Asymmetric pulse effects on pair production in polarized electric fields
AAsymmetric pulse e ff ects on pair production in polarized electric fields Obulkasim Olugh,
1, 2
Zi-Liang Li, and Bai-Song Xie ∗
1, 4 Key Laboratory of Beam Technology of the Ministry of Education,and College of Nuclear Science and Technology,Beijing Normal University, Beijing 100875, China Xinjiang Police College, Urumqi 830011, China School of Science, China University of Mining and Technology, Beijing 100083, China Beijing Radiation Center, Beijing 100875, China (Dated: July 8, 2020)
Abstract
Using the Dirac-Heisenberg-Wigner (DHW) formalism, e ff ects of asymmetric pulse shape on the gener-ation of electron-positron pairs in three typical polarized fields, i.e., the linear, middle elliptical and circularones, are investigated. Two kinds of asymmetries for the falling pulse length, one is compressed and theother is elongated, are studied. It is found that the interference e ff ect disappears with the compression ofthe pulse length, and finally the peak value of the momentum spectrum is concentrated in the center of themomentum space. For the opposite situation by extending the falling pulse length, a multi-ring structurewithout interference appears in the momentum spectrum. Research results exhibit that the momentum spec-trum is very sensitive to the asymmetry of the pulse as well as to the polarization of the fields. It is also foundthat the number density of electron-positron pairs under di ff erent polarizations is sensitive to the asymmetryof electric field. For the compressed falling pulse, the number density can be enhanced significantly over2 orders of magnitude. These results could be useful in planning high power or / and high-intensity laserexperiments. PACS numbers: 12.20.Ds, 03.65.Pm, 02.60.-x ∗ Corresponding author. Email address: [email protected] a r X i v : . [ phy s i c s . a t o m - ph ] J u l . INTRODUCTION In intense electromagnetic fields the vacuum state is unstable and spontaneously to generateelectron-positron pairs. This is known as the Schwinger e ff ect, which is one of the highly non-trivial predictions in quantum electrodynamics (QED) [1–3]. Due to the tunneling nature of theSchwinger e ff ect, this interesting phenomenon is exponentially suppressed and the pair productionrate is proportional to exp( − π E cr / E ), where the corresponding Schwinger critical field strength E cr = m e c / e (cid:126) = . × V / m. The associated laser intensity, e.g. , I = . × W / cm , istoo high and beyond current technological possibilities. Therefore, its detection has remained achallenge for many decades[4]. However, current advances in high-power laser technology [5–7]and the forthcoming available experiments (for example, in view of planned facilities as the Ex-treme Light Infrastructure (ELI), the Exawatt Center for Extreme Light Studies (XCELS), or theStation of Extreme Light at the Shanghai Coherent Light Source) have brought the hope the QEDpredictions enter the realm of observation. On the other hand, by using x-ray free electron laser(XFEL) facilities can in principle get a strong field at about E = . E cr = . × V / m [8] anddrive interest in studying pair production under superstrong fields.Schwinger e ff ect is one of the nonperturbative phenomena in QED, therefore studying the pairproduction in the nonperturbative regime would deepen our knowledge about the relatively lesstested branch of QED. Motivated by this, many exploratory studies of the Schwinger e ff ect basedon a number of di ff erent theoretical techniques have been undertaken, for example, within thequantum kinetic approach [9, 10] and the real time Dirac-Heisenberg-Wigner (DHW) formalism[11–13], WKB approximation [14, 15] as well as worldline instanton technique [16]. In [17] byusing the quantum kinetic approach the momentum spectrum of the produced particles has beencomputed, and spectrum was found to be extremely sensitive to these physical pulse parameters.The concrete description for various approaches in detail and some latest publications can be foundin our recent review for pair production [18].In this paper, we shall further investigated the Schwinger e ff ect by considering asymmetricpulse shape with Gaussian envelope and di ff erent polarizations. We mainly consider asymmetricpulse shape e ff ects on pair production in di ff erent polarization, e.g. , linear polarization, ellip-tic polarization and circular polarization. We will reveal some novel features of the momentumspectra of created pairs for di ff erently polarized electric fields. In this study the real-time Dirac-Heisenberg-Wigner (DHW) formalism is employed. Because the DHW formalism is very e ffi cient2or the calculation of involving circularly [19, 20] or elliptically polarized electric fields [21, 22].This manuscript is organized as follows. In Sec.II, we introduce the model of a backgroundfield. In Sec.III, we introduce briefly the DHW formalism which is used in our calculation forcompleteness. In Sec.IV, we show the numerical results for momentum spectra and analyze theunderlying physics. In Sec.V, we give the numerical results for the pair number density. We endup the paper with a brief summary and discussion in the last section. II. EXTERNAL ELECTRIC FIELD MODEL
We focus on the study of e − e + pair production in di ff erently polarized and time dependentasymmetric electric fields. So the explicit form of the external field is given as E ( t ) = E √ + δ (cid:16) e − ( t /τ ) θ ( − t ) + e − ( t /τ ) θ ( t ) (cid:17) cos( ω t + φ ) δ sin( ω t + φ )0 , (1)where E √ + δ for the field amplitudes, τ and τ are the rising and falling pulse durations, respec-tively, θ ( t ) is the Heaviside step function, and ω the oscillation frequency, φ is the carrier phase,and | δ |≤ E = . √ E cr , ω = . m , and τ = / m , φ =
0, where m is the electron mass. For the fallingpulse length we set the parameter as τ = k τ , where k is the ratio of the falling to rising pulselength. Throughout this paper we use natural units (cid:126) = c = ff ects on pair production in di ff er-ently polarized and time dependent asymmetric electric fields. We mainly consider two di ff erentsituations. One is that the rising pulse length τ is fixed and the falling pulse length τ = k τ becomes shorter with 0 < k ≤
1. The other is that when the rising pulse length τ is fixed and thefalling pulse length τ = k τ becomes longer with k ≥ III. A BRIEF OUTLINE ON DHW FORMALISM
The DHW formalism is an approach to describe the quantum phenomena of a system by aWigner function as the relativistic phase space distribution that has many advantages and prac-tical usages. It has been also further adopted in the studies of Sauter-Schwinger QED vacuum3air production [11, 12]. In the following, we present a brief outline to DHW formalism for acompleteness of paper self-containing.A convenient starting point is the gauge-invariant density operator of two Dirac field operatorsin the Heisenberg pictureˆ C αβ ( r , s ) = U ( A , r , s ) (cid:104) ¯ ψ β ( r − s / , ψ α ( r + s / (cid:105) , (2)in terms of the electron’s spinor-valued Dirac field ψ α ( x ), where r denotes the center-of-mass and s the relative coordinates, respectively. The Wilson-line factor before the commutators U ( A , r , s ) = exp (cid:32) i es (cid:90) / − / d ξ A ( r + ξ s ) (cid:33) (3)is used to keep the density operator gauge-invariant, and this factor depends on the elementarycharge e and the background gauge field A , respectively. In addition, we use a meanfield (Hartree)approximation via replacing the gauge field operator with the background field.The important quantity of the DHW method is the covariant Wigner operator given as theFourier transform of the density operator (2)ˆ W αβ ( r , p ) = (cid:90) d s e i ps ˆ C αβ ( r , s ) . (4)By taking the vacuum expectation value of the Wigner operator, it gives the Wigner function as W ( r , p ) = (cid:104) Φ | ˆ W ( r , p ) | Φ (cid:105) . (5)By decomposing the Wigner function in terms of a complete basis set of Dirac matrices, we canget 16 covariant real Wigner components W = (cid:16) + i γ P + γ µ V µ + γ µ γ A µ + σ µν T µν (cid:17) . (6)According to the Ref. [11, 12] the equations of motion for the Wigner function are D t W = − D x [ γ γ , W ] + im [ γ , W ] − i P { γ γ , W } , (7)where D t , D x and P denote the pseudodi ff erential operators D t = ∂ t + e (cid:82) / − / d λ E ( x + i λ ∇ p , t ) · ∇ p , D x = ∇ x + e (cid:82) / − / d λ B ( x + i λ ∇ p , t ) × ∇ p , P = p − ie (cid:82) / − / d λ λ B ( x + i λ ∇ p , t ) × ∇ p . (8)4nserting the decomposition Eq. (6) into the equation of motion Eq. (7) for the Wigner func-tion, one can obtain a set of partial di ff erential equations (PDEs) for the 16 Wigner components.Furthermore, for the spatially homogeneous electric fields like Eq. (1), by using the characteristicmethod [19], replacing the kinetic momentum p with the canonical momentum q via q − e A ( t ),and the PDEs for the 16 Wigner components can be reduced to 10 ordinary di ff erential equa-tions(ODEs) of the nonvanishing Wigner coe ffi cients w = ( s , v i , a i , t i ) , t i : = t i − t i . (9)For the detailed derivations and explicit form of the 10 equations, one can refer to the Refs. [12,13, 23]. By the way the corresponding vacuum nonvanishing initial values are s vac = − m (cid:112) p + m , v i , vac = − p i (cid:112) p + m . (10)In the following, one can expresses the scalar Wigner coe ffi cient by the one-particle momentumdistribution function f ( q , t ) = Ω ( q , t ) ( ε − ε vac ) , (11)where Ω ( q , t ) = (cid:112) p ( t ) + m = (cid:112) m + ( q − e A ( t )) is the total energy of the electron’s (positron’s)and ε = m s + p i v i is the phase-space energy density. To obtain one-particle momentum distributionfunction f ( q , t ), referring to [19], it is helpful to introduce an auxiliary three-dimensional vector v ( q , t ) v i ( q , t ) : = v i ( p ( t ) , t ) − (1 − f ( q , t )) v i , vac ( p ( t ) , t ) . (12)So the one-particle momentum distribution function f ( q , t ) can be obtained by solving the follow-ing ordinary di ff erential equations including it as well as the other nine auxiliary quantities,˙ f = e E · v Ω , ˙ v = Ω [( e E · p ) p − e E Ω ]( f − − ( e E · v ) p Ω − p × a − m t , ˙ a = − p × v , ˙ t = m [ m v − ( p · v ) p ] , (13)with the initial conditions f ( q , −∞ ) = v ( q , −∞ ) = a ( q , −∞ ) = t ( q , −∞ ) =
0, where the timederivative is indicated by a dot, a ( q , t ) and t ( q , t ) are the three-dimensional vectors correspondingto Wigner components, and A ( t ) denotes the vector potential of the external field.Finally, by integrating the distribution function f ( q , t ) over full momentum space, we obtainthe number density of created pairs defined at asymptotic times t → + ∞ : n = lim t → + ∞ (cid:90) d q (2 π ) f ( q , t ) . (14)5 V. MOMENTA SPECTRA OF THE PRODUCED PARTICLES
In this section, we will report some interesting results for the momenta spectra of the producedparticles with several pulse parameters under typical cases of polarization field such as linear( δ = δ = .
5) and circular ( δ =
1) ones.
A. Linear polarization δ = Firstly, when one keeps the rising pulse length τ fixed but changes the falling pulse length τ = k τ to be shorter with 0 < k ≤
1, the momentum spectra are shown in Fig. 1 for di ff erent k .For k = k is changed to k = .
5, themain peak of the momentum spectrum is shifted to the positive q x and the symmetry distributionof the momentum spectrum is destroyed. This e ff ect is similar to the e ff ect of carrier phase studiedin [17]. Furthermore when k = .
3, the main parts of momentum spectrum appear also to the neg-ative q x beside the positive q x peak, which means the split of momentum spectrum. Therefore, twopeaks are observed. This result is similar to the e ff ect introduced by the frequencies chirp in [23].For the very asymmetric case of k = .
1, the momentum spectrum of the particle is concentratedagain in the center but the oscillation of the momentum spectrum disappears. Finally, it is notedthat the peak value of the momentum spectrum of the pairs is increased from 2 . × − ( k = . × − ( k = . τ is fixed but the falling pulse length τ becomes longerwith k ≥
1, the result of momentum spectrum are shown in Fig. 2. From these figures, we can seethat as field asymmetry increases, the main center peak of momentum spectrum decreases whilesome disconnected ring-like structures with peaks appear and gradually become main ones. Andthis tendency is more striking with larger k .In detail we find that the center maximum value of momentum spectrum decreases until k ≤ k =
10, the maximum value at ring is larger slightly than thatof symmetrical pulse when k =
1. Note that the ring structure in the momentum spectrum is6 k = 1 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 [ ] x q m [ ] x q m [ ] y q m [ ] y q m k = 0 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 k = 0 . 3 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 - · - 5 · - 5 · k = 0 . 1 - 5 · FIG. 1: Momentum spectra of produced e + e − pairs for linear polarization ( δ =
0) at q z = q x , q y )-plane when the rising pulse length τ is fixed but the falling pulse length τ = k τ becomes shorter with0 < k ≤
1. The chosen parameters are E = . √ E cr , ω = . m , and τ = / m , where m is the electronmass. the typical features of the multiphoton pair production mechanism. For example, the inner ringis formed by absorbing four photons, and the outermost obscured structure corresponds to theabsorption of five photons. B. Elliptic polarization δ = . For middle-elliptical polarization case δ = .
5, the result of momentum spectrum for com-pressed pulse cases are exhibited in Fig. 3. From the top left of Fig.3, where k =
1, one can seethat the momentum spectrum is symmetrically distributed for q x axis, and spectrum peak is locatedat q =
0. With k decreasing, we can observe that the distortion of the momentum spectrum occurs,7 [ ] y q m [ ] y q m [ ] x q m [ ] x q m k = 1 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 - 5 · - 5 · k = 2 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 - · - 5 · k = 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 k = 1 0 FIG. 2: Same as in Fig.1 except the falling pulse length τ = k τ becomes longer with k ≥ equivalently, the mirror symmetry about q x is lost. As the peak position is shifted, the maximumvalue of peak is increased. For example, when k = .
5, the main peak shifts along the positive q y direction while when k = .
3, the main peak shifts along the negative q x direction with a littlelarger peak value. For very asymmetric case of k = .
1, the momentum spectrum is concentratedin the surrounding of the center and the main peak almost locates at the center again.Now let us consider the elongated falling pulse cases with k ≥ δ = .
5, the results of momentum spectrum are displayed in Fig. 4. For k = q x is destroyed. The peak position shifts to thepositive and negative q y direction, while the peak value decreases compared to the symmetric case k =
1. For the larger k , the spectrum at the center vanishes gradually with k increasing, and a com-plete ring-like shape appears. The peaks positions are very interesting which form two elongatedstrips by locating at the relative narrower regime of positive and negative q y but relative broaderregime of positive and negative q x . Finally the additional outer ring structure appears again which8 k = 1 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 k = 0 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 [ ] y q m [ ] y q m [ ] x q m [ ] x q m - · - · - 5 · k = 0 . 3 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 - 5 · k = 0 . 1 FIG. 3: Same as in Fig.1 except for elliptic polarization δ = . is a clear signal for multiphoton pair production processes. This can be understood from the factthat with the increases of pulse length k τ , the electric field has enough long duration and changesits direction during the pair creation process. Thus, the created particles may be accelerated intodi ff erent directions depending on the field direction at the time of production. This results in ringstructure of the spectrum. On the other hand, as the pulse duration increases with k , the number ofoscillation cycles within the Gaussian envelope also increases, and there will be more photons con-tributing to pair production by multiphoton absorption mechanism, so the signal of multiphotonpair creation becomes pronounced. However, for the compressed pulse cases, the number of oscil-lation in the envelope is very small, which does not show the standard multiphoton pair productionclearly (although for the small pulse length τ , the Keldysh parameter will be γ = m / eE τ >
1, butit is not strictly a multiphoton process). This explanation is also appropriate for linear and circularpolarization. 9 k = 1 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 k = 2 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 [ ] x q m [ ] x q m [ ] y q m [ ] y q m k = 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 - 5 · - 5 · - · - · k = 1 0 FIG. 4: Same as in Fig.2 except for elliptic polarization δ = . C. Circular polarization δ = For the circular polarization δ =
1, when the pulse length is compressed with 0 < k ≤
1, theresults of momentum spectrum are shown in Fig. 5. From it one can see that in the symmetriccase of k =
1, the momentum spectrum has an obvious ring structure centered around the origin,meanwhile, a weak interference e ff ect is also observed. The ring shape comes from absorbing fourphotons in the multiphoton pair production. We know that the ring radius can be calculated by theenergy conservation by including the e ff ective mass consideration, as | q | = / (cid:112) ( n ω ) + (2 m ∗ ) ,where n is the number of photons participating in the pair creation and m ∗ is e ff ective mass [24].The weak interference e ff ect can be explained by analysing the distribution of turning points insemiclassical picture [23]. The complex-valued turning points are those t p that are obtained by Ω ( q , t p ) =
0, which is responsible for the interference e ff ects of the spectrum.With k decreasing, the peaks of the momentum spectra display a quite rich structure and the10 - · - 5 · - · - · k = 1 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 k = 0 . 7 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 k = 0 . 3 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 [ ] y q m [ ] y q m [ ] x q m [ ] x q m k = 0 . 1 FIG. 5: Same as in Fig.1 except for circular polarization δ = interference e ff ects vanish gradually. When k = .
7, the peak appears in the upper-left side of themomentum spectrum space. When k = .
3, the partial ring structure vanishes and the momentumspectrum becomes distorted. For the very asymmetric case of k = .
1, the peak position located atthe near central region. Note that, for the circular polarization, the peak value of the momentumspectrum is enhanced remarkably by 2 orders of magnitude compared to that in the symmetriccase k = τ becomes longer with k ≥
1. The result of momentum spectrum are shown in Fig. 6. It isobvious that, in this case, momentum distribution at the inner part of the ring vanishes graduallywith k increasing, and the red ring distribution becomes thin with a lacking of interference e ff ect.Finally the additional outer ring shape appears again although it is a little obscure. The red innerring in the momentum spectrum corresponds to 4 photons absorbing, however, the outer ring is forabsorbing 5 photons. 11 k = 1 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 - 5 · - 5 · - · k = 2 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 [ ] y q m [ ] y q m [ ] x q m [ ] x q m - · k = 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0 k = 1 0 FIG. 6: Same as in Fig.2 except for circular polarization δ = f ( q , ∞ ), for the typical polarization δ whenthe rising pulse length τ = / m is fixed and the falling pulse length τ = k τ is compressed or / andelongated. Note that these peaks occur at di ff erent values of the momentum q . f max ( q , ∞ ) at peak ( k =
1) ( k = .
1) ( k = δ = . × − . × − . × − δ = . . × − . × − . × − δ = . × − . × − . × − In Table I we list some corresponding peak values of momentum distribution for di ff erent polar-ization. It is found that, in the compressed cases of the falling pulse, the peak value of momentumspectrums is enhanced but this enhancement decreases as field polarization increases. In the viceversa case, i.e., the falling pulse is elongated, the peak value is enhanced also. However, on one12and, this enhancement increases as field polarization increases, on the other hand, the enhance-ments in the elongated cases are weaker globally compared to the compressed cases. V. NUMBER DENSITY OF PAIR PRODUCTION
In this section, we calculate the change of the pair number density generated in di ff erent po-larization electric fields with asymmetric shape and di ff erent pulse length ratio k . The results areshown in Figs. 7 and 8 for compressed and elongated falling pulse, respectively.It is found that when the falling pulse width is compressed, i.e., 0 < k ≤
1, the number densityof created pairs decreases with the increase of electric field polarization. We find that the numberdensity of electron-positron pairs in di ff erent polarizations increases with the decrease of pulselength ratio value k . For the larger compression it is more obvious especially for those of k = . k = .
1. When the pulse length is compressed, the number density increases by more than twoorders of magnitude for each polarization.Concretely, for the linear polarization the number density increases from 1 . × − when k = . × − when k = .
1. For the elliptical polarization it increases from 8 . × − when k = . × − when k = .
1. For the circle polarization it increases from 7 . × − when k = . × − when k = . δ as wellas the pulse elongation parameter k except that for the linear polarization it has a little decreasewhen the falling pulse elongation is not large but then it still increases with the k becoming largerand larger. This is mainly attributed e ff ects of pulse length on the pair production processes. Forthe linear polarized electric field this pattern is also found in Ref [25], where the authors haveconsidered the single Sauter pulse. It is found that the particle number increases first with theincreasing of pulse length until it reaches τ = . m − , then it decreases and reaches its minimumat τ = m − and finally it increases again slowly, refer to Fig.4 of Ref [25]).From Figs. 7 and 8, one can infer that the number density exhibits polarization dependencefor compressed pulse asymmetry and elongated pulse asymmetry of the field. For compressedpulse asymmetry cases, the number density decreases with the increase of the field polarization,while for the elongated pulse cases, the number density increases with the increase of the fieldpolarization except the case of k =
1. The reason is when k =
1, the number of oscillation cycles13 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 01 0 - 7 - 6 - 5
Number density k L P E P C P
FIG. 7: The number density (in unit of λ − c = m ) of pairs produced in di ff erent polarized electric fields forthe shorten falling length of asymmetric pulse shape with 0 < k ≤
1. The field parameters are the same asin Fig.1. LP, EP and CP with squares, circles and triangles stands for linear δ =
0, elliptical δ = .
5, andcircular δ = - 8 - 7 - 7 - 7 - 7 - 7 - 7 - 7 Number density k L P E P C P
FIG. 8: Same as in Fig.7 except for the elongated falling case with k ≥ of within the envelope pulse is 6, and clean multiphoton pair production signal is not obvious. Forcompressed pulse cases k <
1, the standard multiphoton pair production is even less significant(for the small τ , although the Keldysh parameter γ = m / eE τ >
1, but strictly it is not completemultiphoton process). For the elongation pulse k >
1, as k increases, pair creation is dominated14y multiphoton mechanism, at this time for ω = . m , the corresponding number density forthe circular polarization is greater than that for the middle elliptical polarization, and the latter isgreater than that for linear polarization cases (see also Fig.4 in Ref [21] for a reference).In a word, when the falling pulse length is compressed, the number density can be increasedby two orders of magnitude, however, for the opposite case, i.e., when the falling pulse length isextended, the number density is enhanced only within the half orders of magnitude. Therefore, forasymmetric electric fields with di ff erent polarizations, in order to increase e ff ectively the numberdensity of the produced electron-positron pairs it is better to shorten the falling pulse. Note thatin our previous work for linear polarized cases [26], where it is studied by solving the quantumVlasov equation approach, the similar finding has been presented qualitatively. VI. SUMMERY AND DISCUSSION
In this study, we investigated the e ff ects of asymmetric pulse shape on the momentum spectrumof created electron-positron pairs in strong electric fields for di ff erent polarization scenarios, inthree di ff erent situations of linear, middle elliptical and circular polarized fields on the momentumspectrum of created particles by applying the DHW formalism. The main results for the spectra ofproduced pairs can be summarized as follows.When the falling pulse length is compressed, for linear polarization the produced pairs spectraexhibit a shift and split of peaks. For middle elliptic polarization as well as circular polarizationthe momentum spectrum gets distorted and exhibits shift of peaks. Finally for each di ff erentpolarization the peaks shifted to the central region at the momentum plane, therefor peak valuesenhanced two orders magnitude compared to the symmetric situation.When the falling pulse lengthis elongated, the ring structures appear for di ff erent polarization. It is also noted that for thisasymmetric situation peak values increase with the field polarizations compared to the symmetriccase while it is smaller than that in compressed cases. Some phenomena of the momentum spectraare consistent with the e ff ect of frequency chirp of our previous study [22].We also study the e ff ect of asymmetric falling pulse on the obtained number density. It isfound that the number density decreases or / and increases with the polarization for compressedor / and elongated falling pulse. It is important that, when the falling pulse length is compressed,the number density of the produced pairs can be enhanced significantly more than 2 orders ofmagnitude. 15he results are helpful to understand the influence of pulse length, which is an important param-eter of the external field, and to deepen the understanding of the external pulse structure. Althoughthese results reveal some useful information about the production of e + e − pairs in di ff erent ellip-tical polarization cases, in this study we restricted ourselves to the multiphoton pair creation, sothe asymmetric pulse shape e ff ects for pair creation under the Schwinger mechanism needs to bestudied further for di ff erent polarized field.The other important phenomena observed in our numerical results are the spiral structure inmomentum spectra which has an intrinsic connection with the spin or / and orbital angular momen-tum of field photons as well as the produced electron-positron particles. The theoretical analysisfor this characteristic is not easy and almost ignored completely in present study. However itsabundant information about the rotation degree is very important and helpful to understand theinvolved strong external field interaction with vacuum and the possible application to the futurereal experiment. Acknowledgments
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