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High Power Laser Science and Engineering, (2016), Vol. 4, e41, 7 pages.

licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

doi:10.1017/hpl.2016.42

Numerical investigation and potential tunability scheme

on e

−

and π

−

stimulated pair creation from

vacuum using high intensity laser beams

I. Ploumistakis

, S. D. Moustaizis

, and I. Tsohantjis

Technical University of Crete, Laboratory of Matter Structure and Laser Physics, Chania GR-73100, Crete, Greece

National and Kapodistrian University of Athens, Faculty of Physics, Department of Nuclear and Particle Physics,

GR-15771, Athens, Greece

(Received 21 June 2016; revised 19 September 2016; accepted 27 September 2016)

Abstract

Numerical estimates for electrons and mesons particle–antiparticle creation from vacuum in the presence of strong

electromagnetic ﬁelds are derived, using the complete probability density relation of Popov’s imaginary time method

(Popov, JETP Lett. 13, 185 (1971); Sov. Phys. JETP 34, 709 (1972); Sov. Phys. JETP 35, 659 (1972); Popov and

Marinov, Sov. J. Nucl. Phys. 16, 449 (1973); JETP Lett. 18, 255 (1974); Sov. J. Nucl. Phys. 19, 584 (1974)); (Popov,

Phys. Let. A 298, 83 (2002)), and within the framework of an experimental setup like the E144 (Burke et al., Phys. Rev.

Lett. 79, 1626 (1997)). The existence of crossing point among pair creation efﬁciency curves of different photon energies

and the role of odd/even multiphoton orders in the production rates are discussed. Finally a kind of tunability process

between the two creation processes is discussed.

Keywords: high intensity lasers; multiphoton processes; pair production

1. Introduction

In the presence of strong electromagnetic ﬁelds, vacuum

can be unstable and if a certain ﬁeld strength is exceeded,

electron–positron pair creation can occur

[1, 2]

. This charac-

teristic critical ﬁeld strength is the Schwinger ﬁeld E

)/eλ

' 1.3 × 10

V m

−2

, where m

is the electron

mass, c the speed of light and λ

h/(m

c) is the Compton

wavelength. However, as demonstrated by Schwinger

[1]

, in

order to observe pair creation, the invariant quantities F =

µν

= −

(

− c

), G =

µν

= c

E ·

must be such that

+ G

− F > 0, where F

µν

and

µν



µναβ

αβ

are the electromagnetic ﬁeld tensor and

its dual, respectively. These requirements can be satisﬁed

at an area close to the antinodes of a standing wave or at

the region of a focused laser beam. As the critical ﬁeld

strength corresponds to focal laser intensities of the order

of 10

W cm

−2

, the main question that arises, is whether

an experimental veriﬁcation of the phenomenon is possible.

The rapid development of ultra-intense laser facilities has

rekindled the interest in proposing a possible experimental

Correspondence to: I. Ploumistakis, Technical University of Crete,

Laboratory of Matter Structure and Laser Physics, Chania 73100, Greece.

Email: iploumistakis@isc.tuc.gr

setup as seen in various works

[3–11]

. Theoretical treatment

of pair creation in an oscillating pure electric ﬁeld and

based on the atom ionization theory, was demonstrated

in Brezin and Itzykson

[12]

and Popov’s works

[13–19]

. The

characteristic parameter of those treatments is the rela-

tivistic invariant parameter γ = mcω/eE =

hωE

/mc

which is analogous to the Keldysh parameter. In particular,

Popov

[13–19]

applied the imaginary time method for the case

of oscillating electric ﬁeld such as the one realized at the

antinodes of an electromagnetic standing wave formed by

two coherent counterpropagating laser beams and for which

E  E

and

hω  mc

, distinguishing two important

regimes γ  1 and γ  1. For γ  1 (high electric ﬁeld

strength and low frequency) the adiabatic non-perturbative

tunneling mechanism dominates and the probability density

is expressed as W ∝ exp(−π(E

/E)). For the case of

γ  1 (low electric ﬁeld strength and high frequency)

respectively, the multiphoton mechanism is dominant and

W ∝ (E

/E)

−2n

= 2mc

hω is the multiphoton order

threshold). Additionally in Ref. [20] the imaginary time

method was further analyzed and applied for the cases

of a constant electric ﬁeld and time homogeneous electric

ﬁeld for a single or multiple laser pulses. Also, important

work in pair production has been carried out concerning

2 I. Ploumistakis et al.

the interaction of a high intensity polarized laser beam in

vacuum as seen in Refs. [21–23], leading to observation of

pair production at laser intensities lower than the critical

one. Along these lines, interaction of two circularly polarized

counterpropagating laser pulses

[24]

was shown that pair cre-

ation can become experimentally observable for laser beam

intensities one or two orders of magnitude lower than that of

a single pulse. Finally in Refs. [22, 25, 26] the collision of

multiple electromagnetic pulses is proposed as yet an another

possible experimental scheme, where lower threshold is

required for pair observation. In the ultra-relativistic regime

ξ  1 (ξ = 1/γ ) recollision process of an electron–positron

pair produced by the interaction of a high energy photon

with an intense laser pulse allows relevant high energy

physics effects

[27]

. The concept of the E144 experiment

[28]

and the agreement with theory is mentioned in Ref. [22]

as a unique opportunity for future experiments, in order to

investigate unexplored nonlinear vacuum effects, using laser

intensities to 2–4 order higher than the laser intensity used

for the ﬁrst experimental veriﬁcation of e

−

pair creation

accomplished at SLAC

[28]

. The experiment consisted of two

stages. At ﬁrst a high energy electron beam interacted with a

laser beam via nonlinear Compton scattering producing high

energy γ photons. These high energy photons then interact

with low energy laser photons and electron–positron pairs

are created through Breit–Wheeler process

[29]

. 175 ± 13

positrons were measured in 21962 laser pulses for a n =

5.1 ± 0.2 (see Ref. [28]) multiphoton order process, a result

that is in very good agreement with the theory

[30]

Based on Popov’s treatment we are going to present our

numerical estimates for pair creation efﬁciency on an E144

like experimental setup that can be realized in the near future

by high intensity laser facilities

[31–39]

Continuing with the presentation of Popov’s theory the

probability density is given by Refs. [13–19]

W =

n>n

, (1)

where w

is the nth multiphoton order probability per

Compton volume given by

(2s + 1)

2π

}ω



n − n

∆



1/2

exp

−

2mc

}ω

f −

2 f

∆

(n − n

)

. (2)

In the above relation, s is a factor that is equal to 0

for the case of bosons and 1/2 for the case of fermions,

= 9.1093 × 10

−31

kg (0.5 MeV c

−2

) for electrons or

= 2.488 × 10

−28

kg (139.570 MeV c

−2

) for pions,

= }

e,π

is the electron or pion 4-Compton volume,

7.4 × 10

−59

and 2.16 × 10

−143

s, respectively,

= mc

∆/}ω, ∆ =

πγ ψ(γ )

E(ψ(γ )),

ψ(γ ) = 1/

1 + γ

, (3)

where n

is the threshold multiphoton order for pair produc-

tion to take place and ∆ expresses the effective energy gap

width between the continua and E(.) is the complete elliptic

integral of the second kind. The functions ∆

, ∆

, J

, l, ξ

f , f

, f

are respectively given by

∆

γ ψ(γ )K (ψ(γ )), ∆

γ ψ(γ )E(ψ(γ )),

(4)

l(n−n

[1 + σ (−1)

cos ξ

x] dx,

l = 2



∆

−

∆



, x =

= 2γ ψ(γ )

(n − n

)mc

}ω

f =

πγ

1 +

1 + γ

, f

= πγ ψ(γ )/2 = (1/γ ) f

where K (.) is the complete elliptic integrals of the ﬁrst kind

and σ = 1 for electrons and −1 for mesons (pions)

[13–19]

and p

is the parallel to the electric ﬁeld component of the

momentum p of the created particle.

For the sake of introduction completeness the simpliﬁed

asymptotic formulas for e

−

pair creation will be presented

even though they will not be used in our estimates. For the

two regions of γ that we have mentioned Equation (2) can

be simpliﬁed. In the case γ  1 the spectrum of n

hω of

the n-photon processes is practically continuous giving the

non-perturbative result

[13–19, 40]

3/2

(

E/E

)

5/2

× exp



−π(E

/E)



1 −

+ O(γ

)



while the number of pairs created is given by

N (τ) = 2

−3/2

(

E/E

)

5/2

× exp







−

πE







1 −

















(ωτ /2π ), (5)

τ being the pulse duration. However in the typical multipho-

ton (and of perturbative nature) case γ  1,

)

−5/2



4γ



q(n − n

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