Pair production in asymmetric Sauter potential well
PPair production in asymmetric Sauter potential well
Adiljan Sawut, Sayipjamal Dulat ∗ , and B. S. Xie †
2, 3 School of Physics Science and Technology,Xinjiang University, Urumqi, Xinjiang 830046 China Key Laboratory of Beam Technology of the Ministry of Education,and College of Nuclear Science and Technology,Beijing Normal University, Beijing 100875, China Beijing Radiation Center, Beijing 100875, China (Dated: February 26, 2021)
Abstract
Electron-positron pair production in asymmetric Sauter potential well is studied, where thepotential well has been built as the width of the right edge fixed but the left side of the well ischangeable at different values. We study the momentum spectrum, the location distribution and thetotal pair numbers in this case of asymmetric potential well and compare them with the symmetriccase. The relationship between created electron energy, the level energy in the bound states andthe photon energy in the symmetric potential well is used to the studied problem for the createdelectrons in the asymmetric potential well and its validity is confirmed by this approximation.By the location distribution of the electrons we have also shown the reason why the momentumspectrum has an optimization in the asymmetric well compared with the symmetric one. ∗ [email protected] † [email protected] a r X i v : . [ qu a n t - ph ] F e b . INTRODUCTION The electron-positron pair production has been studied theoretically [1–6] and experimen-tally [7–15] by many authors since Sauter [16] considered pair creation in a static electricfield. In 1951 [17], the pair creation rate in a constant field was calculated by Schwinger andhe pointed out the electric field strength for creating observable pairs, which is E c = 10 V/cm, corresponding to laser intensity of 10 W / cm respectively. With the rapid develop-ments of laser technology and the chirped-pulse amplification (CPA) technology [18, 19],laser intensity could reach 10 W / cm at the present [20], but still far more smaller thanthe critical value of Schwinger effect. However, there is no experiment that directly achievesthe conversion from energy to matter, while it is full of hope in the future.In terms of experiment, Cowan [7] observed positive and negative electron beams in theheavy ion collision experiment in 1986. After a decade, Ahmad [8] produced a single energypositron beam through heavy ion collisions. In these experiments, however, the nuclearreaction caused by the collision of high-energy relativistic ions and the structure change of thehigh-Z nucleus are much greater than the effect of the Coulomb fields‘ superposition of twonuclei. In general, most positrons came from a nuclear reaction rather than a direct vacuumbreakdown. Also, in 1997, Burke et al. [9] used the 46.6 GeV electron beam generatedby Stanford Linear Accelerator (SLAC) to collide with a 10 W/cm laser beam, gainedpair productions. In this experiment, the non-linear Compton scattering of laser photonsand electrons generates high-energy gamma photons [10] , and these high-energy gammaphotons continually interact with the laser to generate positron and electron pairs, calledthe Breit-Wheeler process [11, 12] . In 2009, Chen [13] irradiated a gold target with anultra-strong laser of 10 W/cm , and detected a positron beam of 10 per sphere behind thegold target. In this experiment, laser photons and nuclear scattering generated high-energygamma photons and these high-energy photons interact with high-Z nuclei to generate a largenumber of positrons and electrons, respectively, named the Bethe-Heitler process [14, 15] .In terms of theory, many theoretical ideas have been adopted for solving the non-perturbative and non-equilibrium process in pair production, such as Wentzel-Kramers-Brillouin approximation (WKB) [1, 2] , World-line instanton technique [3, 4] , solvingVlasov equation [5] , Wigner function formalism [6] and so on. In this article, we discussthe pair creation problem in asymmetric potential well with computational quantum field2heory (CQFT) [21–23] which has had a great achievement not only with pair creation, alsowith Zitterbewegung [24] , Klein paradox [25, 26] , relativistic localization problem [24] andso on. Recently more interesting works were performed, which include one of ours for theeffective interaction time mechanism to improve the pair numbers significantly [27]. In thispaper, furthermore, we study the pair creation problem in asymmetric Sauter potential well.We examine it by using the fields which include both the static asymmetric Sauter potentialand the alternating field. It is found that the width of the potential well plays an importantrole in this electron-positron production process.The paper is organized as follows. In Sec.II we discuss the theoretical framework forcomputational quantum field theory. In Sec.III we study the momentum spectrum and thelocation distribution of the created electrons. In Sec.IV we compare the evolution of thenumber of particles created in different types of asymmetric potential well. Finally, in Sec.Va brief summary of the work is given. II. THE THEORETICAL FRAMEWORK FOR CQFT
As for electron and positron we use Dirac equation, we use the atomic units (a.u.) as (cid:126) = m = e = 1 in this whole paper, as for fine structure constant α = 1 /c = 1 / . z direction for the simplicity of our calculationand the model. Here [28, 29] i∂ ˆ ψ ( z, t ) /∂t = [ cα z ˆ p z + βc + V ( z, t )] ˆ ψ ( z, t ) , (1) α z denotes as z component of the Dirac matrix, β denotes unit Dirac matrix, and we onlyfocus on a single spin thanks to there is no magnetic field in our one-dimensional system.In results, the four-component spinor wave function becomes two components and Diracmatrix α z and β are replaced by the Pauli matrix σ and σ , where: σ = , σ = − . (2)The Hamiltonian is H = cσ ˆ p z + σ c + V ( z, t ) , (3)whereby V ( z, t ) is the classical external scalar potential along the z direction. We can3xpress ˆ ψ ( z, t ) as the combine of creation and annihilation operators as follows: (cid:88) p ˆ b p ( t ) W p ( z ) + (cid:88) n ˆ d n ( t ) W n ( z ) = (cid:88) p ˆ b p W p ( z, t ) + (cid:88) n ˆ d n W n ( z, t ) , (4)where (cid:80) p ( n ) stands for the summation over all states with positive and negative energy,and W p ( n ) ( z, t ) = (cid:104) z | p ( n ) ( t ) (cid:105) is the solution of the Dirac equation for the initial condition W p ( n ) ( z, t = 0) = W p ( n ) ( z ), where W p ( n ) ( z ) is the energy eigen-function of the field-freeDirac equation. We also can express the fermion operators as:ˆ b p ( t ) = (cid:88) p (cid:48) ˆ b p (cid:48) U pp (cid:48) ( z, t ) + (cid:88) n (cid:48) ˆ d † n (cid:48) U pn (cid:48) ( z, t ) , (5)ˆ d † n ( t ) = (cid:88) p (cid:48) ˆ b p (cid:48) U np (cid:48) ( z, t ) + (cid:88) n (cid:48) ˆ d † n (cid:48) U nn (cid:48) ( z, t ) , (6)ˆ b † p ( t ) = (cid:88) p (cid:48) ˆ b † p (cid:48) U ∗ pp (cid:48) ( z, t ) + (cid:88) n (cid:48) ˆ d n (cid:48) U ∗ pn (cid:48) ( z, t ) , (7)ˆ d p ( t ) = (cid:88) p (cid:48) ˆ b † p (cid:48) U ∗ np (cid:48) ( z, t ) + (cid:88) n (cid:48) ˆ d n (cid:48) U ∗ nn (cid:48) ( z, t ) , (8)where U pp (cid:48) ( t ) = (cid:68) p (cid:12)(cid:12)(cid:12) ˆ U ( t ) (cid:12)(cid:12)(cid:12) p (cid:48) (cid:69) , U pn (cid:48) ( t ) = (cid:68) p (cid:12)(cid:12)(cid:12) ˆ U ( t ) (cid:12)(cid:12)(cid:12) n (cid:48) (cid:69) , U nn (cid:48) ( t ) = (cid:68) n (cid:12)(cid:12)(cid:12) ˆ U ( t ) (cid:12)(cid:12)(cid:12) n (cid:48) (cid:69) , U np (cid:48) ( t ) = (cid:68) n (cid:12)(cid:12)(cid:12) ˆ U ( t ) (cid:12)(cid:12)(cid:12) p (cid:48) (cid:69) . The time evolution operator of the field ˆ U ( t ) ≡ ˆ T exp ( − i (cid:82) t Hdτ ), ˆ T denotes time-orderoperator and due to the operators at different times may not be commutatived, and it helpsto sort the operators at different times so that the operator at the early time is classified tothe right side of the operator at the late time. The electronic portion of the field operator isdefined as ˆ ψ + e ( z, t ) ≡ (cid:80) p ˆ b p ( t ) W p ( z ) so that the created electrons‘ spatial number densitycan be written as: ρ e ( z, t ) = (cid:104) vac | ˆ ψ + † e ( z, t ) ˆ ψ + e ( z, t ) | vac (cid:105) . (9)The anticommutator relations (cid:110) ˆ b p , ˆ b † p (cid:48) (cid:111) = δ pp (cid:48) and (cid:110) ˆ d n , ˆ d † n (cid:48) (cid:111) = δ nn (cid:48) , the number density ofthe electrons can be rewritten as: ρ e ( z, t ) = (cid:88) n | (cid:88) p U pn ( t ) W p ( z ) | , (10)where U pn ( t ) can be computed with the split operator numerical technique [21]. By inte-grating the Eq.(10), we can obtain the total number of created electrons as: N ( t ) = (cid:90) ρ e ( z, t ) dz = (cid:88) p (cid:88) n | U pn | . (11)4 II. THE MULTI-PHOTON PROCESS
We set the amplitude of potential well V and alternating field V both equal to V = V = 2 c − V ( z, t ) = V S ( z ) f ( t ) + V sin ( ωt ) S ( z ) θ ( t ; t , t + t ) , (12)where S ( z ) = { [tanh ( z − D/ /W ] − [tanh ( z + D/ /W ] } / , (13)and f ( t ) = sin[ πt/ t ] θ ( t ; 0 , t ) + θ ( t ; t , t + t ) + cos[ π ( t − t − t ) / t ] θ ( t ; t + t , t + t ) . (14)Eq.(14) describes how the processes of our potential well to turn on and off, and we bringa step function θ ( t ; t , t ) on. As demonstrated by the total field function, the oscillatingfield doesn’t work during (0 , t ) in order to reduce the trigger effect of the pair production,so we set t = 5 /c for the well establishment of the potential well. Then we turn on theoscillating field during ( t , t + t ), and t = 20 π/c . The total time for our calculation is T = 2 t + t = (10+20 π ) c and divided into N t = 10000 time intervals. We have also dividedthe calculation space into N z = 2048 grid points. When we set these two parameters larger,we will get more precise results in our simulation experiment, meanwhile, the experimenttime also gets longer. Therefore, they should be controlled within a reasonable range. Forthe Eq.(13), D is the width of the potential well, and W , W are widths of the electric fieldat the edge. We set the D = 10 λ e , where the λ e is the Compton wavelength. W = 0 . λ e and keep stable, W has different values, which is from 0 . λ e to 1 . λ e , exactly quater tofive times of the W , respectively. The frequency of the oscillating field is ω = 2 . c .The FIG.1 is the momentum spectrum of the pair production in three different shapesof our potential well. As we compare the black line which represents to the asymmetricpotential well with W = 0 . λ e , W = 0 . λ e , and the red line which represents to thesymmetric well, we can easily notice that the N p > N p < W gets smaller, and the electrons with positivemomentum get more and more increase. On the contrary, blue line which represents to theasymmetric potential well with W = 0 . λ e , W = 0 . λ e , has clear decline in its momentum5 - 6 - 5 - 4 - 3 - 2 momentum distribution m o m e n t u m N P [ 2 p / L ] W = 0 . 3 l e , W = 0 . 1 5 l e W = W = 0 . 3 l e W = 0 . 3 l e , W = 0 . 6 l e o n e - p h o t o n p r o c e s st w o - p h o t o n p r o c e s st h r e e - p h o t o n p r o c e s sf o u r - p h o t o n p r o c e s s FIG. 1: Momentum spectra of the pair creation for one symmetric combined field(red), and twoasymmetric combined field with different widths of the left side(black and blue spectrum). Thefrequency of the oscillating field is ω = 2 . c , other parameters are N z = 2048, N t = 10000, V = V = 2 c − D = 10 λ e , and W is given in the figure. distribution compared with the symmetric potential well (red line). All this three spectrahave one very common feature, which is when N p close to 0 the change is not very clear,but as N p >> z direction. InFIG.2 (a), compare with the symmetric potential well, the asymmetric potential well with W = 0 . λ e , W = 0 . λ e (black line in a) shows there are more electrons in our space.Around z = 0 area, particle number density is little bit higher in the right side than the6 paticles number density z W = 0 . 3 l e , W = 0 . 1 5 l e W = W = 0 . 3 l e ( a ) - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 401 02 03 04 05 0 particles number density z W = 0 . 3 l e , W = 0 . 6 l e W = W = 0 . 3 l e ( b ) FIG. 2: The location distribution of the created electrons, (a) is the comparison of asymmetricpotential well with W = 0 . λ e , W = 0 . λ e (black), and symmetric well with W = W = 0 . λ e (red), (b) is the comparison of asymmetric potential well with W = 0 . λ e , W = 0 . λ e (black),and symmetric well with W = W = 0 . λ e (red). left side or we can say the location distribution is asymmetric when z close to 0, it’s verylogical with our conclusions on the top. But there is also an equal amount of increase onthe both side along the ± z direction. We can explain this phenomenon with the Fouriertransformation, small-scale spatial distribution is related to large momentum and the large-scale spatial distribution is related to small momentum. As in FIG.1, the changing of widthin left side has more effects to the large momentum electrons so that small-scale spatialdistribution has obvious asymmetric values. The large-scale spatial distribution increase inequal amount or we can say they are symmetric, because of our asymmetric potential wellhas few effects to the small momentum electrons as in FIG.1. On the contrary, in FIG.2(b), there is a clear decrease on particle number density when we make the W two timeswider than the W . Thanks to the left side of our potential well keep changing while theright side is being stable, our spectra have optimized when width gets narrower. So in ournext step, we will only discuss W = 0 . W = 0 . λ e shaped one, numerically, the detailedinformation about the momentum spectrum of the pair production in symmetric potentialwell is discussed in the reference [30]. We calculate the eigenvalues of the bound states inthe well with [29]: cp cot ( p D ) = EV cp − cp , (15)7hereby p = (cid:112) c − E /c , p = (cid:112) ( E + V ) /c − c , V = 2 c − , D = 10 λ e . Thisequation is symmetric in ± E , so in our asymmetric well would have some uncertaintiesoccurred. - 7 - 6 - 5 - 4 - 3 - 2 momentum distribution m o m e n t u m N P [2p / L ] W = 0 . 3 l e , W = 0 . 1 5 l e o n e - p h o t o n p r o c e s st w o - p h o t o n p r o c e s st h r e e - p h o t o n p r o c e s sf o u r - p h o t o n p r o c e s s -2 -2 -2 -2 -2 -3 -3 -3 -3 -5 -5 -5 -5 -4 -7 -6 -6 -6 -6 -6 (cid:1)(cid:7) (cid:5)(cid:6) (cid:2)(cid:2)(cid:3)
152 159 164 173 182 193 204 214 (cid:2)(cid:5)(cid:4) (cid:3)(cid:2)(cid:4)
246 260 268 279 291 302 315 (cid:4)(cid:3)(cid:5)
348 366 377 389 402 418 (a ) o n e - p h o t o n p r o c e s s (b ) t w o - p h o t o n p r o c e s st h r e e - p h o t o n p r o c e s s f o u r - p h o t o n p r o c e s s (c ) (d ) momentum distribution m o m e n t u m N P [2p / L ]
FIG. 3: Momentum spectrum of the asymmetric fields, the numerical grids are same as FIG.1
We get the results as follow : E = − . c , E = − . c , E = − . c , E =0 . c , E = 0 . c , E = 0 . c , E = 0 . c , E = 0 . c .The relationship between peak energy in the FIG.1 and the eigenvalue of the bound statesin the symmetric potential well is given in the reference [30], which is E pi = E i + nω . Weverify this relationship whether it works for our asymmetric potential well. In the FIG.3(a), it shows the one-photon process, which means the electron jumps from the well in itsbound states through absorbing one photon with frequency of ω = 2 . c . We test it withcalculation of its energy, E = (cid:112) c p + c , (16)whereby p = 2 N p π/L , where L =2.0 is the length of the numerical grid and the N p isthe pick value we have marked in FIG.3. Take an example of N p = 58, the eigenvalue E = − . c and we calculate the energy which correspond to the N p = 58 with Eq.(16),and we get E p = 1 . c , the relationship is roughly E pi − E i = 1 . c − ( − . c ) =2 . c ≈ ω = 2 . c .We take FIG.3 (c) as an example of our calculation for multi-photon effect which is es-timated that electrons in the bound states absorb three photons and become free particles.According to the values of each peak, we know N p = 254 , , , , , , , i = 1 , , , , , , , . And we calculate the energyof the electrons at every single peak, they can be written as E N p =254 = 5 . c , E N p =260 =6 . c , E N p =268 = 6 . c , E N p =279 = 6 . c , E N p =291 = 6 . c , E N p =302 =6 . c , E N p =315 = 7 . c , E N p =324 = 7 . c . TABLE1: The value of E pi and E i at the peaks i E i − . c − . c − . c . c . c . c . c . c N p
254 260 268 279 291 302 315 324 E pi . c . c . c . c . c . c . c . c With very simple calculation we verify that E pi = E i + 3 ω , also indicate our estimationis fully correct.The changing of the momentum spectrum as we turnover the widths at the two side of thefield is predictable. We have just compared two groups of data, which are W = 0 . λ e , W =0 . λ e and W = 0 . λ e , W = 0 . λ e respectively. Obviously, the momentum distribution ofpositive and negative N p has transferred its corresponding values as we turnover the widthsat each edge. IV. EVOLUTION OF THE ELECTRON NUMBER
In this section we study the pair production in subcritical asymmetric potential well witha subcritical oscillating field, but with different widths of electric fields as we have consideredin the last section. We set V = V = 2 c − ω = 2 . c .In FIG.4. All the curves have only one difference, other parameters are the same. Theright width of the electric field at the edge of the potential well is stable( W = 0 . λ e ),and we keep changing the left side from 0.25 time to 5 times of the right side which arenamely W = 0 . λ e , . λ e , . λ e , . λ e , . λ e , . λ e , . λ e . The curve c represents to thesymmetric potential well, other six curves represent to the asymmetric potential well. Weprefer to take curve c as a comparison with other six curves. As shown in FIG.4, when wekeep the width of right side of the potential well stable, make the left side wider, the rate ofthe electron creation would be slower and slower over time, as in curves d , e , f and g , and9 N(t) t i m e W = 0 . 3 l e , W = 0 . 0 7 5 l e W = 0 . 3 l e , W = 0 . 1 5 l e W = 0 . 3 l e , W = 0 . 3 l e W = 0 . 3 l e , W = 0 . 6 l e W = 0 . 3 l e , W = 0 . 9 l e W = 0 . 3 l e , W = 1 . 2 l e W = 0 . 3 l e , W = 1 . 5 l e abcdefg FIG. 4: The total number N ( t ) for the created electrons in the different well with different widthsgiven very clearly in the figure. The parameters are ω = 2 . c , N z = 2048 , N t = 10000 , V = V = 2 c − , D = 10 λ e , L = 2 . . close to one very specific value as in curve g , which approximately equals to the value of aone-sided potential well. The shape of this potential-well can be written as follow: V ( z, t ) = V S ( z ) f ( t ) + V sin ( ωt ) S ( z ) θ ( t ; t , t + t ) , (17)where S ( z ) = (1 + tanh[ z/W ]) /
2, it is shown that when one side of the potential well isenough wider, and it doesn’t have any effect to the whole system, which we call a criticalvalue of the electric field that can hardly effect to the increasing of the pair production.We realize that compared with the curve c , the yields of the pair production has increasedin curve a and curve b , as the curve a represents to the yields of the electrons in the potentialwell with W = 0 . λ e , W = 0 . λ e and the curve b represents to the yields of the electrons10n the potential well with W = 0 . λ e , W = 0 . λ e . Surely it is logical due to the intensityof the electric field would be stronger as the width gets narrower. As shown in the FIG.4,increase of the yields is evident from W = 0 . λ e to W = 0 . λ e , and it has very weak effectas we make the W narrower, W = 0 . λ e , precisely. That is approximately the maximumvalue of the yields in our asymmetric potential well. V. SUMMARY
We have compared the momentum spectrum, location distribution and the yields of thepair production in different shape of potential well, in other word, symmetric and asym-metric potential well. As for the momentum spectrum in symmetric case, it had obtaineda simple law which is E pi = E i + nω, we find this relationship is also useful for the asym-metric potential well after calculation. It means electrons in the bound states created inthe well, escape from their positions after absorbing one or more photons. The momentumdistribution of the electrons with positive momentum gets obvious increase than the neg-ative momentum electrons when the left side of potential well becomes narrower. On thecontrary, the momentum distribution of the electron with positive momentum gets obviousdecrease than the negative momentum electrons when the left side of potential well becomeswider, as expected. The reason for this effects is explicated by the location distribution asshown in FIG.2.For the yields of the pair production, the growth of N ( t ) gets slower increase when wemake one of the width of the potential well wider, because of the electric intensity getsweaker correspondingly, and close to the value of one-sided potential well. The growthof N ( t ) increases much significantly as the width gets narrower. In general, because ofthe multi-photon effects, the pair production always keeps increasing in any type of ourasymmetric potential well. Acknowledgments
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