Pulse shape effects on the electron-positron pair production in strong laser fields
PPulse shape effects on the electron-positronpair production in strong laser fields
I. A. Aleksandrov , , G. Plunien , and V. M. Shabaev Department of Physics, St. Petersburg State University,7/9 Universitetskaya nab., Saint Petersburg 199034, Russia ITMO University, Kronverkskii ave 49, Saint Petersburg 197101, Russia Institut f¨ur Theoretische Physik, TU Dresden,Mommsenstrasse 13, Dresden, D-01062, Germany
Abstract
The pair-production process in the presence of strong linearly polarized laser fields with a subcycle structure isconsidered. Laser pulses with different envelope shapes are examined by means of a nonperturbative numericalapproach. As generic cases, we analyze two different “flat” envelope shapes and two shapes without a plateau asa function of various characteristic parameters including the carrier-envelope phase, respectively. The resonantRabi oscillations, the momentum distribution of particles created, and the total number of pairs are studied. It isdemonstrated that all these characteristics are very sensitive to the pulse shape.
PACS numbers: 12.20.-m, 12.20.Ds, 11.15.Tk, 42.50.Xa a r X i v : . [ h e p - ph ] M a r . INTRODUCTION The phenomenon of particle-antiparticle production in the presence of strong external electromag-netic fields has been a focus of numerous theoretical investigations since the 1930s [1, 2]. However, ithas been observed only within the multiphoton regime [3], where perturbation theory can be success-fully employed. In the nonperturbative regime this (Schwinger) effect [4] has never been confirmed byexperiments [the characteristic critical field strength is E c = m c / ( | e | (cid:126) ) ≈ . × V/cm, where m and e are the electron mass and charge, respectively]. In this context various scenarios involving laserfields seem to be rather promising. Although a rigorous theoretical formalism based on quantization ofthe Dirac field in the presence of an external electromagnetic background was thoroughly elaborated(see [5] and references therein), many aspects of the e + e − pair-production phenomenon in strong laserfields still need to be elucidated. The present investigation aims at the numerical study of the pulseshape effects on the pair-creation process.Since the exact solutions of the Dirac equation are found analytically for a very few configura-tions of the external background, a detailed analysis can only be provided by means of correspondingnumerical calculations. We perform a numerical integration of the Dirac equation taking into accountthe interaction with the external laser field (i.e. nonperturbatively). The pair-production probabilitiesare evaluated with the aid of the general formalism described in Ref. [5] (see also Refs. [6–9]).If one introduces the adiabaticity parameter ξ = | eE | / ( mcω ) , where E is the peak electric fieldstrength and ω is the carrier frequency, then the tunnelling and multiphoton regimes can be characterizedby ξ (cid:29) and ξ (cid:28) , respectively [10] (the parameter ξ is the inverse of the Keldysh parameter γ [10]).In the present study we focus on the intermediate case ξ = 1 , when the process exhibits a multiphotonbehavior and, at the same time, reflects a nonperturbative nature.A number of previous studies [15–20, 24–27, 29] indicated the importance of the pulse shapeand carrier-envelope phase (CEP) effects on the pair-production process in various conditions. Thismotivated us to further scrutinize these effects regarding the case of a strong individual laser pulsehaving a temporal dependence (the field is assumed to be spatially homogeneous). Such a configurationapproximates the experimental scenario of two counterpropagating linearly polarized laser pulses.We consider two types of temporal profiles. The type-I pulses are characterized by a wide plateau,where the pulse envelope equals unity. The pulses of the type II have no plateau and their envelopesimmediately start to diminish once they reach the maximal value of unity. One of the type-I pulseshapes was examined in detail in Ref. [14], however, in this analysis only the pulse duration was taken2s varying parameter, while the effects of the envelope shape and CEP were not discussed. We shallstudy two different profiles of the type I being characterized by three independent parameters: thepulse duration, the duration of switching on and off (rapidity of the pulse switching), and CEP. It willbe demonstrated that each of these parameters may have a notable impact on the patterns establishedpreviously. Besides, we examine two different profiles of type II: cos and Sauter-like pulses. In thiscase one can separately vary the pulse duration and CEP. It will be shown that the main features of thepair-production process in the case of the type-II pulse shapes considerably differ from those for thetype-I case.In particular, the comparison of these four pulse shapes allows one to reveal several importantfeatures of the resonant Rabi pattern, momentum spectrum of particles created, and total number ofpairs as a function of the laser frequency and pulse duration. Some of these properties were alreadydiscussed in the literature [16, 17]. In Ref. [17] a number of characteristic signatures in the spectrum ofparticles produced were found (only Gaussian envelope was analyzed). In Ref. [16] various Gaussianand super-Gaussian envelope shapes were examined. We extend these studies by providing results basedon an independent numerical approach. We consider the present investigation of the pulse shape effectsto be particularly comprehensive for two reasons. First, we provide the analysis of all the importantcharacteristics of the pair-production process (listed above). Second, we study a broad class of possiblepulse shapes.The paper is organized as follows. In Sec. II we describe the temporal profiles to be analyzed.In Secs. III and IV we discuss the pulse shape effects for type-I and type-II envelopes, respectively.In Sec. V we present results for the total number of pairs produced for various envelope shapes, laserfrequencies, and pulse durations, respectively. Sec. VI contains the analysis of the CEP effects regardingboth type-I and type-II pulses. Finally, in Sec. VII we summarize our results and provide a discussion.Relativistic units ( (cid:126) = 1 , c = 1 ) together with the Heaviside charge unit ( α = e / π , e < )are employed throughout the paper. Accordingly, the Schwinger critical field strength is | e | E c = m ,one relativistic unit of length is (cid:126) / ( mc ) ≈ . × − m, and the unit of time is (cid:126) / ( mc ) ≈ . × − s. 3 I. ENVELOPE SHAPES TO BE EXAMINED
We assume that the external electric field has the following form: E z ( t ) = E ( t ) = E F ( t ) sin( ωt + ϕ ) , E x = E y = 0 , (1)where E is the peak electric field strength, F ( t ) is a smooth envelope function ( ≤ F ( t ) ≤ ), ω isthe carrier frequency, and ϕ describes CEP. First, we consider ϕ = 0 (the CEP effects will be mainlydiscussed in Sec. VI). Within the present study we analyze two envelope profiles having an extendedplateau region (type I) and two forms of an envelope without a plateau (type II). Namely, they have thefollowing expressions: F Ia ( t ) = sin π ( − T/ T − t )2∆ T if − T / − ∆ T ≤ t ≤ − T / , if − T / ≤ t ≤ T / , sin π ( T/ T − t )2∆ T if T / ≤ t ≤ T /
T , otherwise , (2) F Ib ( t ) = 12 (cid:20) tanh 6( t + T /
T / T − tanh 6( t − T / − ∆ T / T (cid:21) , (3) F IIa ( t ) = cos (cid:18) π tT (cid:19) θ ( T / − | t | ) , (4) F IIb ( t ) = 1cosh (8 t/T ) . (5)The external laser pulse contains N c = ωT / (2 π ) cycles of the carrier. For the type-I pulses the switch-ing part contains N s = ω ∆ T /π half cycles. The factors and in Eqs. (3) and (5), respectively,guarantee that the values of ∆ T and T correspond to the duration of the switching part and the totalpulse duration, respectively (in fact, they can slightly differ from and ). In order to make sure that ϕ is the only parameter responsible for the CEP effects, we always choose N c so that N c + N s = 2 k ( k ∈ N ). This leads to sin( ωt + ϕ ) (cid:12)(cid:12)(cid:12) t = t in = − T/ − ∆ T = sin ϕ . (6)Thus, ϕ is the carrier phase at the initial time instant t in . In fact, this specific choice of N c does notimpose any significant restrictions on our computations, while the symmetry of the envelope F ( t ) becomes useful for the necessary derivations. In Fig. 1(a) we display the Ia and Ib envelopes for N c = 2 . and two different values of N s . In Fig. 1(b) the profiles IIa and IIb are depicted for various N c . Note that these envelopes possess the following properties:4 F ( t ) ω t (a) Ia, N s = 2.0Ia, N s = 4.0Ib, N s = 2.0Ib, N s = 4.0 F ( t ) ω t (b) IIa, N c = 4.0IIa, N c = 8.0IIb, N c = 8.0IIb, N c = 16.0 Figure 1. (a) Type-I envelope shapes for N c = 2 . and two different values of N s , (b) Type-II envelope shapesfor various N c . + ∞ (cid:90) −∞ F Ia ( t )d t = + ∞ (cid:90) −∞ F Ib ( t )d t = T + ∆ T, (7) + ∞ (cid:90) −∞ F IIa ( t )d t = 2 + ∞ (cid:90) −∞ F IIb ( t )d t = T / . (8)In the next section the type-I envelope shapes of the external pulse will be analyzed. III. TYPE-I ENVELOPE SHAPES
Solving the Dirac equation in the corresponding external potential on a temporal grid, we evaluateall the necessary pair-production probabilities using the formalism of the in and out complete setsof time-dependent solutions [5] (see Appendix A). All of the envelope shapes described above areanalyzed with the help of the same numerical procedures.Let us introduce a number density of electrons produced per unit volume in momentum space: n ( p ) = (2 π ) V d N (el) p ,r d p , (9)where V is the volume of the system and r = ± denotes the electron spin state (cid:2) in fact, n ( p ) isindependent of r (cid:3) . First, we consider the number density of electrons created at rest n ( p = 0) . For thecase of the type-I envelope shapes this quantity oscillates as a function of N c (see, e.g., Refs. [11–14]).We start with the analysis of the corresponding Rabi oscillations for ξ = 1 . . As will be shown, thereare several pronounced differences between the Ia and Ib pulse shapes regarding this aspect.5 . Resonant Rabi oscillations First, we evaluate the maximal values of the function n ( p = 0) (as a function of N c in the range < N c ≤ ) for various values of ω and N s . In Fig. 2 the corresponding dependences are presentedfor both Ia and Ib envelope shapes. For a given N s one observes a resonant pattern which consists ofa set of equidistant peaks (with respect to the reciprocal ω axis). Each resonance relates to a certainnumber n of laser photons required for pair production ( ω n = 2 m ∗ /n , where n = 1 , , ... and m ∗ isthe laser-dressed effective mass of electrons which is almost independent of ω [14]). In the case of theIa pulse the even resonances disappear for odd values of N s . In the Ib case they are strongly suppressedfor all N s and the picture is almost independent of this parameter. If N s (cid:46) . , the positions of theresonant peaks in the Ia case depend on N s , which means that the effective mass m ∗ depends on thisparameter. Figure 2. Resonant picture as a function of N s for the Ia (left) and Ib (right) envelope shapes. The integer numbersin the white boxes indicate the numbers n for the corresponding n -photon peaks. In order to gain a better understanding of the features described above, we present the correspond-ing resonant Rabi oscillations for the n = 7 and n = 8 resonances and various N s (see Fig. 3). Whilefor the odd resonances in the region N s (cid:38) . the oscillations for the Ia and Ib shapes are identical[Fig. 3(a)], for smaller N s the Rabi frequencies are different [Fig. 3(b)]. Moreover, when one considersan even resonance, this difference becomes dramatic. The Rabi frequency in the case of the Ib envelopeis extremely low [Fig. 3(c)], which does not allow the pair-production probability to reach its maximumdue to the restriction N c ≤ . In fact, for the specific parameters employed in Fig. 3(c) this maximalvalue equals 0.00883 and appears at N c ≈ . In the Ia case the corresponding Rabi frequencies aremuch greater and the maxima equal unity. Thus, all the resonances in Fig. 2(left) are clearly seen.6 (a) Ia, N s = 3.0, n = 7, ω = 0.34892Ib, N s = 3.0, n = 7, ω = 0.34894 n ( p = ) (b) Ia, N s = 1.5, n = 7, ω = 0.36275Ib, N s = 1.5, n = 7, ω = 0.34890 N c (c) Ia, N s = 4.0, n = 8, ω = 0.30532Ib, N s = 4.0, n = 8, ω = 0.30503 ( × Figure 3. Resonant Rabi oscillations for: (a) n = 7 resonance for N s = 3 . and both Ia and Ib pulse shapes,(b) n = 7 resonance for N s = 3 . , (c) n = 8 resonance for N s = 4 . (the values of the Ib case are multiplied by ). To our knowledge, the even peaks of a unit height have not been observed within the previousnumerical studies. This can be explained by the charge-conjugation symmetry [14, 22]. For p = 0 the electron-positron pair must have an odd C parity and, therefore, it can be produced only via anabsorption of odd number of photons (the C parity of a single photon is also odd). However, in the caseof the Ia pulse shape this explanation fails. We suppose that the reason for this is that the external fieldis not monochromatic so it contains photons with various energy k (cid:54) = ω . For instance, in the presenceof two monochromatic fields having the frequencies ω and ω (cid:48) (cid:28) ω, m ∗ the resonant ( n + 1) -photonprocess which relates to nω + ω (cid:48) = 2 m ∗ is not prohibited for even n . Varying the frequency ω , onefinds additional “even” peaks at ω n ≈ m ∗ / , m ∗ / , and so on. This issue will be discussed in moredetail in Sec. VI. B. Momentum distribution
We now turn to the discussion of the spectra of electrons created, i.e. we will consider the function n ( p ) for various p . In Fig. 4 the momentum distribution of electrons created is demonstrated for N c + N s = 20 . (the pulse duration is fixed), ω = 0 . m , two types of the envelope shape (Ia and Ib),and two different values of N s . The electron produced has the longitudinal component of its momentum7 z and the transverse component p ⊥ . We vary both of them. In Fig. 4 we observe a characteristic ring Figure 4. Momentum distribution of electrons created for N c + N s = 20 . , ω = 0 . m , and both Ia and Ib pulseshapes for N s = 4 . (left) and N s = 6 . (right). The pulse duration is T ≈ m − (left) and T ≈ m − (right). structure that can be accounted for by the same resonant condition m ∗ = nω where m ∗ should bereplaced by the laser-dressed effective energy which now depends on p [14]. Each ring correspondsto a certain integer number n . Note that p z is the z component of the gauge invariant momentum andthe center of the momentum distribution does not necessarily coincide with the origin p = 0 . We pointout that the general structure of the momentum distribution is the same for all type-I pulses, whereas itcould be different quantitatively. The pulse shape starts to play a more important role when the numberof cycles in the pulse becomes smaller. For instance, in Fig. 5 we employ N c = 2 . and N s = 4 . .Changing these parameters, one can produce a wide variety of distributions (for the case of a Gaussianenvelope shape this was studied in Ref. [17]).In this section we discussed various features of the pair-creation phenomenon. However, fromthe experimental viewpoint the most important characteristic of the process is the total number of pairscreated. This will be discussed for both type-I and type-II pulse shapes in Sec. V.8 igure 5. Momentum distribution of electrons created for N c = 2 . , N s = 4 . , ω = 0 . m ( T = ∆ T ≈ . m − ), and both Ia (left) and Ib (right) envelope shapes. IV. TYPE-II ENVELOPE SHAPES
In contrast to the type-I envelope pulses, the type-II shapes do not have a plateau region and,therefore, the pulse duration and the way how it is switched on/off are now related. As a result, the Rabioscillations are strongly modified. For the case of the type-II envelopes only odd resonances appear.In Fig. 6 the corresponding Rabi oscillations are presented for various parameters of the external laserpulse. In contrast to the case of the type-I shapes, the pair-production probability now does not reachunity and its maximal value decreases with decreasing ω (i.e., for greater n ). Moreover, the maximumfor the IIa shape is much larger than that for the IIb shape (e.g., for the n = 7 resonance they are . and . , respectively). Besides, we note that the resonant Rabi frequencies in the type-II case aremuch smaller than those for the type-I pulses. In Fig. 6(c) we compare the Rabi oscillations for the IIaand IIb cases and display the latter rescaling the bottom axis by a factor of (i.e., the label N c = 400 corresponds to N c = 800 for the red solid line). We observe that this naive transformation, motivatedby the property (8), does not make the dependences in Fig. 6(c) coincide, which indicates a substantialdifference between the IIa and IIb envelope shapes.In Fig. 7 we display the momentum distribution of electrons created for the IIa pulse with N c =40 . and the IIb pulse with N c = 80 . . These values of N c are chosen in order to keep the parameter T = (cid:82) F ( t )d t the same [see Eq. (8)]. The momentum distribution now considerably differs from thatof the envelope shapes of type I (see Fig. 4). In addition to the ring structure of the maxima, it containsa number of white rings regarding to extremely low probabilities. The distributions for the IIa and IIbenvelope shapes are similar, provided the parameter T is fixed. If one considers these two pulses of thesame duration T , the results will be different (in the next section this will be discussed with respect to9 (a) IIa, n = 7, ω = 0.34806IIb, n = 7, ω = 0.34732 n ( p = ) (b) IIa, n = 9, ω = 0.27033IIb, n = 9, ω = 0.26986 N c (c) IIa, n = 7, ω = 0.34806IIb, n = 7, ω = 0.34732, rescaled Figure 6. Resonant Rabi oscillations for: (a) n = 7 resonance for both IIa and IIb envelope shapes, (b) n = 9 resonance, (c) n = 7 resonance (for the IIb curve the bottom axis scale is chosen so that N c ∈ [0 , ).Figure 7. Momentum distribution of electrons created for ω = 0 . m and both IIa ( N c = 40 . , left) and IIb( N c = 80 . , right) envelope shapes. The pulse duration is T ≈ m − (left) and T ≈ m − (right). the total number of pairs).Again, we make the analogous comparison using smaller values of N c (see Fig. 8). The role of thepulse shape is crucial now. In the IIb case the pair-production probability oscillates with p z and p ⊥ muchfaster than in the IIa case. This indicates that the momentum spectrum of electrons created is extremelysensitive to the envelope shape so the distributions presented here and those from Ref. [17] (for theGaussian pulse shape) are certain to significantly change if one chooses another envelope function (theCEP effects will be discussed in Sec. VI). 10 igure 8. Momentum distribution of electrons created for ω = 0 . m and both IIa ( N c = 2 . , left) and IIb( N c = 4 . , right) envelope shapes. The pulse duration is T ≈ . m − (left) and T ≈ . m − (right). V. TOTAL NUMBER OF PAIRS
In order to evaluate the total number of pairs created, one has to perform an integration over themomentum p : N = 2 (cid:90) d p n ( p ) = 4 π + ∞ (cid:90) −∞ d p z + ∞ (cid:90) d p ⊥ p ⊥ n ( p z , p ⊥ ) , (10)where the factor arises due to the spin projection degeneracy. First, we fix (cid:82) F ( t )d t = T and E = 0 . E c (the value of ξ now varies) for all possible pulse shapes. In Fig. 9 we compare the totalnumbers of pairs for three different shapes of type I. All the curves have large leaps at ω n ≈ m/n ♠ ❛ ① ♠ ❛ ① ❝✶ -6 -5 -4 -3 -2 -1 N ω / m Ia, N s = 2.0Ia, N s = 4.0Ib, N s = 4.0 -3 -2 -1
1 1.4 1.8 2.2 ✶ Figure 9. Total number of pairs as a function of ω for the type-I pulse shapes with different values of N s ( T =50 π ). n -photon channels [16]. While in the region ω ≥ m thesethree curves coincide, in the plateau regions (e.g., . m ≤ ω ≤ . m ) they are different and the totalnumber of pairs is greater for the case of a rapid switching of the external pulse. The oscillations arisesince this time we allow the sum N c + N s to be odd and, therefore, CEP has two possible values — ϕ = 0 and ϕ = π . When N s becomes larger ( N s (cid:38) ), these oscillations disappear. The role of CEPwill be discussed in more detail in the next section.In Fig. 10 we examine the type-II envelope shapes having the same duration T [Fig. 10(left)] andthe same T [Fig. 10(right)]. Since for a given T the parameter T is greater for the type-I pulses (the -6 -5 -4 -3 -2 -1 N ω / m Ia, N s = 4.0, T = 50 π IIa, T = 25 π IIb, T = 12.5 π -6 -5 -4 -3 -2 -1 N ω / m Ia, N s = 4.0, T = 50 π IIa, T = 50 π IIb, T = 50 π Figure 10. Total number of pairs as a function of ω for: (left) type-I and type-II pulses with the same duration,(right) type-I and type-II pulses with T = 50 π . choice of N s , i.e. ∆ T , is not very important here), it is no accident that in this case the number ofparticles produced is larger for almost every ω [Fig. 10(left)]. Nevertheless, in the region . m ≤ ω ≤ . m we observe a quite nontrivial behavior. One may expect that these dependences calculated for agiven value of T should coincide. However, according to Fig. 10(right), the envelopes of the type I foreach value of ω have much more favorable shape. This is in accordance with Ref. [16] where it wasdemonstrated that the “flat” super-Gaussian pulse shape is more advantageous than the Gaussian one.In order to maximize the number of pairs created, it is more reasonable to generate a short pulse withrapid switching parts rather than a slowly varying pulse of long duration.12 I. CEP EFFECTS
In this section we perform the calculations for arbitrary ϕ . Following the same scheme, we startwith the discussion of the resonant structure for ξ = 1 . . It turns out that the resonant pattern in thecase of the Ib envelope shape remains the same no matter which value of CEP is chosen. Nevertheless,in the Ia case for N s (cid:46) . the peaks are shifted differently for different ϕ . In Fig. 11 this is depictedfor ϕ = π/ (left) and ϕ = π/ (right). When N s becomes greater ( N s (cid:38) . ), the positions of Figure 11. Resonant picture for the Ia envelope shape and ϕ = π/ (left) and ϕ = π/ (right). the resonances are similar for all ϕ . However, we observe that the values of N s for which the evenresonances disappear depend on ϕ . In addition to those given by N s = 2 k + 1 ( k ∈ N ) we also have N s = 2 k + 1 − ϕ/π . This can be accounted for by the Fourier analysis of the external time-dependentpulses. Let us introduce the function χ ( k ) = (cid:112) f ( k ) + g ( k ) , (11)where the functions f ( k ) = t out (cid:90) t in d t cos k t (cid:2) A z ( t ) − A z ( t out ) (cid:3) , (12) g ( k ) = t out (cid:90) t in d t sin k t (cid:2) A z ( t ) − A z ( t out ) (cid:3) (13)represent the real and imaginary parts of the Fourier transform of the vector potential, respectively.The function χ ( k ) is gauge invariant and continuous (we discard the singular contributions that arisefrom the nonvanishing parts outside the interval [ t in , t out ] ). It turns out that this function has two13ronounced peaks at k = ± ω and may also be large at k = 0 for some special parameters of theexternal pulse. At other points k ∈ R it is several orders of magnitude smaller. In Fig. 12 we presentthe ratio χ (0) /χ ( ω ) as a function of N s for the type-I pulses with different values of CEP. One observes χ ( ) / χ ( ω ) N s Ia, ϕ = 0Ia, ϕ = π /4Ia, ϕ = π /2Ib, ϕ = 0 Figure 12. The relative contribution of the Fourier transform of the vector potential at zero energy k = 0 for thetype-I envelopes with various CEP as a function of N s . that the “zero-energy” contribution for the Ia case is much larger than that for the Ib envelope shape.However, for several values of N s it also becomes insignificant (we observe relatively sharp minima at N s = 2 k + 1 and N s = 2 k + 1 − ϕ/π , where k ∈ N ). These values exactly correspond to those forwhich the resonant peaks vanish [see Figs. 2(left) and 11]. Thus, we conclude that the appearance ofthe even resonances is due to the nonmonochromaticity of the external field which effectively violatesthe selection rule discussed previously. Since the zero-energy mode in the Ib case is small for all valuesof N s , the even resonances in Fig. 2(right) are always strongly suppressed (the same holds also true forthe type-II envelopes).Another important feature relates to the momentum spectrum of electrons created. As was statedpreviously, the pulse shape does not play any important role when the number of cycles N c is large.A similar conclusion can be drawn regarding CEP. However, for small N c the CEP effects can beeasily revealed. In Fig. 13 we present the momentum distributions for the Ia and IIa pulse shapeswith ϕ = π/ . Comparing these results with those displayed in Figs. 5 and 8, we note that CEP maysubstantially alter the momentum spectrum, especially, in the case of the type-II envelopes. In Ref. [17]it was shown that when the electric pulse with the Gaussian envelope is antisymmetric, i.e. it obeys14 igure 13. Momentum distribution of electrons created for the Ia pulse with N c = 2 . and N s = 4 . (left) andthe IIa pulse with N c = 2 . (right). For both figures ϕ = π/ and ω = 0 . m . E ( − t ) = − E ( t ) [according to Eq. (1), it corresponds to ϕ = 0 ], then the pair-production probabilityoscillates and vanishes at its minima (it is clearly seen in Fig. 8 for the type-II pulses). When the fieldbecomes symmetric ( ϕ = π/ ), these oscillations disappear [17] as can be observed in Fig. 13(right).However, the pulses of the type I do not possess this feature.Finally, we note that we have not found any significant CEP effects on the total number of pairs.Whereas this parameter may be responsible for various very interesting signatures regarding differentialcharacteristics of the pair-production process, it appears not to have a notable impact on the total numberof particles created. VII. DISCUSSION AND CONCLUSION
Within the present study we examined four different pulse shapes for their various parameters. Inorder to provide a comprehensive analysis, we considered several characteristics of the pair-productionprocess: the resonant structure, Rabi oscillations, momentum spectrum of particles created, and totalnumber of pairs. Besides, we studied two different types of the envelope shape (two specific formsfor each type) and the role of CEP, which allowed us to conduct an extensive study. We believe thatthe present investigation considerably expands the previous findings and further clarifies the role of thepulse shape.Our main results can be summarized as follows: • The resonant pattern for p = 0 strongly depends on the shape of the temporal profile. In thecase of the Ib envelopes and the type-II pulses only the odd resonances occur and the resonantstructure is almost independent of N s , while, in contrast to the results of previous studies, in the15a case one can observe the even resonances and the picture is different for different N s in theregion N s (cid:46) . . Moreover, it was found that the peaks with even n disappear for a discrete setof specific values of N s that depend on CEP. This nontrivial property manifests itself due to thefact that the laser pulse is not monochromatic. The interference among modes having differentfrequencies could considerably change the resonant picture. • The Rabi frequencies may be very sensitive to the pulse shape, especially, in the case of thetype-II envelopes. Besides, it was found that the Rabi frequencies are much larger for the type-Icase. • The resonant structure of the momentum distribution of particles created is different for the type-Iand type-II envelope shapes. Nevertheless, when the external pulse contains a large number ofcycles, the distribution is almost independent of the form of the type-I switching function andof the type-II pulse shape. However, for the case of a small number of cycles all the parametersdefining the pulse shape (including CEP) have a great influence on the momentum spectrum. • The total number of pairs produced strongly depends on the envelope shape. For the type-I pulsesit is much greater than for the type-II case. Envelopes with rapid switching functions are moreadvantageous than slowly-varying pulse profiles.In our calculations we employed large values of ω (and, therefore, large E ) for two reasons. First,this allowed us to study the resonant pattern of the n -photon peaks as well as the resonant Rabi oscilla-tions. Second, large values of the laser frequency help one to avoid very time-consuming computations,which is particularly important for the calculations of the total number of pairs created. Although thecorresponding parameters of laser fields are not currently available in the experiment, we suppose thatmost features revealed within the present study (listed above) are likely to be valid for weaker fieldsand lower frequencies as well.In addition, we point out that the pulse shape effects may be significant in the context of dynami-cally assistant pair production [21] (see also Refs. [22, 23] and references therein). This was examinedin Refs. [24–27]. Further analysis of these effects may be a very interesting and important prospect forfuture studies.Finally, one has to note that spatial variations of the external field could play a significant role [28–32]. Although taking them into account appears to be a very difficult task, one of the most important16ubjects of our future investigations is the application of the technique described in Ref. [32] to thecorresponding multidimensional scenarios involving strong laser fields. ACKNOWLEDGMENTS
This investigation was supported by RFBR (Grant No. 16-02-00334) and by Saint PetersburgState University (SPbU) (Grants No. 11.65.41.2017, No. 11.42.987.2016, No. 11.42.939.2016, andNo. 11.38.237.2015). I. A. A. acknowledges the support from the German-Russian InterdisciplinaryScience Center (G-RISC) funded by the German Federal Foreign Office via the German AcademicExchange Service (DAAD), from TU Dresden (DAAD-Programm Ostpartnerschaften), and from theSupercomputing Center of Lomonosov Moscow State University.
Appendix A: Numerical approach
We assume that the external field vanishes for t ≤ t in and for t ≥ t out (cid:2) e.g., for the Ia envelopedefined by Eq. (2), we choose t in = − t out = − T / − ∆ T (cid:3) . Following the rigorous QFT formalismdescribed in Ref. [5], we solve the Dirac equation in order to construct two orthonormal and completesets of time-dependent solutions with given asymptotic behavior: ζ Ψ n ( t in , x ) = ζ Ψ (0) n ( x ) , ζ Ψ n ( t out , x ) = ζ Ψ (0) n ( x ) , (A1)where ζ Ψ (0) n ( x ) and ζ Ψ (0) n ( x ) are the eigenfunctions of the Dirac Hamiltonian considered at t = t in and t = t out , respectively, and ζ is the sign of the corresponding (energy) eigenvalues. It turns out that thesesets of solutions contain all the information about the quantities to be analyzed within the present study.For instance, the mean number of electrons (positrons) created with the given quantum numbers m canbe evaluated via [5] n − m = (cid:88) n G ( + | − ) mn G ( − | + ) nm , (A2) n + m = (cid:88) n G ( − | + ) mn G ( + | − ) nm , (A3)where the G matrices can be defined as the following inner products G ( ζ | κ ) nm = ( ζ Ψ n , κ Ψ m ) , (A4) G ( ζ | κ ) nm = ( ζ Ψ n , κ Ψ m ) . (A5)17he Dirac equation in the presence of an external background reads (cid:0) γ µ (cid:2) i∂ µ − eA µ ( t, x ) (cid:3) − m (cid:1) Ψ( t, x ) = 0 . (A6)We use the conventional substitution Ψ = (cid:2) γ µ (cid:0) i∂ µ − eA µ (cid:1) + m (cid:3) ψ which yields (see, e.g., Refs. [5, 6]) (cid:0) [ i∂ − eA ] − m − ie γ µ γ ν F µν (cid:1) ψ ( t, x ) = 0 , (A7)where F µν = ∂ µ A ν − ∂ ν A µ . Within DA one can choose A = 0 , A x = A y = 0 , A z ( t ) = − t (cid:90) E ( t (cid:48) )d t (cid:48) . (A8)Hence, Eq. (A7) takes the form (cid:2) ∂ t − ∆ + 2 ieA z ( t ) ∂ z + e A z ( t ) + m + ieγ γ E ( t ) (cid:3) ψ ( t, x ) = 0 , (A9)which can be reduced to a scalar equation if one represents the function ψ ( t, x ) as ψ n ( t, x ) = ψ p ,s,r ( t, x ) = e i px v s,r ϕ p ,s,r ( t ) , (A10)where v s,r ( s = ± , r = ± ) are the constant orthonormal eigenvectors of the matrix γ γ = α . Thescalar function ϕ p ,s,r ( t ) is a solution of the following equation: (cid:2) ∂ t + ( p z − eA z ( t )) + p ⊥ + m + iesE ( t ) (cid:3) ϕ p ,s,r ( t ) = 0 , p = ( p ⊥ , p z ) . (A11)By solving this ordinary differential equation, we obtain the matrix elements of the G matrices andemploy Eqs. (A2) and (A3) in order to calculate the spectrum of particles produced. Since the gen-eralized momentum is conserved, the matrices are diagonal: G ( κ | ζ ) mn = δ mn g ( κ | ζ ) n . Note that dueto this relation, in the region t ≤ t in ( t ≥ t out ) each out (in) solution with the sign ζ represents as alinear combination of the only two in (out) solutions with ζ and − ζ , respectively. The correspondingcoefficients contain the necessary matrix elements and can be evaluated by using the function ϕ and itsderivative at the point t = t in ( t = t out ). Thus, it is not necessary to perform the integration accordingto Eqs. (A4) and (A5). [1] F. Sauter, Z. Phys. , 742 (1931).
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