. (7) and (10), we identifythe resonant peaks in Fig. 8(b). It turns out that the main peaks, which can also be found in Figs. 5(a)and 5(c), correspond to the processes with n + = n − . For each value of n + = n − , there is the sameseries of main peaks being enumerated by n (see Table I). The additional peaks in Fig. 8(b) emergeas the resonances with n + (cid:54) = n − . Note that Eq. (10) is symmetric with respect to the interchange n + ↔ n − , so we assume that n + ≥ n − . The resonance condition (10) formally allows the integers n + and n − to also be negative. This, however, in turn, leads to greater values of n , and thus such processesare strongly suppressed in comparison to those displayed in Table I and thus are not indicated here.Note that the spectrum contains only the peaks with an odd sum n + n + + n − . The resonances locatedby means of Eq. (10) and those found numerically coincide at least with . accuracy.13 eries n + – n − n Main peaks (M) – , , ... – , , ...... ...Additional peaks (A) – , , ... – , , ...... ... – , ... – , , ...... ...Table I. The series of the resonant peaks in Fig. 8(b). Each of the M series predicts the main peaks already foundwithin the dipole approximation while all of the A series reproduce the additional ones. If the additional peaks appear already in the DA spectrum, e.g. for ω = 0 . m (see Fig. 6),the results obtained beyond the DA reproduce the same resonant structure. If the DA distributioncontains only odd peaks, the number of resonances doubles beyond this approximation. Although theresonant structure appears mainly owing to the presence of the high-intensity slow field, the modifieddynamics of the weak pulse beyond the dipole approximation gives rise to the additional signatures inthe momentum spectrum. The weak fast pulse now not only lifts the momentum distribution but alsochanges its overall structure assisting the pair production process in the strong field. B. Propagation direction z When only the weak pulse is present, the spectrum contains peaks which can be located usingEq. (5) [see Fig. 9(a)]. However, the resonant values of p z are now not described by the right-handside of Eq. (8) since the p z component of the particle momentum can change due to the absorption ofphotons. Setting p x = p y = 0 and using Eq. (5), one obtains: n + n − ω = p ( n + + n − ) − p z ( n + − n − ) , (11)14here p = (cid:112) m + p z . This leads to p z = n + − n − ω ± n + + n − (cid:115) ω − m n + n − , (12)where n + and n − are positive. This expression allows one to identify the resonances in Fig. 9(a).Whereas in the dipole approximation one observes only one peak at p y = 0 . m , beyond the DA, the -14 -12 -10 -8
0 0.2 0.4 0.6 0.8 1 1.2 n ( p ) p z / m II (a) -8 -6 -4 -2
0 0.2 0.4 0.6 0.8 1 n ( p ) p z / m I+II (b) M M M M M A A A A A A A DAbDA
Figure 9. The momentum distribution of particles created as a function of p z ( p x = p y = 0 ) for the fieldconfigurations II and I+II ( ω = 0 . m ). spectrum becomes more complicated. The two high sharp peaks are associated with the – and – transitions, where, besides n + and n − , we indicate the sign in Eq. (12). Note that both the – and – − resonances correspond to p z = 0 . m because the square root in Eq. (12) vanishes. Moreover,according to Eq. (12), the positions of these “accidentally” degenerate resonances are very sensitivewith respect to the small changes of the fast-pulse frequency ω . It turns out that for ω = 0 . m the expression (12) predicts the – and – − peaks at p z = 0 . m and p z = 0 . m , respectively,which correspond to the peaks in Fig. 9(a). On the other hand, the positions of the other resonanceschange by less than . . Since the external field (1) is, in fact, not monochromatic, it is no accidentthat the – peak splits into two. One could also expect that the structure of the momentum distributionin Fig. 9(a) is quite unstable in the vicinity of p z = 0 . m . Our computations with different envelopefunctions F ( t ) confirm this point. The same holds true when one analyzes the peaks – and – − in the vicinity of p z = 0 . In contrast to the results obtained in the dipole approximation, the evenresonances are now allowed.To further clarify and illustrate the aspects discussed, we present the spectrum for ω = 0 . m (see Fig. 10). The – and – resonances split and form four distinct peaks (the – − peak has a15 -14 -12 -10 -8
0 0.2 0.4 0.6 0.8 1 1.2 n ( p ) p z / m II Figure 10. The momentum distribution of particles created as a function of p z ( p x = p y = 0 ) for the fieldconfiguration II and ω = 0 . m . negative value of p z ), while the positions of the peaks – , – , and – remain almost the same.In the presence of the two pulses [see Fig. 9(b)], the resonant structure can be deciphered as inthe previous subsection. Instead of Eq. (10), one has now E (0 , , p z ) + E (0 , , p z + ( n − − n + ) ω ) = ( n + + n − ) ω + n Ω . (13)This relation does not possess the symmetry n + ↔ n − and provides now a larger variety of resonances.The peaks in Fig. 9(b) are described in Table II. Each n – n + – n − resonance can also be represented asthe ( n − – ( n + + 1) – ( n − + 1) resonance similarly to what is shown in Table I. Although in Table IIthe total number of photons n + n + + n − is always odd, the even peaks can also emerge as was foundin our calculations for other values of ω . C. Electric field direction x The momentum distributions for p y = p z = 0 are depicted in Fig. 11. Their structure can beexplained almost in the same way as it was done for the p y spectrum. In the case of the fast pulse (II),the only difference is that the even resonances are now not forbidden, so the – resonance now leadsto a dramatic rise of the production probability in the vicinity of p x = 0 [Fig. 11(a)]. The peaks – ( – ) and – are less pronounced, for they correspond to higher orders of perturbation theory.The resonant structure in Fig. 11(b) is notably modified in comparison to the results obtained inthe dipole approximation. Apart from the previously found (main) resonances, there are again addi-16 eak n n + n − p z /m (D) p z /m (E) n ( p ) M
651 0 0 0 .
293 0 .
294 1 . × − M
653 0 0 0 .
511 0 .
512 1 . × − M
655 0 0 0 .
663 0 .
663 4 . × − M
657 0 0 0 .
788 0 .
789 9 . × − M
659 0 0 0 .
898 0 .
905 6 . × − A
626 1 0 0 .
096 0 .
093 7 . × − A
628 0 1 0 .
198 0 .
199 1 . × − A
626 1 0 0 .
404 0 .
404 8 . × − A
603 2 0 0 .
607 0 .
606 2 . × − A
628 1 0 0 .
698 0 .
699 1 . × − A
630 1 0 0 .
868 0 .
867 1 . × − A
605 2 0 0 .
946 0 .
948 5 . × − Table II. The list of the resonant peaks discovered beyond the dipole approximation in the p z spectrum [Fig. 9(b)].The p z values derived from Eq. (13) (D) match those found exactly (E). -14 -12 -10 -8
0 0.2 0.4 0.6 0.8 1 1.2 n ( p ) p x / m II (a) -8 -6 -4 -2
0 0.2 0.4 0.6 0.8 1 1.2 n ( p ) p x / m I+II (b)
M M M MA A A A A
DAbDA
Figure 11. The momentum distribution of particles created as a function of p x ( p y = p z = 0 ) for the fieldconfigurations II and I+II ( ω = 0 . m ). tional peaks. The resonance condition now reads: E ( p x , ,
0) + E ( p x , , ( n − − n + ) ω ) = ( n + + n − ) ω + n Ω . (14)By inspection of this equation, we find that the additional peaks correspond to the absorption of one17ast-pulse photon traveling in either direction and n = 627 , ... Alternatively, the resonances canappear as the − (or − ) processes with n = 577 , ... or in even higher orders in n + and n − .Performing the more accurate calculations beyond the uniform-field approximation, we estab-lish that the momentum spectra of particles have in fact a different structure. Nevertheless, the accu-rate quantitative comparison of the two approaches seems complicated. For instance, in Fig. 11(b),the number density evaluated beyond the dipole approximation can be much larger than the dipole-approximation values. In the next section, in order to gain a complete quantitative picture, we computethe total number of pairs. V. TOTAL NUMBER OF PAIRS
In this section we discuss finally the total particle yield and compare the results obtained withinthe uniform-field approximation and beyond it. In particular, we perform the numerical integration ofthe density function n ( p ) : N = 2 (cid:90) n ( p )d p , (15)where the factor appears due to the spin degeneracy. The total number of pairs N represents anextremely important characteristic which has a direct relation to the experiment and is a very usefulindicator in comparison of various computational approaches. On the other hand, the calculation of thisquantity is rather time consuming, especially beyond the dipole approximation where the cylindricalsymmetry is broken by the appearance of the magnetic field. Nevertheless, we carry out the calculationsfor various values of the fast-pulse frequency ω (see Table III). We also evaluate the full enhancementfactor K which is defined as K = N ( I+II ) / [ N ( I ) + N ( II )] , where the particle yield in the case ofthe individual strong pulse is independent of ω . It is seen now that the dipole approximation indeedoverestimates the amount of pairs. Our calculations confirm the other findings of Sec. II. Namely, oneobserves that the enhancement factor is almost insignificant for ω (cid:46) . m . Besides, the individualcontribution of the weak pulse becomes larger than that of the strong pulse for ω (cid:38) . m (DA) and ω (cid:38) . m (bDA). Within the interval . m (cid:46) ω (cid:46) . m , the enhancement factor in the dipoleapproximation can reach a value of about . However, according to the results obtained beyond thisapproximation, the total particle yields are about order of magnitude smaller.18 /m N (DA) N (bDA)II I+II K II I+II K . < − . × − . < − . × − . . < − . × − . < − . × − . .
40 3 . × − . × −
16 1 . × − . × − . .
45 1 . × − . × −
30 1 . × − . × − . .
50 9 . × − . × −
53 2 . × − . × − .
55 6 . × − . × −
94 4 . × − . × − .
60 4 . × − . × −
180 6 . × − . × − .
65 5 . × − . × −
320 8 . × − . × − .
70 1 . × − . × −
22 1 . × − . × − .
75 1 . × − . × −
39 2 . × − . × − .
80 2 . × − . × −
48 2 . × − . × − Table III. The total number of pairs N produced in the presence of the individual fast pulse (II) and both the fastand the strong pulse (I+II) for various values of the fast-pulse frequency ω . The results were obtained in the dipoleapproximation (DA) and beyond it (bDA). The values of N are displayed in units of λ − where λ is the reducedCompton wavelength of the electron ( λ ≈ fm). The particle yield N ( I ) amounts to . × − ( λ − ) . VI. DISCUSSION AND CONCLUSION
Within the present investigation, we examined the main characteristics of the dynamically assistedSchwinger effect going beyond the previously used dipole approximation. In particular, we took intoaccount the coordinate dependence of the fast weak pulse. It turned out that according to these moreprecise calculations, the patterns established in the homogeneous-field approximation cannot always beexpected to provide the real features of the pair-production process. Instead, our results suggest thatone has to take into account the spatial dependence of the external field in order to obtain more accuratequantitative and qualitative predictions.We summarize our main findings below: • The structure of the momentum spectra of particles created becomes significantly different be-yond the dipole approximation. The number of resonant peaks can double, and the momentum19istributions along all three directions x , y , and z become quite different. • Within the dipole approximation, the transversal momentum distribution never contains reso-nances corresponding to an even number of photons absorbed. However, beyond the dipoleapproximation, such peaks do appear unless the momentum along the propagation direction van-ishes ( p z = 0 ). • The momentum spectra obtained in the dipole approximation and beyond it exhibit differentquantitative behavior. While the latter mostly correspond to smaller values of the productionprobability, they can also have higher peaks. In order to accurately predict the quantitative char-acteristics of the spectra, one has to perform the calculations beyond the dipole approximation. • The enhancement of the particle yield due to the dynamical assistance, which is the essence ofthe processes considered in our study, turns out to be overestimated in the dipole approximation.The more precise calculations predict an enhancement factor that is several times smaller togetherwith particle yields that are about order of magnitude smaller.Although the external background considered in the present study incorporates the spatiotemporaldependence of the laser field, further steps towards studying more realistic configurations can also betaken. First, one can examine pulses of a finite size instead of two infinite pulses forming a standingwave. According to the recent studies [16, 17], the coordinate dependence of the envelope functioncan play a very important role, especially in the case of short laser pulses. Besides, in many studies ofvarious scenarios within the dipole approximation, it was demonstrated that the momentum spectra ofparticles and other characteristics can be very sensitive to changes in the shape of the laser pulse (see,e.g., Refs. [9, 15, 23–25]). The analysis of the pulse shape effects beyond the dipole approximation isan important issue to be investigated further.Finally, we stress that the spatial dependence of the external background in the context ofSchwinger pair production was considered so far in a very few studies [16, 17, 20, 26–28]. We ex-pect that multidimensional inhomogeneities should be significant for a much broader class of possiblescenarios. 20 CKNOWLEDGMENTS
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