Numerical approach to the semiclassical method of radiation emission for arbitrary electron spin and photon polarization

We show how the semiclassical formulas for radiation emission of Baier, Katkov and Strakhovenko for arbitrary initial and final spins of the electron and arbitrary polarization of the emitted photon can be rewritten in a form which numerically converges quickly. We directly compare the method in the case of a background plane wave with the result obtained by using the Volkov state solution of the Dirac equation, and confirm that we obtain the same result. We then investigate the interaction of a circularly polarized short laser pulse scattering with GeV electrons and see that the finite duration of the pulse leads to a lower transfer of circular polarization than that predicted by the known formulas in the monochromatic case. We also see how the transfer of circular polarization from the laser beam to the gamma ray beam is gradually deteriorated as the laser intensity increases, entering the nonlinear regime. However, this is shown to be recovered if the scattered photon beam is collimated to only allow for passage of photons emitted with angles smaller than $1/\gamma$ with respect to the initial electron direction, where $\gamma$ is the approximately constant Lorentz factor of the electron. The obtained formulas also allow us to answer questions regarding radiative polarization of the emitting particles. In this respect we briefly discuss an application of the present approach to the case of a bent crystal and high-energy positrons.


I. INTRODUCTION
The semiclassical formalism of Baier, Katkov and Strakhovenko allows for the approximate determination of the spectrum of emitted photons from an ultrarelativistic electron in a virtually arbitrary external electromagnetic field [1]. For numerical applications the formulation with a single time integration as found in [2,3] for the spin and polarization averaged result, is most useful. In this paper we show how the basic result of the semiclassical method with explicit electron spin and photon polarization can also be treated numerically in a similar fashion. We use the obtained formulas in the case of a background plane wave, as the Dirac equation then can be solved analytically [4], to do a direct comparison with the spectrum obtained using the exact solution of the Dirac equation (Volkov states) [4][5][6]. This is the usual approach for such processes [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. We consider the case of a short circularly polarized laser pulse, and find agreement, as expected. The advantage of the presented approach is the possibility of calculating the radiation emission under general circumstances, i.e. also for very complicated field configurations as one only needs the classical trajectory in the external field, which can easily be found numerically for a given field. The presented formulas allow to find the polarization properties of the radiation depending on the spin of the initial and final electron, which also allows to determine if the electrons become polarized. The latter would occur if the spin-flip radiation has a different yield for each of the possible initial spin states, see e.g. [23], i.e. a generalization of the Sokolov-Ternov effect [24] to fields other than that of a permanent magnetic field [25,26]. We briefly demonstrate this in the case of positrons channeling in a bent germanium crystal where one has two kinds of motion superimposed, the oscillatory channeling motion between the bent planes, which in the unbent case would not lead to polarization, along with the motion along the bending arc which leads to transverse polarization of the positrons. When the crystal is strongly bent, i.e. close to the so-called Tsyganov radius [27,28], the polarization as in a magnetic field is obtained, while smaller bending radii lead to smaller degrees of polarization, which the presented method allows to predict.
Below, e indicates the positron charge, and units are used, such that the fine-structure constant α is given by e 2 , whereas the relativistic metric + − −− is employed. We will use Feynman notation to write / a = a µ γ µ , where a µ is a generic 4-vector.

II. SEMICLASSICAL APPROACH
Below, we study the emission by an electron of a single photon in a given background electromagnetic field. The basic result of the semiclassical method of Baier et al. in its most general form for the single-photon radiation probability is expressed as [1] where x µ = {t, x(t)} is the electron 4-position as obtained by the Lorentz force equation in the external field, k µ = ω {1, n}, ω = ε ε ω, ω is the energy of the emitted photon, ε = ε − ω, ε the electron energy, n the direction of emission, and Here, φ i and φ f are the spinors of the initial and final electron state (characterized by the electron 4momentum and the electron spin in its asymptotic rest arXiv:1909.12899v1 [hep-ph] 27 Sep 2019 frame), σ denotes the vector of the Pauli spin matrices, and with being the polarization vector of the emitted photon, v(t) = dx(t)/dt being the electron velocity, and the constants being given by To evaluate the quantity in Eq. (1) we need to carry out the two time integrals v(t)e ik x dt and e ik x dt. However, a direct computation of these integrals converges slowly, and integrations beyond times when the acceleration is different from zero must be included, as explained classically in [29]. From the relations shown in [3], and which are already used there in the case without polarization and spin averaging, it is quite easy to relate these quantities to the quantities whose integrands are proportional to the acceleration. By doing this, we have that where In [3] it is shown in detail how to calculate the electron trajectory and the quantities I and J numerically. In particular it is appropriate to analytically carry out the cancellations between large terms, as in e.g. 1 − n · v because n · v is close to 1 for ultrarelativistic particles. Finally, we may write and therefore we obtain the emission probability as

III. VOLKOV-STATE APPROACH
If the background field is a plane wave, i.e. if the 4vector potential A µ (ϕ) only depends on the phase ϕ = k 0 x, where k 0 = (ω 0 , k 0 ) is the 4-momentum associated with the photons of the plane wave, the corresponding Dirac equation can be solved analytically [4]. Below we assume that the plane wave propagates along the negative z direction and we choose 4-vector potential A µ (ϕ) in the Lorenz gauge where A 0 (ϕ) = A 3 (ϕ) = 0. The positive-energy solution reads where p is the asymptotic 4-momentum of the electron, (we have set the quantization volume equal to 1), where is the classical action of the electron in the plane wave, and where u is a short notation for the constant vacuum bispinor (which is characterized by the electron spin in the corresponding electron rest frame and by the electron 4-momentum p). The leading-order matrix element for single-photon emission is given by where ψ i/f (x) indicates the Volkov state corresponding to the initial/final electron state, and the differential probability of emission is then In the gauge we are working, the 4-potential can be written as where a µ j are two 4-vectors such that a j k 0 = 0 and a 1 a 2 = 0 and where f j (ϕ) are two arbitrary (physically well-behaved) functions. By setting the arbitrary phase in the indefinite integrals in the phase of Volkov states to zero, we introduce the quantities Then, by inserting the expressions of Eqs. (15) and (19) into Eq. (17), we obtain that where we have defined and with j = 1, 2 (we have set µ = (0, )). Now, we can write the functions in Eq. (22) as a Fourier transform where defined for n = 0, 1, 2. When n = 0, the j subscript is superfluous and we will therefore denote this function as A 0 (s, α, β). This function is however problematic as it diverges but it can be regularized by using the identity (see also [9,11,12]) where In this way, we obtain By replacing these expressions in Eq. (22), and carrying out the integration over d 4 x, we can write the amplitude in the form Now we can use the energy delta function to fix s such that and the delta function can be transformed as At this point we would then take the norm-square to obtain the transition probability, however we are then faced with the problem of how to take the square of the delta-function which has the complication that s 0 is a function of the momenta. The correct way to do this, is to consider instead the more realistic case of an initial is the Volkov solution with momentum p i and unindicated fixed spin quantum number. To preserve normalization we must have that |c(p i )| 2 d 3 p i = 1/(2π) 3 . Then the momentum delta-function can be transformed as i indicates the dyadic product between the vectors k 0 and p i ), and so using Sylvester's determinant theorem we obtain Therefore we finally write the transition amplitude in the form Now, in order to find the probability using Eq. (18) we take the norm-square of the above amplitude and, having in mind the case of a narrow wave packet [22], replace |c(p i )| 2 = δ 3 (p i − p i,0 )/(2π) 3 . Analogously as above, we now have a delta-function which we can evaluate by integration over d 3 p f and the transformation of the deltafunction yields a factor of ω0ε f k0p f . Finally, we then obtain the differential emission probability Figure 1. The intensity spectrum ωdP/dω according to the semiclassical approach is shown as solid lines corresponding to the different possibilities of initial and final spins and photon polarizations. The black dotted curves on top of the solid curves indicate the same quantities but calculated using the Volkov-states approach. We have not plotted the curve corresponding to ↑↓ + and ↓↑ − as the corresponding yields are much smaller than the others and the curves would not be visible. Also, we have not plotted ↑↓ − as it coincides with ↓↑ +. In all these cases, the two approaches also agree.
which can now be evaluated numerically. The bispinors in this expression are chosen as [30] where φ are spinors to be chosen as an orthonormal basis of eigenstates of σ · s, with s being the direction of the otherwise arbitrary spin quantization axis in the rest frame of the electron.

IV. DISCUSSION OF RESULTS
The above derivations were carried out without introducing a particular plane-wave pulse. We will now consider a particular choice of the 4-vector potential and carry out the corresponding numerical calculation using the semiclassical method and the Volkov-states method. We set a µ 1 = {0, a x , 0, 0}, a µ 2 = {0, 0, a y , 0}, k µ = {ω 0 , 0, 0, −ω 0 }, and that is, we choose a pulse with envelope d(ϕ) and negative helicity (right-handed) circular polarization [31]. We define the polarizations of the outgoing light as withŷ being the unit vector in the y direction, and According to this choice 1 and 2 are unit vectors orthogonal to each other and to k and such that, if k lies along the z axis, they indicate the polarization along the x and y direction, respectively. The ± basis corresponds to circular polarization with helicity of ±1. As the spin basis we have chosen quantization axis along the z direction such that φ may be chosen as 1 0 T or 0 1 T , Figure 3. Degree of circular polarization of the Compton scattered radiation for nonlinear Compton scattering in a monochromatic wave without collimation, and in the short pulse described in the text with and without angular collimation.
denoted by ↑ and ↓, respectively in the figures. We set a x = a y = mξ/e, where ξ is the classical nonlinearity parameter, which we have set ξ = 1, N = 5, and the electron energy ε = 30 GeV for the Figs. (1), (2) and (3). Since the typical emission angles are small, we write k x = ωθ x and k y = ωθ y and then dΩ = dθ x dθ y . In Fig. (1) we have restricted the angular integration such that |θ x | < (ξ + 3)/γ, where γ is the initial Lorentz factor of the electron, and the same for θ y such that nearly all emitted radiation is included. In this figure we compare the semiclassical approach based on the formulas of Baier, Katkov and Strakhovenko and compare with the results obtained using the Volkov states. The results indicate nearly perfect agreement between the two approaches, which is expected since the motion in a plane wave is intrinsically semiclassical [7]. In Fig. (2) we do the same but restrict the emission angles over a smaller interval (collimation) i.e. |θ x | < 0.1/γ and the same for θ y . In this case the emitted radiation with negative helicity is highly suppressed and therefore we plotted the results on a logarithmic scale. This is expected due to angular momentum conservation along the z axis. Since the electron flipping its spin is unlikely for ultrarelativistic electrons [30], the outgoing light must have opposite helicity as that of the laser field to conserve angular momentum. Finally, the agreement between the semiclassical method and the Volkov-state method in this case indicates an agreement of the two approaches also at the level of angularly resolved spectra.
In Fig. (3) we show how the collimation affects the degree of circular polarization, defined as We compare with the result found in [32], obtained in the case of the monochromatic wave, and see that in the short pulse one reaches a slightly smaller value. However, if one collimates the photon beam, one can achieve circular polarization to a degree close to unity. The method presented here is particularly useful as we require that ξ is of the order of 1 in such a way that the emission of high harmonics is suppressed. Moreover, at ξ ∼ 1 the total probability of emission is of the order of 2παN [1] such that the obtained results are valid even for relatively long pulses as long as multiple photon emission is negligible. At the same time, this also implies that in the situations discussed above one cannot use the often used local constant field approximation, and the semiclassical method presented here is a simple method to obtain accurate values of the degree of polarization which is valid also for external fields of complex spacetime structure [see also Refs. [33][34][35] for an alternative applicable method].

V. POLARIZATION IN A BENT CRYSTAL
Bent crystals can be used to steer an electron or positron beam along a circular arc as investigated in e.g. [36][37][38][39]. Also, the possibility of polarizing an electron/positron beam as in a storage ring through synchrotron radiation was discussed in e.g. [40], where it was assumed that the crystal was bent close to the socalled Tsyganov critical radius which we will define as where d p is the distance between two symmetry planes in the crystal and U 0 is the corresponding potential energy depth. This radius corresponds to the radius at which the strength of the force from the electric field between the planes, estimated as 2U 0 /d p , can no longer provide the necessary centripetal force to sustain the circular motion. Below we consider the motion of a positron between two (110) planes in Germanium such that d p = 2.0 Å and U 0 = 35.73 eV. According to the above discussion, the Tsyganov critical radius is roughly the smallest bending radius at which channeling is still possible in the crystal. In this case the radiation and polarization characteristics are that of the constant magnetic field which produces the same bending radius of the trajectory, and therefore the largest possible polarization is given by 8/ 5 √ 3 [24], when χ 1 [41]. Conversely, when the bending radius becomes large, one must recover the case of the flat crystal, which does not produce any beam polarization. With the presented approach we demonstrate that one can predict the polarization properties for any bending radius R, and not only for the extreme case close to the critical radius. In an experiment the average polarization will depend on the angular distribution of particles when entering the crystal. Thus, we will only apply the approach in the case of a single particle starting with and angle of 0 and a distance of u 0 = 0.083 Å from the plane (this value corresponds to the thermal vibrational amplitude of the nuclei in the crystal lattice). The maximum polarization that can be asymptotically obtained, A, is given by [23,40] where W f i denotes the total transition rate from state i to state f . The quantity W f i for different initial and final spin quantum numbers can be found from Eq. (13) by integrating over angles and photon energies, and by summing over the photon polarization, using a finite piece of trajectory. This formula comes about if it is assumed that the positron has its energy replenished between each radiation emission, as is the case in a synchrotron. With crystals, this would require several thin crystals with accelerating structures in between. We integrated the Lorentz force equation of motion using the electric field obtained from the continuum potential [1,42], such that the electric field in the unbent crystal is along the x direction. We then offset the plane along a circular arc in the xz plane, which at the leading order in the small quantity L/R, where L is the crystal length, means that the bending follows the curve x = z 2 /2R. One may use this approximation as the total deflection angle L/R, is small in a realistic scenario. Due to symmetry, the electric field points along the radius of bending and using Gauss' law one can show that as long as the distance to the plane is much smaller than the bending radius R, the electric field component along the radius of bending is the same as the electric field in the unbent case evaluated at the same distance from the plane. The non-zero components of the electric field are then Here, E cont (x − z 2 /2R) is the electric field obtained from the continuum potential, in the Doyle-Turner approximation [43][44][45], which depends only on the coordinate transverse to the planes (the x coordinate in the considered case). We used a piece of trajectory with roughly 10 periods of oscillation, which was adequate for convergence of the integrals. Moreover, we have integrated over an angular region such that v ⊥ (t) is contained in the region, with an additional angle of 10/γ in each direction. This turned out numerically to be sufficient to cover all of the emitted radiation. In Fig. (4) we show the result for a 50 GeV positron with the mentioned initial conditions. It should be mentioned that this maximum polarization is only achievable under the same circumstances as in a storage ring, i.e. a short piece of crystal where radiation occurs, and subsequently a replenishment of the lost energy so that the particles have the nominal energy before entering a crystal again. It is seen that for a strong bending of the crystal, one approaches the value in the constant field of 8/ 5 √ 3 . While we show only the example of a single trajectory, the method would allow to study radiation reaction in a bent crystal where the effects of polarization of the beam would be essential. We refer the reader to Refs. [46,47] for recent experimental studies of radiation reaction in straight crystals.

VI. CONCLUSION
In conclusion we have presented a method to rewrite the semiclassical formulas of Baier, Katkov and Strakhovenko, which facilities their numerical implementation for arbitrary discrete particles quantum numbers. This then allows for the calculation of radiation emission with arbitrary initial and final electron spins, and with arbitrary polarization of the emitted photon when knowing only the classical trajectory of the electron in the background field. In this way, one does not have to know the Dirac wave function in the background field, which is typically an impossible task for realistic field configurations.
First, we have compared the obtained formulas for a case where a solution of the Dirac equation is known, namely the plane-wave field, and find near perfect agreement between the two methods, corroborating the idea that the motion in a plane wave is intrinsically quasiclassical. As an example, we considered the case of the transfer of circular polarization of the radiation, when an electron beam head-on scatters on a short circularly polarized pulse, with the conclusion that the shortness of the pulse implies a slightly lower degree of polarization as compared to the monochromatic-field case. However, much higher degrees of polarization are observed for the photons emitted approximately along the initial direction of propagation of the electrons, in agreement with angular momentum conservation. Finally, we considered the case of a bent crystal and showed how one can calculate the degree of polarization of the positron beam for an arbitrary bending radius of the crystal.

VII. ACKNOWLEDGEMENTS
For T. N. Wistisen this work was supported by the Alexander von Humboldt-Stiftung.