arXiv:1510.08456v1 [hep-ph] 28 Oct 2015 .King B. Lasers Power High with Polarisation Vacuum Measuring aeytebrfigneo h aum sn h XFEL the using vacuum, a the and beam of scattering, birefringence -photon the of namely XFEL manifestation respectively. facility one the HIBEF magnitude measure the of at to of experiment completion an DESY, orders the at laser three with and coinciding Moreover, six to this 1 npriua rmt icsin ewe hoit and and theorists non-specialist the between for discussions useful experimentalists. promote be particular also which will in provided, hoped of be scattering is will overview photon-photon it scenarios an real experimental various Second, of in signatures predictions predicted for experiments. equivalent the laser essentially planned be of to shown be will scatter photon-photon study to used approaches analytical e polarisation vacuum in interest xeiet uha BMV as such experiments eimai oawal olna ilcrcmtra.Vari material. dielectric nonlinear e polarisation weakly a quantum to the akin fields, medium electromagnetic intense to exposed When Abstract Keywords: suggest scattering. photon-photon been have real that signatures experimental of overview cteiguigjs ihpwrlsrple iseighteen photon-photon lies QED of pulses above for laser magnitude process of power cross-section orders high single just predicted using the the scattering of experimenta on best terms current limit e The in vacuum these scattering”. the of understood “photon-photon the All of be probe pairs investigated. can virtual that theoretically charged been have the of of frequency polarisation e of and multitude electrodynamics wave-vector A and quantum closer. strong-field (QED) TW of multi-hundred phenomena of lasers availability PW increasing The Motivation 1. ril umte o ihPwrLsrSineadEngineer and Science Laser Power High to: submitted Article etefrMteaia cecs lmuhUiest,Pl University, Plymouth Sciences, Mathematical for Centre orsodnet:Eal [email protected] Email: to: Correspondence h iso hswr r w-od is,temain the First, two-fold. are work this of aims The 1 n .Heinzl T. and aumplrsto;poo-htnsatrn,vcu bi vacuum scattering, photon-photon polarisation; Vacuum [1] ff 1 csuigacmiaino ihpwrlsr.Mtvtdb s by Motivated lasers. power high of combination a using ects Wotcllsr hshsgnrtdmuch generated has This laser. optical PW rnstecnraino long-predicted of confirmation the brings 1 [5] n PVLAS and ff [4] ff ects. cso h polarisation, the on ects u eetlaser-cavity recent but , [6] aereduced have [7] dt ofimti rdcino unu lcrdnmc of electrodynamics quantum of prediction this confirm to ed u cee aebe rpsdt esr uhvacuum such measure to proposed been have schemes ous ff plans ing, ects aumi xetdt xii rpriso polarisable a of properties exhibit to expected is vacuum [2,3] ing l muh L A,Uie Kingdom United 8AA, PL4 ymouth, 2015 1 ernec,Heisenberg-Euler refringence, i.1 sabscrdaiecreto htmdfisthe modifies loop’. that appearance ‘ a the correction in through pairs radiative virtual vacuum of in basic photons a of of propagation is diagram Feynman 1, the Fig. in depicted polarisation, Vacuum Polarisation Vacuum Introduction: 2. ttepieo aietcovariance manifest ‘old-fashioned’ of called price is what the on at consideratio energy based emphasises which is theory first perturbation The fect. ), and ( and photons represent 1. 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King renormalisation due to polarisation screening as explained term only. This guarantees that Aext is not altered by the in any standard quantum field theory text [9]. The electric interaction because the Maxwell term will only contain the charge of a increases as one ‘dives’ into its virtual field strength tensor built from the fluctuating fields Aµ, i.e. polarisation cloud, hence with decreasing distance from the Fµν = ∂µAν ∂ν Aµ. particle. As a result, the electric charge becomes scale- In this contribution− we are interested in laser-laser inter- dependent which may be expressed in terms of a distance- actions. In this case, the centre-of-mass energy (even for dependent fine structure constant, α = α(R). At distances x-rays) will always be much lower than the rest large compared to the electron Compton wavelength, R = energy, mc2. Itisthussufficient to work with the low-energy Że = ~/mc, the typical length scale of QED, one has α = effective field theory obtained from the QED Lagrangian e2/4π~c 1/137. However, at the much smaller Compton by ‘integrating out’ the Dirac fields. This can be done by ≃ wavelength of, say, the Z , R = ŻZ = ~/MZc, the employing the functional integral representation of the QED QED coupling α increases to α(ŻZ ) 1/128. vacuum persistence amplitude Z relating in and out vacua: At typical laser energies, the dominant≃ screening particles are indeed pairs of virtual electrons and positrons. Their Z = A ψ ψ¯ exp iSQED[A, ψ, ψ¯] (virtual) presence may be probed by coupling them to ad- Z D D D   ditional photons (see Fig. 2), which may represent either A exp(iSeff[A]) . (2) fluctuating quantum fields or classical background fields ≡ Z D such as provided by lasers. In either case, we are led to In the second step, the fermionic degrees of freedom have consider the probing of vacuum polarisation by “photon- been integrated out by performing a Gaussian integral result- photon scattering”. When large numbers of photons are ing in a fermionic determinant, involved, a classical metaphor of this quantum effect is of charged vacuum pairs forming a polarisable “vacuum iD/ A m exp(iSeff[A]) = exp Tr ln − , (3) plasma” medium with a nonlinear susceptibility and perme- i∂/ m ! ability. An important consequence of this quantum correc- − tion to Maxwell’s equations is the violation of the principle where we have re-exponentiated using Det = exp Tr ln. The of superposition for electromagnetic waves in vacuum. fermionic determinant depends on the photon field A and can only be evaluated analytically for special configurations such as constant fields. Alternatively, one may perform a derivative (i.e. low-energy) expansion [10,11], the leading order of which coincides with the constant field evaluation. For QED this has been done long ago (using different techniques) [12–14] the result being the celebrated Heisenberg- Euler Lagrangian m4 ∞ exp( s) 2 cot coth HE = 2 ds 3− s ab as bs L −8π Z0 s  Figure 2. Probing vacuum polarisation by photon-photon scattering. s2 1+ (a2 b2) , (4) − 3 −  where the dimensionless secular invariants a and b are given 3. Analytical Methods by: The microscopic theory describing laser-matter or laser-laser 1/2 1/2 interactions is QED described by the Lagrangian √ 2 + 2 + √ 2 + 2 a = F G F ; b = F G −F . h E i h E i 1 µν cr cr QED = ψ¯(i∂/ m)ψ Fµν F eψ¯Aψ/ , (1) L − − 4 − These contain the two electromagnetic invariants the separate terms representing the Dirac, Maxwell and = F F µν /4 = (E2 B2)/2 , (5) interaction Lagrangians, respectively. The latter derives F − µν − from ‘minimal substitution’, that is the replacement of the = F µν F˜ /4= E B =0 , (6) G − µν · ordinary by the covariant derivative, i∂ i∂ eA iDA in the free Dirac term, which leads to the→ usual− coupling≡ of with field and dual field strength tensors, electric and mag- µ µ µν ˜µν the photon field Aµ to the Dirac current j = eψγ¯ ψ as netic fields (F , F , E and B, respectively) and the critical µ eψ¯Aψ/ = Aµj . An intense laser field will normally be included as a classical, external background field Aext by the prescription of replacing A A + Aext in the interaction → Measuring Vacuum Interactions with High Power Lasers 3

field strength: 3.1. Scattering Matrix 2 2 3 m In what follows, we will consider a modification of the Ecr = m c /e~ . (7) ≡ e 4-photon scattering amplitude at low energy by assuming that two of the photons involved are stemming from a (Note that we now adopt natural units, ~ = c = 1, for the high intensity laser which is probed by a dynamical photon remainder of this section unless otherwise explicitly stated.) ‘passing through’. This is visualised in Fig. 4 The critical, “Sauter” [15] or “Schwinger” [14] field-strength Ecr is built from the fundamentalconstantsof QED and is the typical field-scale separating weak (E Ecr) from strong- ≪ field (E > Ecr) vacuum polarisation phenomena. The Heisenberg-Euler Lagrangian (4) is equivalent to QED for arbitrary values of the field strength but at energies small compared to mc2. For the foreseeable future, laser experiments will stay well below the critical field strength, hence in the weak-field limit. Thus, to a very good approx- imation, it is sufficient to work with the leading order in a Figure 4. A probe photon (wavy lines) scattering off a classical laser field strength expansion of (4) given by: background (dashed lines) at low energy (so that the Heisenberg-Euler vertex can be employed). (2) c 2 + c 2 , (8) LHE ≃ 1F 2G with dimensionless low-energy constants We assume that an incoming probe photon with four- momentum k and four-polarisation ε scatters off a laser c 2α2 4 1 = . (9) backgrounddescribed by a field strength tensor Fµν resulting ( c2 ) 45m4 × ( 7 ) in an outgoing photon with quantum numbers k′ and ε′. These define effective vertices corresponding to the low- The resulting scattering amplitude is given by the S-matrix energy limit of the diagram in Fig. 2 with the fermion-loop element no longer being resolved, see Fig. 3. ′ ′ ′ ′ ′ ′ ε , k ; out ε, k; in = ε , k Sˆ ε, k Sfi(ε , k ,ε,k) . (10) h | i h | | i≡ ′ ′ Using the leading-orderLagrangian(8), writing Sfi(ε , k ,ε,k) as Sfi(q), the S-matrix element takes on the simple form of a Fourier integral

4 iq·x Sfi(q)= i d xe Sfi(x) , (11) − Z where q = k′ k is the momentum transfer and − ′ ′ ′ ′ Sfi(x)= c1(k ,Fε )(k,Fε)+ c2(k , Fε˜ )(k, Fε˜ ) , (12) Figure 3. The leading order Heisenberg-Euler vertex or photon-photon scattering at low energies. employing the abbreviated scalar products (k,Fε) µν ≡ kµF εν etc. Hence, one may introduce an intensity form factor,

µα,νβ 4 iq·x αµ β,ν αµ β,ν The cross section for the low-energy limit of real photon- W (q) i d xe (c1F F + c2F˜ F˜ ) , photon scattering depicted in Fig. 3 is given by [16]: ≡− Z (13) which is the Fourier transformation of the background in- 973 ω 6 σ = α4 Ż2; ω m tensity distribution. In terms of the latter the scattering 10125π m e   ≪ amplitude may be written as whereas the high-energy limit is given by [16]: S (q)= ε′ k′ W µα,νβ(q)k ε . (14) m 2 fi α µ ν β σ =4.7 α4 Ż2; ω m. ω e   ≫ The results above are reminiscent of elastic electron nucleus The maximum of the cross-section is at the pair-creation scattering, where the scattering amplitude is proportional threshold of colliding photon centre-of-mass energies ω = to the nuclear charge form factor which is nothing but m. the Fourier transform of the nuclear charge distribution. 4 B. King

In photon-photon scattering one is naturally probing an By variation of the quantum action, one can derive the intensity, rather than a charge, distribution. To proceed, one corresponding modified Maxwell equations [22]: has to choose a suitable laser background field, Fµν (x), and calculate its intensity form factor Eq. (13). E = ρvac; B = Jvac + ∂ E, (20) ∇ · ∇ ∧ t in which:

3.2. Polarisation operator ρvac = Pvac; Jvac = Mvac + ∂ Pvac (21) ∇ · ∇ ∧ t An equivalent representation is obtained in terms of a quan- and the vacuum polarisation and magnetisation are: tity aptly called the polarisation operator, denoted Πµν . In its simplest incarnation it is just the mathematical expression ∂ HE ∂ HE P = L ; M = L . (22) for the of Fig. 1, namely vac ∂E vac ∂B 4 µν 2 d p µ 1 ν 1 The wave equations: Π = ie trγ γ γ , (15) − Z (2π)4 /p m (/p k/ m) − − − 2 2 ∂ E E = ρvac[E, B] ∂ Jvac[E, B] (23) t − ∇ −∇∇ − t where the trace tr extends over the Dirac matrices γµ. One 2 2 γ ∂t B B = Jvac[E, B], (24) may generalise this to the polarisation tensor in an external − ∇ ∇ ∧ field Aext, where one trades the free fermion for can be solved using, for example, the method of Green’s interacting ones through the standard minimal substitution functions. p p eAext. Indeed, this method has a long his- tory→[17–20]−as reviewed by [21]. For ourpurposesit is sufficient to just employ the first-order weak-field Heisenberg-Euler 4. Signatures of Vacuum Polarisation Lagrangian (8) once again and rewrite it as The most general vacuum polarisation diagram represents (2) 1 µν an elastic scattering amplitude that relates an incoming HE = AµΠ [Aext]Aν , (16) L 2 ensemble of photons k1,...,kn , which interact in some experimental scenario,| to an outgoingi ensemble of photons with the polarisation tensor thus defining the second-order ′ ′ k ,...,k ′ . In this review, we concentrate on processes term. From (8) one can straightforwardly read off that 1 n that| could bei measured using high power lasers. The fields c1 c2 of these lasers are included in calculations in various ways. Πµν [A ]= k F αµF βν k + k F˜αµF˜βν k , (17) ext 4 α β 4 α β A “monochromatic plane wave” will refer to an infinitely extended wave with no transverse structure, a “beam” will where the background field strength F µν = ∂µAν ∂ν Aµ . ext − ext refer to some inclusion of structure, e.g. a cylinder of To connect this approach with the S matrix formalism we radiation is a “beam”, a “focussed beam” will imply some specialise to forward scattering by setting k = k′ in (12) approximationto a real beam with focal width as a parameter which yields the relation and a “pulse” to a field localised in time with pulse duration as a parameter. Since laser pulse wavelengths are much S (k)= ε′ (k)Πµν (k)ε (k) . (18) fi, fwd µ ν larger than the Compton wavelength, and since expected electric field strengths are much less than the critical Sauter This makes the link between the polarisation operator and field, equivalent to an intensity of the order of 1029 Wcm−2, scattering matrix approaches manifest. the interaction of laser pulses with virtual electron- pairs can be expanded in terms of weak fields. Starting at n = 2 as in Eq. (8), each perturbative order describes a 3.3. Modified Maxwell Equations vacuum 2n-wavemixing process. It is noteworthy that unlike when real electrons and positrons interact with intense laser In standard quantum field theory notion [9], the total Heisen- fields, for virtual electron-positron pairs, the number of laser 4 [23] berg-Euler action, Seff = d x eff, is nothing but the one- photons involved is typically small , which is why the L loop effective (or quantum)R action of QED evaluated at low discussion is mostly in terms of four-wave mixing processes energies where there are no external electron lines. The such as in Fig. 5. This means the vacuum is often compared associated effective Lagrangian is the sum of the classical to a nonlinear optical material with a Kerr-like response [24]. 2 2 Maxwell term M = (E B )/2 and the first quantum Although there is a large overlap with , a correction: L − major difference is that the polarisation of the dielectric (here, the vacuum), can be shaped by the pump laser pulse. eff = M + HE, (19) The majority of suggested signals of vacuum polarisation L L L Measuring Vacuum Interactions with High Power Lasers 5

where Π1,2 = c1,2(k,Tk) are the two nontrivial eigenvalues of the polarisation tensor (17), expressed in terms of the µν µ αν background energy momentum tensor T = F αF . The dispersion relations (28) describe the change in light propagation caused by the energy-momentum density stored in the backgroundfield and have been referred to as modified light-cone conditions [27,28]. They imply group velocities Figure 5. Photons from the pump (dashes) interact with those from the different from the vacuum , c, and hence the probe to produce a pump-dependent vacuum index of refraction. refractive indices (26) different from unity, which can be k,⊥ 2 0 rewritten as nvac = 1+ Π1,2/2ωp, ωp = k c being the probe frequency. can be described by considering how the photons from a The result for the refractive indices has been shown probe laser change due to interaction with a more intense to hold to all perturbative orders using the polarisation [17,28,29] pump laser. The pump laser will also be referred to as the operator and Heisenberg-Euler Lagrangian numer- [30,31] [31] “background” or the “strong field” where appropriate. The ically and analytically . When the pump field is probe laser quantities will often be denoted with subscript space-time dependent as is the case for laser pulses, the p and the pump or strong laser quantities with the subscript effect on the probe is calculated by integrating over the [32] s. The source of probe photons will mostly be a high power inhomogeneousrefractive index of the pump background . 2 There has also been recent work indicating finite-time effects laser, which, satisfying E/Ecr √α(ω/m) , often allows the external field concept to be≫ invoked for the probe [25]. in an inhomogeneous background may leave a detectable [33] Therefore the discussion will include interchangeably effects signal . on probe photons and on the probe electromagnetic field, which assumes the photon-scattering process can be summed Polarisation flip is the underlying physical mechanism of incoherently over the probe photon distribution. We begin by vacuum birefringence. The term is used when an incoming µ reviewing the consequence of real photon-photon scattering photon’s polarisation vector ε is “flipped” to an orthogonal ′ µ at the level of probe laser photons: one ε due to real photon-photonscattering. The flip ampli- tude (for a head-on collision of probe and background) after γ(ω, k,ε(k)) γ(ω′, k′,ε′(k′)). (25) a propagation distance z can be found from the Heisenberg- → Euler forward scattering amplitude (18) and coincides with [34] Three measurable quantities have been highlighted - the the birefringence-induced ellipticity , effect on the probe’s frequency ω, its wave-vector k and its E2 c c e ε′, k S ε, k = s ω z 2 1 , (29) polarisation ε(k) and these will be discussed in turn. 2 p − ≡ h | | i Ecr 2 where ε ε′ = 0. Note the dependence on the difference of the low energy· constants. This implies that a confirmation of 4.1. Effects on probe photon polarisation vacuum birefringence would rule out other versions of elec- trodynamics popular in beyond-the-standard-model Vacuum birefringence refers to the prediction that the [35–37] such as Born-Infeld theory, which has c1 = c2 . From refractive index experienced by a probe propagating through (26), the flip amplitude or ellipticity (29) has the equivalent regions of intense, but weakly-varying strong fields of am- representation [17,26] plitude Es is of the form : (11 3)α E2 n⊥ nk nk,⊥ s (26) e vac vac (30) vac =1+ ∓ 2 , = ωpz − , 45π Ecr 2 where the ( ) indices apply to a probe polarised parallel which is proportional to the difference in refractive indices, (perpendicular)k ⊥ to the strong background. This result may hence the phase shift between different polarisations. be derived from the Heisenberg-Euler quantum equation of Detailed calculations have been performed for photons motion, propagating in an arbitrary plane-wave background [19,34], and the kinematic low-energy limit relevant for laser-based λ µν µ ν µν (∂λ∂ g ∂ ∂ + Π )Aν =0 . (27) experiments was found to be consistent with use of the − Heisenberg-Euler approach for calculating birefringence and [38] A plane wave ansatz for Aν implies two secular equations or ellipticity . A study of the dependency of the flip and dispersion relations, non-flip amplitude on spatial and timing jitter and angle of incidence [39] was performed, with the results also being k2 Π (k) = (gµν c T µν )k k =0 , (28) consistent with a previous similar study in the low-energy − 1,2 − 1,2 µ ν 6 B. King limit [40]. Both studies [39,40] found that modelling the back- where x cos θ = εk and y sin θ = ε⊥. For an x-ray ground as a focussed paraxial Gaussian beam without taking probe counterpropagating with an optical Gaussian pump into account the finite pulse duration led to an order of beam, the rotation angle was found to be the same order of magnitude discrepancy in the number of scattered photons. magnitude as the induced ellipticity [41,42].

Induced ellipticity is a consequence of birefringence as 4.2. Effects on probe photon wavevector pointed out in the previoussubsection, see (29) and (30). The polarisation of a linearly-polarised probe plane wave can be On the photon level, four-wave mixing as depicted in Fig. 5 described with the vector: can be understood as two incoming photons, one from the probe and one from the pump, being scattered to two outgo- k ε cos θ ing photons,one being back into the pump field and the other ⊥ = cos ϕ , (31) ε ! sin θ ! being the signal of the vacuum interaction. Conservation where ϕ is the probe phase. If, over some probe phase ωpz of momentum permits the scattered photons having a wider the and components experience a different refractive transverse distribution than the probeand strong background, k ⊥ k,⊥ k,⊥ index, then when the phase shift δϕ = n ωpz 1, hence allowing one to spatially separate the photon-photon the polarisation changes to: ≪ scattering signal from the large background of pump and probe laser photons. εk cos θ cos θ δϕk cos ϕ = − , (32) On the classical level, a refractive index nvac different from ε⊥ ! "sin θ sin θ δϕ⊥# sin ϕ ! − unity, implies altered transmitted wavevectors via Snell’s and the originally linearly-polarised probe is now elliptically law, and altered transmission T and reflection coefficient R polarised. If the background is constant, the ellipticity can via Fresnel’s law at perpendicular incidence [45]: be written: [41] 4 n 1 n 2 vac vac (36) ⊥ k T = 2 ; R = − . n nvac (1 + n ) 1+ n ! e = ω z vac − sin 2θ , (33) vac vac p 2 If the vacuum refractive index is written as nvac =1+ δnvac, which generalises (30). The induced ellipticity in the inter- the effect on probe transmission O(δnvac) whereas the ∼ action of an x-ray probe plane wave of wavelength λ = effect on reflection O(δn2 ). p ∼ vac 0.4 nm counterpropagating with a Gaussian pump beam of 23 −2 intensity 10 Wcm and wavelength λs = 745nm fo- If the probe beam is considered to be much wider than the cussed to 8 µm was calculated [41] to experience an ellipticity pump background, the region of polarised vacuum can be of e 5 10−9 rad when measured at a distance of 0.25 m considered to “diffract” the probe. It is well-known that the from≈ the· pump-probe collision. By considering the same far-field diffracted field is related to the Fourier transform of pump energy distributed over two pump Gaussian laser the aperture function [46], and via Babinett’s principle, this beams counterpropagating with a Gaussian probe beam, a can be related to an integral over the region of refractive modest improvement of around √2 was found, and the near- index different from unity. We underline the connection field induced ellipticity [42] of this classical analogue to the intensity form-factor of the scattering matrix approach Eq. (13). 2πα Is zeff. zr,pzr,s e = sin 2θ; z ff = , (34) 15 I λ e . z + z cr p r,p r,s Vacuum diffraction was considered in the collision of a with the effective interaction length between the two plane probe and a focussed Gaussian pump beam [41], and Gaussians zeff. depending on the probe zr,p and pump extended to to the collision of focussed Gaussian probe and [47] zr,s Rayleigh lengths. This agrees with the expressions pump beams . The advantage of this signal is that for calculated for a monochromatic probe plane wave counter- increasing scattering angle, while the focussed laser back- propagatingwith a Gaussian pump [43] in the limit z . ground is exponentially suppressed, the scattered photon r,p → ∞ vacuum signal is power-law suppressed. In the detector Polarisation rotation is the macroscopic consequence of plane then, the number of scattered photonscan be calculated coherent polarisation flipping at the photon level. The effect in “measurable” regions, where the signal to noise ratio on the transverse photon polarisation states in Eqs. (31) and is much larger than unity. One interesting scenario was (32) has the consequence that the polarisation angle θ will calculated of colliding two parallel, highly-focussed Gaus- rotate as the initially linearly-polarised probe acquires an sian pump beams with a wide weakly-focussed Gaussian ellipticity. The ellipse traced out by the probe field vector probe beam, such that the photons scattered in the two can be seen to be [44]: slit-like polarised regions around the pump beams would interfere and hence together form an all-optical double-slit x2 2xy cos(δϕ⊥ δϕk)+ y2 = sin2(δϕ⊥ δϕk), (35) experiment [47]. For the case of two colliding Gaussian − − − Measuring Vacuum Interactions with High Power Lasers 7

Vacuum reflection refers to the back-scattering of photons in real photon-photon scattering. Static magnetic inhomo- geneities of the form of a Lorentzian, Gaussian and oscillat- ing Gaussian have been studied [53] and more recently static electromagnetic inhomogeneities but most significantly scat- tering in a Gaussian beam [54], although calculations for pulses of a finite duration are still to be performed.

4.3. Effects on probe photon frequency

Figure 6. Predicted diffracted electric field in a collision of two counterpropagating Gaussian beams. Adapted from [44]. pulses, the dependency of the diffracted photon signal on Figure 7. Parametric frequency up-shifting (left) and down-shifting (right) experimental parameters such as the total beam power, can occur between pump and probe through the vacuum interaction. spatial and timing jitter, angle of collision, pulse duration, [40] probe wavelength and focal width has been carried out . The frequency of probe photons can change via interaction With 10 PW total laser power split into pump and probe with the polarised vacuum. However, this effect is much focussed optical pulses, of the order of a few photons more difficult to measure experimentally because of the lim- were predicted to be diffracted into measurable regions on ited range of energy and momenta for which it is permitted. a detector place 1 m from the interaction centre. These Suppose via the four-photon interaction, two photons from [48] results were verified in a study by different authors , who the strong pump background merge with a probe photon. used a different beam model. The diffraction paradigm Then via energy-momentum conservation: was extended from single and double slits to a “diffraction ff ff ′ ′ grating” of having a probe beam di ract o a regular series ωp + ωs,1 + ωs,2 = ω ; kp + ks,1 + ks,2 = k , (37) of pump beams [49]. Only on positions of the detector where the Bragg condition: but at the same time, the photon must be real to propa- ′2 ′ ′ θ gate to the detector so ω = k k . This constrains nq =2k sin , · p 2 the allowed frequencies, momenta and angles that can be combined. Similar relations occur for Raman and Brillouin for integer n, probe wavenumber kp, wavenumber of the scattering [55], except all the waves here are electromagnetic. pump beam structure q and angle between incoming and diffracted probe θ, is there constructive interference of the Vacuum parametric frequency-shifting has been calcu- signal of scattered photons. Since the addition of diffracted lated for special beam configurations. Combining three waves occurs at the level of the field, and since the number monochromatic plane waves at right angles, whose wave- of photons scattered depends upon the total diffracted lengths are 800 nm, 800 nm and 400 nm, was predicted field squared, there is an enhancement in such a set-up to produce a signal that is spatially and frequentially (at proportional to the square of the number of modulation 267 nm) separated from the background [56]. For respective periods. Alternatively, rather than using many beams, a beam powers 0.1 PW, 0.1 PW and 0.5 PW, taking the in- single, wide-angle beam diffracting with itself at the focus teraction region to be cuboidal, on average 0.07 photons has also been studied [50], with the conclusion that the would be frequency-upshifted per collision of the beams, number of diffracted photons increases exponentially with which is predicted to be larger than the Compton-scattering the angular aperture. Since only the near-field signal was background. A signature of the frequency-shiftingfour-wave presented, more work is required to determine measurability mixing process on the number of total measurable diffracted in this scheme. photons for a collision of two ultra-short focussed Gaussian pulses was also calculated [40]. For 10 PW total beam power The idea of using the diffracted photons’ flipped polarisa- split into a probe with wavelength 228 nm and duration 2 fs, tion as well as their altered wavevector in an experimental as the duration of the 910 nm pump is reduced to 1 fs, the measurement was explored for the wide-angled single-beam total number of diffracted photons is predicted to change by set-up [50], a single propagating Gaussian beam taking into around 20 %, equal to one photon per shot. Calculations account higher orders in a Hermite-Gauss expansion [51] and beyond the paraxial approximation recently performed [57] has been most recently applied to the upcoming HIBEF for two co-propagating beams of different frequencies in- experiment [52]. cident on a parabolic mirror suggest 1-10 PW laser beams 8 B. King are required to observe vacuum frequency mixing, although the method of detecting the signal needs to be given more attention.

Vacuum high-harmonic generation can take place if the colliding laser pulses have the same frequency. Then via the four-wave mixing process in Eq. (37), if ωp = ωs,1 = Figure 9. Vacuum high-harmonic generation of the nth harmonic of the ωs,2 = ω, the signal of the vacuum process has a frequency probe via a chain of six-photon scattering. ω′ = 3ω and so is at the third harmonic of the probe. By considering six-, eight- and in general 2n-wave mixing as depicted in Fig. 8, it can be seen that a harmonic spectrum Photon splitting as depicted in Fig. 10, is sometimes for the vacuum interaction can be produced. As each extra thought of as the opposite of high-harmonic generation, but unlike harmonic generation, the emitted photons can have a continuum of energies. If one considers splitting

Figure 8. Vacuum high-harmonic generation of the nth harmonic of the probe via 2n-photon scattering. Figure 10. An incoming probe photon can split into k outgoing ones, due to interaction with the background. interaction between the virtual pair and a laser photon is to two photons via four-wave mixing then via energy and weighted at the amplitude level with a factor E/Ecr 1, higher harmonics are in general exponentially suppressed.≪ momentum conservation, one possibility is: Nevertheless, the harmonic spectrum produced by a standing ′ ′ ′ ′ wave formed of two monochromatic pump laser beams was ωp + ωs = ω1 + ω2; kp + ks = k1 + k2, (39) calculated for subcritical (E < Ecr) strengths where higher ′ 2 harmonic orders j were found [58] to follow the hierarchy where now two constraints on these equations are (ω1,2) = ′ ′ 4j k k . The continuum of allowed energies and the (E/Ecr) . In a set-up involving three beams, the minimum 1,2 · 1,2 power of each laser required to scatter one photon was found possibility for a wide angular distribution of emitted photons to be: makes this process worthy of study. The process has been comprehensively studied for a probe photon propagating 1/3 2/3 through a plane wave background of arbitrary form and λ w0 1fs 1 fs P 33.5 GW, (38) [66] min 1 nm 1 nm τ τ polarisation , which was found to depend on the two ≈ ! c ! 2 parameters η = ωpωs/m and χ = (ωp/m)(E/Ecr). for typical beam cross-sectional dimension w0, interaction Two events per hour were predicted using 108 250 MeV duration τ and coherence time τc. The most likely fre- tagged photons per second almost counterpropagating with quency of the scattered photon is, however, the fundamental 100 fs 1015 Wcm−2 1 keV XFEL beams separated by 93 ns. harmonic. The intensity at which a single focussed laser Alternatively, two events per hour were also predicted using pulse will begin to produce harmonics via self-interaction 108 100 MeV tagged photons counterpropagating with a [59] has been studied , with the conclusion that a pulse of 1 Hz 1 eV optical pump of intensity 1025 Wcm−2. The 1000 nm photons focussed within a cone of angle 0.1 rad conclusion was that a different experimental set-up must −2 will produce one photon per period at 5 1027Wcm . A ff · be considered if this e ect is to be observed in the near recent calculation of an alternative route to high-harmonic future [66]. generation through having many scattering events involving low numbers of photons [60–65] (as in Fig. 9), has recently ff been suggested to be more efficient. For the collision 4.4. E ects on probe pulse form of a Gaussian probe at much higher frequency than the In addition to the effects on single photons, one can consider 3 4 background, if the parameter (64α/105π)(Es Ep/Ecr)ωpτs, the consequence of real photon-photon scattering on the where τs is the duration of the pump, can be made close to propagation of an ensemble of photons. A probe laser pulse unity, harmonic generation will dominate, with the spectrum can be understood as a superposition of photons with a range displaying a power-law behaviour and the appearance of a of frequencies and phases. From the study of nonlinear corresponding electromagnetic shock [31]. dispersive media, it is well known that a refractive index that Measuring Vacuum Interactions with High Power Lasers 9 depends on a probe’s intensity directly or indirectly can lead background. The wave-equation for the probe can be recast to pulse shape effects [55]. In particular for the interaction as a nonlinear Schr¨odinger equation [55] with the conse- with vacuum, probe pulse effects can occur if the next-to- quence that the pulse envelope becomes spacetime depen- leading order effect of a probe-dependent refractive index is dent, even if assumed initially homogeneous. Unlike typical taken into account. optically-nonlinear dispersive media, the nonlinearity of the vacuum is “formed” by the pump laser background, Nonlinear phase shift is a term used to denote the relative which is then probed by a second pulse. Even when the difference in phases between parts of a probe beam that leading-order effect on the probe is a nonlinear refractive have experienced different vacuum refractive indices. For index that is independent of the probe pulse, because of a constant refractive index, the relative phase difference its effect on the pump’s evolution, it can indirectly effect compared to a unitary refractive index is: the probe’s propagation. This interplay between a Gaussian probe distribution propagating through a radiation gas has

δφ = (nvac 1)ωpz, been demonstrated to lead to self-focussing and collapse − of the probe into “photon bullets”, thereby driving acoustic [71] where ωpz is the phase over which δφ has been accrued. waves as demonstrated in Fig. 11. Depending on initial For two counterpropagating initially monochromatic plane parameters, probe collapse can occur before or after the waves, with the envisaged ELI parameters of 800 nm wave- critical Schwinger limit is reached [72]. length, 1025 Wcm−2 intensity, 10 fs duration and 10 µm focal spot diameter, a phase shift of the order of δφ ≈ 10−7 rad has been calculated [30,67]. This nonlinear phase shift can be enhanced by using multiple crossings of the interacting beams. For Nr reflections from plasma mirrors of reflectivity Rmir of two beams crossing each other at an [68] angle θc, the gain factor has been calculated to be :

θ Nr+1 sin4 c Rn . 2 ! mir Figure 11. Cerenkov-like radiation (right) generated by pulse collapse into nX=0 photon bullets (left) against longitudinal z and transverse r co-ordinates of an initially Gaussian pulse of central wavenumber k0. Reproduced with The measurement of this phase shift using Fourier imaging permission [71]. has also been explored [69].

Vacuum self-focusing is an analogue to the well-known plasma self-focussing or “Benjamin-Weir” instability [55] in 4.5. Finite time effects which there is positive feedback between a refractive index Similar to the case for regular plasmas, there are effects on increasing the intensity of a pulse via focussing, and a higher the probe when propagating through regions of the polarised intensity resulting from that focussing in turn increasing the “vacuum plasma” that do not persist long enough to be refractive index. Mutual channelling of counterpropagating directly detected. laser pulses and large-scale focussing have been considered, [63] but either YW powers are predicted as necessary or inten- Photon acceleration is well-known from plasma physics [73] [70] sities above critical , before which vacuum pair-creation and corresponds to the frequency downshift (upshift) as would have set in. In considering the idealised geometry of probe photons traverse an increasing (decreasing) plasma a Gaussian plane wave probe pulse counterpropagating and gradient. The possibility of measuring this effect in vacuum interacting via six-wave mixing with a much slower varying has been considered for a probe photon propagating almost pump, the probe-dependent refractive index: parallel with a pump pulse [74], with a frequency up (down) 2 α E 8 64 E Ep shift occurring at the rear (front) of the pump beam. nk s s vac =1+ 2 + (40) π E "45 105 Ecr Ecr # cr High harmonic generation can also occur due to the was predicted to lead to the generation of a shock wave, a inhomogeneity of the pump pulse background, in an effect signature of self-focussing, when the phase difference due distinct from standard vacuum high harmonic generation. to the probe-dependent refractive index tended to a quarter- For a probe pulse counterpropagating with a slowly-varying wavelength [31]. background, this is predicted to occur at finite time during overlap of the probe and pump pulses at an order earlier (via Pulse collapse is predicted to occur for high-intensity four-photon scattering), than for those photons that reach probe pulses propagating through an even higher-intensity a detector (via six-photon scattering) [75]. This finite-time 10 B. King signal disappears when the probe and pump pulses are well- cavity with a resistance of 1 nΩ and a resonant, vacuum- separated again, but is calculated to dominate the signal of mixing frequency of 13.2 µeV. This idea was refined [78] and frequency-shifted photons when the pulses overlap in this the prediction made that 18 photons can be produced by a 2 set-up if (E /Ecr) ω τ 1 for strong-pulse duration τ . magnetic field of around 0.28 T in a cylindrical cavity of s p s ≪ s length 2.5 m, radius 25 cm and quality factor 4 1010. Gradient-dependent vacuum refractive index is a way × to describe the addition to the standard predicted vacuum refractive index that occurs when the pump laser is time- Real plasmas already have a refractive index different varying. This has been calculated for a probe propagating from unity, and this can combine with the shift of the through the electric/magnetic antinode of a pump standing refractive index due to vacuum polarisation and lead to wave [33]. The change in vacuum refractive index ∆n can vac an enhancement. The system of equations by Akhiezer be written in the form: and Polovin [79] for the propagation of a circularly-polarised E plane wave through a cold collisionless plasma was updated nk,⊥ p nk,⊥ ′ (41) ∆ vac (ϕ)= ′ vac (ϕ). Ep to include the vacuum current in Maxwell’s equations and also take into account collisions [80]. For the collisionless In a set-up of two colliding plane waves with no transverse case, the modified refractive index of the combined system structure, it was shown that this term is a surface term and was found to be: is zero initially and finally, when the probe and background 1 are well-separated [75]. The contact term was also noted n = n2 + δn⊥ (1 n2 )2 r pl 4 vac − pl in a recent study of polarisation flipping in arbitrary plane [34] ⊥ ⊥ waves . Although it has been suggested this part of with npl the plasma refractive index and δn = n 1 as vac vac − the interaction could be a useful probe of dark matter defined in Eq. (26). Another detectable signal of photon- particles [33], a consistent finite-time calculation has yet to be photon scattering has been calculated to exist when an performed to establish the nature of this effect. overdense plasma channel is subjected to an intense laser beam [81]. In addition, the altered dispersion relation for elec- 4.6. Non-perfect vacua tromagnetic waves due to vacuum polarisation effects in a strongly-magnetised cold plasma has been calculated [82–84], In any realistic experiment, the vacuum will be synthetic and which is particularly relevant for the dynamics of strongly- hence imperfect. Residue particles in interaction chambers magnetised stars. will also be affected by intense laser pulses and can produce a source of background that may obscure the measurement of real photon-photon scattering. The Cotton-Mouton effect, 5. Summary in which a dilute gas becomes birefringent in the presence of an electromagnetic wave is just one such example [76]. There has been a proliferation of labels to describe polar- ff In light of this, various proposals have been considered isation e ects of the quantum vacuum due to intense laser that instead use an altered vacuum to enhance the signal pulses. However, as we have discussed, all of these are mani- of vacuum polarisation. festations of the QED prediction that real photons can scatter off one another. The commonality of the main approaches of describing real photon-photon scattering, through cal- Resonant Cavities can be employed in order to increase culation of the polarisation operator, scattering matrix el- the sensitivity of whatever eigenfrequencies are resonant ements and Heisenberg-Euler-modified Maxwell equations, for that particular cavity [45]. For example, a cavity can be has been made manifest. Many signals of this long-predicted designed such that the frequency that is generated by vacuum phenomenon, whether at the level of individual photons or four-wave mixing of two modes of the cavity, is resonant. at the level of electromagnetic fields, have been calculated This idea has been studied for the TE01 modes of such a and found measurable in experiments using high-intensity cavity and the growth of the mixing signal in the form of the laser pulses. This implies that the first measurement of real longitudinal standing-wave magnetic field, found to increase photon-photon scattering will finally be performed in the linearly with time [77] as near future.

itV 2 ∗ B3(t)= B1 B2 , 2ω3 References for source magnetic standing wave strengths B1,B2 and 1. C. Danson, D. Hillier, N. Hopps, and D. Neely, coupling constant V . The vacuum signal was predicted to be Petawatt class lasers worldwide, High Power Laser detectable if an electric field 2 10−8 times the critical Science and Engineering 3 (2015). Schwinger field was employed× with a superconducting 2. M. Marklund and P. K. Shukla, Nonlinear collective Measuring Vacuum Interactions with High Power Lasers 11

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