Table Of Contentff
Modulation e ect in multiphoton pair production
Ibrahim Sitiwaldi1 and Bai-Song Xie ∗1,2
1Key Laboratoryof Beam Technologyand MaterialsModificationof theMinistryof Education,
and College of Nuclear Science and Technology,
Beijing Normal University, Beijing 100875, China
7
2Beijing Radiation Center, Beijing 100875, China
1
0
2 (Dated: January 16,2017)
n
a Abstract
J
3 Weinvestigatetheelectron-positron pairproductionprocessinanoscillatingfieldwithmodulatedampli-
1
tudeinquantumkineticformalism. Bycomparingthenumberdensityinfieldwithandwithoutmodulation,
]
h
wefind that the pair production rate can be enhanced by several orders when the photon energy just reach
p
-
t thethresholdwiththehelpofshiftedfrequencyduetomodulation. Wealsodetectthesameeffectinapulse
n
a
u train with subcycle structure. We demonstrate that the frequency threshold can be lowered by frequency
q
[ ofpulse-train due to modulation effect. Wealso find that the momentum distribution for N-pulse train can
1
reachN2timesthesinglepulseatthemaximumvalueandthenumberdensityasafunctionofpulsenumber
v
6
8 followsthepowerlawswithindex1.6whenthemodulation effectismaximized.
5
3 keywords: pairproduction, modulation effect,pulsetrain
0
.
1
0 PACSnumbers:12.20.Ds,05.20.Dd
7
1
:
v
i
X
r
a
∗Correspondingauthor.Emailaddress:[email protected]
1
I. INTRODUCTION
The electron-positron pair production from vacuum under the strong external field is one of
remarkablepredictionsofthequantumelectrodynamics(QED). Afterpioneeringworks ofSauter
[1],HeisenbergandEuler[2],andSchwinger[3]alargenumberofinvestigationsarededicatedto
study on vacuum pair production through employing different methods such as proper time tech-
nique[4–6],Wentzel-Kramers-Brillouin(WKB)approximation[7],worldlineinstantontechnique
[8, 9], quantum field theoretical simulation [10, 11] as well as the quantum kinetic method [12–
15]. The experimental verification of vacuum pair productionis stillremaining unavailableso far
due to Schwinger threshold of external electric field E = m2c3/e~ = 1.32×1018V/m (m and −e
cr
denote mass and charge of electron, respectively.), which is too high to achieve in the laboratory
at present.
With the rapid development of laser technology in recent years, it is being expectable that the
field strengthofexperimentalfacilitymaybemoreapproachingto theSchwingerthresholdin the
nearfuture,forexample,theEuropeanextreme-light-infrastructure(ELI)programisnowadvanc-
ing[17]. Motivatedbythis,variousschemeshavebeenproposedtosupporttheupcomingpossible
experiments in recent years. A strongly enhanced pair production rate was presented in dynami-
callyassistedSchwingermechanismwherearapidoscillatingelectricfieldissuperimposedontoa
slowlyvaryingone[18–20]. Thestudyontime-domainmulti-slitinterferenceeffectinalternating
sign N-pulse electric field [21] showed that the maximum of central longitudinal momentum can
reach N2 timesthesinglepulsevalue. Thepairproductioninashortpulsewithsubcyclestructure
was studied [22] to present momentum spectrum extremely sensitive to subcycle dynamics. The
accurate study on pair production in a pulsed electric field with subcycle structure [23] detected
the signature of effective mass of electrons and positrons in the given strong electric field, where
theeffectivemass is a littlehigherthan real mass so that thefrequency thresholdrises a littlecor-
respondingly. These findings and schemes advance greatly the topic of pair production in strong
field, moredetailscan beseen in[24–30].
In present letter we introduce a scheme where the amplitude of a high frequency oscillating
electric field is modulated. Amplitudemodulating change the dynamics of oscillating field in the
following two aspects. On the one hand the frequency of oscillating field is shifted up by mod-
ulation frequency, which may have a positive influence on the production rate, on the other hand
the field strengthis decreased to suppress theproduction rate in multiphotonregime. Therefore it
2
is desirable to show overall influence of the amplitudemodulation on pair production process. In
this paper, we study the production rate and momentum distribution for different parameters in a
high frequency oscillating field with amplitude modulated by a sinusoidal signal. The study is to
highlight how the modulation effect influences the pair production. Furthermore, we investigate
howthemodulationeffectcanlowerthefrequencythresholdinamorerealisticfieldconfiguration-
pulsetrainwithsubcyclestructure. WeemployquantumVlasovequationinthekineticformalism
and allquantitiesareworkingon thenatural units(~ = c = 1)in thispaper.
The paper is organized as follows. In Sec.II we introduce the quantum Vlasov equation for a
completeness. In Sec.III we get the numerical results and necessary theoretical analysis. In the
lastsectionwe provideabriefconclusion.
II. THEORETICALFORMALISMBASEDONQUANTUMVLASOVEQUATION
The spectral information of created particles in an external field is encoded in the distribution
function f(p,t). Theequationofmotionfor f(p,t)canbederivedfromcanonicalquantizationwith
fully quantized spinor and the electromagnetic field as a classical background [14]. We are only
interestedinsubcriticalfieldstrengthregimeE ≪ E wherecreatedparticledensityissolowthat
cr
thecollisioneffectandselfconsistentfieldcurrentduetocreatedparticlescanbeneglected. Since
the achievable spatial focusing scale is orders of magnitude larger than the Compton wavelength
of electrons we ignore any spatial dependence, and we also ignore the magneticfield. With these
simplifications,thequantumVlasovequationfor f(p,t)reads:
1 t
f˙(p,t) = q(p,t) dt′q(p,t′)[1−2f(p,t′)]cos[2Θ(p,t′,t)], (1)
2 Z
−∞
where f(p,t) accounts for both spin directions due to absence of magnetic fields. Here, q(p,t) =
eE(t)ε /ω2(p,t) and Θ(p,t′,t) = tω(p,τ)dτ with quantities as the electron/positron momentum
⊥ t′
R
p = (p ,p ),transverseenergy squaredε2 = m2+ p2, thetotalenergy squaredω2(p,t) = ε2 + p2,
⊥ k ⊥ e ⊥ ⊥ k
and the longitudinal momentum p = P − eA(t). This equation may be expressed as a linear,
k 3
first order, ordinary differential equation system (ODEs) [15] for the convenience of numerical
treatment:
1
f˙(p,t) = q(p,t)g(p,t), (2)
2
g˙(p,t) = q(p,t)[1−2f(p,t)]−2ω(p,t)w(p,t), (3)
w˙(p,t) = 2ω(p,t)g(p,t). (4)
3
The term g(p,t), i.e. theintegral part of Eq.(1)constitutesan importantcontributionto thesource
ofpairproduction,wherethequantumstatisticscharacter isrepresentedbytheterm[1−2f(p,t)]
duetothePauliexclusiveprinciple. Thetermw(p,t)denotesacounteringtermtopairproduction,
which is associated to the pair annihilation in pair creation process to some extent. The last one
of ODEs means that the more pairs are created, the more pairs are annihilated probably in pair
creation process. Notethatthestudiedsystemhas atypicalnon-Markoviancharacter.
By numerically solving this ODEs with the initial conditions f(p,−∞) = g(p,−∞) =
w(p,−∞) = 0, we can obtain spectral information f(p,t) of the created particles for any given
spatially homogenous, time-dependent electric field. The number density n(t) of created particles
can beobtaineasilyfrom f(p,t):
d3p
n(∞) = 2 f(p,∞), (5)
Z (2π)3
where the factor 2 comes from the degeneracy of electrons. It is need to remind that the particle
interpretation of f(p,t) is invalid in the presence of external field, it can be considered as the
distributionfunctionforreal particlesonlyat asymptotictimest = ±∞.
III. NUMERICALRESULTS
In order to reveal the physical mechanism briefly as well as for the simplicity of numerical
calculations, we just consider the problem in the one-dimensional momentum space for created
electron-positronpair.
A. Pairproductioninahighfrequencyoscillatingfieldwithamplitudemodulatedbyasinusoidal
signal
The spatially homogenous, time-dependent electric field in a given direction can be expressed
as E(t) = −A˙(t) = (0,0,E(t)). The transverse momentum of created particles which is perpen-
dicular to electric field is fixed as p = 0 as mentioned above. We introduce a high frequency
⊥
oscillating field (carrier) with frequency ω and amplitude E modulated by a sinusoidal signal
c 0
withmodulationfrequency ω (ω ≪ ω ),
m m c
1+cos(ω t)
E (t) = (1− M m )E sin(ω t), (6)
M 0 c
2
4
where 0 ≤ M ≤ 1 denotes modulation degree (M = 0 for no modulation and M = 1 for full
modulation)as displayedinFig.1.
E0
ME0
)
(tM
E 0
-E0
t
FIG.1: Highfrequency fieldwithamplitudemodulated byasinusoidal signalwithmodulation degree M.
Now Eq.(6) can be expanded as E (t) = (1 − M)E sin(ω t) − ME sin(ω t) − ME sin(ω t),
M 2 0 c 4 0 + 4 0 −
whereω = ω ±ω . Itindicatesthatmodulatingamplituderesultsinfrequencyshift,fromwhich
± c m
onecanexpectapossiblepositiveinfluenceonthepairproductionrate. Ontheotherhandtheaver-
age strength (power) of electric field is suppressed by a factor of 2π/ωm E2 (t)dt/ 2π/ωm E2(t)dt =
0 M 0 0
R R
3(4−M)2+1 duetomodulation,whichmayhaveanegativeinfluenceonthepairproductionrate.
8 3 3
To investigate the overall influence of amplitude modulation on the pair production rate, we
compareproducednumberdensitywithandwithoutmodulationforfullfrequencyspace. Wefind
that the pair production rate can be enhanced due to modulation for carrier frequency near the
frequencythresholdofmultiphotonprocess. Wediscusstheresultscorrespondingtothree-photon
regimehere,withoutlosinggenerality. InFig.2wedisplaytheincreasingfactorn /n asafunction
1 0
ofmodulationfrequencyω forcarrierfrequenciesω = 0.64mandω = 0.65mwithfieldstrength
m c c
E = 0.1E , where n and n denote the produced numbers density for full modulation M = 1
0 cr 1 0
and no modulation M = 0, respectively. To avoid trivial effects we switch electric field on and
off exponentiallywithin timet = 100πτ (τ = 1/m denotes the electron Compton time), and
switch 0 0
chosethedurationtimelongenought = 1000πτ wheretotalcalculationtimeis 2t +t . The
d 0 switch d
modulationfrequency isscanned withastepof0.002m.
5
1000
100
n0 10
n/1
1 c=0.64m
c=0.65m
0.1
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
m/m
FIG.2: (coloronline). Theincreasing factorn /n asafunctionofmodulation frequency ω fortwovalue
1 0 m
ofcarrierfrequency ω = 0.64m(blacksolidline)andω = 0.65m(reddashedline).
c c
Itcanbeseenthattheincreasingfactorn /n stronglydependsonmodulationfrequency. There
1 0
are two main peaks for each case where the production rate is enhanced by several orders due to
modulation. The first peaks for ω = 0.64m and ω = 0.65m are at ω = 0.044m and ω =
c c m m
0.028m with increasing factor 72 and 65, respectively. Both of this peaks satisfy a relation as
ω + 2ω = 2.006m. The second peaks for ω = 0.64m and ω = 0.65m are at ω = 0.088m
c + c c m
and ω = 0.056m with increasing factor 1364 and 445, respectively. These peaks also satisfy
m
a relation as 2ω + ω = 2ω + ω = 2.006m. This result indicates that the pair production
c + + −
rate can be enhanced due to modulation when the gap between carrier frequency and threshold
is supplemented by modulation frequency. The threshold value here is 2.006m instead of 2m can
be explained by effective mass m [23], where 2m = 2.006m agree with the result of analytic
∗ ∗
calculation: m = m 1−hAµA ie2/m2 ≈ 1.0023m.
∗ µ
p
In Fig.3 we display produced number density as a function of the modulation degree for two
cases corresponding to the main peaks in Fig.2. There are no clear enhancement before modu-
lation degree M reachs 0.5 due to the fact that the amplitude of upshifted frequency component
E M/4 is not strong enough to participate the multiphoton process. The produced number den-
cr
sity dramatically increase when M exceeds 0.6. It indicates that the shifted frequency component
begin to play a role. The production rate maximized at M = 0.9 and M = 0.85 are seen for two
cases, respectively. This non-monotonic behaviour can be explained by the competition between
decreasing totalfield strengthand increasingcomponentofupshiftedfrequency.
6
7
6
5 c=0.64m, m m
c=0.65m, m m
-4 10] 4
n [ 3
2
1
0
0.0 0.2 0.4 0.6 0.8 1.0
M
FIG.3: Producednumberdensityasafunctionofmodulation degree M fortwocasesofcarrierandmodu-
lationfrequencies ω = 0.64m,ω = 0.088m(black solidline)andω = 0.65m,ω = 0.056m(reddashed
c m c m
line). Otherparameters arethesamewithinFig.2.
25 25
M=0 M=0.2
4 ] 20 20
- 0
1
) [ 15 15
p//
(
f
10 10
5 5
0 0
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
300
25 M=0.4 M=0.8
250
-4 0]20 200
1
p) [//15 150
(
f
10 100
5 50
0 0
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
p// /m p// /m
FIG.4: Thelongitudinal momentumdistribution fordifferentmodulation degrees.
7
The longitudinal momentum distributions for different modulation degrees are displayed in
Fig.4, where carrier frequency ω = 0.65m and modulation frequency ω = 0.056m, other pa-
c m
rameters are the same as in Fig.2. For M = 0, i.e. the case without modulation, the momentum
distribution is peaked at p = ±0.83m, satisfying the energy-momentum relation for four-photon
k
process, 4ω = 2 p2 +m2. As M increases, for example, when M = 0.2 and M = 0.4, the
c k
q
peaks at p = ±0.83m become lower and some new peaks appear around of p = 0, which satisfy
k k
the energy-momentum relation in threshold value (p ≈ 0) with effective mass for three-photon
k
processes 2ω + ω = 2ω + ω = 2m . The small peaks at p = ±0.83m are still correspond-
c + + − ∗ k
ing to the energy-momentum relation for four-photon processes but with 4ω = 2 p2 +m2 and
− k
q
4ω = 2 p2 +m2, respectively. As M increases further almost only one peak around p = 0 is
+ k k
q
remained,forexample,seethecaseof M = 0.8,whichindicatesthattheholeprocessisdominated
completely by the three-photon processes due to modulation. This result also agrees with thefact
inFig.3 thatthenumberdensityreach themaximumafter M exceeds 0.8.
B. Themodulationeffectamongpulsesinapulsetrain
Nowweconsidera N-pulsetrainwithsubcyclestructurewherethefieldisasuperpositionofa
serial pulses as E(t) = N E (t) as displayed in Fig.5 when N = 3. Here the nth pulse E (t) can
n=1 n n
P
berepresented as:
En(t) = E0e−(t−nτT2m)2 sin(ωct), (7)
whereE denotesfieldstrengthofeachpulse,ω denotesthecarrierfrequency,τdenotesthepulse
0 c
durationandT denotesthetimedelayofpulsetrainandalsoitisassociatedtothepeakpositions
m
of nth pulsefield as nT . Notethat we fix theτ foreach pulseso that the modulationby Gaussian
m
envelopforeachpulsearethesamebutthedifferentdelaytimeT nowplaysaroleofmodulation
m
of pulse train. Thus we can define a pulse-train frequency as ω = 2π/T as the modulation
m m
frequency ofpulse-trainwhen N > 1. Obviouslythereis nomodulationofpulse-trainforasingle
pulsewhen N = 1.
8
0.10
0.05
Ecr
E(t)/ 0.00
-0.05
-0.10
0 50 100 150 200 250 300
t/
FIG.5: Theillustration of pulse train with subcycle structure when N = 3 for ω = 0.65m, ω = 0.056m,
c m
τ= 16τ and E = 0.1E .
0 0 cr
We display the produced number density as a function of carrier frequency in a pulse train
with subcycle structure for one pulse (N = 1) and ten pulses (N = 10) in Fig.6 with parameters
ω = 0.056m,i.e. T = 112.2τ , E = 0.1E , τ = 16τ .
m m 0 0 cr 0
8
7 10-pulse 66
1-pulse(multipled by 10) 0. 7
8
6 49 0.6
6
0.
5
-5 n [10] 4 0.631
3
2
1
2 0.6
1
0
0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70
c/m
FIG. 6: (color online). The produced number density as a function of carrier frequency in a pulse train
with subcycle structure for one pulse when N = 1 (dashed line) and ten pulses when N = 10 (solid line),
respectively.
The curve for one pulse (amplified by 10 times for convenience) is peaked at the ordinary
frequency threshold ω = 0.67m for the three-photon process as one may expect. However, the
c
curve for ten pulse is peaked at several places with the same distance. We suppose this field
can also be composed of three single oscillating fields with frequencies ω = ω ± ω and ω
± c m c
9
analogous to the case in the last subsection. The peak at ω = 0.668m is corresponding to the
c
ordinary frequency threshold of three-photon process, with the relation 3ω = 2m (we take the
c ∗
effective mass m = 1.002m in this subsection), and the peak at ω = 0.649m is corresponding to
∗ c
2ω + ω = 2ω + ω = 2m , where the carrier frequency is lower than threshold frequency by
c + + − ∗
0.019mwhilethenumberdensityisaboutstillof80%ofthatincaseofthresholdfrequency. These
peaks at ω = 0.631m, ω = 0.612m and ω = 0.687mare also correspondingto ω +2ω = 2m ,
c c c c + ∗
3ω = 2m and 2ω +ω = 2ω +ω = 2m , respectively. The difference in peak heights can be
+ ∗ c − − + ∗
explained by different strength of each frequency component as we expanded in last subsection.
This result indicates that the frequency threshold can be lowered by a shifted frequency due to
modulationamongpulsesinapulsetrain.
The produced number density as a function of pulse number N for ω = 0.631m and ω =
c m
0.056m is displayed in Fig.7. As a comparison, the results for alternative modulationfrequencies
ofω = 0.046mandω = 0.066mare shownalso. Otherparameters are thesameas inFig.6.
m m
m=0.056m
1 m=0.066m
-5 0] m=0.046m
1 Fitted line
n [
0.1
1 2 3 4 5 6 7 8 910
N
FIG. 7: (color online). The number density in a pulse train with subcycle structure as a function of pulse
number N forω = 0.65mwithdifferentmodulation frequencies ω ,thefittedlineisn = n N1.6.
c m N 1
The number densities for three values of ω are the same when N = 1. This is not surprising
m
since that for a single pulse the modulation among pulses is absent and the different T just shift
m
the position of peak field. However, as the pulse number increases, the number density is greatly
increased for three sets of ω but the increasing for ω = 0.056m is most remarkable because
m m
the ω +2ω = 2m holds in this case in comparisons with either ω = 0.046m or ω = 0.066m.
c + ∗ m m
Thusitrepresentsatypicalsignificanceofmodulationeffectforthepairproductioninmultiphoton
regime. Especiallythenumberdensityasafunctionofpulsenumberfollowsthepowerlawswith
10
