Ponderomotive effects in multiphoton pair production

The Dirac-Heisenberg-Wigner formalism is employed to investigate electron-positron pair production in cylindrically symmetric but otherwise spatially inhomogeneous, oscillating electric fields. The oscillation frequencies are hereby tuned to obtain multiphoton pair production in the nonperturbative threshold regime. An effective mass as well as a trajectory-based semi-classical analysis are introduced in order to interpret the numerical results for the distribution functions as well as for the particle yields and spectra. The results, including the asymptotic particle spectra, display clear signatures of ponderomotive forces.


I. INTRODUCTION
Electron-positron pair production in strong electric fields, the Sauter-Schwinger effect, is a long-standing theoretical prediction [1] which still awaits experimental verification. Multiphoton pair production, on the other hand, has already been observed in a laboratory [2]. The dynamically assisted Sauter-Schwinger effect [3,4] exploits the idea that a combination of a low with a high frequency ("multi-photon") laser pulse will lead to pair production rates which are significantly larger than the sum of the rates for the two separate pulses. Together with the arising capabilities of high-intensity laser technology, see, e.g., refs. [5,6], such a combination of laser pulses will make experimental tests in this regime of nonlinear QED in the near future possible.
Besides on the technological progress future experimental tests and their interpretations will depend substantially on more reliable calculations which include the inhomogeneities of the electromagnetic fields as they typically occur in the focus of crossing laser beams. Computations for inhomogeneous and time-dependent fields (i.e., beyond the previously well-studied case of timedependent but homogeneous electric fields) have been started a few years ago and have seen steady progress [7][8][9][10]. These studies are, however, still far from providing a satisfactory understanding of the effects which originate from the finite spatial extension of the considered laser pulses.
As the particle creation rate depends on the laser intensity, see, e.g., ref. [11] for a discussion in the context of the planning of the XFEL, one would naïvely expect that better focusing and therefore higher local field intensities * christian.kohlfuerst@uni-jena.de † reinhard.alkofer@uni-graz.at lead always to an increase of the particle yield. However, the dynamics in high-intensity laser fields is much more complex and results not only in strong but also non-monotonic dependencies of the yield on the characteristics of the laser field [4,8,10]. Other quantities like, e.g., distribution functions and particle spectra, display a rich structure with sometimes surprisingly large sensitivities to small changes of the field parameters. Clearly such a situation calls for the search of concepts which are able to allow an, at least qualitative, understanding. One possibility to quantify the effects associated with a spatially inhomogeneous field is based on the notion of an effective mass which an electron acquires in a background electromagnetic field [12][13][14][15][16][17][18]. This parameter, as every effective mass, reflects the "integrated" collective interactions of a particle with its surroundings. It thus provides the possibility of a drastic simplification but nevertheless allows the coarse-grained description of highly intricate effects. Although this might be an oversimplification with respect to details of the resulting spectra, the concept of an effective mass works astonishingly well, a fact which can be attributed to the unique conditions in high-intensity laser experiments [19]. Correspondingly, the idea of an effective mass has been applied recently to multiphoton pair production [20,21]. In these studies, however, the employed electric fields were homogeneous thereby greatly simplifying the process under investigation.
In the following we will discuss particle creation in inhomogeneous fields and introduce to this end a more general concept for the effective mass and relate it via a semi-classical analysis to ponderomotive forces. (NB: A ponderomotive force is a nonlinear force that a "classical" charged particle experiences in an inhomogeneous oscillating electromagnetic field, cf. ref. [22] and references therein.) We will concentrate in a first step hereby on multiphoton pair production in the nonperturbative threshold regime. As is evident from the dis-cussion above, in the parameter regions of interest the effects of a laser pulse's finite spatial extension cannot be neglected. (For a more detailed discussion of this issue, see, e.g., ref. [23].) Hereby one should distinguish two aspects: First, how does the finiteness of the fields' extension influence the pair production process, and second, how does a spatially inhomogeneous field alter the subsequent electron/positron dynamics. To keep the calculational complexity in a manageable range we will assume cylinder symmetry of the electric field. The spatial dependence can be inferred, e.g., from a gauge potential which is directed along a direction and is inhomogeneous only w.r.t. the same direction. This has the advantage that the corresponding magnetic field vanishes. However, one then clearly has no propagating waves. Nevertheless, such a configuration provides some, although simplistic, model of the focus of two counter-propagating lasers.
This paper is organized as follows: In Sect. II several aspects of multiphoton pair production in the nonperturbative threshold regime underlying our work will be briefly summarized: In subsection II A a short description of the Dirac-Heisenberg-Wigner (DHW) formalism for the case of an electric field with cylinder symmetry is provided; subsection II B deals with the numerical treatment via Fourier transform; and in subsection II C the form of the electric field employed later on is given. In sect. III two interpretational concepts are introduced and discussed: the effective mass and a trajectory-based semiclassical analysis. In sect. IV numerical results, ordered according to the longitudinal and transverse momentum distributions, are presented and discussed. Sect. V contains the conclusions of the presented study.
Furthermore, we use = c = 1 throughout this paper.

A. DHW Formalism
The present study is based on the DHW formalism, a relativistic phase-space approach, which has been developed for the case of pair production in refs. [24]. Hereby the electron is treated as a quantum field but the laser pulse is approximated by its mean-field. Given the magnitudes of the electric field needed in pair production this is a justified approximation.
A convenient starting point is provided by the gaugeinvariant density operator of the system, C αβ (r, s) = U (A, r, s) ψ β (r − s/2) , ψ α (r + s/2) , (1) in terms of the electron's spinor-valued Dirac field ψ α (x). Hereby r denotes the center-of-mass and s the relative coordinate. The Wilson line factor U (A, r, s) = exp i e s is introduced to render the density operator gaugeinvariant. It depends on the elementary charge e and the background gauge field A. Treating the background field in Hartree approximation, i.e., no path ordering is needed, and in a given Lorentz frame and gauge the background gauge field A (x, t) is a fixed cnumber valued function. The covariant Wigner operator, reflects thus the quantum fluctuations of the electron but not the one of the laser field. The main implication of the Hartree approximation for the electromagnetic field becomes evident when taking the vacuum expectation value of the covariant Wigner operator to obtain the covariant Wigner function In its equation of motion the electromagnetic field can be factored out, e.g., which allows to resolve the otherwise infinite BBGKY hierarchy of correlation functions. Being a Dirac-matrix valued quantity the Wigner function is best decomposed into 16 covariant Wigner coefficients (7) where the corresponding transformation properties are made evident by the notation. Working in a definite frame it proves advantageous to project on equal times which yields the equal-time Wigner function and by an analogous decomposition to eq. (7) the corresponding equal-time Wigner coefficients ×, Ô, Ú 0,x,y,z , . . ..
Exploiting cylindrical symmetry, which will be assumed throughout the following, and keeping the electric field homogeneous in transversal direction results in a reduced system of differential equations 1 : 1 The coefficients given in the following are obtained by linear combinations of the equal-time Wigner coefficients. The quantity Ú, for example, is defined as a linear superposition of Úz and tensor components Øxz and Øyz. The details of the derivation can be found in ref. [23].
where the pseudo-differential operator D t reads We refrained from putting a spatial index on the electric field E as by construction it is oriented in the z-direction. The advantage of a representation with Wigner coefficients lies in the fact, that it allows to identify × as mass, Ú 0 as charge and Ú as current density [24]. In order to perform calculations within the DHW formalism we employ vacuum initial conditions: where ω = ω (p z , p ρ ) = m 2 + p 2 z + p 2 ρ is the oneparticle energy. It is convenient to subtract these initial conditions, and therefore we define where Û is a placeholder for any Wigner component.
Additionally, we define the particle number density per unit volume in momentum space Consequently, the total particle yield per unit volume is defined via N = dp z dp ρ N (p z , p ρ ) .

B. Fourier transform and numerical treatment
Eqs. (9)- (12) can be solved numerically without any further approximations or truncations [8,23]. The challenging part is the non-locality of the pseudo-differential operator (13) which can be nevertheless treated if the electric field E (z, t) can be Taylor expanded and integrated sufficiently efficient. The differential operator (13) splits naturally in two parts: (18) To apply the operator ∆ on the (subtracted) Wigner components we Fourier transform and inverse Fourier transform ∆Û v w.r.t. to the variable p z , i.e., we employ Taylor expand the electric field, use and resum then to obtain the generic form (for more details see ref. [23]): . (21) Due to the fact, that a Fourier transform takes into account all points in a domain, the introduction of global basis functions turned out to be favorable compared to the finite-difference-method used in ref. [8]. Hence, we have equidistantly discretized spatial and momentum directions, respectively, turning the system of partial differential equations (9)-(12) into a high-dimensional (N z ×N pρ ×N pz ) system of ordinary differential equations with t as the only continuous parameter.
Furthermore, we choose periodic boundary conditions in z and p z , and respectively. Additionally, we set These choices do not influence the numerical results as long as the chosen discretized domain is sufficiently large, and the number of grid points in every direction is sufficiently high.
C. Model for the fields As stated above, if the spatial dependence of the electric field is inferred from a gauge potential which is directed along a direction and is inhomogeneous only w.r.t. the same direction the corresponding magnetic field vanishes. In addition, two time scales are needed to tune a pulse of finite duration to the multiphoton regime. These requirements are fulfilled by the ansatz for t ∈ [−πτ /2, πτ /2], and E = 0 otherwise. Hereby the critical field strength E 0 = m 2 /e has been factorized out for convenience. Non-perturbative multiphoton pair production is probed if one choses the product ωτ > 1 and a Keldysh parameter of γ = ω/mε > 1 [23].
As we want to investigate how focusing influences the particle distribution rate we chose a well-localized electric field with a Gaussian shape of width λ:

III. SEMI-CLASSICAL ANALYSIS
In Sect. IV it will become obvious that the dependencies of observables on the field parameters are of an astonishing complexity. In order to obtain an interpretation and such an understanding of the results we will analyze them referring to the concepts developed in this section.
Introducing a generalized effective mass concept and relating it to arising ponderomotive forces is a central aspect of this paper. We discuss the improvements compared to previous definitions of the effective mass in the context of pair production in section III A. Moreover, based upon refs. [22,27], we draw a connection between a spatially dependent effective mass and ponderomotive forces. However, an interpretation of the final particle distribution on the basis of an effective mass becomes more involved for spatially inhomogeneous fields due to the position-dependence of the gradient. Hence, we rely on a semi-classical trajectory-based model, which allows us to determine the overall scheme by simple means.

A. Effective mass and ponderomotive forces
Various studies have used effective masses to simplify intermediate calculations [14,15,28,29]. It was suggested only recently to employ the concept of an effective mass to determine directly observable quantities regarding particle creation, see ref. [20].
In case of a monochromatic plane wave, the effective mass takes the form [12]: More general definitions have been proposed in ref. [29]. We, however, adopt essentially but modify slightly the definition above and parameterize the effective mass as follows Similarly to refs. [20,29], we cope with the temporal finiteness of the pulse by averaging over one field oscillation around t = 0 only; this approximation is well justified for long, flat-topped multicycle pulses due to the minor influence of the envelope function.
The relativistic ponderomotive force then yields, see ref. [22], where γ 0 is the Lorentz factor and v 0 denotes the velocity of the quasi-particle.
We do not use equation (30) directly to calculate reference values. However, as equation (30) describes an effective force, where all short scale contributions are "integrated out", it primarily serves as a tool helping to interpret the results obtained from solving the system (9)- (12). Analyzing equation (30) analytically, we can deduce that the term ∇ x m * is the decisive factor in order to understand the effective force on the particles. In turn, we expect that all particles are forced from strong-field towards weak-field regions. Furthermore, in the case under consideration we primarily expect a boost in the parallel momentum. If particles are created with vanishing transversal momentum, ponderomotive forces only act upon them in direction parallel to the applied field.

B. Trajectory-based semi-classical model
The virtue of the DHW method, namely to take into account various effects in one common approach, obstructs the analysis of its results. Hence, we introduce a trajectory-based model in order to overcome these difficulties.
Contrary to Schwinger pair production the formation time in multiphoton pair production is quite long. For the sake of simplicity, we still want to assume that particles are created at points in space-time, where the electric field takes on its maximal values. 2 Moreover, for the case of n-photon pair production we expect that the initial particle momenta p can be derived via energy conservation laws [20,30]: These assumptions are sufficient to determine the trajectory of a particle in an external field as they provide all needed initial conditions. Due to the form of the electric field and especially due to the absence of a magnetic field, spin-effects will be ignored at this point. Hence, we employ the relativistic Lorentz equation to analyze the particle's trajectory in the external field.
Here, F µν is the electromagnetic field strength tensor, U ν the four-velocity and P µ the four-momentum. This method is a convenient yet powerful tool to analyse the dynamics of pair production. However, it should be understood as a simple approximation, which clearly cannot replace a full quantum field theoretical treatment of the process. In ref. [23] its usefulness is demonstrated for a variety of field configurations.

IV. NUMERICAL RESULTS FOR MULTIPHOTON PAIR PRODUCTION IN THE THRESHOLD REGIME
In the following we discuss the solutions of the PDEs (9)- (12) and compare the outcome with the results obtained from the trajectory-based approach. We analyze the distribution function for parallel particle momenta and subsequently the total production rate.
For the pulse length we have chosen a value of τ = 100 m −1 for all calculations. Such a pulse length is sufficient to capture all essential features of multiphoton pair production.

A. Parallel momentum distribution
In this subsection we only analyze particle spectra where p ρ = 0. In Fig. 1 a typical spectrum for multiphoton-dominated particle creation is displayed. Especially for quasi-homogeneous fields, λ = 1000 m −1 , the characteristic multiphoton peaks are easily distinguishable. Here, the peak in the middle stems from a 3-photon process and the side maxima are related to 4-, and 5-photon pair production.
However, comparing the relative peak sizes with results obtained from calculations within homogeneous fields one finds that the strong peak around p z = 0 is now more pronounced. Nevertheless, we recover the quantum kinetic limit [31] when evaluating the results for λ → ∞ at z = 0.
Going beyond this limit reveals that a decrease of the pulse's spot size leads to dramatic changes in the distribution function. In case of, e.g. λ = 10 m −1 , the dominant peak in the particle momentum spectrum takes on a much wider form which can be related to quantum interferences. It is known from atomic ionization [32] that n-photon peaks can be interpreted as a result of particle trajectories adding up to the interference pattern observable. Around λ = 10 m −1 the finite size of the laser pulse seems to prevent a coherent superposition. In turn, the corresponding interference pattern become disturbed resulting in the broadened distribution. A quantitative can be related to n-photon pair production. The stronger the focus the stronger the ponderomotive forces, thus the higher the final particle momentum.
comparison between the trajectory-based approach and the DHW formalism is summarized in Tab. I. At an even smaller focus size, λ = 2.5 m −1 , we observe so-called peak splitting. This effect can be understood in the context of ponderomotive forces. Concentrating on the 3−photon peak, particles are created with close to vanishing momentum p z and subsequently follow the oscillations of the background field. Due to the spatial inhomogeneity these particles then drift to low-intensity regions in space. As the applied electric field is symmetric in z there are, however, two equally likely options: either the particles are accelerated in z or in −z direction. Hence, the two peaks at nonzero final momentum. I: Trajectory analysis of particles within an external field. The results are obtained by evaluating the relativistic Lorentz force equation for particles seeded at t0 = 0, z0 = 0 in a field of strength ε = 0.5, length τ = 100 m −1 , frequency ω = 0.7 m and spatial extent λ. The different initial momenta pz,0 correspond to different n-photon processes. The final momenta p z,f are obtained at asymptotic times. For comparison we provide the results from a DHW calculation pDHW. Particles with non-zero initial momenta acquire only an additional push due to ponderomotive forces. The strength of this boost depends on the initial conditions, but seems to be non-monotonic following the results in Tab. I. We interpret the increase in momentum as a consequence of particle acceleration out of the strong field region within one half-cycle. If the spatial extent of the background field is sufficiently small these particles are basically unaffected by the following field oscillations and keep their respective momentum.
A finite spatial extent also affects the particle production rate. Fig. 2 shows the reduced particle yield for different photon frequencies and various spot sizes. In the case of 3-photon pair production (blue line in Fig. 2) the introduction of a spatial extent only lowers the particle production rate. However, in case of photon frequencies of ω = 0.7 m and a field strength of ε = 0.5 a 3-photon pair production process is not possible due to energy conservation laws, see equation (31) and ref. [20]. The fact, that we still observe a peak at p z ∼ 0 in the particle spectrum, see Fig. 1, is a hint towards a dynamically assisted tunneling process, where absorbing 3 photons lowers the energy barrier and, in turn, increases the likelihood of a tunneling process to happen. Additionally, the 4-photon creation process contributes towards the yield in Fig. 2 (dashed red line). All in all, it seems as if especially particles created via the assistance mechanism [4] benefit from a small spot size and the resulting strong ponderomotive forces leading to an overall increase in the reduced yield. Concerning the drop-off for small values of the spatial extent λ: If the spatial extension of the field is of the order of the Compton wavelength the total electric field energy becomes too small to produce a sizable amount of particles [8].

B. Transversal momentum distribution
Following the discussion on particle acceleration parallel to the applied field we now turn our attention towards electrons/positrons, which are created with nonvanishing transversal momentum. For demonstration purpose we have chosen a slightly higher field frequency. In this way, the implications of ponderomotive forces in the particle momentum spectrum become more evident. The Keldysh parameter for the field used is γ = 1.68 indicating a process in the crossover regime with multiphoton dominance [23]. The comparison between a flat (λ = 100 m −1 ) and a sharp peak (λ = 2 m −1 ) is illustrated in Fig. 3 with focus on a 3-photon process. The figure at the top shows a situation close to a result obtained via a purely timedependent electric field [21,33]. Particles created via photon absorption obtain a well-defined total momentum (31). Additionally, the resulting ring-like structure is superimposed by quantum interferences [23,30].
A smaller spatial extension of the laser focus leads to an increase in strength of ponderomotive forces, clearly visible in Fig. 3 in the form of an acceleration in p z - evaluated for particles seeded at local maxima of the field, t0 and z0 = 0, in a field of strength ε = 0.5, length τ = 100 m −1 , frequency ω = 0.84 m and spatial extent λ. The initial momenta were chosen to be pz,0 = 0.665 m and pρ,0 = 0 resembling the photon peak positions in Fig. 3. direction. In similarity to the previous case, the bunch of particles at p z = 0, p ρ = 0.7 m is boosted in either p z -direction leading to peak splitting. Moreover, the interference pattern diminishes and the sharp peaks signalizing n-photon processes are washed out. Once again, we analyze the particle trajectories to understand this line broadening. Constraining ourselves to the center of the laser pulse we seed the particles at local maxima of the applied electric field and calculate the particles final momenta in dependence of the spatial extent, see Tab. II. In the case of a broad spatial extent all particles acquire the same final momenta. In case of a small extent, however, the finiteness of the pulse plays a crucial role because then different particle creation times lead to different behaviour. Due to the fact, that we seed the particles at maxima of the field only, we obtain an upper and a lower limit for the particles' final momenta. Assuming that electrons and positrons are created also at intermediate times, we expect that their respective final momenta are distributed in between these two limits. It is therefore not surprising that the peaks in the momentum spectrum appear much wider compared to the quasi-homogeneous case.
The smaller boost for higher transversal momenta is a consequence of relativistic mechanics as can be seen by comparison with the effective ponderomotive forces (30). For the sake of completeness, we provide the data obtained through a trajectory analysis in Tab. III.

V. CONCLUSIONS
We have presented numerical solutions describing multiphoton pair production for oscillating, spatially inhomogeneous electric fields in the DHW formalism. Spatial inhomogeneities introduce effective ponderomotive forces, which directly affect the particle momentum distribution and subsequently the total yield. Moreover, we have shown that these forces can be understood via a generalized effective mass concept.
With the aid of a semi-classical trajectory-based model we produced reference values to analyse our findings regarding new phenomena connected with a finite spatial pulse size: Peak splitting and line broadening. As for the first effect, we note that the peaks split due to strong ponderomotive forces altering the particle momentum spectrum. Line broadening happens because the particle spectra properties which are characteristic for multiphoton pair production erode and instead of sharp lines one obtains broad bunches.
In summary, we presented here further evidence how important it is to take spatial inhomogeneities of the fields underlying pair production processes into account. Therefore, further investigations aiming at an understanding of matter creation from fields will have, at least, to include spatial variations of the fields. Given sufficient computational resources the DHW formalism can be readily extended such that inhomogeneous magnetic fields can be included. This would imply a major step towards understanding multiphoton pair production in realistic scenarios further closing the gap between theory and experiment.