Pair production in differently polarized electric fields with frequency chirps

Electron-positron pair production in strong electric fields, i.e., the Sauter-Schwinger effect, is studied using the real-time Dirac-Heisenberg-Wigner formalism. Hereby, the electric field is modeled to be a homogeneous, single-pulse field with subcritical peak field strength. Momentum spectra are calculated for four different polarizations - linear, elliptic, near-circular elliptic or circular - as well as a number of linear frequency chirps. With details depending on the chosen polarization the frequency chirps lead to strong interference effects and thus quite substantial changes in the momentum spectra. The resulting produced pairs' number densities depend non-linearly on the parameter characterizing the polarization and are very sensitive to variations of the chirp parameter. For some of the investigated frequency chirps this can provide an enhancement of the number density by three to four orders of magnitude.


I. INTRODUCTION
Electron-positron (e + e − ) pair production in strong electric fields, also known as the Sauter-Schwinger effect, is a long-standing theoretical prediction [1][2][3], which is, however, not yet experimentally verified, for a recent review see, e.g., [4]. The pair production rate is hereby exponentially suppressed and proportional to exp(−πE cr /E) as long as the electric field is of the order of or smaller than the critical field, E cr = m 2 e c 3 /e ≈ 1.3 · 10 18 V/m. The related laser intensity, e.g., I = 4.3 × 10 29 W/cm 2 for 1µm light, is beyond current technological possibilities but the progress in high-intensity laser technology [5][6][7] might make experimental tests possible in the next decade, especially in view of planned facilities as the Extreme Light Infrastructure (ELI), the Exawatt Center for Extreme Light Studies (XCELS), or the Station of Extreme Light at the Shanghai Coherent Light Source. On the other hand, the already operating X-ray free electron laser (XFEL) at DESY in Hamburg can in principle achieve near-critical field strength as large as E ≈ 0.1 E cr , see, e.g., [8]. Triggered by the technical design report of the XFEL numerical estimates of the achievable number densities and of the resulting momentum spectra have been performed within the quantum kinetic approach at the beginning of the millenium [9,10] but many studies of the Sauter-Schwinger effect based on a number of different theoretical techniques have been undertaken in the last and in this century, for a guide to the literature we refer to the recent review [4].
Amongst the contributions from theorists towards an experimental verification of nonperturbative ultra-strong field pair production the dynamically assisted Sauter-Schwinger effect [11] deserves special mentioning. It exploits the idea that a combination of a low with a high frequency laser pulse leads to e + e − pair production rates which are by several orders of magnitude larger than the sum of the rates for the two separate pulses. Herein we report on a study which extends this idea by exploiting time-dependent frequencies, i.e., frequency chirps. We focus on linear chirps but allow then for different types of polarization. For simplicity we study pair production in a single-pulse field with a Gaussian envelope: where E 0 / √ 1 + δ 2 is the amplitude of the electric field, τ denotes the pulse duration and ω the oscillation frequency at t = 0. For completeness we kept the carrier phase φ (which is known to 2 have a significant effect on the momentum spectra of the produced pair [12,13]) in this expression, however, it will be set to zero in the following. The main interest in this study is the dependence on the chirp parameter b. Note that a non-vanishing b can be interpreted as a time-dependent effective frequency, ω eff = ω + bt. The effect of the chirp parameter b on the time dependence of the electric field is displayed in Fig. 1. The parameter δ with −1 ≤ δ ≤ 1 describes the ellipticity of the electric field, δ = 0 corresponds to linear and δ = 1 to circular polarization. For the here presented study the Dirac-Heisenberg-Wigner (DHW) formalism adapted to pair production [14,15] is used. This choice is motivated due to its efficiency for calculations involving circularly or elliptically polarized electric background fields, see, e.g., [16,17], in which pair production in rotating circularly polarized electric fields has been investigated, or [18] for elliptically polarized fields.
3 At this point a remark with respect to the chosen polarizations for this study is in order. On the experimental side, due to limitations in instruments, it is much harder to produce a perfect circularly polarized field than an elliptically polarized and/or linearly polarized field. For lowintensity laser fields a polarization of up to ±0.93 have been achieved experimentally [19]. Due to this we include calculations for a near-circular elliptic polarization, i.e., for δ = 0.9.
In addition, we note that high-intensity laser pulses are also obtained through the chirped laser pulse amplification technique [20]. Therefore the study of pair production in fields with frequency chirps is well-motivated even besides the here found amplification similar to the one of the dynamically assisted Sauter-Schwinger effect.
Throughout this paper natural units = c = 1 are used. Furthermore, three of the five parameters characterizing the electric field stay fixed: where m is the electron mass. The Keldysh adiabaticity parameter is thus γ = 4.25 For the chirp parameter b we investigate several cases in the interval 0 ≤ b ≤ 0.06 m 2 , and for the polarization four different values of δ are chosen. We are aware that the pulse length is hardly sufficient to provide a clean multi-photon signal, and that a value of b = 0.06 m 2 is already to large to be classified as a "normal chirp", however, the goal of the present exploratory study is a qualitative understanding of the influence of chirps on the produced number densities of pairs and the related momenta spectra for different polarizations, and to this end the chosen parameter sets are very suitable.
This paper is organized as follows: In Sec. II we introduce briefly the DHW formalism to make the presentation reasonably self-contained. In Sec. III we present our numerical results for the number densities for different chirp parameters and different polarizations. In Sec. IV we summarize briefly for four polarizations how generic properties of the momentum spectra change for an increasing chirp parameter. In Sec. V we re-analyze the spectra within a semi-classical treatment and discuss in how far the momentum spectra can be qualitatively understood. In the last section we present our conclusions.

II. THEORETICAL DESCRIPTION: THE DHW FORMALISM
The here presented study employs the DHW formalism which is a relativistic phase-space approach. It has been further developed for the case of Sauter-Schwinger pair production in refs. [14,15]. Within this method the electron is treated as a quantum field but the laser pulse is approximated by its mean-field which is justified by the magnitude of the used electric field.
To make this paper self-contained we briefly review the formalism. To this end we start from the gauge-invariant density operator of the system, in terms of the electron's spinor-valued Dirac field ψ α (x), and r denotes the center-of-mass and s the relative coordinate. The Wilson line factor U (A, r, s) = exp i e s renders the density operator gauge-invariant. Note that this factor depends on the elementary charge e and the background gauge field A. The background field is treated in mean-field (Hartree) approximation, i.e., and, because in a given Lorentz frame and gauge the background gauge field A (x, t) is a fixed c-number valued function, no path ordering is needed. The covariant Wigner operator, thus includes the electron's quantum fluctuations but not the one of the electric field.
The simplification introduced by the mean-field approximation for the electromagnetic field becomes apparent when one considers the vacuum expectation value of the covariant Wigner operator to obtain the covariant Wigner function W (r, p) = Φ|Ŵ (r, p) |Φ .
In the equation of motion of this correlation function the electromagnetic field factors out: This in turn allows to terminate the in general infinite hierarchy of correlation functions.

5
As the Wigner function is a Dirac-matrix valued quantity it can be decomposed into 16 covariant Wigner coefficients Hereby, the related spin and parity properties are made evident by the notation. As the modeling of the electric field already indicates we work in a definite frame. Correspondingly, one can project on equal times which yields the equal-time Wigner function and by an analogous decomposition to eq. (9) the corresponding equal-time Wigner coefficients s, p, v 0,x,y,z etc..
As the equations of motions for the Wigner coeffecients are quite lengthy we refrain from repeating the respective formula here. Their explicit form as well as detailed derivations can be found in [15,21]. A decisive advantage of employing Wigner coefficients is given by the relation of s with the mass, of v 0 with the charge, and of v as current density in the case without electric field [14,15]. Correspondingly, one chooses vacuum initial conditions as starting values. The non-vanishing values are In general, the equations of motions for the Wigner coefficients are integro-differential equations. Their numerical solution is due to the non-local nature of the respective pseudo-differential operators very challenging, see, e.g., [15,[21][22][23]. For the homogeneous electric field (1) studied here these equations can be reduced to ordinary differential equations [16]. To this end we note first that then at most ten out of the sixteen Wigner coefficients are non-vanishing: Second, the kinetic momentum p is related to the canonical momentum q via and thus time-dependent. In a next step one expresses the scalar Wigner coefficient by the oneparticle distribution function f (q, t). The latter is related to the phase-space energy density, Hereby, Ω(q, t) = p 2 (t) + m 2 = m 2 + (q − eA(t)) 2 is the electron's (resp., positron's) energy.
In addition, it is helpful to define an auxiliary three-dimensional vector v(q, t): The one-particle momentum distribution function f (q, t) can be then be obtained by solving the following ten ordinary differential equations for f (q, t) and the nine auxiliary quantities where as usual the dot is a shorthand for the time derivative. Together with the initial conditions

this set of equation is a well-defined and
numerically straightforward solvable initial value problem.
The number density of created pairs is obtained by integrating the distribution function f (q, t) over all momenta at asymptotically late times t → +∞:

III. NUMERICAL RESULTS FOR NUMBER DENSITIES
As already stated the carrier phase is chosen to be φ = 0 leaving studies similar to the one presented here but with non-vanishing carrier phase for future investigations. Herein, we examine the main results for the number density of the produced particles for several chirp parameters for the different polarization.
The number densities as a function of the polarization parameter δ are shown in Fig. 2. They clearly display the expected symmetry when mirroring δ → −δ. This then implies that the case of linear polarization, δ = 0, provides an extremum in the number density. As one immediately sees from Fig. 2 the maximum which is present at small chirps becomes a minimum at larger values First, with increasing chirp the relative variation in the number density becomes much smaller.
For vanishing chirp the ratio of the number density for linear polarization to the one for circular polarization is more than a factor of two. At b = 0.06 m 2 the largest number density deviates from the smallest one (assumed at δ ≈ ±0.6) by less than three per mille. Second, with increasing chirp the peak number density increases significantly. This effect is most pronounced when increasing 8 from b = 0.02 m 2 to b = 0.03 m 2 for which the number density increases by more than a factor of ten for all polarizations.  Table I.
Here, a technical remark is in order. For some large values of the chirp parameter b, e.g., around b = 0.05 m 2 , some irregular changes of the numerical results for the number densities for each polarization are observed. As these have to be very likely attributed to instabilities of the numerical procedure we are refraining from displaying these exceptional points here. Nevertheless, this issue will be clarified in future investigations.
A. Momentum spectra for linear polarization δ = 0 First of all, we note that in the case of linear polarization the electric field is oriented only along the x-axis, and the momentum spectra possess correspondingly a rotational symmetry around the q x -axis. For the linear polarized (δ = 0) pulse the momentum spectra in q x and q y for q z = 0 are plotted in Fig. 4.. For vanishing chirp, b = 0, the results agree with the ones of a previous investigation [24]. For non-vanishing chirp parameters the main result is, besides the expected lower symmetry of the spectra, the appearance of strong interference effects leading to several maxima and minima of the pair production rate as a function of momenta.
As can be seen in the upper panel of Fig. 4 small chirp parameters lead to small variations in the spectrum: There is a slight enhancement in the height of the peak, a small shift towards positive q x , and a broadening as well as the loss of one of the reflection symmetries of the peak. For larger frequency chirps, b ≥ 0.02m 2 , the momentum spectra display some remarkable structures as can be seen from the lower row in Fig. 4. For b=0.02m 2 the spectrum possesses two peaks at negative and positive q x , respectively. For b=0.03m 2 the main peak is located at negative q x . For b=0.04m 2 the peak goes back to positive momenta, and at b=0.06m 2 again to negative momenta. In the latter case strong interference effects are visible, note especially the ring-like structure.
In all the cases shown in Fig. 4 the spectra are symmetric w.r.t. to reflection of q y . (NB: For nonvanishing q z the spectra would be symmetric w.r.t. to a rotation around the q x -axis.) Therefore, the effects of chirps can be understood in more detail by plotting the number density for q y = q z = 0 as a function of q x . In the left panel of Fig. 5(a) one sees that for vanishing chirp the peak is located at q x = 0. As b increases, the peak is very slightly shifted to positive q x . For b=0.02m 2 one sees now not only the two prominent maxima but also additional but less pronounced ones. At b=0.06m 2 the momentum spectrum displays quite complicated interference patterns but note also the change in the height of the peaks, enhanced from 2.94 × 10 −5 (b = 0) to 3.7 × 10 −2 ( b = 0.06m 2 ).
As we will argue in Sec. V the effects of the frequency chirp on the spectra can be explained by a semi-classical analysis based on the WKB approximation. As the related effective potential changes with the frequency chirp the numerically observed drastic effects are plausible.
B. Elliptical polarization δ = 0.5 As a next case we consider the momentum spectra for an elliptically polarized electric field, δ = 0.5., see its direction during the pair creation process. Thus, the particles may be accelerated into different directions depending on the field direction at the time of production. These findings are very similar to the results of the strong-field ionization of helium using an elliptically polarized laser pulses [25] and the effects of electric field polarizations on pair production from vacuum [18].
Otherwise, one sees also drastic effects of the chirp for relatively small chirp parameters, and especially the ring form is distorted to a spiral one. For very large chirp parameters, on the other hand, the spectrum shows in this case less structure.
In contrast to the previously discussed linear polarization in the case of the near-circular elliptic polarization the spectra loose their symmetry in both of q x and q y . The patterns observed in Fig. 7 will become clearer when discussing the case of perfect circular polarization. For the circularly polarized (δ = 1) the spectra are shown in Fig. 8. The b = 0 spectrum, shown at the top and most left, displays a ring-shaped maximum centered around the origin. It exhibits the weak interference pattern or/and oscillation between the hole and outer ring along negative values of q y [16,18,24]. This might be interpreted within a semi-classical analysis by means of an effective scattering potential [26], see the next section. The outer ring structure results from multiphoton pair creation [16], and in the strong field limit its radius can be determined by the energy conservation to be at |q| = 1/2 (nω) 2 − 4m 2 where n is the number of photons participating in the pair creation, cf. ref. [27].
Again the spectra are very sensitive to chirps even for relatively small chirp parameters. This includes the distortion of the ring structure, the appearance of spirals, and, last but not least, a significant increase of the one-particle distribution function. The characteristic shape of the spectra could be helpful in an experimental identification of pair production. E.g., at b = 0.03 m 2 one clearly identifies an Archimedian spiral which is going to start from almost the central region slightly shifted to negative q x values. For very large chirp parameters, b ≥ 0.05 m 2 , the spiral structure is fading away, and the spectra become less structured.
To summarize this section, we have obtained quite some detailed information how the spectra of the produced pairs change for a given polarization when frequency chirps from relatively modest to quite large ones are considered. The positions of the global extrema of the one-particle distribution function, i.e., the peaks of the spectra display a quite rich structure. Common to all the considered cases is the strong increase in peak values for increasing chirp parameters which is easily understood from the effective frequency, ω eff = ω + bt, of the field, respectively, by the The peak values for the one-particle distribution function at late times f (q, ∞). Note that these peaks occur at different values of the momentum q, see the discussion above.  Table II.

V. SEMI-CLASSICAL ANALYSIS
In this section we will employ a semi-classical analysis to obtain a qualitative explanation of the effects of chirps on the spectrum. Hereby we follow the WKB method outlined in ref. [28].
These authors considered a Sauter pulse, i.e., a gauge potential A(t) ∝ tanh ωt, for which the turning points of the analogue semi-classical scattering potential can be determined analytically by solving the condition Ω(q, t p ) = m 2 + (q − eA(t p )) 2 = 0. In this case a single complexconjugated pair dominate, and the spectra of the produced particles can be qualitatively explained (see Fig. 3 in Ref. [28]). The WKB result for the created number of pairs of momentum q is hereby given by where t 1 and t 2 are dominant turning points closed to the real t axis.
In case of A(t) ∝ 1/(ω(1 + ω 2 t 2 )) one obtains two pairs of complex turning points which then explain interference effects [29]. The production rate was then estimated to be a sum of two terms which takes the form where the + and − signs refer to bosonic and fermionic pair production, respectively. Hereby, the K 1,2 are obvious generalizations of the K above, and α represents some respective phase, for details see ref. [29]. The interference term in Eq. (20) is responsible for the oscillations in the spectra. As the integrals K, respectively, K 1,2 , are almost linear in t the dominant contribution originates from the terms involving turning points which are closest to the real t axis.
For the gauge potential of the electric field given in eq. (1) there exists an infinite number of complex turning points which can be obtained only numerically. In addition, as for non-vanishing chirp parameters the momentum spectra are due to the strong interference effects peaked at quite different momenta we follow here ref. [26] and select for each parameter set representative values for the momenta to evaluate the turning points.
For the case of linear polarization, these turning points are shown in Fig. 9. For circular polarization the turning points are depicted in Fig. 11 for several values of chirp parameters. As we can see, for b = 0, one of the important difference from the cases of linear polarization and elliptic polarization shown in Fig. 9(a) is that now the dominant contribution comes from the two central turning points because they are equally distant from the real axis. The two turning point pairs explain the (weak) interference effect observed for circular polarization and b = 0. Increasing now the chirp parameter leads now to quite a distinctive pattern of turning points as for the other two polarizations, and explains thus the differences in response to the chirp parameter for circular versus the linear or a more general polarization.

VI. SUMMARY AND CONCLUSIONS
Within the real-time DHW formalism we studied the effect of linear frequency chirps for four polarizations, namely linear, elliptic, near-circular elliptic and circular polarization. The main results for the number densities and spectra of produced pair can be summarized as follows: For an electric field with linear polarization field the produced pairs' spectra exhibit a shift and split of peaks as well as strong interference effects as the chirp parameter increases. The most complex pattern for increasing chirps occurs, not unexpectedly, for an elliptic polarization. For the near-circular elliptic and the circular polarization the ring form of the spectrum present at vanishing chirps is distorted, spiral structures appear and eventually, for very large chirp parameters, the peak is shifted to the central region.
The most important finding, however, is the very strong increase in number densities when the chirp parameter is increased. For vanishing and small chirps we have verified the known differences in number densities for different polarizations, with the largest number density achieved by a linearly polarized field. Also quite unexpected, this effect goes away for larger chirps, and the number densities for different polarizations become degenerate.
In this exploratory study we restricted ourselves to a quite large value of the electric field and a quite short pulse duration. To verify or falsify a possible interpretation of the increase in the number of produced pairs in terms of an onset of multi-photon pair production, a study with much longer pulses would be necessary. Given the steep increase in the produced number pair densities and the related improved potential for an experimental observation the effort of a study employing smaller values of the electric field and much longer pulse times is certainly justified. 20