Volkov States and Non-linear Compton Scatter- 7 ing in Short and Intense Laser Pulses 1 0 2 Daniel Seipt n Lancaster University, Physics Department, Lancaster LA1 4YB, United Kingdom & a The Cockcroft Institute Daresbury Laboratory, Warrington WA4 4AD, United Kingdom J 8 DOI: will be assigned ] h The collision of ultra-relativistic electron beams with intense short laser pulses makes p possible to study QED in the high-intensity regime. Present day high-intensity lasers - m mostly operate with short pulse durations of several tens of femtoseconds, i.e. only a few optical cycles. A profound theoretical understanding of short pulse effects is important s a not only for studying fundamental aspects of high-intensity laser matter interaction, but l also for applications as novel X- and gamma-ray radiation sources. In this article we give p abriefoverviewofthetheoryofhigh-intensityQEDwithfocusoneffectsduetotheshort . s pulse duration. The non-linear spectral broadening in non-linear Compton scattering due c to the short pulse duration and its compensation is discussed. i s y h 1 Introduction & High-Intensity Parameters p [ Rapidadvancesintera-andpetawatt-classlasertechnologyallowtoexplorelight-matterinter- 1 actions in the experimentally uncharted high-intensity regime. Theoretical predictions include, v for instance, intensity dependent non-linearities in high-energy photon emission and pair pro- 2 ductionprocesses[1],ornon-linearquantumvacuumopticseffectssuchasvacuumbirefringence 9 6 and photon splitting [2–5]. Eventually one expects the spontaneous creation of particle anti- 3 particle pairs from the vacuum (so-called Sauter-Schwinger effect) for field strengths on the 0 order of1 E m2/e 1.3 1018V/m [6–9]. s . ∼ ∼ × 1 When an electron interacts with a high-intensity laser, the laser pulse can be described as a 0 plane wave. Plane waves are null-fields, i.e. both field invariants vanish: E2 B2 =E B=0, 7 which means no pairs can be produced from vacuum via the Sauter-Schwi−nger mech·anism). 1 Themajorityofmodernhigh-intensitylasersystemsproduceslaserpulseswithwavelengthson : v theorderof1µm,andwithpulsedurationsontheorderoftensoffemtoseconds,i.e. onlyafew Xi optical cycles. We therefore shall investigate in this article the important effects of the short pulse duration in high-intensity laser matter interactions. r a Let us first recall some of the properties of (transverse) plane waves, which we assume to propagate along the negative z axis, and with frequency ω = (1eV). Their wave vector is O a light-like four-vector, k2 = ω2 k2 = 0, and the field strength is a univariate function, − Fµν = Fµν(k x). In this case, φ = k x = ωx+, which tells us that the field depends solely · · on the light-front variable x+ = t + z (cf. Appendix A). In addition, Maxwells equations 1We use natural Heaviside-Lorentz units with (cid:126) = c = 1 and fine structure constant α = e2/4π (cid:39) 1/137. Scalarproductsbetweenfour-vectorsaredenotedas(kp)=k·p=kµpµ=k0p0−k·p. SFHQ2016 1 in vacuum require that Fµν is transverse, k Fµν = 0. The field can be represented by a µ transverse vector potential, Aµ = (0,A (k x),0), with k A = 0, and parametrized as A = ⊥ ⊥ · · A g(φ/∆φ)Re[(cid:15)e−iφ] with a complex polarization vector ε, and an envelope function g with 0 pulseduration∆φ. Theinteractionofaprobeelectronwithfour-momentumpµ withtheplane wave field can be characterized by the following dimensionless gauge and Lorentz invariant parameters: (i) the classical nonlinearity parameter a =eA /m [10], (ii) the quantum energy 0 0 (cid:112) parameter b = (kp)/m2, and (iii) the quantum efficiency parameter χ = e (F p)2/m3 = 0 e · a b . For a pulsed laser field one has in addition the dephasing parameter a2∆φ [11], which 0 0 0 represents the ratio of the laser peak intensity and the laser bandwidth a2∆φ I/(∆ω/ω). 0 ∝ Theclassicalnonlinearityparametera representsthelaserenergydensityseenbytheprobe 0 electronandcanberelatedtothelaserintensityviaa2 =7.309 10−19I[W/cm2]λ2[µm]. With 0 × presentlasertechnologyoneisabletoreachlightintensitiesontheorderofI =2 1022 W/cm2, × which corresponds to a 100 at λ=800nm typical for Ti:Sa laser systems [12,13]. With the 0 ∼ upcominggenerationofmulti-petawattlasersoneexpectstoreachintensitiesupto1023...1025 W/cm2 andevenbeyond2. Theparameteralsoa representstheworkdonebythefieldinone 0 wavelength λ¯ of the wave in units of the electron mass a0 = eEm0λ¯. The limit a0 →∞, refers to both the limit of infinite intensity at fixed frequency and the static limit ω 0 (λ ) at → → ∞ fixed intensity. For a 1 the probabilities of processes happening in the laser field approach 0 (cid:29) those in a constant crossed field [1] and are described by the quantum efficiency parameter χ e alone. The regime a 1 is also referred to as “quasi-static” or “tunnelling” regime, i.e. 1/a 0 0 (cid:29) serves as the Keldysh parameter of the electron laser interaction. The quantum efficiency parameter χ is the value of the electric field experienced by the e particle in its rest frame in units of the Sauter-Schwinger field strength, χ = 2γE/E , with e s E =m2/e. Thismeansthatχ canbelargeforultra-relativisticparticleswithγ 1colliding s e (cid:29) with a laser pulse, although the electric field in the laboratory frame is much weaker than E s due to the Lorentz transformation of the transverse electric field. 2 Classical Dynamics of an Electron in a Laser Pulse Before we dive into the quantum theory of the laser matter interaction let us start with a clas- sical description. The classical picture is valid whenever both the quantum energy parameter b 1 and the quantum efficiency parameter χ 1. According to the classical theory of 0 e (cid:28) (cid:28) electrodynamics, the dynamics of point-like charged particles in a given external field configu- ration, described by the field strength tensor F = ∂ A ∂ A , is governed by the Lorentz µν µ ν ν µ − force equation [14] duµ m =Fµν(x)u , (1) dτ ν where m is the electron mass and we have conveniently absorbed the electron charge into the background field vector potential eA A. The electron’s four-velocity uµ = dxµ/dτ is the → tangent vector to the particle’s world line xµ(τ), parametrized by proper time τ. Due to the anti-symmetry of the field strength tensor the four-acceleration duµ/dτ is always orthogonal to the four-velocity, and the four-velocity is normalized as u u=1. · 2For instance, the Vulcan laser at RAL-CLF (UK) (http://www.clf.stfc.ac.uk/CLF/), the ELI project (CzechRepublic,Hungary,Romania)http://www.eli-beams.eu,ortheOMEGAEP-OPALproject(Rochester, USA). 2 SFHQ2016 In general this is a non-linear differential equation as the field strength has to be taken along the world line of the particle to be solved for Fµν =Fµν(x(τ)). The solution of (1) can be quite a difficult task. For instance, the motion of an electron in standing waves can show chaotic behaviour [15]. Luckily, the dynamics of an electron in a plane wave laser field is one of the few exactly solvable cases. 2.1 Solution of the Lorentz Force Equation in Plane Waves The solution of the Lorenz force equation in a plane wave field becomes particularly simple whenemployinglight-frontcoordinates(cf.AppendixA),asonecanfindaverysimplerelation between the laser phase φ = ωx+ and the particle’s proper time τ. By dotting the laser wave-vector into the equation of motion (1) we immediately find d (k u) = 0. That means dτ · k u = const. is a constant of motion, i.e. the u+ component of the electron’s four-velocity is co·nserved u+ =u+. From this we find that the relation between the laser phase φ, light-front 0 time x+ and proper time τ can be expressed as dφ =k u=ωu+ x+ =u+τ. (2) dτ · ⇒ That means one can replace the proper time derivative by a light-front time derivative in the equations of motion. Since the normalized laser vector potential has only transverse compo- nents, aµ = eAµ/m = (0,0,a (x+)), the transverse components of the Lorentz force equation ⊥ can be written as du da ⊥ = (k u) ⊥ , dτ − · dφ which gives u +a = const = u because of (2). Because the four-velocity is normalized, ⊥ ⊥ ⊥,0 u2 =1, the fourth component u− of the velocity can be obtained simply via 1+u2(φ) 1+[u a (φ)]2 u−(φ)= ⊥ = ⊥,0− ⊥ u+ u+ The particle world line xµ is obtain by integrating the four-velocity again (cid:90) (cid:90) dx+ xµ = dτ uµ = uµ. (3) u+ For later use we here also define the kinetic four momentum of the electron as πµ = muµ. The solution for the kinetic momentum can also be represented in a covariant way as πµ(x+)=Λµ (x+)pν, (4) ν where pν is the initial value of the momentum, and Λµ (φ)=(eX)µ with ν ν (cid:90) x+dz+ kµAν kνAµ Xµν = Fµν(z+)= − p+ k p · has the form of a Lorentz transformation [16,17]. In particular, the minus-component of the kinetic electron momentum can be written as 2p A A2 π− =p−+ · − . (5) p+ SFHQ2016 3 2.2 Radiation Back-Reaction Whentheelectroninteractswithahigh-intensitylaserpulsetheback-reactionoftheemissionof radiationontheelectronmotion, theso-calledradiationreaction(RR),canbecomesignificant. The radiation power is given by the Larmor formula, P = 2αu˙2. Whenever the emitted − 3 radiation energy αγ2ωa2∆φ becomes comparable to the electron’s initial energy γm, the equations of motio∼n (1) hav0e to be amended by a radiation force term Fµ: R mu˙µ =Fµνu +Fµ. (6) ν R In the radiation dominated regime, characterized by the parameter R = αχ a (cid:38) 1, the C e 0 electron loses a large fraction of it’s kinetic energy in one cycle of the background laser field, and the RR force Fµ becomes comparable to the Lorentz force. This regime could be reached, R for instance by colliding a 500 MeV electron beam with a laser pulse with a = 200, which 0 corresponds to a peak intensity of 1023W/cm2. Many different forms for the RR force Fµ have been suggested since the seminal works R of Lorenz and Abraham more than a century ago [18–21]. A particularly popular form of the RR force (which is often used in numerical simulations of laser-matter interaction at high intensity [22–24]) is the so-called Landau-Lifshitz force [25]. It reads (cid:20) (cid:21) 2α 1 1 Fµ = (uα∂ Fµν)u + FµνF uα+ (Fανu )2uµ . (7) R,LL 3m α ν m να m ν The equations of motion (6) with the LL radiation force (7) also possess closed form analytic solutionsinthepresenceofplanewavebackgrounds[26]. Bydottingagainthelaserwave-vector k into the equations of motion we find d(k u) 2α · = (a(cid:48) a(cid:48))(k u)3 dτ 3m · · which can be integrated to find a (non-linear) relation between the laser phase φ and proper time τ [cf. (2)] allowing to integrate (6) [26]. Finally, we note that the classical RR is a continuous process, in contrast to quantum RRwhichisstochastic. QuantumRR effectsare importantwheneveralready theemissionof a singlephotoncanaltertheelectrontrajectorysignificantlyduetothemomentumrecoil[27–29]. The transition from the classical to the quantum RR regime is characterized by χ (cid:38) 1 and e R =αa (cid:38)1 [27]. Q 0 3 High-intensity QED in the Furry Picture In high-intensity QED the laser field is usually described as a background field3. This back- ground field has to be treated to all orders, which can be achieved by going to the Furry inter- action picture in which the background field is treated as part of the unperturbed system [34]. For this we need to know the solutions of the Dirac equation [i∂/ A/(φ) m]Ψ(x)=0 (8) − − 3However,thedevelopmentofavalanchetypeQEDcascadesatextremeintensitiescouldcauseadepletionof thelaser,turninganinitiallystrongfieldweak,andrenderingthebackgroundfieldapproximationinvalid[30–33]. SeealsoA.M.Fedotov’scontributioninthisvolume. 4 SFHQ2016 in a given background A describing the laser field. The solutions Ψ(x) of (8) in a univariate null-field background A = A(φ) are called Volkov wave functions [35]. We use here the Dirac slash notation p/=pµγµ to denote scalar products between four-vectors and the Dirac matrices γ . µ 3.1 Derivation of Volkov States LetusnowexplicitlycalculatetheVolkovwavefunctions. Althoughthederivationcanbefound in textbooks (e.g. in [36]) will be present it here in some detail to work out why a null-field backgroundisrequiredinordertosolveEq.(8)andindicatepossibleextensionstomoregeneral backgrounds. In the first step on transforms (8) into a second order differential equation by multiplying with the adjoint Dirac operator i∂/ A/+m from the left [36], yielding − (cid:20) (cid:21) 1 (i∂ A)2 m2 σµνF (x) Ψ(x)=0, (9) − − − 2 µν with σµν = i[γµ,γν] as the commutator of gamma matrices. The so-called Pauli interaction 2 term 1σµνF (x)describestheinteractionofthehalf-integerelectronspinwiththebackground 2 µν field. The general idea is to seek a solution of (9) in the form Ψ(x) = e−i(px)Ω(φ)u with a constant four-vector pµ, and where Ω is a (4 4) Dirac matrix depending only on the laser × phase φ=k x and u denotes a Dirac bi-spinor. · When switching off the background field, the quantum numbers pµ represent the four- momentumofafreeparticle. Withthebackgroundfieldpresent,theyrepresenttheasymptotic electronmomentumoutsidethelaserpulse. Becausethreemomentumoperatorsp1 = i∂/∂x1, − p2 = i∂/∂x2 and p+ = 2i∂/∂x− commute with the Dirac-operator, the corresponding quan- − tum numbers are conserved. In addition, p− =(p2 +m2)/p+ is determined by the mass shell ⊥ condition p2 = m2. Similarly, the Dirac bi-spinors are the the free Dirac spinors u for (on- p,r shell) momentum p, fulfilling (p/ m)up,r =0, and where r =1,2 is the spin quantum number. − We choose here and in the following the normalization u¯ u =2mδ . p,r p,r(cid:48) rr(cid:48) Bypluggingtheaboveansatzforthewavefunctioninto(9)oneobtainsanequationforthe unknown matrix function Ω. Because Eq. (9) is a second order differential equation, in general the equation for Ω can be expected to be a second order as well. However, the coefficient of the second order term turns out to be k2. That means for a light-like univariate background field with wave vector k2 =0 the equation for Ω is of first order. Note that this is not true for space-like (k2 <0) or time-like (k2 >0) wave vectors. The first case, for instance, corresponds to magnetic undulators or the wave propagation in a medium with refractive index n > 1. r The latter case may refer to time dependent electric fields, or the propagation of waves in a medium with refractive index n < 1, e.g. a plasma. No general solutions of relativistic r wave equations exist in background fields with k2 =0. Recent attempts to find (approximate) (cid:54) solutions can be found in Refs. [37–40]. The situation becomes even more complicated for bi- variate backgrounds, for instance for the electron dynamics in two counter-propagating laser beams [41]. For the Volkov problem, with k2 = 0, we eventually find the following first order ordinary differential equation for the hitherto unknown matrix function Ω: dΩ (cid:16) (cid:17) 2i(kp) = 2(pA) A2 ik/A/(cid:48) Ω. (10) dφ − − SFHQ2016 5 This equation can be easily integrated yielding (cid:18) k/A/ (cid:19) Ω (φ)= 1+ e−ifp(φ), (11) p 2(kp) f (φ)=(cid:90) dφ (cid:2)2p A(φ) A2(φ)(cid:3)= p−x+ + 1(cid:90) dx+π−(x+) (12) p 2k p · − − 2 2 · where we used that k/A/ is a nilpotent operator of grade 2, i.e. (k/A/)n≥2 =0, which again holds because the background field has a light-like wave vector with k/k/ = k2 = 0 and (kA) = 0. The non-linear phase f can be written as an integrals of the minus component of the classical p kinetic four-momentum (5) by using that φ=ωx+. Note that the Volkov wave functions (14) are normalized using a covariant light-front normalization,4 1(cid:90) (Ψ¯ Ψ )= dx−dx⊥Ψ¯ (x)γ+Ψ (x)=2p+(2π)3δ(p(cid:48) p)δ (13) p(cid:48),r(cid:48)| p,r 2 p(cid:48),r(cid:48) p,r − r(cid:48)r and where p stands for (p+,p⊥), i.e. δ(p)=δ(p+)δ(p1)δ(p2). It is also convenient to define the so-called Ritus matrices E (x)=e−i(px)Ω (φ), such that p p the Volkov states can be written as Ψ (x)=E (x)u . (14) p,r p p,r The Ritus matrices can decomposed as Ep = V(φ)e−iSHJ, where SHJ denotes the classical Hamilton-Jacobiactionofaclassicalparticleinthebackgroundfield. ThatmeansVolkovwave function is an exact semiclassical wavefunction, i.e. while being an exact solution of the wave equation (8) the action does not contain quantum corrections (in powers of (cid:126)). The matrix V(φ) is the bi-spinorial representation of the Lorentz transformation Λµ (φ) that generates the ν classical orbits (4), and which is defined by V−1γµV =Λµ γν. ν The Lorentz transformation V(φ) transports the spinor u along the classical trajectory [16, p,r 17]. TheRitusmatrices,analyticallycontinuedtooff-shellmomentap2 =m2,havethefollowing (cid:54) orthogonality and completeness properties: (cid:90) (cid:90) d4xE¯ (x)E (x)=(2π)4δ(p(cid:48) p), d4pE (x(cid:48))E¯ (x)=(2π)4δ(x x(cid:48)). p(cid:48) p p p − − Thewavefunctions(14)representthepositiveenergysolutionsΨ Ψ(+) of(8), i.e.they p,r p,r ≡ describe the propagation of electrons in a background field. In order to describe positrons as well (for instance to calculate the probabilities for pair production in a laser field) we need the negative energy solutions of (8). They can be obtained from (14) via the transformation p p, i.e. Ψ(−)(x) Ψ (x), where the notation for negative energy spinors is as usual p −p → − ≡ u v [43,44]. −p p ≡ 4 The scalar product between any two spinor wave functions is defined in a Lorentz invariant way as [42] (Ψ¯1|Ψ2) = (cid:82)σdσµΨ¯1(x)γµΨ2(x), with σ being a hypersurface in Minkowski space and dσµ the infinitesimal normal vector thereupon. The hypersurface can be expressed in general curvilinear coordinates ξµ(x), where ξ0 = const. defines the hypersurface and ξ1,ξ2ξ3 parametrize σ, such that dσµ = dξ∂0x(µx)√−gdξ1dξ2dξ3 with thedeterminantg ofthemetrictensorinthecoordinatesξµ. Choosingalight-frontsurfacex+=const.yields (13). 6 SFHQ2016 60 S[Ep] 60 T13[Ep]−iT23[Ep] 1.0 1.0 0.8 0.8 40 40 0.6 0.6 20 0.4 20 0.4 0.2 0.2 mt 0 0.0 mt 0 0.0 0.2 0.2 − − 20 0.4 20 0.4 − − − − 0.6 0.6 40 − 40 − 0.8 0.8 − − − − 1.0 1.0 60 − 60 − − 60 40 20 0 20 40 60 − 60 40 20 0 20 40 60 − − − − − − mz mz Figure 1: Contour plot of (the real part of) the scalar and tensor projections of the Ritus matrices E (x) in the (z-t) plane for a circularly polarized laser pulse with a = 2 and pulse p 0 duration ∆φ=20. 3.2 Properties of Volkov States The properties of the Volkov states can be studied by investigating the Ritus matrix function E (x)astheycontainalltheinformationontheinteractionwiththelaserfield. SincetheE (x) p p are 4 4 matrices it is reasonable to study the different projections onto the basis elements of × theCliffordalgebra[45]. Itturnsoutthatonlythescalarandanti-symmetrictensorprojections yield non-zero results: 1 [E ]= trE (x)=exp iS (x) , (15) S p 4 p { p } 1 i µν[E ]= trσµνE (x)= (Aµkν Aνkµ)exp iS (x) , (16) T p 4 p 2k p − { p } · where σµν = i [γµ,γν], and tr denotes the trace over the Dirac matrix indices. 2 The real parts of the scalar projection [E ] and the combination of tensor projections p S 13 i 23 are exhibited in Figure 1 for an electron with an energy of 50GeV propagating T − T head-on through a strong laser pulse with a = 2 and a pulse duration ∆φ = 20. The two 0 projectionsareshownintheframewheretheelectronisinitiallyatrest. Inthatframe,thefree electron wave function outside the laser pulse behaves as exp ip x =exp imt . In the ∝ {− · } {− } case of the scalar projection of the Ritus matrices the effect of the laser pulse is a local tilt of the electron wave fronts, see left panel of Fig. 1. This behaviour corresponds to the build-up of an intensity dependent ponderomotive quasi-momentum inside the laser pulse [46]. The tensor projection of the Ritus matrices shown in the right panel of Fig. 1 are non-zero only in regions where the laser pulse is present. They correspond to the k/A/ term in the Volkov states, i.e. the Pauli interaction term. InthefollowingweconstructwavepacketsfromtheVolkovwavefunctionsandshowthatthe motion of the centroid of the packet follows is closely related to the classical trajectory [47,48]. SFHQ2016 7 100 50 mt 0 50 − 100 − 100 50 0 50 100 − − mz Figure 2: Contour plot of a normalized scalar Volkov wave packet in the (t-z) plane. For comparison, theclassicaltrajectoryisdepictedassolidblackcurve. Thelaserpulsepropagates between the two black dotted lines. We will restrict the discussion to the scalar part of the Volkov wave function which describes charged spin-0 bosons. As a simple example it suffices to construct a one-dimensional wave packet as a superposition of Volkov waves with a light-front momentum distribution h (p+) + but all having the same transverse momenta p =0, ⊥ (cid:90) dp+ χ(x)= h (p+) [E (x)]. (17) 2p+ + S p Suchawavepackethasafinitetemporalduration,butremainsinfinitelyextendedinthetrans- verse coordinates. In the following we choose a Gaussian distribution h+ = e−(p+−p+0)2/2∆2 with Gaussian width ∆ = 0.05m and with p+ = m. The laser pulse parametNers are the same 0 as for Figures 1. This construction provides a localized wave packet in the (t-z) plane with a minimum Gaussian size at t = z = 0. This wavepacket is shown in Figure 2 in the (t-z) plane. Thecentroidofthewavepacketfollowstheclassicaltrajectoryz(t)(blackcurve),which shows the close correspondence between the classical trajectory solutions and the Volkov wave functions. In addition, the Dirac-Volkov current Ψ¯ γµΨ πµ(φ) jµ = p,r p,r = =uµ(φ), (18) Ψ¯ Ψ m p,r p,r coincides with the classical four-velocity uµ. Let us now investigate the momentum space spectrum of the Volkov states and find rela- tions to the classical motion. The Ritus matrices E (x) have a spectral representation as a p 8 SFHQ2016 superposition of plane waves (cid:90) d(cid:96) E (x)= e−i(p+(cid:96)k) x ((cid:96)) (19) p 2π · Ep with the spectral components ((cid:96))=K ((cid:96))+ ma0k/(cid:2)/(cid:15)K ((cid:96))+/(cid:15)∗K ((cid:96))(cid:3). (20) Ep 0 2(kp) + − which are be represented by the three scalar functions (cid:26)K ((cid:96))(cid:27) (cid:90) (cid:26) 1 (cid:27) 0 = dφ exp i(cid:96)φ if (φ) . (21) K±((cid:96)) g(φ)e∓i(φ+φˆ) { − p } The variable (cid:96) parametrizes the amount of light-front laser momentum k− which is exchanged between the electron and the background field. It can be interpreted as some continuous analogue to the number of exchanged laser photons. Let us now investigate the limit of infinitely long monochromatic plane wave background fields (IPW), formally achieved by setting g 1. In that case the background is a periodic → function such that the Floquet theorem applies [49]: The solution of the Dirac equation (8) in a periodic background takes the form Ψ (φ)=e−iq xΦ(φ), (22) p · where Φ(φ+2π)=Φ(φ) has the same periodicity as the background, and qµ =pµ+ m2a20kµ is 4k p calledthequasi-momentum. TheFourier-zeromode,i.e.thenon-periodicpartofthenon·-linear phase f in (12), has been absorbed into the definition of the quasi-momentum. p In the limit of an IPW background the momentum distribution functions (21) degenerate to a delta comb K ((cid:96)) g→1 (cid:88)∞ δ((cid:96) n m2a20)K (n) (23) j −−−→ − − 4k p j n=−∞ · with support at discrete momentum values qµ +nkµ. For arbitrary polarization of the back- ground field the coefficients K (n) are related to generalized Bessel functions (see e.g. [50,51]), j and they turn into ordinary Bessel functions of the first kind in the case of circular polariza- tion. TheVolkovwavefunctioninanIPWbackgroundappearsasaninfinitesumoverdiscrete momentum states ∞ (cid:88) E (x)= e−i(q+nk) x IPW(n), (24) p · Ep n=−∞ whicharealsocalledZel’dovichlevels[52]. Thelevelstructurefurnishesaneasyinterpretationof strong-field phenomena. For instance, the appearance of harmonics in the non-linear Compton spectra or the resonant singularities in second-order strong-field scattering processes can be seen as transitions between different Zel’dovich of the incident and final state Volkov electrons. Modifications of this level structure due to radiative corrections, i.e. the electron self-energy have been calculated as well [53]. SFHQ2016 9 Figure3: SpectralcomponentsoftheVolkovstatewithasmall(large)rangeofspectralcompo- nents (cid:96) contributing in the left (right) panels. In the left panel the black vertical lines indicate the positions of the Zel’dovich levels at (cid:96) = n+ ma20 in the case of an infinite monochromatic 8ωγ plane wave. One can make a clear connection between these Zel’dovich levels and the peaks in the Volkov spectrum for a pulsed field. In the right panel a large number of spectral com- ponents contribute the width of each level is larger than their separation which makes a clear identification of individual Zel’dovich levels difficult. If one now starts from the well known case of infinite monochromatic plane waves (with the discrete level structure) and looks at modifications due to finite pulse duration one finds a broadening of the Zel’dovich levels. There are two different contributions to the level broad- ening: (i) a bandwidth broadening ∆φ−1 due to the finite laser bandwidth (ii) a non-linear, intensity dependent broadening ma20 due to the gradual change in the laser intensity in a ∼ ωγ pulsed field. This second mechanism is related to the ponderomotive, i.e. slowly varying, part of the non-linear phase f , and can be seen as a gradual build-up of an intensity-dependent p quasi-momentum as the laser intensity increases [46]. Depending on the bandwidth and the peakintensityeitherofthetwomechanismscanbedominatingthebroadeningoftheZel’dovich levels. The broadened Zel’dovich levels are shown in Fig. 3. The scalar spectral components K ((cid:96)) of a Volkov state also have a close relation to the j classical electron motion in a laser pulse. The effective range of values of (cid:96) is determined by the classical dynamics [51]. To be specific, the maximum and minimum values of the minus component of the kinetic four-momentum determine the cut-off values of (cid:96) via (cid:96) = max/min (π− p−)/k−. Beyondthosecut-offvaluesthefunctionsK ((cid:96))droptozeroexponentially max/min− j fast. Thereasonisthatonlyfor(cid:96) <(cid:96)<(cid:96) thephaseintegralsin(21)possesrealstationary min max phase points. Also the local minima and maxima of π−, which appear during the course of the laser pulse, lead to pronounced structures in the Volkov state spectral component K ((cid:96)). + Thisiscausedbyasuddenchangeinthenumberofstationaryphasepointsrelatedtofold-type caustics. All the local extremal points of π− are indicated by red lines in Figure 1, connecting them to pronounced structures in the spectral components K ((cid:96)). One could possibly observe j those spectral features of the Volkov states by using laser-assisted Compton scattering of X- rays [54]. 10 SFHQ2016