Effects of polarization and high harmonics of two-color fields on dynamically assisted pair production
EE ff ects of polarization and high harmonics of two-color fields ondynamically assisted pair production Obulkasim Olugh, Zi-Liang Li, and Bai-Song Xie ∗
1, 3 Key Laboratory of Beam Technology of the Ministry of Education,and College of Nuclear Science and Technology,Beijing Normal University, Beijing 100875, China School of Science, China University of Mining and Technology, Beijing 100083, China Beijing Radiation Center, Beijing 100875, China (Dated: June 10, 2019)
Abstract
Electron-positron pair production in dynamically assisted two-color electric fields is investigated for var-ious polarizations. Momentum spectrum and number density of the created pairs are examined carefully,in particular, the e ff ects of polarization and high harmonics of two-color fields are exhibited. For only sin-gle strong field, the interference e ff ects of momentum spectrum would vanish when polarization is high,however, for the dynamical assisted two-color fields, the interference e ff ects would be more and moreremarkable with polarization. The multiple peaks of momentum spectrum in elliptic or / and circular polar-ization are observed and explained. It is found that there exists a strong nonlinear dependence of the numberdensity on the high harmonics of two-color fields, for example, the number density can be enhanced sig-nificantly over 7 − ff ect on number density is gradually weaken as the high harmonics increases, however, aweak nonlinearity relation appears again if the high harmonics exceeds a single-photon threshold. PACS numbers: 12.20.Ds, 03.65.Pm, 02.60.-x ∗ [email protected] a r X i v : . [ qu a n t - ph ] J un . INTRODUCTION A strong background electric field causes the quantum electrodynamic (QED) vacuum a de-cay accompanied by the electron-positron ( e − e + ) pair production, which is known as the Sauter-Schwinger e ff ect [1–3]. This remarkable prediction of QED is a nonperturbative process thathas not been experimentally observed yet [4] because the pair production rate exp( − π E cr / E )is exponentially suppressed for the electric field E smaller than the Schwinger critical fieldstrength E cr = m e c / e (cid:126) = . × V / cm, where the corresponding critical laser intensity I cr = . × W / cm . On the other hand, however, the high intense electromagnetic field canbe achieved by high power lasers, and with recent advance of high-intensity laser technology [5–7], the laser intensity of the order of 10 W / cm are expected by Extreme Light Infrastructure(ELI) [8] in current construction; and in the planned facilities as the Exawatt Center for ExtremeLight Studies (XCELS), the Station of Extreme Light at the Shanghai Coherent Light Source, onemay expect the experimental tests of Sauter-Schwinger e ff ects will be feasible in the near future.Meanwhile, the already operating X-ray free electron laser (XFEL) at DESY in Hamburg can getnear-critical field strength as large as E ≈ . E cr [9], which is enough to produce a considerablenumber of e − e + pairs [10, 11].Beside a nonperturbative characteristic, Sauter-Schwinger pair production is also a nonequilib-rium process with a typical non-Markovian e ff ect [12]. Over past a few decades, many theoreti-cal works have been performed based on a number of di ff erent theoretical methods to cope withsuch di ffi culties, more details can be seen in review [4, 13]. Among them a dynamically assistedSchwinger mechanism, which combines two laser fields with low-frequency strong field and high-frequency weak field, was proposed by Sch ¨utzhold et al. [14], where the pair production rate isincreased significantly due to lowering the Schwinger critical limit by 2-3 orders of magnitude.From then on, many interesting findings in pair production researches have been achieved [15–22]. Two important features of studies on the e − e + production are impressive recently. One is thatwith more realistic field parameters by choosing an appropriate pulse shape. The other is that thepolarized e ff ect has to be considered for the studied problem. Experimentally, for example, a po-larization of up to ± .
93 is already achieved [23] for low-intensity electric fields. Moreover, whilethe number density of created pairs is the main concern on the study, the momentum spectra ofpairs could be helpful for understanding the dynamics of the problem in point of view theoreticallyas well as experimentally. 2herefore, motivated by these advances and interests, in this work, we shall extend involvedstudy by considering a realistic laser electric field with envelop pulse and without perfect lin-ear / circular polarization. The main research is focused on the polarized electric field e ff ects ondynamically assisted pair production. The real-time Dirac-Heisenberg-Wigner (DHW) formalism[24–26] is adapted, as a very e ffi cient theoretical approach which has been used extensively fornumerical calculations of pair production, for instances, in rotating circularly polarized electricfields [28–31], elliptic polarized background fields [32] and also recent work for the frequencychirp e ff ects in di ff erently polarized fields [33]. In addition, it is noted that the field ellipticitye ff ects on pair production has also been studied by other approaches beside DHW [34–36] forplane-wave fields as well as for time-dependent electric fields.We focus on the study of e − e + pair production in dynamically assisted fields with various po-larizations. The electric field is considered as the combination of strong but slowly varying field, E s ( t ) and a weak but rapidly changing field, E w ( t ). So the explicit form of the external field, E ( t ) = E s ( t ) + E w ( t ), is given as E ( t ) = E s √ + δ exp (cid:32) − t τ (cid:33) cos( ω t + ϕ ) δ sin( ω t + ϕ )0 + E w √ + δ exp (cid:32) − t τ (cid:33) cos( b ω t + ϕ ) δ sin( b ω t + ϕ )0 , (1)where E s , w / √ + δ is the amplitude of the electric field, τ denotes the envelop pulse length, ω denotes the oscillating frequencies, ϕ is the carrier phase, b is the high harmonics order of theweak field to strong field and | δ |≤ b ≥
9. Such purely time dependent electric fieldEq.(1) could be considered as the approximation of standing wave formed by two coherent contourpropagating laser beams with the e ff ects of spatial part neglected for each di ff erent polarization.In this study the characteristic fixed field parameters are chosen as: E s = . √ E cr , E w = . √ E cr , ω = . m and τ = / m , where m is the electron mass. Throughout this papernatural units of (cid:126) = c = E w ( t ) is set to zero,the electric field Eq.(1) reduces to the one studied in [32].It is well known that the Keldysh adiabaticity parameter is defined as γ = m ω/ eE [37], where E and ω are the strength and frequency of the external electric field, such that the Schwinger (tun-neling) and multiphoton pair creation can be characterised by γ (cid:28) γ (cid:29)
1, respectively. Forthe given parameters in this paper the Keldysh adiabaticity parameters are thus γ s = . √ + δ γ w = . b √ + δ for a single either strong or weak field, respectively.A typical case of how the high harmonics b ω a ff ects the time depended electric fields is dis-played in Fig. 1(lower panel) for linear polarization δ = b =
9. For the high harmonics b ω ,we examined several cases in the interval 0 . m ≤ b ω ≤ . m , and for the polarization we choosefour di ff erent δ as typical studied situations. By the way, we are aware that in some circumstancesthe high harmonics b ω is too large to be lies in the normal regime. However, in this study, sinceour main interest is the influence of the high harmonics on the momentum spectrum and pair pro-duction rate for di ff erent polarization, it is valuable to examine the e ff ect of a large b ω on numberdensity, which corresponds to the single photon absorption or even above the threshold 2 m .In the following, by using DHW formalism, we numerically compute the momentum spectrumand the number density of the produced pair for several values of the high harmonics b ω of laser - 2 0 0 0 2 0 0 - 0 . 1 50 . 0 00 . 1 5 - 2 0 0 0 2 0 0 - 0 . 0 1 50 . 0 0 00 . 0 1 5 - 2 0 0 0 2 0 0 - 0 . 1 6 50 . 0 0 00 . 1 6 5 E(t) ( ) s E t t [ 1 / m ]
E(t) ( ) w E t
E(t) ( ) ( ) s w
E t E t + FIG. 1: The time dependence of the electric field E ( t ) in units of the critical field for the linearly polarized( δ =
0) case. The chosen parameters are E s = . √ E cr , E w = . √ E cr , ω = . m , and τ = / m where m is the electron mass. The upper panel is for the single strong field E s ( t ) and the middle is for theweak field E w ( t ) when b =
9. The lower panel displays for the dynamically assisted field E ( t ) with a highharmonics b ω = . m . ϕ = ff erent high harmonics and di ff erentpolarizations. We end our paper with a brief conclusion in the last section. II. A BRIEF OUTLINE ON DHW FORMALISM AND WKB APPROXIMATION
The DHW formalism is a relativistic phase-space quantum kinetic approach [24] that has beennow widely adopted to study the pair production from QED vacuum in strong background field.In the following, we present a brief review of the DHW formalism.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 e s (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 mean-field (Hartree)approximation via replacing gage field operator by 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)and taking the vacuum expectation value of the Wigner operator gives the Wigner function 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)5ccording to the Ref. [25, 26] the equations of motion for the Wigner function are D t W = − (cid:126) D (cid:126) x [ γ (cid:126)γ, W ] + im [ γ , W ] − i (cid:126) P { γ (cid:126)γ, W } , (7)where D t , (cid:126) D (cid:126) x and (cid:126) P denote the pseudodi ff erential operators D t = ∂ t + e (cid:82) / − / d λ (cid:126) E ( (cid:126) x + i λ(cid:126) (cid:53) (cid:126) p , t ) · (cid:126) (cid:53) (cid:126) p ,(cid:126) D (cid:126) x = (cid:126) (cid:53) (cid:126) x + e (cid:82) / − / d λ(cid:126) B ( (cid:126) x + i λ(cid:126) (cid:53) (cid:126) p , t ) × (cid:126) (cid:53) (cid:126) p ,(cid:126) P = (cid:126) p − ie (cid:82) / − / d λλ(cid:126) B ( (cid:126) x + i λ(cid:126) (cid:53) (cid:126) p , t ) × (cid:126) (cid:53) (cid:126) p . (8)Inserting the decomposition Eq. (6) into the equation of motion Eq. (7) for the Wigner function,one can obtain a system 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 characterizedmethod [28], replacing the kinetic momentum p with the canonical momentum q via q − e A ( t ),and the partial di ff erential equation(PDE) system for the 16 Wigner components can be reducedto ten ordinary di ff erential equations(ODEs). And the nonvanishing Wigner coe ffi cients are: w = ( s , v i , a i , t i ) , t i : = t i − t i . (9)Because due to the Wigner coe ff ecients equations of motions are quite lengthy thus here we refrainrepeating the respective formula form. For the detailed derivations and explicit form we refer thereader to [26, 27]. 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. In order to precisely calculate to one-particlemomentum distribution function f ( q , t ), referring to [28], it is helpful to introduce an auxiliarythree-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)6o the one-particle momentum distribution function f ( q , t ) can be obtained by solving the follow-ing ordinary di ff erential equations,˙ 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)For the sake of keeping a self-sustained, let us turn to introduce the semiclassical approxima-tion for spinor QED. The e ff ects of the field polarization on momentum distribution, especiallythe interference pattern can be qualitatively and quantitatively interpreted within a semiclassicalanalysis by means of an e ff ective scattering potential based on the WKB approximation [38]. Theapproximated production number for the created fermionic particles takes the form (cf.[38]) N spinorq ≈ (cid:88) t p e − K ( p ) q + (cid:88) t p (cid:44) t p (cid:48) α ( p , p (cid:48) ) q )( − p − p (cid:48) e − K ( p ) q − K ( p (cid:48) ) q , (15)where t p are the turning points by Ω ( q , t p ) = (cid:112) m + [ q − e A ( t )] = K ( p ) q = (cid:12)(cid:12)(cid:12)(cid:12)(cid:82) t p t ∗ p Ω ( q , t ) dt (cid:12)(cid:12)(cid:12)(cid:12) , and α ( p , p (cid:48) ) q = (cid:82) t p (cid:48) t p Ω ( q , t ) dt is the phase accumulation related to the di ff erent turning point pairs, fora more detailed discussion, see Ref [38]. As was discussed in [38], the first term is describedas every distinct pair of turning points for pair production, whereas the second one is interferenceterm of di ff erent turning points which is responsible for the oscillation in the momentum spectrum.For example, if a single pair of turning point dominates, then there is no interference pattern inmomentum distribution. Therefor the WKB result for the created number of pairs for momentum q is given by N spinorq ≈ exp( − K ) , K = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t t dt Ω ( q , t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (16)where t and t are dominant turning points closed to the real t axis. This is the case that was studiedin Ref. [39]. For two pairs of complex turning points, the production rate was then estimated to be7 sum of two terms which takes the form N spinorq ≈ e − K + e − K − α ) e − K − K , (17)which is then explain interference e ff ects in the momentum spectra. III. MOMENTUM SPECTRA
By using the DHW method, we compute the momentum spectrum of the produced particles forseveral values of the high harmonics b ω when the carrier phase is chosen as ϕ =
0. Note that themomentum spectra of the created pairs is highly sensitive to the change of carrier phase ϕ [40, 41],however, it is not our main concerned issue in present paper. Instead our main concerns are fortypical polarizations as the linear δ =
0, the near-linear δ = .
1, the middle elliptic δ = . δ = A. Single strong field
By solving Eq. (13) numerically, we can obtain the created particles distribution functions.Fig. 2 depicts the two-dimensional momentum distribution in the ( q x , q y ) plane for the weakfrequency strong field E s ( t ) when E w ( t ) =
0. From Fig. 2 (LP case), one can see a strongoscillation characteristic appeared in spectra for the linearly polarized case. And the peak positionof distribution is also located at the center of momentum spectrum, which agrees with results ofRefs. [32, 40, 42]. For the nonzero polarization, however, the momentum spectrum looses itsoscillation structure gradually and its peak shifts along the positive q y direction with a decreasingof peak value when polarization parameter δ increases until to that a ringlike shape appearing inspectrum for δ ∼
1, see Fig. 2 (CP). To understand that the momentum spectrum are not peakedaround q x / y = q y axis for the δ (cid:44) E x ( t )component of the electric field has to be accelerated or / and decelerated along the y direction bythe E y ( t ) component. Due to the even and odd function properties of x and y field components,thus, the momentum spectrum exhibit the symmetric / asymmetric behavior in q x / q y directions.For the given electric field Eq. (1), there exists an infinite number of turning points, andthe related dominant turning points are numerically obtained when the e ff ective scattering po-tential Ω ( q , t p ) = (cid:112) m + [ q − e A ( t )] is zero for certain q . From the Fig. 2, one can infer8 [ ] x q m [ ] y q m - · - · - · - · [ ] y q m [ ] x q m L P - 1 0 1- 0 . 50 . 00 . 5 - 1 0 10 . 61 . 21 . 8 - 2 - 1 0 1 2 3- 3- 2- 10123
C P
FIG. 2: Momentum spectra of created e − e + pairs in the( q x , q y )plane(where( q z = E s ( t ) with di ff erent polarizations. From top left to bottom right the values of polarizationparameters are δ = , . , . ,
1, respectively. The other field parameters are E s = . √ E cr , ω = . m ,and τ = / m . that, for nonzero polarization parameters, the oscillation or interference in momentum spectrumvanish. Because pair production in a strong field is a non-Markovian poroses [12], the cosinefunction in equation (15) has a time integral which consequently causes the accumulated phases α ( p , p (cid:48) ) q = (cid:82) t p (cid:48) t p Ω ( q , t ) dt to depend on the complete earlier history; so even the small changes of theelectric field parameters can easily change relative phases of the amplitude naturally. Thus we canunderstand that the variation of the field polarization can easily change dominant turning pointslocation in the complex time plane.To clarify the above discussion, we plot the location of the complex conjugate pair of turning9 -
100 0 100 200 - - - H t L I m H t L - -
100 0 100 200 - - - H t L I m H t L - -
100 0 100 200 - - - H t L I m H t L - -
100 0 100 200 - - - H t L I m H t L FIG. 3: Contour plots of | Ω ( q , t ) | in the complex t plane, showing the location of turning points where Ω ( q , t ) =
0. These plots are for the strong field E s ( t ) with di ff erent polarizations. The other field parametersare E s = . √ E cr , ω = . m , and τ = / m . From top left to bottom right the values of polarizationparameters are δ = , . , . ,
1, respectively, and the momentum values (in units of m ) are ( q x = , q y = , ( q x = , q y = . , ( q x = , q y = . , ( q x = , q y = . points in the complex time t plane, which is shown in Fig. 3. When δ =
0, the location of pairsof turning points agrees with the result [42]. As the field polarization increases, the positions ofdominating pair of turning points close to the real t axis also change. Obviously turning pointsdeparts from the real time t axis when one goes from linear to circular polarization. Therefore,the interference e ff ects between those pairs of turning points become weaker and weaker and evenvanishing with field polarization increases, which results in vanishing of oscillatory structure ofthe spectrum. In Fig. 3, it is evident that the location of turning points in the case of circular10olarization, see Fig. 2 (CP), are far away from real axis and also the fewer number of pointsexist compare to other cases. Consequently, the interference between these turning points becomeweaker as the increasing filed polarization for the few-cycle electric field in Eq. (1), which is onlyappropriate for the weak frequency but strong field, i.e. , for the E s ( t ). [46]As a comparison study, let us now turn to the results of the momentum spectrum for dynami-cally assisted two-color field for di ff erent polarization and di ff erent high harmonics. B. Two-color field when δ = and δ = . The particle momentum spectra under dynamically assisted field is shown in Fig. 4 for linearpolarization case δ =
0. One can see that the momentum spectrum exhibits strong and rapidoscillation structure. Obviously, there exist some discrete side maxima at the large momentumregime. We think it is due to the impact of the high-frequency but weak field with high harmonics b ω superimposed to the strong field. To our knowledge, in the presence of a non-perturbative field E s , the produced pairs are continuously accelerated, and created particle momentum is mainlydetermined by its creation time. At the earlier time it is created then it has to be accelerated at thelonger time and finally it gets the higher longitudinal momentum. This prediction is also supportedby the Refs. [15, 21]. As high harmonics b ω increases it is more remarkable. Meanwhile thepeak values of momentum distribution are increased to 2 and 4 orders larger compared to linearlypolarized case of a single strong field, respectively, when b ω ≈ . m and b ω ≈ m . Theseamplification results mainly attribute to the dynamically assisted Sauter-Schwinger e ff ect.For the interference e ff ects of created particles in spectra is again due to the interaction betweenthe complex conjugate pairs of turning points. These dominate pairs of turning points distributiongradually approaches real t axis more and more with b ω and finally the di ff erent sets of turningpontes are almost equidistant from real t axis. This results in the complicated and pronouncedoscillation structures of the momentum spectra.For the near-linearized case δ = .
1, the momentum spectrum is exhibited in Fig. 5. Com-parable to the single strong field case, in the case of two-color fields, the peak value of spectrumincreases and the range of spectrum expands in momentum space. And this tendency is morestriking with b ω . Meanwhile, the interference e ff ects also become stronger with b ω . For example,for small b ω , the interference pattern appear only along the negative q y , while for large b ω , theinterference e ff ects occur also along the positive q y .11 b = 9 - 2 - 1 0 1 2- 0 . 50 . 00 . 5 - · - · - · - · b = 1 1 - 2 - 1 0 1 2- 0 . 50 . 00 . 5 b = 1 5 - 3 - 2 - 1 0 1 2 3- 0 . 50 . 00 . 5 [ ] y q m [ ] y q m [ ] x q m [ ] x q m b = 2 0 FIG. 4: Momentum spectra of created e − e + pairs in the ( q x , q y ) plane(where( q z = δ =
0) combined field E ( t ). The high harmonics b ω = . m , . m , . m , m from topleft to bottom right, respectively. And the other field parameters are E s = . √ E cr , E w = . √ E cr , ω = . m , and τ = / m . C. Two-color field when δ = . For middle-elliptical polarization case δ = .
5, the result of momentum spectrum is shown inFig. 6. From the top left of Fig.6, when b ω = . m , one can see that the e − e + pairs locates mainlyin the regime of positive q y while a weak interference e ff ect appears in the regime of negative q y .With the increase of b ω , we can observe multiple peaks appear at the same time. On the otherhand, stronger interference e ff ects at the negative q y plane occur, see the lower panel of Fig. 6.When b ω = m , the spectrum has the 14 peaks.12 b = 9 - 2 - 1 0 1 2 - 0 . 50 . 00 . 5 b = 1 1 - 2 - 1 0 1 2- 0 . 50 . 00 . 5 b = 1 5 - 3 - 2 - 1 0 1 2 3- 0 . 50 . 00 . 5 [ ] x q m [ ] x q m [ ] y q m [ ] y q m - · - · - · - · b = 2 0 FIG. 5: Momentum spectra of created e − e + pairs in the ( P x , P y ) plane(where( P z = δ = .
1) combined field E ( t ). The high harmonics b ω = . m , . m , . m , m from top left to bottom right, respectively. And the other field parameters are E s = . √ E cr , E w = . √ E cr , ω = . m , and τ = / m . These results are very similar to the findings of the Ref. [43]. These multiple peaks are referredto as shell structures. In general the peak pattern is dominated by the strong but slowly varyingfield E s , while the second weak but rapidly changing field E w is mainly responsible for theappearances of additional peaks, which are not visible in the case of a single strong field E s alone.Therefore the observed multiple peaks form a shell structure exhibiting a lifting pattern whenthe b ω is large and the middle-ellipticity is applied which destroy the symmetry of momentumdistribution along q y .In particular, when b ω = m , a pearl-necklace-like pattern is observed in the spectrum with13 b = 9 - 2 - 1 0 1 2- 1012 b = 1 1 - 2 - 1 0 1 2- 1012 b = 1 5 - 3 - 2 - 1 0 1 2 3- 2- 1012 [ ] y q m [ ] y q m [ ] x q m [ ] x q m - · - · - · - · b = 2 0 FIG. 6: Momentum spectra of created e − e + pairs in the ( P x , P y ) plane(where( P z = δ = .
5) combined field E ( t ). The high harmonics b ω = . m , . m , . m , m from top left to bottom right, respectively. And the other field parameters are E s = . √ E cr , E w = . √ E cr , ω = . m , and τ = / m . more pronounced interference pattern. The interference e ff ects are even evident at the positive q y .For this case, there would be more complex conjugate turning points pairs having approximatelythe same distance to the real t axis. Thus, the distinct interference e ff ects occur in the momentumspectrum.By comparing with the single strong field of middle-elliptical polarization shown inFig.2(0.5EP), we find that, in the case of two-color field, the momentum spectrum becomes largerwith b ω . Finally, the peak values of spectra are enhanced from 5 . × − (for single strong field E s ) to 5 . × − (two-color field when b ω = m ). Note that in this case the position of spectrum14eak often appears at the positive q y plane and q x = - 2 - 1 0 1 2- 2- 1012 b = 9 - 2 - 1 0 1 2- 2- 1012 - · - · - · - · b = 1 1 - 2 - 1 0 1 2- 2- 1012 [ ] y q m [ ] y q m [ ] x q m [ ] x q m b = 1 5 - 2 - 1 0 1 2- 2- 1012 b = 2 0 FIG. 7: Momentum spectra of created e − e + pairs in the ( P x , P y ) plane(where( P z = δ =
1) combined field E ( t ). The high harmonics b ω = . m , . m , . m , m fromtop left to bottom right, respectively. And the other field parameters are E s = . √ E cr , E w = . √ E cr , ω = . m , and τ = / m . D. Two-color field when δ = For the circular polarized case δ = .
0, the momentum spectra is shown in Fig. 7 for di ff erent b ω . For b ω = . m , we can observe that 8 peaks are centered around the origin with an exhibitionof the weak interference pattern at the bottom of the negative q y . As b ω increases, the numberof multiple peaks of the momentum spectrum increases also and the separated adjacent peaked15 - H t L - - - H t L - - H t L - - - H t L ** * * ** * * *** * * * * ** ** FIG. 8: The corresponding vector potential A ( t ) for time dependence of the electric field E ( t ) in units of thecritical field for the circularly polarized ( δ =
1) case. Left panel is vector potential for strong field E s ( t )without superimposed weak field. This right displays vector potential for the dynamically assisted field E ( t )with a high harmonics b ω = m . The black stars give the predicted pair production peaks as given by Eq.(19). The chosen parameters are E s = . √ E cr , E w = . √ E cr , ω = . m , and τ = / m where m is the electron mass. - -
100 0 100 200 - - H t L I m H t L - -
100 0 100 200 - - H t L I m H t L FIG. 9: Contour plots of | Ω ( q , t ) | in the complex t plane, showing the location of turning points where Ω ( q , t ) =
0. These plots are for circular polarized ( δ =
1) combined field E ( t ) when high harmonics b ω = m . The other field parameters are E s = . √ E cr , E w = . √ E cr , ω = . m , and τ = / m .From left to right the momentum values (in units of m ) are ( q x = , q y = . , ( q x = , q y = − . b ω = . m , there appears a complete ring-like form in the momentum spectrum. Meanwhileit is found that there is always ( b −
1) peaks when b ω increases until to 0 . m with the stronginterference signature at the negative q y plane. Moreover in all cases of b ω ≤ . m the maximumof peaks persist at the positive q y regime.For the circular polarization, if the Gaussian envelop is disregarded, then the homogeneouselectric field (1) strength is simplified as | E ( t ) | = E cr (cid:113) E s + E w + E s E s cos[( b − ω t ] , (18)which has an periodical number of local maxima at t k = K π ( b − ω , K ∈ Z . Hereby, N = | b − | is thenumber of maxima for this field. This also holds for the pulsed field where | t k | < τ , cf. Ref. [29].In general, most pairs are expected to appear at those times corresponding to the local maxima offield. Then those produced pairs are subject to acceleration by the electric field, and the gainedmomenta are q = (cid:90) ∞ t e E ( t ) dt = A ( t ) − A ( ∞ ) = A ( t ) . (19)It is therefore not surprising, why those b − b ω = m , strong interference e ff ects are visible at the bottom ofthe negative q y plane and a sub ring structure appears at the inner part of the multiple peaks ring inthe momentum distribution. In contrast to situations b ω ≤ . m , when b ω = m , three di ff erentaspects are remarkably found: (1) the interference e ff ects at the positive q y plane with the innerring appears; (2) overall b =
20 but not b − =
19 peaks exist; and (3) the 20 th peak as themaximum one locates at the negative q y but not the positive q y plane, see Fig. 7( b = q y plane that results in a resonances e ff ect and maybe it leads to an additional peak besidethe b − A ( t )for the circular polarization ( δ =
1) in Fig. 8. On the one hand we find that multiple peaks appearsin the vector potential (right) for the dynamically assisted field E ( t ) with a high harmonics b ω = m as compared to the vector potential (left) for only strong field E s ( t ) without superimposedweak field. On the other hand, from the right panel of Fig. 8 we also find that 19 (star markers)maximum peak strengths appears in the large outer ring. However, it is found that, at the marked17ositions by black crosses, the spirales of created pair particle meet at two di ff erent times withsame final vector potential (momenta). By comparing the right panel of Fig. 8 with Fig. 7( b =
20) we observe that the produced pairs are almost close to the prediction momenta accordancewith Eq. (19) so that the corresponding interference pattern is observed at the black cross points.Therefor it provides a potential explanation for the number of pair production peaks in outer ringmomentum spectrum with strong interference pattern around at q y = − . m in negative q y at thesame time, it also interprets why the interference e ff ects at the positive q y plane around q y = . m of the inner ring. Because particles created at two di ff erent times simply end up at the same finalmomentum that leads to the enhancing of the total yields, which is the reason why maximum peakappears around at q y = − . m , see Fig. 7( b = q x direction and the left-moving particles along the negative q x direction carry di ff erentphase with the circular distribution. For this circular polarization case, the number of rotation cy-cles increase with high harmonics b ω , and the ends of the momentum spectrum distribution meetagain in negative q y momentum plane and display a ring-shape [28]. Consequently, the interfer-ence term appears in the sum of corresponding quantum mechanical wave functions with di ff erentphases. Therefor, we observe interference e ff ects at the negative q y plane.On the other hand we can also understand the interference structure which change and increasethe attribution in the momentum distribution from the semiclassical turning points, shown in Fig.9, for b ω = m in the circular polarized δ =
1. As we can see, for example when b ω = m , oneof the important di ff erence from the case of circular polarization for the single strong field shownin Fig. 3(d) is that now the dominant contribution comes from appearance of more turning points.In fact, as b ω increases, the number of cycles within the pulse duration also increases, thus moreturning point pairs come close to the real t axis. Therefore, the interference e ff ects become moreand more significant. From the Fig. 9 one can infer that for b ω = m more pairs of turning pointsappears for ( q x = , q y = − .
90) as compared to ( q x = , q y = . q y occurs stronger interference e ff ects with smaller width, buthigher peaks, see Fig. 7( b = q y is higher than that of the most positive q y , the total amount of particles around of the peak at negative case is still lower than that aroundof the peak at the opposite positive case of q y plane.18 V. NUMBER DENSITY
The number density is calculated by scanning over the field polarization for di ff erent b ω , shownin Fig. 10. It is seen that the number density enhances as the high harmonics b ω increases for eachpolarization, and can reach its maximum at δ = b ω ∼ . m , cf the upper panel of figure.In particular for the small b ω , the number density decreases more sharply compared to the large b ω as δ approaches to 1.One possible physical reason is that for small high harmonics b ω , as pair production rate issmaller for weak fields, the created pairs number density is dominated by the electric-field compo-nent with a large peak field strength. In addition, the polarized electric field given in Eq. (1) witha small value of δ ∼ δ ∼
1, and therefore results in a high number density.From the lower panel of Fig. 10, we can find that, for large 2 . m ≤ b ω ≤ . m , the relativeincrement of number density with b ω is very small and it reaches the maximum at δ = .
1, 0 . . . δ = b ω = . m , however, its maximum at δ = b ω , we can conclude that the nonlinear relation between the numberdensity and the field polarization.In order to better understand the relation between the number density and b ω , we depict thenumber density by scanning over the field high harmonics b ω for di ff erent polarizations in Fig. 11.For the single strong field from Fig. 11, we can see that with the increase of the field polarizationthe number density is monotonically decreasing. In the case of single weak field, for each di ff erentpolarization, the curves have large leaps at b ω ≈ m / n indicating the appearance of additional n -photon channels. Due to the e ff ective mass e ff ects of the electron or positron in the strong field,cf. Ref. [44], it is no surprising that the number density is actually peaked slightly above thisexpectation b ω ≈ m / n and this deviation becomes more significant for higher b ω .From Fig. 11 one can easily infer that for dynamically assisted two-color filed E ( t ) a com-bination of E s ( t ) and E w ( t ) can significantly enhance pair production relative to the separateapplication of each field for each di ff erent polarization. As b ω increases, the created pairs num-ber density increases for each polarization. When 0 . m ≤ b ω ≤ . m , the number density isdistinguished for each polarization, and as the polarization parameter | δ | increases, the createdpairs number density is decreased, therefore, in this region for the linear polarization the producedparticle number density is higher than other polarized ones. While in the region b ω > . m , we19 (cid:1) Number density - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 00 . 0 0 1 60 . 0 0 1 80 . 0 0 2 00 . 0 0 2 2 (cid:1)
Number density
FIG. 10: The number density (in units of λ − c = m ) of created particles as a function of the field polarization δ for the di ff erent high harmonics b ω ≥ . m . The other field parameters are E s = . √ E cr , E w = . √ E cr , ω = . m , and τ = / m . can see that for di ff erent polarized fields, the e + e − pair number density is indistinguishable as b ω further increases. Thus in this region the polarization e ff ect on number density is gradually weakenas b ω . We also see that the high harmonics b ω for maximum pair production has been shifted tolarger at about b ω = . m ( b =
45) compared to the single weak field E w ( t ) (multiphoton) case.Similar e ff ect is also observed in [45] by using the computational quantum field theory approach.Consequently, we can get the optimum number density as in the circular polarization ( δ =
1) when b ω = . m , after that it decreases nonlinearly with b ω .Finally, in Fig. 12, we display an approximate enhancement factor N (1 s + w ) / [ N s + N w ] ofthe number density for the dynamically assisted two-color filed as function of high harmonics20 . 5 1 . 0 1 . 5 2 . 0 2 . 51 E - 1 61 E - 1 51 E - 1 41 E - 1 31 E - 1 21 E - 1 11 E - 1 01 E - 91 E - 81 E - 71 E - 61 E - 51 E - 41 E - 3 w E Number density
E 1 + E 2 ( L P ) E 1 + E 2 ( E P ) E 1 + E 2 ( C P ) E 2 ( L P ) E 2 ( E P ) E 2 ( C P )E 1 ( L P ) E 1 ( E P ) E 1 ( C P ) b (cid:1) FIG. 11: The number density (in units of λ − c = m ) of created particles as a function of the high harmonics b ω for the di ff erent polarization δ = δ = . δ = E ( t ),strong field E s ( t ), and weak field E w ( t ), respectively. The other field parameters are E s = . √ E cr , E w = . √ E cr , ω = . m , and τ = / m . b ω , where N s and N w denotes the value of the number density in the case of single strong andsingle weak field for each di ff erent polarizations, respectively, and N (1 s + w ) is for two-color filed.One can observe that for small b ω ≤ . m , the enhancement factor is quite distinguished foreach polarization with the small enhancement and the enhancement factor increases with the fieldpolarization. While in the region 0 . m ≤ b ≤ . m , very clearly visible is the suppression in theenhancement factor when one goes from linear to circular polarization and for large values region1 . m ≤ b ≤ . m the enhancement can reach serval orders of magnitude for each polarization.When b ω > m , the e ff ects of high harmonics parameters b ω on the approximate enhancementfactor is almost negligible for each di ff erent polarization. V. SUMMERY AND CONCLUSION
In this study, pair production is studied in dynamically assisted two-color electric fields fordi ff erent polarization scenarios using the real-time DHW formalism. We have examined featuresof dynamically assisted pair production in four di ff erent situations of linear, near-linear, middleelliptical and circular polarized fields on the momentum spectra and number density of createdparticles. 21 . 5 1 . 0 1 . 5 2 . 0 2 . 511 01 0 01 0 0 01 0 0 0 0 Enhancement factor
L PE PC P b (cid:1) FIG. 12: The approximate enhancement factor of the number density for each di ff erent polarization respec-tively δ = δ = . δ = N (1 s + w ) / [ N s + N w ] as a function of thehigh harmonics b ω . The other field parameters are E s = . √ E cr , E w = . √ E cr , ω = . m , and τ = / m . For single strong field pulse, the interference e ff ect in the momentum distribution vanisheswith the increase of field polarization. While for the dynamically assisted two-color pulse, astrong significant enhancement is seen in the spectrum for the particles cereaed from vacuumwhen appropriate high harmonics is applied, and interference e ff ects appear with multiple peaksin elliptic polarization as well as in circular polarization. That could be understood semiclassicallyas the appearance of new turning point pairs due to the additional superimposed weak field. Andwith the enhancement of high harmonics of two-color fields, interference e ff ects become strongerand the number of multiple peaks also increase for each polarization.For the number density, it is also found that it exhibits distinctive polarization dependence forsmall and large high harmonics of two-color fields. When high harmonics is small, the numberdensity decreases with polarization from linear to circular; while for a larger high harmonics, thenumber density is sensitive to the high harmonics degree and its polarization dependence exhibitsa strong nonlinear characteristic. So the maximal number densities are achieved for either linear,circular or elliptical polarization when di ff erent larger high harmonics are applied. It is also notedthat for the single strong field, number density decreases with polarization varying from linear tocircular. By comparing the number density in each individual field with the combined two-color22elds, it is demonstrated that the production rate can be enhanced significantly for each polariza-tion due to the combination of two fields for certain high harmonics parameter region. Finally thepolarization e ff ect on number density is gradually weaken as the high harmonics increases whilea weak nonlinearity relation appears again if the high harmonics of two-color fields exceeds asingle-photon threshold.We hope that our studies reveal some useful information about pair creation processes in di ff er-ent elliptically polarization scenarios with dynamically assisted two-color field. In this study, weonly consider fixed pulse length scale, the results by considering the di ff erent pulses with di ff erentduration scales would be reported elsewhere. Acknowledgments
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