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  • Open Access

Effect of time-dependent inhomogeneous magnetic fields on the particle momentum spectrum in electron-positron pair production

Christian Kohlfürst
Phys. Rev. D 101, 096003 – Published 8 May 2020
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Abstract
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Article Text
  • INTRODUCTION
  • MOTIVATION
  • DHW FORMALISM
  • NUMERICAL SOLUTION TECHNIQUES
  • RESULTS
  • CONCLUSION AND OUTLOOK
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Electron-positron pair production in spatially and temporally inhomogeneous electric and magnetic fields is studied within the Dirac-Heisenberg-Wigner formalism (quantum kinetic theory) through computing the corresponding Wigner functions. The focus is on discussing the particle momentum spectrum regarding signatures of Schwinger and multiphoton pair production. Special emphasis is put on studying the impact of a strong dynamical magnetic field on the particle distribution functions. As the equal-time Wigner approach is formulated in terms of partial integro-differential equations an entire section of the manuscript is dedicated to present numerical solution techniques applicable to Wigner function approaches in general.

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    • Received 19 January 2020
    • Accepted 17 April 2020

    DOI:https://doi.org/10.1103/PhysRevD.101.096003

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Particle productionQuantum electrodynamicsSchwinger effect
    1. Techniques
    Phase space dynamicsPlasma kinetic theory
    Particles & FieldsPlasma Physics

    Authors & Affiliations

    Christian Kohlfürst*

    • Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany

    • *c.kohlfuerst@hzdr.de

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    Issue

    Vol. 101, Iss. 9 — 1 May 2020

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    Images

    • Figure 1
      Figure 1

      Illustration of the two different temporal envelopes used in the manuscript. For pulse lengths of τ=25  m−1 (Gaussian) and τ=75  m−1 (super-Gaussian) we find that around τ≈125  m−1 both fields take on similar values.

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    • Figure 2
      Figure 2

      Sketch of the integration scheme employed to find solutions for the set of equations of motion. At every time step ti derivatives with respect to coordinates x and momenta p are calculated via pseudospectral methods. Using a Fourier transform FTx,p derivative operators are converted into multiplicative factors. Advancing in time by one step Δt is done in coordinate-momentum space.

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    • Figure 3
      Figure 3

      Particle distribution functions f(px) (top) and f(pz) (bottom) normalized by the expected production rate in local constant field approximation. Due to the fact, that the field is polarized along ex quantum interferences show up in parallel direction px only. The smaller λ the higher the magnetic peak field strength and, in turn, the stronger particles are accelerated along pz-direction. As for λ=3  m−1 particle bunches do not occupy the same region in phase-space any more, the interference pattern vanishes. Both simulations feature a temporal pulse length of τ=25  m−1 and a frequency of ω=0.2  m.

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    • Figure 4
      Figure 4

      Nonlinearly scaled particle momentum spectrum f(px,pz) for large (top) and small (bottom) spatial extent λ. Electric field strength eϵ=0.5  m2, pulse length τ=25  m−1 and field frequency ω=0.5  m are fixed. Due to the presence of a strong field and the high photon energies particles are predominantly created along ellipses. For a vanishing magnetic field (top) multiphoton peaks as well as a pronounced interference pattern are clearly visible. A strong magnetic field disturbs these patterns and additionally applies a strong force in perpendicular direction ±pz.

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    • Figure 5
      Figure 5

      Normalized radial distribution function f¯(ϑ) for various n-photon channels for a configuration with field frequency ω=0.5  m, field strength eϵ=0.5  m2 and pulse length τ=25  m−1. The higher the photon number n the more minima/maxima in the spectrum are visible. In particular at ϑ=±π/2 we have alternating extrema. The variable ϑ rotates clockwise with starting point (px,n,0).

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    • Figure 6
      Figure 6

      Particle distribution f(px,pz) for a background field featuring a field strength of eϵ=0.2  m2, a super-Gaussian envelope with pulse length τ=75  m−1, a field frequency of ω=0.2  m and a spatial extent of λ=1000  m−1 (top) as well as λ=20  m−1 (bottom). The spectrum shows a Schwinger-like distribution (within white dashed area) as well as multiphotonlike patterns (ellipses).

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    • Figure 7
      Figure 7

      Particle distribution f(px,pz) for a background field featuring a field strength of eϵ=0.5  m2, a super-Gaussian envelope with pulse length τ=75  m−1, a field frequency of ω=0.2  m and a spatial extent of λ=1000  m−1 (top) as well as λ=20  m−1 (bottom). The main structure (cigar shaped area) is superposed by multiple ellipses stemming from multiphoton pair production giving rise to additional quantum interferences.

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    • Figure 8
      Figure 8

      Normalized radial distribution function f¯(ϑ) for various above-threshold signals in momentum space for a background field with peak field strength eϵ=0.5  m2, pulse length τ=75  m−1 (super-Gaussian envelope) and frequency ω=0.2  m. Due to the high intensities the threshold for pair production is increased thus particles acquire less kinetic energy. Here, the photon numbers n correspond to effective energies of E20=0.65  m, E21=1.06  m, E22=1.36  m and E23=1.60  m.

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    • Figure 9
      Figure 9

      Normalized particle distribution as a function of parallel momentum px. The oscillations in f(px) are composed of multiple frequencies, which vanish the smaller λ gets and thus the stronger the applied magnetic field becomes. Parameters: eϵ=0.5  m2, τ=75  m−1 (super-Gaussian envelope), ω=0.2  m.

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    • Figure 10
      Figure 10

      Normalized particle distribution function f¯(pz) for multiple values of λ (smaller λ corresponds to higher magnetic field strength). The change in the distributions with increased magnetic field strength can be understood as a two-step process. For weak magnetic fields particles that have already been accelerated by the electric field are deflected by the magnetic field in direction of pz. The stronger the magnetic field becomes the stronger the effect. At a certain limit (λ∼10  m−1) the particle rate drops considerably and a strong magnetic field prevents a clear signal to form. Additionally, for high magnetic field strengths the symmetry in pz is broken. Field parameters: eϵ=0.5  m2, τ=75  m−1 (super-Gaussian envelope), ω=0.2  m.

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    • Figure 11
      Figure 11

      Comparison of the particle distribution functions f(px,pz=0) and f(px=0,pz) for a homogeneous background field of Sauter type with peak field strength eϵ=0.5  m2 and pulse length τ=10  m−1. Simulation (DHW) and analytical result are in good agreement (MRE below 0.1%).

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    • Figure 12
      Figure 12

      Convergence of the total particle number N as a function of the number of grid points per direction. The deviation from the result obtained through an estimate employing a locally homogeneous approximation is displayed in terms of the mean relative error MREN. The field configuration is given by λ=20  m−1, eϵ=0.5  m2, τ=75  m−1 (super-Gaussian envelope) and ω=0.2  m.

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    • Figure 13
      Figure 13

      Particle distribution function f(px,pz) in a high-resolution simulation (top) in comparison with the result obtained through a low-resolution computation (bottom) for a background field with spatial extent λ=20  m−1, peak field strength eϵ=0.5  m2, pulse length τ=75  m−1 (super-Gaussian envelope) and field frequency ω=0.2  m. Above a field-specific threshold for the grid size, details in the interference pattern become more pronounced while spurious signals and numerical artifacts vanish.

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    • Figure 14
      Figure 14

      Particle momentum distribution as function of parallel (px) and perpendicular (pz) momentum for various spatial extents λ. From top to bottom: λ=1000  m−1, λ=20  m−1, λ=10  m−1, λ=5  m−1 and λ=3  m−1. Particles are created due to the Schwinger effect and subsequently accelerated by electric and magnetic fields. The smaller λ the stronger the magnetic field thus the more particles are pushed to nonvanishing transversal momenta. Further parameters: eϵ=0.2  m2, τ=25  m−1 and ω=0.2  m.

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    • Figure 15
      Figure 15

      Particle spectrum as a function of parallel (px) and perpendicular (pz) momentum for various spatial extents λ. Particles are created mainly via the Schwinger effect. The smaller λ the stronger the applied magnetic fields thus the stronger the distortion. Additionally, quantum interference effects vanish. Parameters: eϵ=0.5  m2, τ=25  m−1, ω=0.2  m and, in terms of appearance, λ=1000  m−1 (top), 20  m−1, 10  m−1, 5  m−1 and 3  m−1 (bottom). To improve readability we only show absolute values in the last plot.

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    • Figure 16
      Figure 16

      Left: Particle distribution f(px,pz) for various values of the spatial extent λ (and therefore the magnetic peak field strength). We have chosen a peak field strength of eϵ=0.5  m2, a pulse length of τ=25  m−1, a field frequency of ω=0.5  m and a spatial envelope parameter of λ=(1000,10,5,1.5)  m−1 (top to bottom). Schwinger as well as multiphoton pair production are visible (strong peaks, elliptical shape of the distribution function). A strong magnetic field can deform the particle structure and break pz symmetry. Right: Particle distribution f(px,pz) for various values of the spatial extent λ=(1000,50,20,10)  m−1 (top to bottom) for field strength eϵ=0.2  m2, pulse length τ=75  m−1 (super-Gaussian envelope) and field frequency ω=0.2  m. Due to the appearance of multiple strong subcycles in the electric field signatures of multiphoton pair production are clearly visible (ring superposed by quantum interferences). A strong magnetic field can destroy the rings, but the multipeak structure is still retained. To improve readability we only show absolute values in the last plot.

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    • Figure 17
      Figure 17

      Particle distribution f(px,pz) in the intermediate regime for various values of the spatial extent λ=(1000,100,50,35,20,10,5)  m−1 (left to right, top to bottom) for field strength eϵ=0.5  m2, pulse length τ=75  m−1 (super-Gaussian envelope) and field frequency ω=0.2  m. The stronger the magnetic field the more the particles are accelerated in transversal direction pz. To improve readability we only show absolute values in the last plot.

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