Phase-space analysis of the Schwinger effect in inhomogeneous electromagnetic fields

Phase-space analysis of the Schwinger effect in
inhomogeneous electromagnetic fields

Christian Kohlfürst C.Kohlfuerst@gsi.de Helmholtz-Institut Jena, Fröbelstieg 3, 07743 Jena, Germany Theoretisch-Physikalisches Institut, Abbe Center of Photonics,
Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany
August 2, 2019
Abstract

Schwinger pair production in spatially and temporally inhomogeneous electric and magnetic fields is studied. The focus is on the particle phase-space distribution within a high-intensity few-cycle pulse. Accurate numerical solutions of a quantum kinetic theory (DHW formalism) are presented in momentum space and, with the aid of coarse-graining techniques, in a mixed spatial-momentum representation. Additionally, signatures of the carrier-envelope phase as well as spin-field interactions are discussed on the basis of a trajectory-based model taking into account instantaneous pair production and relativistic single-particle dynamics. Although our simple semi-classical single-particle model cannot describe every aspect of the particle production process (quantum interferences), essential features such as spin-field interactions are captured.

I Introduction

The creation of matter via light is one of the most striking features of strong-field QED Schwinger:1951nm (); Heisenberg:1935qt (); Sauter:1931zz (). Even though multiphoton pair production has been measured experimentally Burke:1997ew (), the prominent Schwinger effect still waits for an experimental verification Heinzl:2008an (); Ringwald:2001ib (). The recent effort, however, that is put into the research field, c.f. upcoming laser facilities Lasers (), could bring it closer to detection Marklund:2008gj (); mStar:FELs (); Turcu:2015cca ().

Due to the advent of lasers that can probe the relativistic regime, there has been a substantial activity in studying strong-field QED in recent years Dunne:2004nc (); review (). With the recent advances of a measurement of light-by-light scattering of quasi-real photons Light () the research field is expected to attract even more attention. In this regard, Schwinger pair production represents the perfect show case, because it is a non-perturbative effect that inevitably unites the highly relativistic regime with the quantum regime.

Moreover, theoretical approaches that have been initially developed in the last century, improved substantially in the last twenty years. In turn, this progress paved the way for investigations that simply were not possible with previous techniques Hebenstreit:2011wk (). Especially the broad usage of contemporary numerical methods is worth mentioning here, because it enabled to establish kinetic theories Kluger:1992md (); Rafelski:1993uh (); Vasak:1987um (); BB (); Smolyansky:1997fc ().

In order to lay a solid foundation for the discussion on the Schwinger effect we employ the Dirac-Heisenberg-Wigner (DHW) formalism Hebenstreit:2008ae (); Vasak:1987um (); BB (). The main advantage of the DHW formalism is that it automatically combines quantum electrodynamics with notions familiar from statistical physics Akkermans:2011yn (); Hebenstreit:2009km (). Its versatility allows to incorporate temporal Smolyansky:1997fc (); Temp (); Blinne () as well as spatial inhomogeneities Kohlfurst:2015zxi (); Hebenstreit:2011wk (); Berenyi:2013eia (); Kohlfurst:2015niu (). On top of that, the DHW formalism gives access to the complete phase-space distribution of the created particles. In the present work, we take advantage of this feature to calculate various particle distributions without being limited to a momentum or a spatial representation. Nevertheless, we also search for characteristic signatures in the particle momentum spectrum.

Performing simulations for the complete phase-space includes solving a coupled system of partial differential equations incorporating, at least in principle, an infinite series of differential operators. Hence, to perform this ambitious task advanced numerical methods are required. In Sec. III.1 we shortly summarize the most important aspects of the DHW formalism and state the equations of motion. We proceed by presenting the solution strategies applied to the problem, where the focus is on providing a detailed insight into the technical implementation, see Sec. III.2. In Sec. III.3 we introduce coarse-graining methods and discuss their advantage when analyzing phase-space quasiprobabilities.

In addition, we perform a comparison with an effective theory for the particle production rate in Sec. IV. The creation of particles is investigated in analogy to the formation of quark pairs via constant chromoelectric fields, cf. the flux-tube model in Refs. Casher:1979gw (). As a result of further particle interactions with the external field, we obtain an approximate distribution function in phase-space. The big advantage of this approach is that we get an analytic estimate for the production rate and a direct access to the particle dynamics as we can switch on/off any forces and interaction terms. To put it simple, it gives the opportunity to understand the outcome of a complex quantum field theoretical investigation on the basis of a semi-classical model.

Furthermore, this comparison facilitates to acquire a more comprehensive picture of the Schwinger effect accompanied by the introduction of a comparatively simple model for the field. The pair production process is then studied and interpreted on the basis of a phase-space approach and a semi-classical effective action approach. Despite the simplicity of the employed electric and magnetic fields, nontrivial particle distributions emerge, which, in turn, can be well understood in terms of a semi-classical picture.

We exemplarily calculate the phase-space distribution of electrons and positrons in a mixed spatial-momentum representation in Sec. V.1. We demonstrate, that by applying coarse-graining techniques the interpretability of the data can be greatly improved. In Sec. V.2 we thoroughly examine particle distributions in momentum space. The focus is on comparing strongly inhomogeneous fields with spatially nearly homogeneous fields. We further study the influence of strong magnetic fields on the particle distribution, see Sec. V.3. Simultaneously, we investigate the impact of the carrier envelope phase of few-cycle pulses analyzing various field configurations and discussing the emerging interference patterns.

Ii Schwinger pair production

The focus of this work is entirely on Schwinger pair production; separating virtual charged particles with the aid of an electric field to create real matter Dunne:2004nc (); Sauter:1931zz (). The relevant scale of the process is given by the mass of the participating particles; Compton time and Compton length of an electron are s and m, respectively. Moreover, as the field has to provide the rest energy of the particle-antiparticle pair, the field strengths needed are of the order of

(1)

The picture given above is only valid, if the temporal variation of the field is much smaller than the duration of the formation process, . More specifically, in order to avoid undesirable artifacts due to absorption processes (multi-photon pair production Kohlfurst:2014 (); Ruf:2009zz (), dynamically assisted pair production PhysRevLett.101.130404 ()) we have to make sure, that the photon energies are sufficiently small. However, an investigation of Schwinger pair production within a phase-space approach using laser pulse lengths ( as ) is computationally expensive Blinne (). Hence, we introduce a model for the field, that describes a given realistic situation reasonably well still capturing all essential features.

The corresponding model of choice is given by

(2)

Here, determines the electric field strength, sets the temporal scale and specifies the spatial scale. The parameters and control the pulse structure. Specifically, the parameter should not be confused with a photon energy. Rather it should be seen as a control parameter ensuring a few-cycle pulse and determining the ratio between electric and magnetic field strength.

Within this work, the electric and magnetic fields are derived from Eq. (2) reading

(3)
(4)

In this way the homogeneous Maxwell equations are automatically fulfilled and both fields fall off sufficiently fast at asymptotic times, see Fig. 1 and Fig. 2. Moreover, it is convenient to work with fields of the form of Eqs. (3) and (4), because the relevant phase-space can be drastically reduced. To be more specific, particle dynamics can be confined to the -plane (by fixing ). Additionally, the magnetic field as well as the particle spin point in -direction, thus they can be treated as scalar quantities Kohlfurst:2015zxi ().

Figure 1: Density plot of an electric field (top) and its corresponding magnetic field (bottom) as functions of space and time for . This configuration features a single prominent peak in the electric field. Particles created at the main peak are exposed to steep field gradients of the magnetic field. Further parameters: , , and .
Figure 2: Density plot of an electric field (top) and its corresponding magnetic field (bottom) as functions of space and time for . This configuration features two peaks of equal strength in the electric field. The magnetic field is strongest between these peaks. Further parameters: , , and .

Iii Phase-space formalism

The main advantage of the DHW formalism is its generality as it incorporates all features of quantum field theory. The downside is that only few analytical results are known up to date, thus one has to implement a numerical scheme in order to solve the governing equations Berenyi:2013eia (); Kohlfurst:2015zxi (); Kohlfurst:2015niu (); Hebenstreit:2011wk (). This, however, turns out to be only feasible for selected field configurations due to limitations in available computer power.

iii.1 Theoretical foundations

The DHW formalism has been developed in Refs. Vasak:1987um (). Moreover, in Refs. BB (); Kohlfurst:2015zxi () one can find additional information covering all essential features of quantum kinetic approaches. Nevertheless, we want to allow for a gentle introduction into this work. Hence, we define the covariant Wigner operator and proceed going through all the important steps, that are necessary to obtain the equations of motion in the end. Throughout this paper we use natural units and express all quantities in terms of the electron mass. In the following, we use for the mass term.

In principal, a phase-space approach can take into account all quantum effects on the basis of the full QED Lagrangian

(5)

where and . We have introduced the vector potential , the electromagnetic field strength tensor and the spinor fields and .

The foundation of this phase-space approach is the density operator

(6)

supported by the Wilson line factor ensuring gauge invariance

(7)

Here, we introduced the center-of-mass coordinate and the relative coordinate . A Fourier transform with respect to the relative coordinate then yields the covariant Wigner operator

(8)

In order to derive the equations of motion in the DHW formalism we combine the Dirac equation

(9)
(10)

with derivatives of the Wigner operator (8). In turn, we obtain two coupled operator equations

(11)
(12)

with the pseudo-differential operators

(13)
(14)

Before we proceed by taking the vacuum expectation value of Eqs. (11) and (12), we implement a simplification of Hartree type (mean-field)

(15)

transforming the operator-valued electromagnetic field strength tensor to a C-number field. Hence, terms of the form simply become

(16)

As a result, we obtain an equation of motion for the covariant Wigner function

(17)

and subsequently, as we are interested in a time-evolution formalism, for the equal-time Wigner function

(18)

In order to make calculations in inhomogeneous fields feasible we have to reduce the available phase-space. Hence, we abandon the idea of deriving a complete -dimensional formalism and concentrate on a quasi -dimensional subspace instead. This greatly simplifies the following derivations as we are now allowed to work with a reduced Wigner function . Decomposing it into Dirac bilinears yields111The non-consecutive indices are chosen to put emphasis on the idea of working in a subspace of the whole phase-space, see Refs. Hebenstreit:2011wk (); Kohlfurst:2015niu (); Kohlfurst:2015zxi () for comparison.

(19)

Following Ref. BB () we can interpret as mass density, as charge density and as a current density vector. The definition above represents only one of various possible reductions corresponding to a -spinor derivation of the Wigner function using a QED Lagrangian as basis, cf. Ref. Kohlfurst:2015niu (); Kohlfurst:2015zxi (). In this case, the underlaying -spinor formulation inherently favors one spin direction.

Eventually, we obtain a coupled set of equations of motions for the Wigner coefficients, cf. Refs. Kohlfurst:2015niu (); Kohlfurst:2015zxi (),

(20)
(21)
(22)
(23)

with the pseudo-differential operators

(24)
(25)
(26)
(27)
(28)

As we have restricted the available phase-space volume to the -plane, spatial and momentum vectors are given via

(29)

In the following we incorporate vacuum initial conditions

(30)

into the system of equations (20)-(23) by switching to modified Wigner components Hebenstreit:2011wk ()

(31)

In this way, Eqs. (20)-(23) are turned into a set of inhomogeneous partial differential equations. The particle number density

(32)

as well as the charge density

(33)

are defined for asymptotic times ().

Furthermore, the particle’s momentum distribution per unit volume

(34)
(35)
(36)

as well as the particle’s position-momentum distribution per unit volume

(37)

are derived from Eq. (32).

iii.2 Solution strategy

The equations of motion (20)-(23) are solved numerically in the vicinity of background fields given by Eqs. (3) and (4). As the fields are homogeneous in , the domain is three-dimensional. No further truncation is applied, cf. see Refs. Kohlfurst:2015niu (); Kohlfurst:2015zxi () for alternative strategies. Nevertheless, we enhance numerical stability at reduced computational costs by introducing a transformation of variables Kohlfurst:2015zxi () of the form

(38)
(39)

The quantities and give the length in - and -direction, respectively. The parameters and control the strength of the deformation, with corresponding to the identity transformation. In turn, the differential operators are transformed accordingly

(40)
(41)

Due to the fact, that we have already taken care of the initial conditions (31) all reduced Wigner functions vanish for high , and . This allows us to artificially demand periodic boundary conditions in spatial and momentum coordinates, thus transforming the flat phase-space domain to a toroidal domain.

The transformed system of PDEs is then solved by taking advantage of the method of lines. More precisely, the domain in phase-space is equidistantly discretized leaving the time variable as the only continuous parameter. To account for the boundary conditions we simply set , where is the total number of grid points in . The same procedure is applied to the variables and . This discretization allows to solve the differential equation with spectral methods on a Fourier basis at given time Boyd (); Tref ().

The general procedure to calculate derivatives is then given by

(42)

where denotes a Fourier transformed quantity. Evaluating the pseudo-spectral differential operators (24)-(28) is more involved due to the appearance of derivatives as arguments of functions. That is, when the essential advantage of the pseudo-spectral method comes into play. Exemplary for any non-local terms in Eqs. (24)-(28), we write

(43)

for a generic modified Wigner component. Applying pseudo-spectral methods we obtain

(44)

Due to the special form of the vector potential (2), (i) the time dependency can be factored out and (ii) the integral can be performed analytically, eventually leading to

(45)

Hence, we have successfully transformed a non-local differential operator into a simple multiplicative factor. Computational costs can then be reduced further by applying anti-aliasing procedures, e.g. termination of the highest wave numbers Boyd (). Additionally, the inhomogeneous source terms do not need to be solved spectrally. A Taylor expansion in the momentum variables (up to eight order) turned out to be completely sufficient for the fields under investigation in this article. In order to perform the time integration, we rely on a Dormand-Prince Runge-Kutta integrator of order 8(5,3) NR ().222Technical aspects: Computations were performed on Supermicro Servers. The calculations were done in parallel cumulating in a total CPU time of ( ) and ( ), respectively. The grid size in phase-space was ( ) and ( ), respectively.

iii.3 Coarse graining

The way the Wigner function is defined, an interpretation in terms of real observables is technically only allowed if they are given either in momentum or in spatial coordinates moyal_1949 (). In case of mixed representations, the Wigner method yields only quasi-probabilities which, in turn, makes discussions generally vague. To overcome this systematic handicap, we have implemented a coarse graining technique assuming that the method-induced effects vary on a smaller scale than the underlying physics.

Coarse graining methods are an important tool in order to study, e.g., chemical processes Chemie (). Proper application of coarse-grained modeling significantly reduces the number of degrees of freedom, while the relevant information is retained. This enabled the study of time-evolutions of large complex structures, cf. polymer melting Poly () or molecular dynamics BioMol ().

A related coarse graining technique has already been introduced in Ref. Rafelski:1993uh () to study the relativistic classical limit of the DHW formalism. In this work, a Gaussian-type smearing function was introduced

(46)

with the coarseness parameters . The convolution of the Wigner function (19) with Eq. (46) then yields coarse grained versions of the phase-space functions. Moreover, it could be shown, that these functions give the correct classical limit. We expand the idea to obtain meaningful results also in the quantum regime. In contrast to the previous work, we apply the smearing function only at asymptotic times. In this way, we do not introduce further truncations, while simultaneously eliminating the constraint on working in either a spatial or a momentum representation.

Hence, we implement the coarse graining as a post-processing step within the phase-space approach. As we are mainly interested in particle distributions in -space, we first calculate and then apply the discrete smearing operator

(47)

for every phase-space coordinate . Indices run over and , where and determine the domain. Additionally, the operator is normalized to unity preserving the total yield. For the sake of simplicity, we assumed a quadratic stencil leading to and .

Iv Single-particle trajectory analysis

An alternative approach towards studying momentum resolved particle production is given via analysis of the trajectories of “randomly” created particle pairs. Here, we present a semi-classical model combining effective field theory with classical equations of motion.

Instead of dealing with the full QED Lagrangian (5) we introduce the Heisenberg-Euler Lagrangian Heisenberg:1935qt () at this point

(48)

The quantities and play a decisive role as they are connected to the Lorentz invariants

(49)
(50)

Hence, they can be expressed as

(51)

Analysis of Eq. (48) for constant perpendicular fields, , reveals three different possibilities Dunne:2004nc (): If then and pair production is not possible. If vanishes, then there are no quantum corrections at all. Only in case of the formation of particles is allowed.

As we are interested in pair production we concentrate on the case and . This automatically implies, that and . 333There has been a notational revision compared to Refs. Kohlfurst:2015niu (); Kohlfurst:2015zxi (). Nevertheless, the “effective field amplitude”, that has been introduced in these works, yields the same information. In turn, we obtain

(52)

Following Ref. Schwinger:1951nm () and assuming that is sufficiently smaller than the critical field strength we extract an estimate for the formation rate of single electron-positron pairs Casher:1979gw ()

(53)

with the transversal momentum . 444Generally, the probability for pair production to happen is given in terms of a vacuum decay rate Cohen:2008wz (); Hebenstreit:2008ae (); Narozhnyi (); Lin:1998rn (), which does not allow for a momentum resolved investigation.

The relations above have been derived for constant electric and magnetic fields. Nevertheless, these equations are known to hold also for slowly varying fields with corrections governed by , where denotes the variation scale of the field Galtsov (). Hence, having fields of the form of Eqs. (3) and (4) in mind, we will use Eq. (53) in a locally constant field approximation as a probability weight to resemble an instantaneous source term for particle production in an inhomogeneous background field

(54)

To be more specific, we first create a sample of random variables . Then we test each tuple for the likeliness of pair production at time at coordinate and initial transversal momentum by comparing with a random variable . If the coordinates are accepted for further treatment.

After the particles have been created they are deflected by the electromagnetic background field. In the simplest approximation, the electrons/positrons are assumed to follow “classical” trajectories, where, e.g., radiation effects can be ignored. Moreover, we want to assume that these particles do not interact with each other, thus allowing to inspect their trajectories one-by-one. At this point, we give up quantum mechanical phase information. Hence, we trade the ability to describe e.g. quantum interferences for an easier numerical implementation. However, this approach still helps to properly set up the parameters for a full DHW calculation.

The actual calculation of the particle trajectories is done via a modified relativistic Lorentz force equation Silenko:2007wi (); MengWen ()

(55)

where is the four-velocity, the electromagnetic field strength tensor and is an additional model-specific, spin-dependent force. Based upon the analysis provided in Ref. MengWen () we decided to choose a Foldy-Wouthuysen-like model Silenko:2007wi (). Omitting the anomalous magnetic moment of the electron we define Silenko:2007wi ():

(56)

with the kinetic momentum operator . Performing the Foldy-Wouthyusen transformation yields

(57)

with the spin operator , the polarization operator and . The time evolution equation for the momentum operator is given by

(58)

Plugging in Eq. (57) then yields

(59)

In the semi-classical limit the operators are transformed to classical quantities. Accordingly, the evolution equation for the momentum operator becomes an equation of motion for a particle in an external electromagnetic background field

(60)

with the classical spin vector and the Lorentz factor . 555 In general, the particle spin precesses in an external field. This dynamics is governed by the Thomas-Bargmann-Michel-Telegdi equation Thomas:1927yu (), which can be derived in a similar fashion. Omitting the anomalous magnetic moment the spin motion is approximately given by Silenko:2007wi ()

Due to the special form of the vector potential (2) and due to the focus on lower dimensional configurations the equations can be reduced to a system of two coupled equations. In particular the spin force term can be greatly simplified assuming that the spin term points in -direction and, hence, does not precess. Eventually, we introduce a relation between relativistic velocity and momentum to obtain a complete set of differential equations

(61)
(62)
(63)

The spin term reduces to and the Lorentz factor is given by . All in all, we have reduced the problem of determining the particle dynamics to solving an ordinary differential equation. The corresponding initial conditions are given by the coordinates of a particle created at .

Solving the equations of motion (61)-(63) for particles gives us a total of data points in phase space. As we have sampled the initial conditions randomly, meaningful statements can only be made by evaluating the data collectively. Hence, we use a kernel density estimation method parzen1962 (); Rosenblatt () in order to transfer the individual data points into a smooth density function. In this way, we can make statements on the particle distributions even allowing for a comparison with the results obtained via solving the DHW equations. As this single-particle trajectory method is not designed to study phase information it cannot describe quantum interference effects. However, we can search the particle phase-space for overlapping data points. If, for example, two particles created at totally different times are found close to each other in phase-space, we would expect quantum effects to play a decisive role.

We want to emphasize, that our semi-classical model should not be confused with semi-classical methods to compute Schwinger pair production SemiClass (). The latter uses semi-classical approximations to obtain a quantum production rate in inhomogeneous fields. Our model, however, uses the constant field production rate locally and then extracts phase-space information from the subsequent classical trajectories. In fact, both approaches can in principal be combined yielding an improved estimate for the production process.

V Results

In the following we present results obtained from numerically solving the DHW equations (20)-(23). We interpret the particle distributions and highlight features that are connected with the applied electric and magnetic fields. Moreover, we discuss the particle densities on the basis of the semi-classical trajectory model.

v.1 Coarse-grained particle distribution

Figure 3: Density plots of the particle distribution function obtained from a DHW calculation at time without (top) and with (bottom) averaging in phase space. Overall, the characteristic oscillatory pattern of the Wigner distribution vanishes upon averaging, while signatures of particle creation remain intact. Parameters: , , , and as well as and .
Figure 4: Qualitative comparison of the absolute values for the coarse-grained positron distribution (top) with a smooth single-particle positron density (bottom) where the spin . The characteristic bulks as well as the asymmetric distribution are featured in both plots. The data sets are normalized to unity. The structure at in the first plot are remnants of the full Wigner function. Parameters: , , , , and as well as and .

In Fig. 3 a comparison between the particle distribution function obtained from the DHW calculation before and after averaging is displayed. Discussing the direct output from the DHW calculation we can discriminate two different structures: small smooth lines, that form an “X” and a rapidly oscillating pattern around . This makes it obvious that an interpretation in terms of a probability density is not possible, because the distribution function yields large negative values. Applying coarse graining procedures, we basically filter out these highly oscillatory parts. Although we inevitably lose information, we gain a clearer picture of actually measurable quantities. Hence, apart from minor fluctuations (at the level), we can in principle transform the quasi-probability function into a probability density.

Fig. 3 illustrates a further feature of the phase-space formalism. It seems as if four particle bulks are clearly separated from each other. This picture, however, is misleading, because it displays particles and antiparticles as if they were sharing the same momentum variables, here . This is not correct in QFT, because antiparticles are generally defined with reverse momenta. Hence, in order to obtain a meaningful result in terms of real particles one has to identify the proportion of the antiparticle distribution and reverse the signs of the respective momentum coordinates.

Given the phase-space particle distribution as well as the charge distribution we can discriminate particles from antiparticles. Let us write

(64)
(65)

to account for the electron/positron distribution. Besides minor fluctuations, we observe in Fig. 4 the formation of two bulks, which are, however, not equally dense. To understand this asymmetry, we discuss the results obtained from the trajectory model.

As we have derived the Wigner equations of motion (20)-(23) for one spin direction only, we have also fixed the spin in our model. With this point in mind, we are able to reproduce the particle distribution function in Fig. 4. The greatest strength of the trajectory model is, that it allows to examine the contributions of fields individually. Evaluating, for example, the semi-classical equations of motion (61)-(63) without the spin-interaction term yields a perfectly symmetric particle distribution. In the next step, we performed two additional calculations, where we switched off the term in scenario (i) and the term in scenario (ii). In the latter, we could nicely reproduce the results given in Fig. 4, thus we conclude that the magnetic field gradient is mainly responsible for breaking the -symmetry.

v.2 Particle distribution in momentum space

Signatures of a spin-field interaction show up also in the distribution functions in momentum space. Solving the DHW equation (20)-(23) for large spatial extent ( ) and thus weak magnetic field leads to a distribution that is strongly confined in -direction, see Fig. 5.

With the aid of the semi-classical model, the distribution can be understood assuming that particles are created with essentially vanishing initial longitudinal momentum. Subsequently, as the electric field points in -direction, particles are accelerated in -direction only. As the magnetic field is approximately zero throughout the entire computation a conversion from longitudinal momentum to transversal momentum is impossible. It is therefore not surprising, that the particle density exhibits a single peak in momentum space. Furthermore, the electric field varies only slightly in the spatial directions, thus the distribution nicely resembles results obtained from calculations in spatially homogeneous fields Hebenstreit:2008ae ().

One advantage of the DHW formalism is, that it allows to study pair production in strongly varying, spatially inhomogeneous fields. An example is given in Fig. 5, where and . This configuration yields, as a side effect, also a strong magnetic field. Again, particles are created around the main peak of the electric field and subsequently accelerated due to the strong force on the charged particles. In turn, depending on the time of creation, the particles acquire a different momentum . The earlier a particle pair is created the more it is accelerated by the electric field (here in negative -direction). However, by contrast with the previous consideration, here the magnetic field is strong enough to transfer a substantial amount of momentum from to . Such a conversion of momentum is not possible for particles produced at later times ( stays close to zero at first). Due to the fact, that these particles are nearly unaffected by the magnetic field they cannot be pushed away from the strong field region. In turn, they are basically only accelerated by the second minor peak in the electric field (in positive -direction). Hence, the “” shaped structure of the particle distribution.

Additionally, the second peak in time of the magnetic field shows a strong spatial gradient. Hence, the magnetic field directly interacts with the particle spin, similarly to the Stern-Gerlach experiment. (Anti-)fermions have non-zero spin, thus depending on their spin alignment they feel an additional force. As we performed calculations for only one spin-direction, this force pushes particles only in one -direction. Eventually, analyzing a collection of particles, this leads to a net force breaking -symmetry.

We have illustrated the smooth particle densities on the basis of the trajectory model in Fig. 6. Studying trajectories essentially supports our interpretation of Fig. 5. However, in contrast to the semi-classical analysis a computation based on the DHW formalism describes quantum interferences. This explains the differences in Figure 5 and Figure 6. The DHW results show a strong interference pattern, visible as additional side maxima. This feature is clearly missing in the semi-classical calculations, where one obtains only the peaks without any interference pattern.

Figure 5: Density plots of the particle distribution function (DHW calculation) in momentum space for , , , and (top) or (bottom) . Particles created at early times are strongly accelerated towards negative momentum by the main peak of the electric field and then deflected in -direction by the magnetic field. Particles created at late times only acquire substantial momentum (towards positive ), because of the minor peak in the electric field. Due to the presence of (i) a favored spin direction and (ii) a strong magnetic field gradient, the asymmetry in can be attributed to direct spin-field interactions.
Figure 6: Smooth density histogram of the single-particle distribution as function of the momenta at final time . A total of approximately trajectories have been evaluated, interpolated and normalized. The stronger the magnetic field the more momentum can be transferred from to . In turn, the particle bulk becomes widely distributed in -direction. Background field parameters: , , , , (top) or (bottom) .

v.3 The envelope phase

In the following, we apply our findings to field configurations exhibiting more than one prominent peak in the electric field. Solving the DHW equation (20)-(23) for various , we are able to display a series of density plots in Fig. 7 demonstrating the influence of the envelope phase in a few-cycle pulse. The first picture illustrates the particle density for a field configuration with one dominant peak in the electric field (general parameters: , and ). Due to the shorter interaction time compared to the previous discussion, see Sec. V.2, quantum interferences play only a minor role. The overall particle distribution, however, is qualitatively the same.

The picture changes drastically when considering a non-vanishing envelope phase . As already investigated in Refs. Akkermans:2011yn (); Hebenstreit:2009km (), an electric field exhibiting multiple peaks in time can act as a double-slit experiment. The same seems to be true for spatially inhomogeneous fields. Particles created at different peaks in time can still occupy the same phase-space volume, thus leading to quantum interferences.

Figure 7: Series of density plots of the particle distribution function obtained from a DHW calculation. The envelope phase is varied (top to bottom) ; for fixed , , and . The interference pattern is a characteristic feature of pair production in electric fields with multiple prominent peaks.
Figure 8: Smooth density histogram of the single-particle distribution within a semi-classical trajectory analysis broken down into contributions from the first (top) , the second (middle) and both (bottom) peaks in the electric field for . Particles produced at the first peak are greatly accelerated and dispersed by the magnetic field. Nevertheless, both peaks produce the same total number of particles. Background field: , , and .

Choosing, for example, a phase of results in an electric field, that features one prominent and one smaller peak in time. According to the semi-classical model, the greater peak at produces a larger amount of particles. These particles, however, are exposed to a strong magnetic field leading to the “wings” in the particle density. Nevertheless, a substantial fraction of particles is nearly unaffected by the magnetic field. These particles, together with the particles created at the second peak, eventually culminate in a distribution function superimposed by wave peaks and troughs. The most prominent example, however, is given in case of , where both extrema in the electric field show the same absolute value. Although the magnetic field still deflects many particles the interference pattern dominates the distribution.

For the sake of completeness, we have also displayed the result of a configuration with in Fig. 7. Basically, the situation is opposite to the case with . Now, most particles are created after the magnetic field vanished, thus they are only affected by the electric field. As a result the distribution is more confined and the quantum interferences play only a minor role. Note, that we have neither normalized the distribution function nor the electric peak field strength. It is therefore not surprising, that the peak numbers change drastically.

The trajectory analysis approach makes the perfect tool in order to analyze the particle distributions in Fig. 7. To support our interpretation we have therefore exemplarily evaluated a total of approximately trajectories for a configuration with within the semi-classical approach, see Fig. 8 for an illustration. The electric background field is antisymmetric in time, see Fig. 2, thus particle production mainly takes place around the two different peak positions. If particle pairs are created at the first peak in time, they are exposed to very strong electric and magnetic fields, where especially the latter is accelerating particles into the -direction. By contrast, the magnetic field has nearly vanished at the time the second peak in the electric field starts to produce particles. As a result, nearly no conversion in momentum takes place. Although the particles are distributed differently, both bulks contribute evenly towards the total production rate. This is not surprising, because the peaks in the electric field differ only by a sign.

Figure 9: Particle spectra for Schwinger pair production as function of the momenta and . The interference pattern for arises due to the appearance of multiple strong peaks in the electric field. Top : The spectrum for configurations with and appear on top of each other. Bottom : Symmetry in is broken, because we take into account only one spin direction within a DHW calculation. Parameters: , , and .

Interference patterns and a shift towards higher momentum also show up in the distributions and , see Eq. (36). Fig. 9 serves as an example, where we have integrated out one momentum coordinate, respectively. We used the same parameters as in Fig. 7; , , and .

Analyzing the particle distribution as a function of the momentum we observe remarkable agreement with results obtained within the QKT framework Hebenstreit:2009km (). This is insofar astonishing as firstly QKT describes pair creation in purely time-dependent electric fields, while we have taken spatial inhomogeneities as well as a strong magnetic field into account. Secondly, the particle density in the whole momentum space does not look familiar at all, see Fig. 7.

Nevertheless, the function displays all essential features of Schwinger pair production. Particles created within a single-peak electric field show a smooth distribution function. In case of multiple peaks, however, the electric field acts as if it were a double-slit experiment in time. Another interesting aspect, that still holds for moderately varying fields , is given by the distributions and . Although their representation in full momentum space is different, their distributions as functions of cover each other. This could be related to the fact, that both field configurations still possess the same field energy as well as the same general structure. To be more specific, the relation holds.

A completely different picture is drawn by the function . Not only is the link between field configurations of type and not visible, also any obvious signature of an interference pattern is integrated out. Nevertheless, all configurations display a shift to higher momentum , although the strength of the effect varies.

Vi Summary

Based on numerical solutions within the DHW approach we have discussed the Schwinger pair production process in spatiotemporally inhomogeneous few-cycle background fields. The DHW formalism provides access to all phase-space informations. Employing advanced numerical methods, we have been able to compute particle momentum spectra as well as spatial-momentum distribution functions in order to thoroughly investigate how spatial and temporal variations in the electric and magnetic fields affect the particle distribution. Furthermore, we have introduced a semi-classical model on the basis of an effective theory for the particle production rate supplemented by a rigorous trajectory analysis. This model served as a supporting tool providing an additional point of view and facilitating our interpretation of time-resolved Schwinger pair production.

Our main goal was to investigate particle creation in the vicinity of an additional strong and inhomogeneous magnetic field. We have found remarkable signatures of quantum interferences and spin-field interactions. Additionally, we observed the formation of characteristic patterns strongly depending on the carrier-envelope phase of the background fields. To sum up, the inhomogeneous magnetic field turns out to be a decisive factor towards understanding pair production under realistic conditions.

We have introduced various strategies enabling us to perform calculations within a phase-space formalism without any additional truncations opening up the possibility to perform calculations for more realistic field configurations. Correspondingly, the trajectory-based model can be easily extended to more advanced field configurations, too. Moreover, due to the fact that one has full control of the particles in the semi-classical model, it should be possible to expand it such that one can take electron-electron interactions as well as radiation reaction effects into account. The inclusion of phase information is conceptually more difficult considering that one probably wants to keep the simple and easy-to-use form of the approach. Nevertheless, a combination of both methods appears to be promising particularly with regard to future challenges in the research field.

Acknowledgments

We want to thank Holger Gies and Alexander Blinne for many fruitful discussions. We are very grateful to Holger Gies for comments on the manuscript. The work of CK is funded by the Helmholtz Association through the Helmholtz Postdoc Programme (PD-316).

Computations were performed on the “Supermicro Server 1028TR-TF” in Jena, which was funded by the Helmholtz Postdoc Programme (PD-316).

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