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Electromagnetic and gravitational radiation from the coherent oscillation of

electron-positron pairs and fields

Wen-Biao Hana and She-Sheng Xueba Shanghai Astronomical Observatory, 80 Nandan Road, Shanghai, 200030, China.

b ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy

Physics Department and ICRA, University of Rome La Sapienza, Piazzale Aldo Moro 5, I–00185 Rome, Italy.

(Dated: Received version August 2, 2018)

Integrating equations of particle-number and energy-momentum conservation and Maxwell fieldequations, we study the oscillation and drift of electron and positron pairs coherently with fields afterthese pairs are produced in external electromagnetic fields. From the electric current of oscillatingpairs, we obtain the energy spectrum of electromagnetic dipole radiation. This narrow spectrumis so peculiar that the detection of such radiation can identify pair production and oscillation instrong laser fields. We also obtain the energy spectrum of gravitational quadrapole radiation fromthe energy-momentum tensor of oscillating pairs and fields. Thus, we discuss the generation ofgravitational waves on the basis of rapid development of strong laser fields.

PACS numbers: 52.27.Ep; 52.40.Db; 04.30.-w

Introduction. Positron and electron pairs are pro-duced from the vacuum in a constant electromagneticfield and the production rate is sizable when the fieldstrength reaches the critical value (Ec = 1.3×1016V/cm);see Refs. [1, 2]. To reach this critical value in labora-tory, based on recent advanced laser technologies, thereare many ongoing experiments: x-ray free-electron laser(XFEL) facilities [3], optical high-intensity laser facili-ties such as Vulcan or ELI [4], and SLAC E144 usingnonlinear Compton scattering [5] for details, see Refs. [6–8]. This leads to the physics of ultrahigh intensity laser-matter interactions in the critical field [10].

We focus on the backreaction and screening effects ofelectron and positron pairs on external electric fields thatlead to the phenomenon of plasma oscillation: electronsand positrons moving back and forth coherently with al-ternating electric fields. In a constant electric field Eext,the phenomenon of plasma oscillations is studied in twoframeworks: (1) the semiclassical QED with a quantizedDirac field and classical electric field [11, 12]; and (2) thekinetic description using the Boltzmann-Vlasov equation(or equations of particle-number and energy-momentumconservation) and the Maxwell equations [13, 14]. InRef. [11], two frameworks are discussed. The first frame-work is semiclassical, where quantized fermion fields ψsatisfy the Dirac equation in an external classical poten-tial Aµ, which satisfies the Maxwell equation coupling tothe mean value of charged fermion current. These equa-tions are numerically integrated in (1+1)-dimensionalcase. The second framework is classical — the descrip-tion of particle distribution or density is adopted, andthe kinetic equation of the Boltzmann-Vlasov for parti-cle density and the Maxwell equation for fields are nu-merically integrated. The results obtained in two frame-works are in good quantitative agreement [11], for details,see Refs. [2]. In this paper, we adopt the second frame-work to investigate the plasma oscillations of electron and

positron pairs in the (2+1) space-time with the presenceof both electric and magnetic fields. We obtain not onlythe frequencies of plasma oscillations, but also the oscil-lating pattern of electron and positron pairs in the (2+1)space-time. In addition, we obtain the energy spectra ofelectromagnetic and gravitational radiation from plasmaoscillations.

PLASMA OSCILLATION.

In 1931 Sauter [15] and four years later Heisenbergand Euler [16] provided a first description of the vacuumproperties in constant electromagnetic fields. They iden-tified a characteristic scale of strong field Ec = m2

ec3/e~,

at which the field energy is sufficient to create electronpositron pairs from the vacuum. In 1951, Schwinger [17]gave an elegant quantum-field theoretic reformulation oftheir result in the spinor and scalar QED framework (seealso [18]). The special attention was given for the pres-ence of magnetic fields [19]. In the configuration of con-stant electromagnetic fields, the pair-production rate perunit volume is given by

Γ

V=αε2

π2

∑

n=1

1

n2

nπβ/ε

tanhnπβ/εexp

(

−nπEc

ε

)

, (1)

where the two Lorentz invariants ε and β are

ε ≡√

(S2 + P2)1/2 + S, β ≡√

(S2 + P2)1/2 − S. (2)

In terms of the two Lorentz invariants, the scalar S ≡(E2 − B2)/2 = (ε2 − β2)/2, and the pseudoscalar P =E · B = εβ. In order to focus on studying the phe-nomenon of plasma oscillations, as a model for quanti-tative calculations, we postulate an initial configurationof constant electromagnetic fields: (i) The electric andmagnetic fields are perpendicular to each other (E ⊥ B).

2

(ii) Their amplitudes are different (|E| > |B| 6= 0) in thelaboratory frame, i.e., the rest frame of electron-positronpair production. For this electromagnetic configurationP = 0 and leading term (n = 1), Eq. (1) yields

S =m4

e

4π3

(

2S

E2c

)

exp

[

−πEc

(2S)1/2

]

, (3)

where the critical field Ec ≡ m2e/e and me (−e) is the

electron mass (charge). Note that Eq. (3) is valid onlyfor S > 0, i.e., |E| > |B| and E ⊥ B. In this case β = 0and ε2 = 2S, Eq. (3) is equivalent to the case for a purelyelectric field E = 2S. Equation (3) approaches zero as|B| approaches |E|+ 0−. As shown below, we have cho-sen an electric field strength E that is significantly largerthan the magnetic one B; otherwise, Eq. (3) would ap-proximately vanish for S ≈ 0 and P = 0, analogouslyto the field configuration of a monochromatic laser beam(plane wave S = P = 0). We will also discuss the sit-uation in which electromagnetic fields are parallel. It isan important issue for future investigations how initialconfigurations are dynamically generated from the out-set. We adopt ~ = c = 1 and Compton units of lengthλC = ~/mec, time τC = ~/mec

2, energy scale mec2 and

critical field strength Ec.In the kinetic description for plasma fluids of positrons

(+) and electrons (−), with single-particle spectra p0± =

(p2± + m2

e)1/2, we define the number densities n±(t,x)

and “averaged” velocities v±(t,x) of the fluids:

n± ≡

∫

d3p±

(2π)3f±, v± ≡

1

n±

∫

d3p±

(2π)3

(

p±

p0±

)

f±, (4)

where f± = f±(t,p±,x) is the distribution function inphase space. The four-velocities of the electron andpositron fluids are Uµ

± = γ±(1,v±), the Lorentz factor

γ± = (1 − |v±|2)−1/2, and the comoving number densi-

ties n± = n±γ−1± . The collisionless plasma fluid of elec-

trons and positrons coupling to electromagnetic fields isgoverned by the equations of particle-number and energy-momentum conservation and the Maxwell equations:

∂(

n±Uµ±

)

∂xµ= S;

∂T µν±

∂xν= −Fµ

σ(Jσ± + Jσ

±pola), (5)

∂Fµν

∂xν= −4π(Jµ

cond + Jµpola + Jµ

ext), (6)

where we have an external electric current Jµext =

(ρext,Jext), electron and positron fluid currents Jµ± =

±en±Uµ±, and energy-momentum tensors

T µν± = p±g

µν + (p± + ǫ±)Uµ±U

ν±, T µν

m =∑

±

T µν± . (7)

Here the pressure p± and energy density ǫ± are related bythe equation of state p± = p±(ǫ±) in the fluid comovingframe. In the laboratory frame, the electron and positron

energy density p0± ≡ T 00± and momentum density pi± ≡

T i0± are given by p0± = (ǫ± + p±v

2±)γ

2± and p± = (ǫ± +

p±)γ2±v±. The conducting four-current density is

Jµcond ≡ e(n+U

µ+ − n−U

µ−), ∂µJ

µcond = 0, (8)

and the polarized four-current density Jµpola =

∑

±Jµ±pola

with Jµ±pola =

(

ρ±pola,J±

pola

)

defined by [14]

F νµJ

µ±pola = Σν

±, Σν± =

∫

d3p±

(2π)3p0±pν±A, (9)

where A is related to Eq. (3) by S =∫

d3p±/[(2π)3p0±]A.

Fµν and T µνem are the field strength and the energy-

momentum tensor of electromagnetic fields.We now assume external electromagnetic fields Eext =

Eextz and Bext = Bextx, where Eext and Bext are con-stant fields in space and time. As will be shown be-low, in this system, the electron-positron fluid velocities[Eq. (4)] have z and y components v± = (vy±y+ vz±z) inthe y − z plane, and the total electromagnetic fields areE = Eyy + Ezz and B = Bxx, which are the superposi-tion of two contributions:

Ez = Eext + Ez(t, y, z), Ey = Ey(t, y, z);

Bx = Bext + Bx(t, y, z), By = 0, (10)

where the space- and time-dependent Ez,y(t, y, z) and

Bz(t, y, z) are the electromagnetic fields created by themotion of electron and positron pairs.We adopt the approximations p± ≈ 0, ǫ± ≈ men±,

and ǫ± = ǫ±γ2± when the pair number density is not very

large for E ≃ Ec. Using Eqs. (4) and (9), we obtain

Jz,ypola ≈

Ez,y

E2(meγ±S) , J

0±pola ≈

vz±Ez + vy±Ey

E2meγ±S,

where E2 = E2z + E2

y . The total electric current andcharge densities of the electron-positron fluid are com-posed by Eqs. (8) and (9) as

Jz = e+n+vz+ + e−n−v

z− + Jz

+pola + Jz−pola, (11)

Jy = Jz(z → y) and ρ =∑

±(e±n± + J0

±pola), where thepositron and electron charge e± ≡ ±e.It turns out to be a (1 + 2)-dimensional problem in

space-time coordinates (t, y, z). Equations (5) and (6)are reduced to (i) the particle-number and energy con-servation,

∂n±

∂t+∂n±v

z±

∂z+∂n±v

y±

∂y= S, (12)

∂ǫ±∂t

+∂pz±∂z

+∂py±∂y

= e±n±vz±Ez+e±n±v

y±Ey+meγ±S;

(ii) the momentum conservation,

∂pz±∂t

+∂pz±v

z±

∂z+∂pz±v

z±

∂y= e±n±Ez−e±n±v

y±Bx+EzJ

0±pola

3

with (z ↔ y,Bx → −Bx); (iii) Maxwell equations∇·E =4πρ, ∇ ·B = 0,

∂Ez

∂t+∂Bx

∂y= −4πJz,

∂Ez

∂y−∂Ey

∂z= −

∂Bx

∂t, (13)

with (z ↔ y,Bx → −Bx). The pair-production rate[Eq. (3)] can be approximately used for varying elec-tromagnetic fields [Eq. (10)], provided E(t, y, z) andB(t, y, z) created by electron-positron pair oscillationsvary very slowly compared with the rate of electron-positron pair productions O(mec

2/~). This is justifiedif the inverse adiabaticity parameter [22] η = me

ωp

EEc

≫ 1,

where ωp is the frequency of plasma oscillations.

We are in the position of numerically integrating thebasic equations (12) and (13). The initial conditions(t = 0) are given by the constant electromagnetic fieldsEz = Eext and Bx = Bext. To simplify numericalintegrations, we assume the (z − y) homogeneity thatthe electron-positron fluid quantities and electromagneticfields are independent of y and z. As a result, Eqs. (12)and (13) are reduced to ordinary differential equations,and Eq. (13) leads to Bx = 0, i.e., the magnetic field Bx

of Eq. (10) is a constant in space and time. The initialcondition Ey = 0 leads to the solution Ey = 0 and Jy = 0for t 6= 0, because vz− = −vz+ and vy− = vy+ > 0. This isverified in the following numerical calculations.

To illustrate the plasma oscillations of pairs and fields,we consider two cases: (i) Eext = Ec and Bext = 0.1Ec;(ii) Eext = Ec and Bext = 0.3Ec. Due to the presence ofthe magnetic field Bx, pairs are not only oscillating upand down in the z direction, as first shown in Ref. [11],but they also move in the y direction. In Fig. 1, weshow the trajectory and velocity of pairs produced atz = y = 0 and t = 0. When Bx 6= 0 and dvy ∼ evzBxdt,vy increases for vz > 0 and decreases for vz < 0. In thecase of Bx being small enough compared with Ez , vy doesnot change its sign (see Fig. 1, Bx = 0.1Ec) in the periodof one circle oscillation in the z direction; therefore pairsmove forward in the y direction. When Bx = 0.3Ec, vychanges its sign (see Fig. 1, Bx = 0.3Ec); therefore pairsalso oscillate back and forth, while they are moving in they direction. In contrast to the case Bx = 0, the negativeEz amplitude is smaller than the positive Ez amplitude(see Fig. 1). The reason is that vy increases in the phaseof positive decreasing vz when Ez < 0; i.e., the electricenergy goes to the kinetic energy of the motion in the y

direction.

In Fig. 3, we plot (i) the electric current density of pairsJz as a function of the time, which is the source of electro-magnetic radiation, and (ii)the total energy-momentumtensor of pairs and fields T µν = T µν

m +T µνem as functions of

the time, which are the sources of gravitational radiation.

Before ending this section, we would like to presentsome discussions on the role of magnetic fields. In theparticular initial configuration of fields E ⊥ B and |E| >

0 2000 4000 6000 8000−600

−400

−200

0

200

400

600

y

z

positronelectron

0 1000 2000 3000 4000 5000−500

0

500

y

z

positronelectron

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

vy

vz

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

vy

vz

0 2000 4000 6000 8000 10000−1

−0.5

0

0.5

1

t/τC E

z

0 2000 4000 6000 8000 10000−1

−0.5

0

0.5

1

t/τC

Ez

FIG. 1: With initial conditions Ez = Ec, Bx = 0.1Ec (left)and Bx = 0.3Ec (right), we plot the electron and positrontrajectories and velocities vy vs vz and the electric field Ez vsthe time t from t = 0 to t = 104.

|B| considered in this paper, by integrating Eqs. (3), (5)and (6), we show the oscillating electric field strength(see Fig. 1), and the number and current densities ofpairs (see Fig. 2) are suppressed by magnetic fields, com-pared with their counterparts in the absence of magneticfields. However, we cannot conclude that such magneticsuppression is generally true. For example, when electro-magnetic fields are parallel (E × B = 0 and |E| > |B|),Eq. (1) yields (see Ref. [19])

Γ

V≃α|B||E|

πcoth

(

π|B|

|E|

)

exp

(

−πEc

|E|

)

, (14)

indicating that the pair-production rate receives an en-hancement (π|B|/|E|) coth(π|B|/|E|) to the prefactor,compared with the rate in the absence of magnetic fields[19, 20] (see also [1, 2]). It is worthwhile to study thephenomenon of plasma oscillations by numerically inte-grating Eqs. (5), (6) and (14) consistently with the initialconfiguration of parallel electromagnetic fields [21].

Electromagnetic and gravitational radiation. Weattempt to study electromagnetic and gravitational radi-ation generated, respectively, by the electric current andenergy-momentum tensor of pairs and fields. Supposethat we observe this radiation in the wave zone; that is,at distances much larger than the dimension R of theplasma oscillations, and also much larger than ωR2 and

4

0 2000 4000 6000 8000 100000

0.1

0.2

0.3

0.4

0.5

t/τC

n

B=0B=0.1B=0.3B=0.5B=1.1

0 2000 4000 6000 8000 10000−3

−2

−1

0

1

2

3x 10

−3

t/τC

Jz

B=0.1B=0.3

FIG. 2: The pair number density n± and current density Jz

vs the time t for Ez = Ec and different Bx field values.

0 5000 10000−0.01

−0.005

0

0.005

0.01

t/τC

Jz

B=0

0 2000 4000 6000 8000 10000−15

−10

−5

0

5

10

15

t/τC

T00

,Tzz

Tzz

T00

FIG. 3: Ez = Ec and Bx = 0.0. The charged current densityJz vs time t (left). The total energy-momentum tensor T

00

and Tzz vs time t (right).

1/ω, where ω is the typical frequency of radiation.For definiteness we think of the electric current and

energy-momentum tensor of the plasma oscillations oc-curring in the volume V and for a finite interval of timeT , so that the total energy radiated is finite. Thus, theelectromagnetic energy radiated per unit solid angle perfrequency interval is given by [23]

d2Eem

dωdΩ= 2

∣

∣

∣

∣

∫

V

d3x′∫

T

dt′eiωt′−ikx′

[

∂Jz(x′, t′)

∂t′

]∣

∣

∣

∣

2

. (15)

The gravitational energy radiated per unit solid angle perfrequency interval is then given by [24]

d2Egrav

dωdΩ= 2Gω2

[

T µν∗(k, ω)Tµν(k, ω)−1

2|T ν

ν(k, ω)|2]

,

Tµν(k, ω) =

∫

V

d3x′∫

T

dt′ Tµν(x′, t′)eiωt′−ikx′

(16)

where |k| = ω and T µν(x′, t′) = T µνm (x′, t′) + T µν

em(x′, t′).We consider ωR ≪ 1 and e−ikx′

≈ 1 for dipole elec-tromagnetic radiation in Eq. (15), and for quadrapolegravitational radiation in Eq. (16). In the calculationsof Eq. (16), we set Bext = 0 and Eext = Ec, and thenthe nonvanishing components are T 00 = T 00

m + T 00em and

T zz = T zzm +T zz

em. Using the approximation of spatial ho-mogeneity in Eqs. (15) and (16), we can factorize out thevolume V =

∫

Vd3x′, in which the total energy density

T 00 = T 00m + T 00

em = E2ext/(8π) is conserved (see Fig. 3).

Let T and V also be the time and volume of strongfields Eext & Ec created by coherent laser beams. Se-

0.05 0.1 0.15 0.210

0

102

104

106

108

1010

ω

dE

/dΩ

2×105τC

105τC

2×104τC

104τC

0 0.05 0.1 0.15 0.2 0.2510

−46

10−44

10−42

10−40

10−38

10−36

ω

dE

/dΩ

2×105τC

105τC

2×104τC

104τC

FIG. 4: Ez = Ec and Bx = 0.0. By factoring V2 out, the

electromagnetic radiation (left) of Eq. (15) and the gravita-tional radiation (right) of Eq. (16) are plotted as functions ofthe frequency ω, for different T values .

lecting different T values, in Fig. 4 we plot the elec-tromagnetic and gravitational radiation spectra (15) and(16) with V2 factored out. These two energy spectra arenarrow, and the locations (ωpeak) of their peaks are re-lated to the coherent oscillation frequency (ωp) of pairsand fields, which depend on T and Eext (see Ref. [25]).The peculiar energy spectrum of electromagnetic radia-tion is clearly distinguishable from the energy spectra ofthe bremsstrahlung radiation, electron-positron annihila-tion and other possible background events. Therefore, itis sensible and distinctive to detect such peculiar radia-tive signatures to identify the production and oscillationof electron-positron pairs in strong laser fields. As shownin Fig. 4, gravitational radiation is much smaller thanthe electromagnetic one for the reason that the gravi-tational coupling Gm2

e = 2.5 × 10−45 is much smallerthan the electromagnetic coupling e2 = 1/137. In orderto achieve a sizable radiation intensity from the plasmaoscillation, the volume V of oscillating pairs and strongelectric fields should be large enough and/or the strengthof strong fields should be enhanced (Eext & Ec) to in-crease the pair density. It is worthwhile to point out thatFig. 4 shows the numerical results of Eqs. (15) and (16)being consistent with the approximate relation betweenEqs. (15) and (16) in the ultrarelativistic limit of chargedparticles moving in external electromagnetic fields [26].

Gravitational waves from an inflationary cosmos [29]are in the high frequency band (108 − 1011Hz). Gravi-tational waves originating from some sources in groundlaboratories are also in this frequency band, and severalproposals have been made to detect high-frequency grav-itational waves up to 5 GHz [30]. Gravitational wavesgenerated from the high-energy particle beam [26, 31]in the ground experiments of the Stanford Linear Col-lider and LHC have much higher frequencies of O(1023)Hz. The frequency of gravitational wave discussed hereis O(1019−20) Hz, i.e., O(100−1)KeV, or sub nanometerO(10−(9−10)) cm. It is not clear whether such gravita-tional waves could ever be detected, or have observableeffects. One would have to build an atom-sized gravita-tional wave detector to response incoming gravitational

5

wave with such high frequencies (for some more details,see Ref. [32])

To end this paper, we remark again that the intensityof electromagnetic radiation emitted by the plasma os-cillations is tens of orders of magnitude larger than theirgravitational radiation; therefore any detectable signalis enormously more likely to result from the electromag-netic interaction. The prospect of detecting gravitationalradiation of such ultrahigh frequencies looks dim. Never-theless, our theoretical investigation of the gravitationalradiation from the electron-positron plasma oscillationwould be useful for the study of gravitational radiationemitted from particles and antiparticles in the very earlyUniverse.

Acknowledgements: Wen-Biao Han is supported byNSFC Grant No.11273045.

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[21] We are planning to do this investigation to understandthe role of magnetic fields on the pair-production andplasma oscillations.

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