Exact classical and quantum dynamics in background electromagnetic fields

2.1 MB·

by  Tom Heinzl

Exact classical and quantum dynamics in background electromagnetic fields
Tom Heinzl1,∗ and Anton Ilderton1,†
1Centre for Mathematical Sciences, University of Plymouth, PL4 8AA, UK
Analytic results for (Q)ED processes in external fields are limited to a few special cases, such
as plane waves. However, the strong focussing of intense laser fields implies a need to go beyond
the plane wave model. By exploiting Poincaré symmetry and superintegrability we show how to
construct,andsolvewithoutapproximation,newmodelsoflaser-particleinteractions. Weillustrate
the method with a model of a radially polarised (TM) laser beam, for which we exactly determine
theclassicalorbitsandquantumwavefunctions. Includinginthiswaytheeffectsoftransversefield
structure should improve predictions and analyses for experiments at intense laser facilities.
Recent years have seen a flurry of experimental and conserved quantities. We will also illustrate and validate
7 theoreticalactivityininvestigatingthephysicsofintense the method by applying it to a toy model of a radially
1
laserfields. Suchfieldsinteractstronglywithanycharged polarized (TM) beam, solving for both the classical par-
0
or polarisable matter present, which necessitates a non- ticle trajectories and quantum wave functions exactly.
2
perturbative approach. This has traditionally [1–3], and We will see that these are as analytically tractability as
n
near universally, been made possible by modelling the forplanewavebackgrounds,butcapturebothtransverse
a
J laser field as a plane wave [4]. Within this model one field effects and pair production. Further examples will
1 can address longitudinal and temporal pulse shape ef- be provided in a separate article.
3 fects [5, 6], the detailed analysis of which has been a We begin with the classical problem. Let P =
µ
central issue in the last decade [7–11]. The same model mx˙ +eA bethecanonicalparticlemomentainaback-
µ µ
h] provides the quantum interaction rates for all widely- ground(e.g.laser)fieldAµandletξµgenerateaPoincaré
p used particle in cell (PIC) codes [12]. Thus the plane transformation, ξµ ≡ aµ + ωµνxν, where aµ and ωµν
- wave model is a crucial input for current strong field (ω +ω =0)parametrizethefourtranslationsandsix
p µν νµ
physics. However, the model cannot capture intensity Lorentz transformations of the 10 dimensional Poincaré
e
h gradient effects stemming from the transverse profile of group. Thekeyquestiontoanswerishowthebackground
[ a laser pulse [13, 14]. Such effects may become sizeable affects the 10 corresponding Noether charges ξ.P which

for strong focussing [15], precisely the method by which would be conserved for a free particle. Dotting ξ into
1 µ
v current field strengths are reached. the equations of motion mx¨µ =eFµνx˙ν shows that
6
Itisthusnecessarytogobeyondtheplanewavemodel: d
6 ξ.P =ex˙µL A , (1)
1 this is a long-standing, challenging problem. One op- dτ ξ µ
9 tion is to resort to suitable approximations, such as as-
thus the change in time of the free Noether charges
0 suming high centre-of-mass energy in laser particle col-
1. lisions [16, 17], or one could try to simplify the defining is given by the Lie derivative of the background field,
L A = ξ.∂A +A ∂ ξν. Suppose then that A is a
0 equations, e.g. using reduction of order [18, 19]. How- ξ µ µ ν µ µ
7 ever, it would clearly be advantageous to also have exact symmetric gauge field [25], meaning that for a given ξ
1 the Lie derivative vanishes up to gauge transformations:
solutions, in particular as these are required for a non-
:
v perturbative quantum Furry picture [20]. We will show
L A (x)=∂ Λ(x). (2)
i here how to generate such solutions. ξ µ µ
X
In this case (1) can be integrated and implies the exis-
r Recallthatchargemotioninaplanewaveisintegrable,
a tence of a conserved quantity Q,
i.e. exactly solvable, due to the existence of three con-
served momenta [21], stemming from the three transla-
Q:=ξ.P −eΛ=constant. (3)
tional symmetries of a plane wave. Below we will show
for the first time that the plane wave has an even larger Thus we have the important result that any Poincaré
numberofconservationlaws,whichrenderchargemotion symmetry of the background yields a conserved quan-
maximallysuperintegrable [22,23]. Ingeneralthisimplies tityinthechargedynamics. Withsufficientlymanysuch
that part, or all, of a classical orbit may be determined symmetries we obtain an integrable system. The precise
algebraically and, crucially, all known such systems are requirements are formulated in the Hamiltonian picture.
exactly solvable quantum mechanically [24], as is indeed ThecovariantHamiltonianofparticledynamicsvanishes,
thecaseforplanewaves[4]. Thisinspiresourapproachto though, due to reparametrisation invariance [26], so we
constructingnewexactlysolvablemodelsforlaser-matter must gauge fix by choosing a time parameter to describe
interactions: wesearchforsuperintegrablesystems. This dynamical evolution [27]. All choices lead to equiva-
is not to say that we search blindly; we will give below lent results, but for the fields we consider the most suit-
a method which automatically generates systems with able choice is, as could be expected from the plane wave
2
case [6, 28], light-front time x+ = t+z. Phase space is Itspolarisationstructuremaybeemployedtoincreasefo-
then six dimensional, spanned by the canonical coordi- cussing [35]. The TM mode in question has radially (az-
natesx− =t−z, x⊥ =(x,y)andthecorrespondingcon- imuthally) polarised electric (magnetic) fields transverse
jugate momenta P = (P −P )/2, P = (P ,P ). The tothebeampropagationdirection[36–38],suggestingro-
− t z ⊥ x y
Hamiltonian and equal time (x+) Poisson brackets are tational symmetry under, say, L . For a doughnut mode
z
the transverse fields go like ∼ x⊥exp−|x⊥|2, hence van-
(P −eA )2+m2 ish on-axis, rise linearly with x⊥ close to the axis, and
H ≡P = ⊥ ⊥ +eA ,
+ 4(P −eA ) + (4) haveamaximumoff-axisbeyondwhichtheydecreasera-
− −
{x−,P }=1, {xi,P }=δi . dially. Whilethemagneticfieldisstrictlytransverse,the
− j j
electricfieldhasalongitudinalcomponent,whichhases-
The evolution of any quantity Q is now determined by sentiallythesametemporaldependenceasthetransverse
fields but is suppressed relative to them (in the parax-
dQ ∂Q ial approximation by the usual small focussing parame-
= −{Q,H}. (5)
dx+ ∂x+ ter for a Gaussian beam [38]); this connection between
transverseandlongitudinalprofilessuggestsretainingin-
Forthedynamicstobeintegrable,theremustexistthree
variance under T . We also retain invariance under P .
independentchargesQ whichareconserved,dQ /dx+ = j −
i i A field with the four symmetries {P ,T ,L } is
0, andininvolution, {Q ,Q }=0foralli,j. Ifthereare − j z
i j
additionalQ ,notnecessarilyininvolution,thenthedy-
k F =(x n −n x )E(n.x),
µν µ ν µ ν
namics is superintegrable [22, 23]. The key point is that (9)
(cid:0) (cid:1) (cid:0) (cid:1)
E=E(x+) x,y,x+ , B=E(x+) y,−x,0 .
asourceofsuchQisgivenby(2). (Theremaybefurther
conservedquantitiescorrespondingtonon-Poincarésym-
The transverse electric/magnetic field has ra-
metries of phase space.) We illustrate these ideas with
dial/azimuthal polarisation, as for a TM beam. The
theplanewavecase. Letn bealight-likevector,n2 =0,
µ
profile E is common to the transverse and longitudinal
andlj bethetwomutuallyorthogonalspace-likevectors
µ fields, and the latter is suppressed by the additional
obeying lj.n=0. A plane wave is then described by
factor of x+ near the temporal peak of the field. (If we
eF =(nlj −ljn) f(cid:48)(n.x), eA =ljf (n.x), (6) takeE tobepeakedattheorigin, thenx+E(x+)<E(x+)
µν µν j µ µ j close to the peak.) Further, the relativistic invariants
of (9) obey E.B = 0 and E2 − B2 > 0 in agreement
wheref isanarbitraryprofile. Thefield(6)issymmetric
forξ equalton orlj;thisimplies,choosingcoordinates with [38]. The field (9) therefore captures some essential
µ µ µ
features of a TM doughnut mode near the beam axis,
such that n.x=x+, that the three particle momenta P
−
including the polarisation structure, the local rise of the
and P are conserved. These are in involution, giving
⊥
transverse fields, and the suppression of the longitudinal
integrability. However it does not previously seem to
field. (See [39] for a similar approximation based on
have been observed that charge motion in plane waves
a vortex beam.) In addition, the fields (9) obey the
is superintegrable, because F is also symmetric under
µν
twonullrotationsT ,forwhichξj =(nlj−ljn) xν,i.e. homogeneous Maxwell equations. A potential with
j µ µν
L A =0 for the four Poincaré generators above is
ξ µ
T ≡ξj.P =xjP+−x+Pj . (7)
j (cid:0) x.x (cid:1)f(n.x)
eA = n −x , (10)
(Recall P+ ≡ 2P in lightfront coordinates.) The po- µ 2n.x ν ν n.x
−
tential in (6) is invariant up to a gauge term as in (2),
where f(cid:48)(x) := xE(x). The background (9) is not in
yielding the two conserved quantities, for g(cid:48) ≡f ,
j j general source-free, but in this proof-of-principle inves-
tigation the key point is the solvability of the charge
Q =T +g (x+)=constant, (8)
j j j dynamics, to which we now turn. The four quantities
{P ,T ,L } are conserved, and the first three are in
as can be verified using (5). Together {P ,P ,Q } are − ⊥ z
− j j involution, implyingsuperintegrabilityofthechargemo-
five independent, conserved quantities, three in involu-
tion. (There is a fifth conserved quantity corresponding
tion, so particle motion is superintegrable. The trans-
to a non-Poincaré symmetry; this will be discussed else-
verse particle orbit in a plane wave can then be derived
where.) The conservation of {P ,T } reduces the equa-
algebraically just by rearranging (8), see [29]. − ⊥
tions of motion from second to first order; we read off
Before moving on to QED, we present a new example
of superintegrable charge motion using a toy model of a
P =const. =⇒ mx˙+ =p+−eA+(x+), (11)
radially polarised TM beam (also called a “doughnut” or −
x⊥−x⊥
TEM mode[30,31]). Suchafieldcanbegeneratedas T =const. =⇒ mx˙⊥ =p+ 0 −eA⊥, (12)
01∗ ⊥ x+
thesuperpositionoftwolinearlypolarisedHermite-Gauss
TEM modes[32],asrealisedexperimentallyin[33,34]. where p+ and x⊥ are the conserved quantities. Equation
01 0
3
(11) is immediately solved by quadrature:
1
x+
(cid:90) ds 0.8
m =τ , (13)
p+−eA+(s) 0.6
0.4
which allows us to trade τ for x+, as in the plane wave
0.2
case. Equation (12) can now be solved to give the trans-
verse orbit as a function of x+, also as for plane waves, -3 -2 -1 1 2 3x+/w0
|E⟂|at|x⟂|=w0
x+ E∥
(cid:90) ds
x⊥(x+)=u⊥x+−p+x⊥x+ , (14)
0 0 s2(p+−eA+(s))
whereu⊥ canberelatedtotheinitialtransversevelocity.
0
Havingdeterminedx+ andx⊥,adirectintegrationyields
x− via the mass-shell condition,
x+
(cid:90) dx⊥dx⊥ (cid:18) dτ (cid:19)2
x− = dx+ + . (15)
dx+ dx+ dx+
Asforplanewaves,theexactsolutiontotheequationsof
motionisexpressedintermsofthepotentialandintegrals
over it. These integrals can be performed analytically in
certaincases,alsoasforplanewaves. Wegiveanexplicit
example,suchthattheelectricfieldhasaLorentzprofile,
1 E
E= (x,y,x+) 0 , (16)
w0 (1+x+2/w02)2 FIG. 1. Upper left: The transverse and longitudinal electric
field profiles (16). Upper right: density and vector field plot
where E0 has units of the electric field. w0 is a length of the transverse fields at fixed x+. Lower panel: Electron
which, in this near-axis model, can be interpreted as the trajectories at γ =−1/2. A circular distribution of electrons
remnant of the doughnut waist, see the density plot in inthetransverseplane,initiallyatrestwhenthefieldsvanish,
Fig. 1. Charge dynamics in this field are controlled by is drawn toward the symmetry (z) axis and focussed to a
single point as time evolves. The electrons are expelled from
an (inverse) adiabacity parameter γ := eE w /(2p+).
0 0 this point as the fields switch off.
Taking E > 0, we note that γ is negative/positive for
0
electrons/positrons. Working with dimensionless ϕ :=
x+/w ,therelation(13)betweenx+andτ becomes,writ-
√0 out from the focal point. These behaviours are due to
ing 1−γ =:µ
the transverse field structure: in a plane wave, all elec-
p+ τ γ(cid:20)π (cid:18)ϕ(cid:19)(cid:21) trons would be pushed in the same transverse direction.
=ϕ+ +tan−1 , (17) Thus, even in our simplified field model, we can access
m w µ 2 µ
0
physics to which the plane wave approximation is blind.
for −∞ < γ < 1. If γ > 1, the ϕ-range accessible It would be interesting to examine profiles E(x+) with
√
by the particle is restricted to either (−∞,− γ−1) or multiple field oscillations, which may allow an analytic
√
( γ−1,∞) because there are zeros of the denomina- investigation of particle trapping as well as focussing.
tor in (13)–(14) [40]. The corresponding expressions are We now extend our method to the Dirac and Klein-
given in the appendix. Turning to the transverse coordi- Gordon equations in a background A , in order to ob-
µ
nates, the explicit solution is, for 0<γ <1, tain Furry picture wave functions [20] for use in QED
calculations. In order to give a unified description we
(cid:20) (cid:18) (cid:18) (cid:19)(cid:19)(cid:21)
γ ϕ π ϕ
x⊥ =x⊥+u⊥ϕ+x⊥ 1+ +tan−1 . (18) present the scalar and spinor cases together. It may be
0 0 0 µ2 µ 2 µ
verified directly from the equations of motion that the
field theory version of (1) is
The expression for x− is similar. The electron orbits
are illustrated in Fig. 1 and clearly exhibit particle fo-
∂ (cid:2)(Dµφ)†L φ+c.c.(cid:3)=ejµL A ,
cussing: electrons are drawn toward the beam axis and µ ξ ξ µ
(19)
arefocussedtothesamespatialpositionatthesametime ∂ (cid:2)iψ¯γµL ψ(cid:3)=ejµL A .
µ ξ ξ µ
(while positrons are repelled from the axis, see the ap-
pendix). As the fields switch off, the electrons are flung Here the scalar and spinor Lie derivatives corresponding
4
to Poincaré transformations are yielding the Volkov solutions. Let us now add to these
results, turning to the TM beam model (9).
L φ=ξ.∂φ, L ψ =ξ.∂ψ+ 1∂ ξ (cid:2)γµ,γν(cid:3)ψ . (20) For simplicity we look for a scalar φ which is a simul-
ξ ξ 2 µ ν
taneous eigenvector of the three commuting operators
The U(1) currents in (19), jµ = ψ¯γµψ for spinors and {P−,Tj}. As for a plane wave, this reduces the Klein
jµ =iφ†Dµφ+c.c. for scalars, are conserved, ∂ jµ =0. Gordon equation to an ODE in x+, which is solved by
µ
If the potential is symmetric, L A = ∂ Λ, then from
ξ µ µ (cid:0) (cid:1)
exp −iS
(19) we also have conserved Poincaré currents, φ(x)= , (23)
(cid:112)
x+ p+−eA+(x+)
Jµ =(Dµφ)†[L φ+ieΛφ]+c.c.,
ξ ξ
(21) where S is the classical Hamilton-Jacobi action obeying
Jµ =iψ¯γµ(cid:2)L ψ+ieΛψ(cid:3).
ξ ξ (∂ S−eA )2 =m2,
µ µ
We now employ this symmetry to solve for φ or ψ. To
x+
this end we note that a sufficient condition guaranteeing p+x.x p+|x⊥|2 (cid:90) p+2|x⊥|2+m2s2
S := + 0 + ds 0 .
conservation of (21) is 2 x+ x+ 2s2(p+−eA+(s))
(iL −eΛ)φ=λφ, (iL −eΛ)ψ =λψ , (22) It is worth highlighting the similarities and differences
ξ ξ
between (23) and the Volkov solutions [4]. The classical
with real λ, for then ∂ jµ ∝ ∂ jµ = 0. We then impose action appears in the exponent in both cases, and con-
µ ξ µ
(22) for as many “iL − eΛ” as possible – these must tains three degrees of freedom. Re-exponentiating the
ξ
clearly be mutually commuting, which is the analogue of denominator in (23), we see that the exponent differs
theclassicalinvolutioncondition. Theideaisthat,asall from(minusitimes)theclassicalactionbyanimaginary
known maximally superintegrable classical systems are part, representing quantum corrections. The Volkov so-
also solvable quantum mechanically [23, 24], this proce- lution,ontheotherhand,isequaltotheexponentofthe
dure will identify enough of the field structure to make classical action, i.e. receives no quantum corrections.
the Dirac/ Klein-Gordon equation soluble. This is ex- A key difference is the non-trivial dependence of (23)
actly what happens for plane waves – imposing (22) for on the transverse coordinates, due to the transverse field
thethreecommutingderivatives{∂ ,∂ }reducese.g.the structure. Here it is revealing to compare the Fourier
− ⊥
Klein-Gordon PDE to a soluble first order ODE in x+, transforms of (23) and the Volkov solution,
x+
(cid:90) e−ip.x (cid:90) (cid:20) i (cid:90) eA+(s) (cid:18) p+2x⊥x⊥(cid:19)(cid:21)
φ(x)= d¯p d2x⊥Φ(x⊥,p+) exp ix⊥p − ds m2+ 0 0 , (24)
(cid:112)p+−eA+(x+) 0 0 0 ⊥ 2p+ p+−eA+(s) s2
−∞
x+
(cid:90) e−ip.x (cid:20) i (cid:90) (cid:21)
φ (x)= d¯p √ Φ(p ,p+) exp − ds 2eA⊥(s)p −e2A⊥(s)A (s) , (25)
Volkov p+ ⊥ 2p+ ⊥ ⊥
−∞
in which Φ is an arbitrary function in both cases, p is by the above symmetries in pair production, nonlinear
µ
on-shell and d¯p is the on-shell measure. It is now simple Compton scattering, and other quantum processes. On
to read off the past asymptotic behaviour of the fields: thisnote,letusaddressthefeasibilityofcalculatingwith
they become free, with Φ being the initial wave packet. thesolutions(24). RecallthatusingVolkovsolutionsone
Another difference is the presence of p+ −eA+ in de- canalwaysperformthex− andx⊥ integralsateachQED
nominators,ratherthanjustp+ asappearsintheVolkov vertex exactly, yielding three delta functions. In the TM
solution (25). This is due to the longitudinal field. The case, the x− integrals will still yield delta functions, and
denominator can vanish which, as for a purely longitudi- since φ is Gaussian in x⊥ we will still be able to per-
nal electric field [41, 42], reflects the fact that the field formalltransverseintegralsexactly(toyieldGaussians).
can spontaneously produce pairs, as E2−B2 >0, recall HencescatteringcalculationsintheTMbackgroundwill
thediscussionbelow(9). Thusourexactresultsdescribe allow as much analytic progress as the plane wave case.
classical dynamics, quantum corrections, and pair pro- This will be pursued in a future paper, but as a first
duction effects. exampletheappendixcontainsacalculationoftheinflu-
Itwillbeveryinterestingtoinvestigatetheroleplayed ence of the TM field on wavepacket spreading.
5
In conclusion, we have shown how to construct new, [17] A.DiPiazza,Phys.Rev.Lett.117(2016)no.21,213201
exactly solvable models of classical and quantum charge [arXiv:1608.08120 [hep-ph]].
dynamics in background fields, based on superintegra- [18] E. Raicher, E. Raicher, S. Eliezer, and A. Zigler, Phys.
Lett.B750,76(2015),arXiv:1502.03558[physics.plasm-
bility. This finally provides a way to go beyond the
ph].
plane wave model in laser-matter interactions, without
[19] T. Heinzl, A. Ilderton, and B. King, Phys. Rev. D94,
approximation. We have demonstrated the feasibility of
065039 (2016), arXiv:1607.07449 [hep-th].
the method using a first example in which the effects of [20] W. H. Furry, Phys. Rev. 81, 115 (1951).
transversefieldstructure,towhichtheplanewavemodel [21] L.LandauandE.Lifshitz,TheClassicalTheoryofFields
is blind, can be analytically investigated. This also adds (Butterworth-Heinemann,Oxford,1987)(CourseofThe-
to the mathematical literature of known superintegrable oretical Physics, Vol. 2).
[22] S. Wojciechowski, Phys. Lett. A 95, 279 (1983).
systems. We hope that our exact analytic method will
[23] W. Miller, Jr., S. Post, and P. Winternitz, J. Phys. A
lead to new and improved predictions and analyses for
46, 423001 (2013).
upcoming high intensity laser experiments. Of course
[24] P. Tempesta, A. V. Turbiner, and P. Winternitz, J.
our method is not limited to laser fields; another target Math. Phys. 42, 4248 (2001).
for investigation is the effect of spatial inhomogeneities [25] R.Jackiw,Diverse Topics in Theoretical and Mathemat-
in magnetic fields [43]. ical Physics (World Scientific, Singapore, 1995).
[26] P. A. M. Dirac, Can. J. Math. 2, 129 (1950).
We thank Ben King and David McMullan for use- [27] P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949).
ful discussions. This project has received funding from [28] B.L.G.Bakkeretal.,Nucl.Phys.Proc.Suppl.251-252,
165 (2014), arXiv:1309.6333 [hep-ph].
theEuropeanUnion’sHorizon2020researchandinnova-
[29] T.HeinzlandA.Ilderton,Superintegrablerelativisticsys-
tion programme under the Marie Skłodowska-Curie grant
tems in spacetime-dependent background fields, to ap-
No. 701676.
pear.
[30] A. E. Siegman, Lasers (University Science Books, Mill
Valley, CA, 1986) www.uscibooks.com.
[31] R.Menzel,Photonics.LinearandNonlinearInteractions
of Laser Light and Matter (Springer,Berlin,Heidelberg,
∗ [email protected] 2007).
† [email protected] [32] H. Kogelnik and T. Li, Appl. Opt. 5, 1550 (1966).
[1] H. R. Reiss, J. Math. Phys. 3, 59 (1962). [33] R.Oron,S.Blit,N.Davidson,A.A.Friesem,Z.Bomzon,
[2] N. B. Narozhnyi, A. Nikishov, and V. Ritus, Zh. Eksp. and E. Hasman, Appl. Phys. Lett. 77, 3322 (2000).
Teor. Fiz. 47, 930 (1964), [Sov. Phys. JETP 20, 622 [34] S. Quabis, R. Dorn, and G. Leuchs, App. Phys. B 81,
(1965)]. 597 (2005).
[3] L.S.BrownandT.W.B.Kibble,Phys.Rev.133,A705 [35] R.Dorn,S.Quabis, andG.Leuchs,Phys.Rev.Lett.91,
(1964). 233901 (2003).
[4] D. Volkov, Z. Phys. 94, 250 (1935). [36] D. Pohl, Appl. Phys. Lett. 20, 266 (1972).
[5] T. W. B. Kibble, Phys. Rev. 138, B740 (1965). [37] L. W. Davis and G. Patsakos, Opt. Lett. 6, 22 (1981).
[6] R. A. Neville and F. Rohrlich, Phys. Rev. D3, 1692 [38] K. T. McDonald, arXiv:0003056 [physics.optics].
(1971). [39] I. Bialynicki-Birula, Phys. Rev. Lett. 93, 020402 (2004),
[7] M. Boca and V. Florescu, Phys. Rev. A80, 053403 arXiv:physics/0403078 [physics].
(2009). [40] R. P. Woodard, Nucl. Phys. Proc. Suppl. 108 (2002),
[8] T. Heinzl, D. Seipt, and B. Kämpfer, Phys. Rev. A81, arXiv:hep-th/0111282 [hep-th].
022125 (2010), arXiv:0911.1622 [hep-ph]. [41] T.N.Tomaras,N.C.Tsamis, andR.P.Woodard,JHEP
[9] T. Heinzl, A. Ilderton, and M. Marklund, Phys. Lett. 11, 008 (2001), arXiv:hep-th/0108090 [hep-th].
B692, 250 (2010), arXiv:1002.4018 [hep-ph]. [42] A.Ilderton,JHEP09,166(2014),arXiv:1406.1513[hep-
[10] A.DiPiazza,C.Müller,K.Hatsagortsyan, andC.Kei- th].
tel, Rev. Mod. Phys. 84, 1177 (2012), arXiv:1111.3886 [43] H. Gies, F. Karbstein, and N. Seegert, New J.Phys. 15,
[hep-ph]. 083002 (2013), arXiv:1305.2320 [hep-ph].
[11] V. Dinu, T. Heinzl, A. Ilderton, M. Marklund, and
G. Torgrimsson, Phys. Rev. D90, 045025 (2014),
arXiv:1405.7291 [hep-ph].
[12] A. Gonoskov, S. Bastrakov, E. Efimenko, A. Ilderton,
M. Marklund, I. Meyerov, A. Muraviev, A. Sergeev,
I. Surmin, and E. Wallin, Phys. Rev. E 92, 023305
(2015).
[13] T. W. B. Kibble, Phys. Rev. Lett. 16, 1054 (1966).
[14] T. W. B. Kibble, Phys. Rev. 150, 1060 (1966).
[15] C. Harvey, M. Marklund, and A. R. Holkundkar, Phys.
Rev. Accel. Beams 19, 094701 (2016), arXiv:1606.05776
[physics.plasm-ph].
[16] A. Di Piazza, Phys. Rev. Lett. 113, 040402 (2014),
arXiv:1310.7856 [hep-ph].
6
x⟂/w
0
1.5
e+,γ=0.5
1.0
n 0.5
o
siti
o x+/w
p -2 -1 1 2 0
al
niti -0.5
I
-1.0
-1.5
FIG. 2. Transverse motion of positrons at γ = 1/2. The FIG. 3. Dynamics of a quantum wave packet of Klein-
red curve shows a typical example of the local energy den- Gordonsolutions. Colour(darktolightblue)showstheden-
sity (E2+B2)/2 on the positron’s trajectory (not to scale), sity of the Gaussian wave packet, which is centred on the
illustrating that despite the transverse rise of the field, the classical trajectory (solid line, initial conditions p+ = m,
particles are asymptotically free. This can be verified from x⊥0 = w0). The dotted line shows ± one stan√dard deviation
the asymptotic expansions of (17)–(18). σ around the classical trajectory (σ0 = w0/ 2). The wave
packetspreadsincoordinatespacewhenthefieldisweak,but
can be narrowed by the fields of the TM beam.
Appendix: positron motion
which is centred on the classical orbit x⊥ as in (14), and
cl
Forcompletenesswegiveheretheextensionoftheclas- where the variance of the spreading wavepacket is
sicalmotionresults(17)-(18)topositrons. For0<γ <1
the explicit expressions are as in the text, and the or- (x+−x++x+x+I)2
σ2(x+)=σ2(1−x+I)+ 0 0 , (29)
bits are illustrated in Fig. 2 above. The explicit expres- 0 p+2σ2
0
sions differ though for γ > 1. For x+ we have, writing
√
ν := γ−1 with
p+ τ =ϕ− γ coth−1(cid:20)ϕ(cid:21), (26) I ≡(cid:90)x+ds 1 eA+(s) . (30)
m w0 ν ν s2p+−eA+(s)
x+
0
and for the transverse coordinates
The width obeys σ(x+) = σ and for A → 0 recovers
(cid:18) (cid:20) (cid:21)(cid:19) 0 0 µ
γ ϕ ϕ
x⊥ =x⊥+u⊥ϕ−x⊥ 1− coth−1 . (27) the usual free-field spreading of the wave packet,
0 0 ν2 ν ν
σ2(x+)(cid:12)(cid:12)(cid:12) =σ2+ (x+−x+0)2 . (31)
(cid:12) 0 p+2σ2
A=0 0
Appendix: Wave packet spreading
Hence when the field is weak the wave packet spreads
as in quantum mechanics. The wave packet can though
As a first illustration of a quantum calculation, we
be focussed by the TM field, as shown in Fig. 3. Note
show how the TM beam influences the spread of a wave
that the derivation of these results requires performing
packet of Klein-Gordon solutions. This ‘first quantised’
Gaussian integrals only.
calculation is only meant to demonstrate the feasibility
of using the wave functions (24); we concentrate only on
the transverse coordinates and the exponential part of
the wave function, neglecting pair production.
We choose Φ in (24) such that the real part of the
scalar φ is, at initial time x+, Gaussian in the trans-
0
verseco-ordinates,centredonpositionx⊥andwithwidth
0
(standard deviation) σ . For x+ > x+, the real part of
0 0
the wave packet has the form,
(cid:20) (x⊥−x⊥)2(cid:21)
φ(x)∼exp − cl , (28)
2σ2