Electron-positron pair production in ultrastrong laser fields

Xie Bai Song; Li Zi Liang; Tang Suo

doi:10.1016/j.mre.2017.07.002

Volume 2 Issue 5

Sep. 2017

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Xie Bai Song, Li Zi Liang, Tang Suo. Electron-positron pair production in ultrastrong laser fields[J]. Matter and Radiation at Extremes, 2017, 2(5). doi: 10.1016/j.mre.2017.07.002

Citation:

Xie Bai Song, Li Zi Liang, Tang Suo. Electron-positron pair production in ultrastrong laser fields[J]. Matter and Radiation at Extremes, 2017, 2(5). doi: 10.1016/j.mre.2017.07.002

Xie Bai Song, Li Zi Liang, Tang Suo. Electron-positron pair production in ultrastrong laser fields[J]. Matter and Radiation at Extremes, 2017, 2(5). doi: 10.1016/j.mre.2017.07.002

Citation:

Xie Bai Song, Li Zi Liang, Tang Suo. Electron-positron pair production in ultrastrong laser fields[J]. Matter and Radiation at Extremes, 2017, 2(5). doi: 10.1016/j.mre.2017.07.002

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Electron-positron pair production in ultrastrong laser fields

doi: 10.1016/j.mre.2017.07.002

Xie Bai Song^{1, 2
,
,},
Li Zi Liang^{1, 3},
Tang Suo^{1, 4}

1.
aCollege of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China
2.
bBeijing Radiation Center, Beijing 100875, China
3.
cSchool of Science, China University of Mining and Technology, Beijing 100083, China
4.
dMax-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

More Information

Corresponding author: *Corresponding author. College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China. E-mail address: bsxie@bnu.edu.cn (B.S. Xie).
Received Date: 2017-02-24
Accepted Date: 2017-07-10
Publish Date: 2017-09-15

Abstract

Abstract

Electron–positron pair production due to the decay of vacuum in ultrastrong laser fields is an interesting topic which is revived recently because of the rapid development of current laser technology. The theoretical and numerical research progress of this challenging topic is reviewed. Many new findings are presented by different approaches such as the worldline instantons, the S-matrix theory, the kinetic method by solving the quantum Vlasov equation or/and the real-time Dirac–Heisenberg–Wigner formalism, the computational quantum field theory by solving the Dirac equation and so on. In particular, the effects of electric field polarizations on pair production are unveiled with different patterns of created momentum spectra. The effects of polarizations on the number density of created particles and the nonperturbative signatures of multiphoton process are also presented. The competitive interplay between the multiphoton process and nonperturbation process plays a key role in these new findings. These newly discovered phenomena are valuable to deepen the understanding of pair production in complex fields and even have an implication to the study of strong-field ionization. More recent studies on the pair production in complex fields as well as beyond laser fields are briefly presented in the view point of perspective future.
- Strong field physics,
- Vacuum pair production,
- Nonlinear quantum electrodynamics

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References(129)

References

[1]	P. A. M. Dirac, The quantum theory of the electron. roc. R. Soc. London, Ser. A 117 (1928) 610–624.10.1098/rspa.1928.0023
[2]	C. D. Anderson, The positive electron. Phys. Rev. 43 (1933) 491C494.10.1103/physrev.43.491
[3]	F. Sauter, Ber das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs. Z. Phys. 69 (1931) 742–764.10.1007/bf01339461
[4]	W. Heisenberg, H. Euler, Consequences of Dirac's theory of positrons. Z. Phys. 98 (1936) 714–732.10.1007/bf01343663
[5]	J. Schwinger, On gauge invariance and vacuum polarization. Phys. Rev. 82 (1951) 664–679.10.1103/physrev.82.664
[6]	N. B. Narozhnyi, A. I. Nikishov, The simplest processes in the pair creating electric field. Yad. Fiz. 11 (1970) 1072
[7]	V. S. Popov, Pair production in a variable external field (quasiclassical approximation). Sov. Phys. JETP 34 (1972) 709.
[8]	E. Brezin, C. Itzykson, Pair production in vacuum by an alternating field. Phys. Rev. D. 2 (1970) 1191–1199.10.1103/physrevd.2.1191
[9]	T. Tajima, G. Mourou, Zettawatt-exawatt lasers and their applications in ultrastrong-field physics. Phys. Rev. Spec. Top.--Accel. Beams 5 (2002) 031301.10.1103/physrevstab.5.031301
[10]	G. A. Mourou, T. Tajima, S. V. Bulanov, Optics in the relativistic regime. Rev. Mod. Phys. 78 (2006) 309–371.10.1103/revmodphys.78.309
[11]	V. Yanovsky, V. Chvykov, G. Kalinchenko, P. Rousseau, T. Planchon, et al., Ultra-high intensity- 300-TW laser at 0.1 Hz repetition rate. Opt. Express 16 (2008) 2109–2114.10.1364/oe.16.002109
[12]	ELI: http://www.eli-beams.eu/.
[13]	XCELS: http://www.xcels.iapras.ru/.
[14]	HiPER: http://www.hiper-laser.org/.
[15]	L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields. Elsevier, Oxford, 1975.
[16]	E. J. Moniz, D. H. Sharp, Radiation reaction in nonrelativistic quantum electrodynamics. Phys. Rev. D 15 (1977) 2850–2865.10.1103/physrevd.15.2850
[17]	C. Harvey, T. Heinzl, M. Marklund, Symmetry breaking from radiation reaction in ultra-intense laser fields. Phys. Rev. D 84 (2011) 116005.10.1103/physrevd.84.116005
[18]	C. Harvey, M. Marklund, Radiation damping in pulsed Gaussian beams. Phys. Rev. A 85 (2012) 013412.10.1103/physreva.85.013412
[19]	V. B. Berestetskii, E. M. Lifshitz, L. P. Pitaevskii, Quantum Electrodynamics. Elsevier Butterworth-Heinemann, Oxford, 1982.
[20]	W. Dittrich, M. Reuter, Effective Lagrangians in Quantum Electrodynamics. Springer, Heidelberg, 1985.
[21]	W. Dittrich, H. Gies, Probing the Quantum Vacuum. Springer, Heidelberg, 2000.
[22]	A. Ringwald, Pair production from vacuum at the focus of an X-ray free electron laser. Phys. Lett. B 510 (2001) 107–116.10.1016/s0370-2693(01)00496-8
[23]	http://www.xfel.eu/.
[24]	R. Alkofer, M. B. Hecht, C. D. Roberts, S. M. Schmidt, D. V. Vinnik, Pair creation and an X-ray free electron laser. Phys. Rev. Lett. 87 (2001) 193902.10.1103/physrevlett.87.193902
[25]	C. D. Roberts, S. M. Schmidt, D.V.Vinnik, Quantum effects with an X-ray free-electron laser. Phys. Rev. Lett. 89 (2002) 153901.10.1103/physrevlett.89.153901
[26]	A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, C. H. Keitel, Extremely high-intensity laser interactions with fundamental quantum systems. Rev. Mod. Phys. 84 (2012) 1177–1228.10.1103/revmodphys.84.1177
[27]	I. K. Affleck, O. Alvarez, N. S. Manton, Pair production at strong coupling in weak external fields. Nucl. Phys. B 197 (1982) 509–519.10.1016/0550-3213(82)90455-2
[28]	S. P. Kim, D. N. Page, Schwinger pair production via instantons in strong electric fields. Phys. Rev. D 65 (2002) 105002.10.1103/physrevd.65.105002
[29]	G. V. Dunne, C. Schubert, Worldline instantons and pair production in inhomogenous fields. Phys. Rev. D 72 (2005) 105004.10.1103/physrevd.72.105004
[30]	G. V. Dunne, Q. H. Wang, Multidimensional worldline instantons. Phys. Rev. D 74 (2006) 065015.10.1103/physrevd.74.065015
[31]	R. Schützhold, H. Gies, G. Dunne, Dynamically assisted Schwinger mechanism. Phys. Rev. Lett. 101 (2008) 130404.10.1103/physrevlett.101.130404
[32]	B. S. Xie, M. Mohamedsedik, S. Dulat, Electron-positron pair production in an elliptic polarized time varying field. Chin. Phys. Lett. 29 (2012) 021102.10.1088/0256-307x/29/2/021102
[33]	C. Muller, A. B. Voitkiv, N. Grun, Differential rates for multiphoton pair production by an ultrarelativistic nucleus colliding with an intense laser beam. Phys. Rev. A 67 (2003) 063407.10.1103/physreva.67.063407
[34]	C. Müller, A. B. Voitkiv, N. Grün, Nonlinear bound-free pair creation in the strong electromagnetic fields of a heavy nucleus and an intense X-ray laser. Phys. Rev. Lett. 91 (2003) 223601.10.1103/physrevlett.91.223601
[35]	C. Deneke, C. Muller, Bound-free e+e− pair creation with a linearly polarized laser field and a nuclear field. Phys. Rev. A 78 (2008) 033431.10.1103/physreva.78.033431
[36]	H. Y. Hu, C. Müller, C. H. Keitel, Complete QED theory of multiphoton trident pair production in strong laser fields. Phys. Rev. Lett. 105 (2010) 080401.10.1103/physrevlett.105.080401
[37]	A. Ilderton, 2011. Trident pair production in strong laser pulses. Phys. Rev. Lett. 106, 020404.10.1103/physrevlett.106.020404
[38]	H. Y. Hu, B. S. Xie, X. H. Guo, H. Y. Wang, Electron-positron pair production in an arbitrary polarized ultrastrong laser field. Commun. Theor. Phys. 58 (2012) 863–871.10.1088/0253-6102/58/6/13
[39]	S. S. Bulanov, Pair production by a circularly polarized electromagnetic wave in a plasma. Phys. Rev. E 69 (2004) 036408.10.1103/physreve.69.036408
[40]	C. K. Dumlu, G. V. Dunne, Stokes phenomenon and Schwinger vacuum pair production in time-dependent laser pulses. Phys. Rev. Lett. 104 (2010) 250402.10.1103/physrevlett.104.250402
[41]	E. Akkermans, G. V. Dunne, Ramsey fringes and time-domain multiple-slit interference from vacuum. Phys. Rev. Lett. 108 (2012) 030401.10.1103/physrevlett.108.030401
[42]	Z. L. Li, Z. L. Lu, B. S. Xie, Multiple-slit interference effect in the time domain for boson pair production. Phys. Rev. D 89 (2014) 067701.10.1103/physrevd.89.067701
[43]	E. Strobel, S. S. Xue, Semiclassical pair production rate for rotating electric fields. Phys. Rev. D 91 (2015) 045016.10.1103/physrevd.91.045016
[44]	D. B. Blaschke, A. V. Prozorkevich, C. D. Roberts, S. M. Schmidt, S. A. Smolyansky, Pair production and optical lasers. Phys. Rev. Lett. 96 (2006) 140402.10.1103/physrevlett.96.140402
[45]	F. Hebenstreit, R. Alkofer, G. V. Dunne, H. Gies, Momentum signatures for Schwinger pair production in short laser pulses with a subcycle structure. Phys. Rev. Lett. 102 (2009) 150404.10.1103/physrevlett.102.150404
[46]	A. Nuriman, B. S. Xie, Z. L. Li, D. Sayipjamal, Enhanced electron-positron pair creation by dynamically assisted combinational fields. Phys. Lett. B 717 (2012) 465–469.10.1016/j.physletb.2012.09.060
[47]	N. Abdukerim, Z. L. Li, B. S. Xie, Effects of laser pulse shape and carrier envelope phase on pair production. Phys. Lett. B 726 (2013) 820–826.10.1016/j.physletb.2013.09.014
[48]	Z. L. Li, D. Lu, B. S. Xie, L. B. Fu, J. Liu, B. F. Shen, Enhanced pair production in strong fields by multiple-slit interference effect with dynamically assisted Schwinger mechanism. Phys. Rev. D 89 (2014) 093011.10.1103/physrevd.89.093011
[49]	C. Kohlfürst, H. Gies, R. Alkofer, Effective mass signatures in multiphoton pair production. Phys. Rev. Lett. 112 (2014) 050402.10.1103/physrevlett.112.050402
[50]	A. Blinne, H. Gies, Pair production in rotating electric fields. Phys. Rev. D 89 (2014) 085001.10.1103/physrevd.89.085001
[51]	Z. L. Li, D. Lu, B. S. Xie, Effects of electric field polarizations on pair production. Phys. Rev. D 92 (2015) 085001.10.1103/physrevd.92.085001
[52]	Z. L. Li, D. Lu, B. S. Xie, B. F. Shen, L. B. Fu, J. Liu, Nonperturbative signatures in pair production for general elliptic polarization fields. Europhys. Lett. 110 (2015) 51001.10.1209/0295-5075/110/51001
[53]	J. W. Braun, Q. Su, R. Grobe, Numerical approach to solve the time-dependent Dirac equation. Phys. Rev. A 59 (1999) 604–612.10.1103/physreva.59.604
[54]	P. Krekora, Q. Su, R. Grobe, Klein paradox in spatial and temporal resolution. Phys. Rev. Lett. 92 (2004) 040406.10.1103/physrevlett.92.040406
[55]	P. Krekora, Q. Su, R. Grobe, Relativistic electron localization and the lack of zitterbewegung. Phys. Rev. Lett. 93 (2004) 043004.10.1103/physrevlett.93.043004
[56]	P. Krekora, Q. Su, R. Grobe, Klein paradox with spin-resolved electrons and positrons. Phys. Rev. A 72 (2005) 064103.10.1103/physreva.72.064103
[57]	P. Krekora, K. Cooley, Q. Su, R. Grobe, Creation dynamics of bound states in supercritical fields. Phys. Rev. Lett. 95 (2005) 070403.10.1103/physrevlett.95.070403
[58]	C. C. Gerry, Q. Su, R. Grobe, Timing of pair production in time-dependent force fields. Phys. Rev. A 74 (2006) 044103.10.1103/physreva.74.044103
[59]	M. Jiang, W. Su, Z. Q. Lv, X. Lu, Y. J. Li, , et al., Pair creation enhancement due to combined external fields. Phys. Rev. A 85 (2012) 033408.10.1103/physreva.85.033408
[60]	S.Tang, B. S. Xie, D. Lu, H. Y. Wang, L. B. Fu, J. Liu, Electron-positron pair creation and correlation between momentum and energy level in a symmetric potential well. Phys. Rev. A 88 (2013) 012106.10.1103/physreva.88.012106
[61]	F. Hebenstreit, J. Berges, D. Gelfand, Simulating fermion production in 1+1 dimensional QED. Phys. Rev. D 87 (2013) 105006.10.1103/physrevd.87.105006
[62]	V. Kasper, F. Hebenstreit, J. Berges, Fermion production from real-time lattice gauge theory in the classical-statistical regime. Phys. Rev. D 90 (2014) 025016.10.1103/physrevd.90.025016
[63]	N. Mueller, F. Hebenstreit, J. Berges, Anomaly-induced dynamical refringence in strong-field QED. Phys. Rev. Lett. 117 (2016) 061601.10.1103/physrevlett.117.061601
[64]	F. Gelis, N. Tanji, Formulation of the Schwinger mechanism in classical statistical field theory. Phys. Rev. D 87 (2013) 125035.10.1103/physrevd.87.125035
[65]	D. L. Burke, R. C. Field, G. Horton-Smith, , J. E. Spencer, D. Walz, et al., Positron production in multiphoton light-by-light scattering. Phys. Rev. Lett. 79 (1997) 1626–1629.10.1103/physrevlett.79.1626
[66]	C. Bamber, S. J. Boege, T. Koffas, T. Kotseroglou, A. C. Melissinos, et al., Studies of nonlinear QED in collisions of 46.6 GeV electrons with intense laser pulses. Phys. Rev. D 60 (1999) 092004.10.1103/physrevd.60.092004
[67]	L. V. Keldysh, Ionization in the field of a strong electromagnetic wave. Sov. Phys. JETP 20 (1965) 1307–1314.
[68]	A. M. Perelomov, V. S. Popov, M. V. Terentév, Ionization of atoms in an alternating electric field. Sov. Phys. JETP 23 (1966) 924–934.
[69]	A. M. Perelomov, V. S. Popov, M. V. Terentév, Ionization of atoms in an alternating electric field: II. Sov. Phys. JETP 24 (1967) 207–217.
[70]	V. I. Usachenko, V. A. Pazdzersky, J. K. McIver, Reexamination of high-energy above-threshold ionization (ATI): an alternative strong-field ATI model. Phys. Rev. A 69 (2004) 013406.10.1103/physreva.69.013406
[71]	Y. V. Vanne, A. Saenz, Exact Keldysh theory of strong-field ionization: residue method versus saddle point approximation. Phys. Rev. A 75 (2007) 033403.10.1103/physreva.75.033403
[72]	C. Bula, K. T. McDonald, E. J. Prebys, C. Bamber, S. Boege, et al., Observation of nonlinear effects in Compton scattering. Phys. Rev. Lett. 76 (1996) 3116–3119.10.1103/physrevlett.76.3116
[73]	J. Reinhardt, W. Greiner, Quantum electrodynamics of strong fields. Rep. Prog. Phys. 40 (1977) 219–295.10.1088/0034-4885/40/3/001
[74]	R. P. Feynman, An operator calculus having applications in quantum electrodynamics. Phys. Rev. 84 (1951) 108–128.10.1103/physrev.84.108
[75]	C. Schubert, Perturbative quantum field theory in the string-inspired formalism. Phys. Rep. 355 (2001) 73–234.10.1016/s0370-1573(01)00013-8
[76]	V. S. Popov, M. S. Marinov, E+E− pair production in variable electric field. Yad. Fiz. 16 (1972) 809
[77]	C. K. Dumlu, Multidimensional quantum tunneling in the Schwinger effect. Phys. Rev. D 93 (2016) 065045.10.1103/physrevd.93.065045
[78]	D. M. Volkov, Über eine Klasse von Lösungen der Diracschen Gleichung. Z. Phys. 94 (1935) 250–260.
[79]	H. R. Reiss, Absorption of light by light. J. Math. Phys. 3 (1962) 59–67.10.1063/1.1703787
[80]	A. I. Nikishov, V. I. Ritus, Quantum processes in the field of a circularly polarized electromagnetic wave. Sov. Phys. JETP 19 (1964) 769–796.
[81]	W. Greiner, J. Reinhardt, Quantum Electrodynamics, fourth ed. Springer-Verlag, Berlin Heidelberg, 2009.
[82]	M. Mohamedsedik, B. S. Xie, S. Dulat, Analytical study of pair production rate from vacuum in an elliptic polarized field by a two-level transition technique. Commun. Theor. Phys. 57 (2012) 422.
[83]	S. Schmidt, D. Blaschke, G. Röpke, S. A. Smolyansky, A. V. Prozorkevich, et al., A quantum kinetic equation for particle production in the Schwinger mechanism. Int. J. Mod. Phys. E 7 (1998) 709.10.1142/s0218301398000403
[84]	S. Schmidt, D. Blaschke, G. Röpke, A. V. Prozorkevich, S. A. Smolyansky, et al., Non-markovian effects in strong-field pair creation. Phys. Rev. D 59 (1999) 094005.10.1103/physrevd.59.094005
[85]	J. C. R. Bloch, V. A. Mizerny, A. V. Prozorkevich, C. D. Roberts, S. M. Schmidt, et al., Pair creation: back reactions and damping. Phys. Rev. D 60 (1999) 116011.10.1103/physrevd.60.116011
[86]	F. Hebenstreit, F. Fillion-Gourdeau, Optimization of Schwinger pair production in colliding laser pulses. Phys. Lett. B 739 (2014) 189.10.1016/j.physletb.2014.10.056
[87]	M. Pont, M. Gavrila, Stabilization of atomic hydrogen in superintense, high-frequency laser fields of circular polarization. Phys. Rev. Lett. 65 (1990) 2362–2365.10.1103/physrevlett.65.2362
[88]	B. S. Xie, Classical intense-field stabilization analysis by four-dimensional mapapproximation. Phys. Lett. A 242 (1998) 79–82.10.1016/s0375-9601(98)00155-8
[89]	C. Kohlfürst, Electron-positron pair production in structured pulses of electric fields, arXiv:1212.0880.
[90]	C. Kohlfürst, M. Mitter, G. Von Winckel, F. Hebenstreit, R. Alkofer, Optimizing the pulse shape for Schwinger pair production. Phys. Rev. D 88 (2013) 045028.10.1103/physrevd.88.045028
[91]	I. Bialynicki-Birula, P. Górnicki, J. Rafelski, Phase-space structure of the Dirac vacuum. Phys. Rev. D 44 (1991) 1825–1835.10.1103/physrevd.44.1825
[92]	F. Hebenstreit, R. Alkofer, H. Gies, Schwinger pair production in space- and time-dependent electric fields: relating the Wigner formalism to quantum kinetic theory. Phys. Rev. D 82 (2010) 105026.10.1103/physrevd.82.105026
[93]	F. Hebenstreit, R. Alkofer, H. Gies, Particle self-bunching in the Schwinger effect in spacetime-dependent electric fields. Phys. Rev. Lett. 107 (2011) 180403.10.1103/physrevlett.107.180403
[94]	C. Kohlfürst, R. Alkofer, On the effect of time-dependent inhomogeneous magnetic fields in electron–positron pair production. Phys. Lett. B 756 (2016) 371.10.1016/j.physletb.2016.03.027
[95]	G. R. Mocken, M. Ruf, C. Müller, C. H. Keitel, Nonperturbative multiphoton electron-positron–pair creation in laser fields. Phys. Rev. A 81 (2010) 022122.10.1103/physreva.81.022122
[96]	M. Ruf, G. R. Mocken, C. Müller, K. Z. Hatsagortsyan, C. H. Keitel, Pair production in laser fields oscillating in space and time. Phys. Rev. Lett. 102 (2009) 080402.10.1103/physrevlett.102.080402
[97]	C. K. Dumlu, G. V. Dunne, Interference effects in Schwinger vacuum pair production for time-dependent laser pulses. Phys. Rev. D. 83 (2011) 065028.10.1103/physrevd.83.065028
[98]	A. N. Pfeiffer, C. Cirelli, M. Smolarski, D. Dimitrovski, M. Abu-samha, et al., Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms. Nat. Phys. 8 (2012) 76–80.10.1038/nphys2125
[99]	A. N. Pfeiffer, C. Cirelli, A. S. Landsman, M. Smolarski, D. Dimitrovski, et al., Probing the longitudinal momentum spread of the electron wave packet at the tunnel exit. Phys. Rev. Lett. 109 (2012) 083002.10.1103/physrevlett.109.083002
[100]	C. Hofmann, A. S. Landsman, A. Zielinski, C. Cirelli, T. Zimmermann, et al., Interpreting electron-momentum distributions and nonadiabaticity in strong-field ionization. Phys. Rev. A 90 (2014) 043406.10.1103/physreva.90.043406
[101]	S. S. Schweber, An Introduction to Relativistic Quantum Field Theory. Harper & Row, New York, 1962.
[102]	W. Greiner, Relativistic Quantum Mechanics: Wave Equations, third ed. Springer-Verlag, Berlin Heidelberg, 2000.
[104]	C. Müller, K. Z. Hatsagortsyan, C. H. Keitel, Muon pair creation from positronium in a linearly polarized laser field. Phys. Rev. A 78 (2008) 033408.10.1103/physreva.78.033408
[105]	C. Müller, C. Deneke, C. H. Keitel, Muon-pair creation by two X-ray laser photons in the field of an atomic nucleus. Phys. Rev. Lett. 101 (2008) 060402.10.1103/physrevlett.101.060402
[106]	I. V. Sokolov, N. M. Naumova, J. A. Nees, G. A. Mourou, Pair creation in QED-strong pulsed laser fields interacting with electron beams. Phys. Rev. Lett. 105 (2010) 195005.10.1103/physrevlett.105.195005
[107]	S. Tang, M. Ali Bake, H. Y. Wang, B. S. Xie, QED cascades induced by high energy γ photon in strong laser field. Phys. Rev. A 89 (2014) 022105.10.1103/physreva.89.022105
[108]	C. P. Ridgers, C. S. Brady, R. Duclous, J. G. Kirk, K. Bennett, et al., Dense electron-positron plasmas and ultraintense rays from laser-irradiated solids. Phys. Rev. Lett. 108 (2012) 165006.10.1103/physrevlett.108.165006
[109]	M. H. Thoma, Colloquium: field theoretic description of ultrarelativistic electron-positron plasmas. Rev. Mod. Phys. 81 (2009) 959–968.10.1103/revmodphys.81.959
[110]	Y. S. Huang, Quantum-electrodynamical birefringence vanishing in a thermal relativistic pair plasma. Sci. Rep. 5 (2015) 15866.10.1038/srep15866
[111]	C. Schneider, R. Schützhold, Prefactor in the dynamically assisted Sauter-Schwinger effect. Phys. Rev. D 94 (2016) 085015.10.1103/physrevd.94.085015
[112]	A. Blinne, E. Strobel, Comparison of semiclassical and Wigner function methods in pair production in rotating fields. Phys. Rev. D 93 (2016) 025014.10.1103/physrevd.93.025014
[113]	N. Abdukerim, Z. L. Li, B. S. Xie, Enhanced electron–positron pair production by frequency chirping in one- and two-color laser pulse fields. Chin. Phys. B 26 (2017) 020301.10.1088/1674-1056/26/2/020301
[114]	I. Sitiwaldi, B. S. Xie, Modulation effect in multiphoton pair production. Phys. Lett. B 768 (2017) 174–179.10.1016/j.physletb.2017.02.050
[115]	Q. Wang, J. Liu, L. B. Fu, Pumping electron-positron pairs from a well potential. Sci. Rep. 6 (2016) 25292.10.1038/srep25292
[116]	I. A. Aleksandrov, G. Plunien, V. M. Shabaev, Electron-positron pair production in external electric fields varying both in space and time. Phys. Rev. D 94 (2016) 065024.10.1103/physrevd.94.065024
[117]	K. Fukushima, T. Hayata, Schwinger mechanism with stochastic quantization. Phys. Lett. B 735 (2014) 371.10.1016/j.physletb.2014.06.060
[118]	F. Gelis, K. Kajantie, T. Lappi, Chemical thermalization in relativistic heavy ion collisions. Phys. Rev. Lett. 96 (2006) 032304.10.1103/physrevlett.96.032304
[119]	P. M. Saffin, A. Tranberg, Real-time fermions for baryogenesis simulations. J. High Energy Phys. 7 (2011) 66.
[120]	V. V. Skokov, P. Lvai, Transverse and longitudinal momentum spectra of fermions produced in strong SU(2) fields. Phys. Rev. D. 78 (2008) 054004.10.1103/physrevd.78.054004
[121]	F. Cooper, J. F. Dawson, B. Mihaila, Casimir dependence of transverse distribution of pairs produced from a strong constant chromoelectric background field. Phys. Rev. D 78 (2008) 117901.10.1103/physrevd.78.117901
[122]	J. F. Dawson, B. Mihaila, F. Cooper, Dynamics of particle production by strong electric fields in non-Abelian plasmas. Phys. Rev. D 81 (2010) 054026.10.1103/physrevd.81.054026
[123]	D. Gelfand, F. Hebenstreit, J. Berges, Early quark production and approach to chemical equilibrium. Phys. Rev. D 93 (2016) 085001.10.1103/physrevd.93.085001
[124]	P. V. Buividovich, M. V. Ulybyshev, Numerical study of chiral plasma instability within the classical statistical field theory approach. Phys. Rev. D 94 (2016) 025009.10.1103/physrevd.94.025009
[125]	N. Müller, S. Schlichting, S. Sharma, Chiral magnetic effect and anomalous transport from real-time lattice simulations. Phys. Rev. Lett. 117 (2016) 142301.10.1103/physrevlett.117.142301
[126]	F. Gelis, N. Tanji, Quark production in heavy ion collisions: formalism and boost invariant fermionic light-cone mode functions. J. High Energy Phys. 2 (2016) 126.
[127]	Gordon W. Semenoff, Konstantin Zarembo, , Holographic schwinger effect. Phys. Rev. Lett. 107 (2011) 171601.10.1103/physrevlett.107.171601
[128]	J. Sonner, Holographic Schwinger effect and the geometry of entanglement. Phys. Rev. Lett. 111 (2013) 211603.10.1103/physrevlett.111.211603
[129]	M. Ghodrati, Schwinger effect and entanglement entropy in confining geometries. Phys. Rev. D 92 (2016) 065015.
[130]	M. R. Jia, Z. L. Li, C.Lv, F. Wan, B. S. Xie, Pair production in strong SU(2) background fields. Front. Phys. 12 (2017) 121101.10.1007/s11467-016-0615-0