Extremely high-intensity laser interactions with fundamental quantum systems
aa r X i v : . [ h e p - ph ] A p r Extremely high-intensity laser interactions with fundamental quantumsystems
A. Di Piazza, ∗ C. M¨uller, † K. Z. Hatsagortsyan, ‡ and C. H. Keitel § Max-Planck-Institut f¨ur Kernphysik,Saupfercheckweg 1,69117 Heidelberg,Germany (Dated: April 26, 2012)
The field of laser-matter interaction traditionally deals with the response of atoms,molecules and plasmas to an external light wave. However, the recent sustained tech-nological progress is opening up the possibility of employing intense laser radiation totrigger or substantially influence physical processes beyond atomic-physics energy scales.Available optical laser intensities exceeding 10 W/cm can push the fundamental light-electron interaction to the extreme limit where radiation-reaction effects dominate theelectron dynamics, can shed light on the structure of the quantum vacuum, and cantrigger the creation of particles like electrons, muons and pions and their correspondingantiparticles. Also, novel sources of intense coherent high-energy photons and laser-based particle colliders can pave the way to nuclear quantum optics and may even allowfor potential discovery of new particles beyond the Standard Model. These are themain topics of the present article, which is devoted to a review of recent investigationson high-energy processes within the realm of relativistic quantum dynamics, quantumelectrodynamics, nuclear and particle physics, occurring in extremely intense laser fields. PACS numbers: 12.20.-m, 32.80.-t, 52.38.-r, 25.20.-x
CONTENTS
I. Introduction 2II. Novel radiation sources 3A. Strong optical laser sources 31. Next-generation 10 PW optical laser systems 42. Multi-Petawatt and Exawatt optical lasersystems 4B. Brilliant x-ray laser sources 5III. Free electron dynamics in a laser field 6A. Classical dynamics 6B. Quantum dynamics 7IV. Relativistic atomic dynamics in strong laser fields 9A. Ionization 10B. Recollisions and high-order harmonic generation 12V. Multiphoton Thomson and Compton scattering 14A. Fundamental considerations 14B. Thomson- and Compton-based sources of high-energyphoton beams 17VI. Radiation reaction 18A. The classical radiation dominated regime 20B. Quantum radiation reaction 21 ∗ [email protected] † [email protected]. New address: In-stitut f¨ur Theoretische Physik I, Heinrich-Heine-Universit¨atD¨usseldorf, Universit¨atsstraße 1, 40225 D¨usseldorf, Germany ‡ [email protected] § [email protected] VII. Vacuum-polarization effects 23A. Low-energy vacuum-polarization effects 231. Experimental suggestions for direct detection ofphoton-photon scattering 242. Polarimetry-based experimental suggestions 263. Low-energy vacuum-polarization effects in aplasma 27B. High-energy vacuum-polarization effects 27VIII. Electron-positron pair production 29A. Pair production in photon-laser and electron-lasercollisions 30B. Pair production in nucleus-laser collisions 31C. Pair production in a standing laser wave 32D. Spin effects and other fundamental aspects oflaser-induced pair creation 34IX. QED cascades 35X. Muon-antimuon and pion-antipion pair production 38A. Muon-antimuon and pion-antipion pair production inlaser-driven collisions in plasmas 38B. Muon-antimuon and pion-antipion pair production inhigh-energy XFEL-nucleus collisions 39XI. Nuclear physics 40A. Direct laser-nucleus interaction 401. Resonant laser-nucleus coupling 402. Nonresonant laser-nucleus interactions 42B. Nuclear signatures in laser-driven atomic andmolecular dynamics 42XII. Laser colliders 43A. Laser acceleration 43B. Laser-plasma linear collider 44C. Laser micro-collider 45XIII. Particle physics within and beyond the Standard Model 47 A. Electroweak sector of the Standard Model 47B. Particle physics beyond the Standard Model 47XIV. Conclusion and outlook 49Acknowledgments 50List of frequently-used symbols 50References 50
I. INTRODUCTION
The first realization of the laser in 1960 (Maiman,1960) is one of the most important technological break-throughs. Nowadays lasers are indispensable tools forinvestigating physical processes in different areas rang-ing from atomic and plasma physics to nuclear and high-energy physics. This has been possible mainly due to thecontinuous progress made along two specific directions:decrease of the laser pulse duration and increase of thelaser peak intensity (Mourou and Tajima, 2011). On theone hand, multiterawatt laser systems with a pulse du-ration in the femtosecond time scale are readily availablenowadays and different laboratories have succeeded in thegeneration of single attosecond pulses. Physics at the at-tosecond time scale has been the subject of the recentreview Krausz and Ivanov, 2009. In this review it hasbeen pointed out how pulses in the attosecond domainallow for the detailed investigation of the electron mo-tion in atoms and during molecular reactions. The pro-duction of ultrashort pulses is strongly connected withthe increase of the laser peak intensity. This is not onlybecause temporal compression evidently implies an in-crease in intensity at a given laser energy, but also be-cause higher intensities allow, in general, for controllingfaster physical processes, which in turn can be exploitedfor generating correspondingly shorter light pulses.Not long after the invention of the laser, available in-tensities were already sufficiently high to trigger non-linear optical effects like second harmonic generation.It is, however, only after the experimental implementa-tion of the Chirped Pulse Amplification (CPA) technique(Strickland and Mourou, 1985) that it has been possi-ble to reach the intensity threshold of 10 -10 W/cm corresponding to electric field amplitudes of the sameorder as the Coulomb field in atoms. At such inten-sities the interplay between the laser and the atomicfield significantly alters the electron’s dynamics in atomsand molecules and this can be exploited, for example,for generating high-frequency radiation in the extreme-ultraviolet (XUV) and soft-x-ray regions (high-orderharmonic generation or HHG) (Agostini and DiMauro,2004; Protopapas et al. , 1997). HHG as well as atomicprocesses in intense laser fields have been recently re-viewed in Fennel et al. , 2010; Teubner and Gibbon, 2009;and Winterfeldt et al. , 2008, with specific emphasis on the control of high-harmonic spectra by spatio-temporalshaping of the driving pulse (Winterfeldt et al. , 2008), onharmonic generation in laser-plasma interaction (Teub-ner and Gibbon, 2009) and on the dynamics of clustersin strong laser fields (Fennel et al. , 2010).By increasing the optical laser intensity to the or-der of 10 -10 W/cm , another physically importantregime in laser-matter interaction is entered: the rel-ativistic regime. In such intense electromagnetic fieldsan electron reaches relativistic velocities already withinone laser period, the magnetic component of the Lorentzforce becomes of the same order of magnitude of the elec-tric one, and the electron’s motion becomes highly non-linear as a function of the laser’s electromagnetic field.Although the increasing influence of the magnetic forcecauses a suppression of atomic HHG in the relativistic do-main, the highly nonlinear motion of the electrons in suchstrong laser fields is at the origin of numerous new effectsas relativistic self-focusing in plasma and laser wakefieldacceleration (Mulser and Bauer, 2010). In the recent re-views Ehlotzky et al. , 2009; Mourou et al. , 2006; andSalamin et al. , 2006, different processes occurring at rel-ativistic laser intensities are discussed. In particular, inEhlotzky et al. , 2009, QED processes like Compton, Mottand Møller scattering in a strong laser field are covered, inMourou et al. , 2006, technical aspects and new possibili-ties of the CPA techniques are reviewed together with rel-ativistic effects in laser-plasma interaction as, for exam-ple, self-induced transparency and wakefield generation,while in Salamin et al. , 2006, spin-effects as well as rel-ativistic multiphoton and tunneling recollision dynamicsin laser-atom interactions are reviewed. Also in the sameyear another review was published on nonlinear collectivephoton interactions, including vacuum-polarization ef-fects in a plasma (Marklund and Shukla, 2006). Whereas,the physics of plasma-based laser electron acceleratorsis the main subject covered in Esarey et al. , 2009 andMalka, 2011, with a special focus on the different phasesinvolved (electron injection and trapping, and pulse prop-agation) and on the role of plasma instabilities in theacceleration process. Finally, in Ruffini et al. , 2010 dif-ferent processes related to electron-positron ( e + - e − ) pairproduction are reviewed with special emphasis on thoseoccurring in the presence of highly-charged ions and inastrophysical environments.In the present article we address physical processesthat mainly occur at optical laser intensities mostly largerthan 10 W/cm , i.e., well exceeding the relativisticthreshold. After reporting on the latest technologicalprogress in optical and x-ray laser technology (Sec. II),we review some basic results on the classical and quan-tum dynamics of an electron in a laser field (Sec. III).Then, we bridge to lower-intensity physics by reviewingmore recent advances in relativistic ionization and HHGin atomic gases (Sec. IV). The main subject of the re-view, i.e., the response of fundamental systems like elec-trons, photons and even the vacuum to ultra-intense radi-ation fields is covered in Secs. V-X. As will be seen, suchhigh laser intensities represent a unique tool to investi-gate fundamental processes like multiphoton Comptonscattering (Sec. V), to clarify conceptual issues like radi-ation reaction in classical and quantum electrodynamics(Sec. VI) and to investigate the structure of the quantumvacuum (Sec. VII). Also, other fundamental quantum-relativistic phenomena like the transformation of purelight into massive particles as electrons, muons and pi-ons (and their corresponding antiparticles) can becomefeasible and can even limit the attainability of arbitrar-ily high laser intensities (Secs. VIII-X). Finally, we willalso review recent suggestions on employing novel high-frequency lasers and laser-accelerated particle beams todirectly trigger nuclear and high-energy processes (Secs.XI-XIII). The main conclusions of the article will be pre-sented in Sec. XIV.Units with ~ = c = 1 and the space-time metric η µν = diag(+1 , − , − , −
1) are employed throughoutthis review.
II. NOVEL RADIATION SOURCES
In this section we review the latest technical and ex-perimental progress in laser technology. We will discussoptical and x-ray laser systems separately. The latterare especially useful for e + - e − pair production, for di-rect laser-nucleus interaction, as well as as probes forvacuum-polarization effects (see in particular Secs. VII,VIII and XI). For overviews of feasible accelerators alsoof relevance for the present review see, e.g., Esarey et al. ,2009; Malka, 2011; Nakamura et al. , 2010; and Wilson,2001 and the relevant original literature as quoted in therespective sections. A. Strong optical laser sources
As has been mentioned in the Introduction, since theinvention of the CPA technique (Strickland and Mourou,1985) laser peak intensities have been boosted by sev-eral orders of magnitude. Another amplification tech-nique called Optical Parametric Chirped Pulse Ampli-fication (OPCPA), based on the nonlinear interactionamong laser beams in crystals, was suggested almost atthe same time as the CPA and proved to be promising aswell (Piskarskas et al. , 1986). As a result of the increasein available laser intensities, exciting perspectives havebeen envisaged in different fields spanning from atomic toplasma and even nuclear and high-energy physics (Feder,2010; Gerstner, 2007; Mourou, 2010; Tajima et al. , 2010).The group of G. Mourou at the University of Michi-gan (Michigan, USA) holds the record so far for thehighest laser intensity ever achieved of 2 × W/cm (Yanovsky et al. , 2008), while no experiments have beenperformed at this intensity yet. This record intensity hasbeen reached when the HERCULES laser was upgradedto become a 300 TW Ti:Sa system, amplified via CPAand capable of a repetition rate of 0 . f / . µ m. This experimental achievement on the laser in-tensity pushed the capabilities of a multiterawatt laseralmost to the limit.The 1-PW threshold has been already reached andeven exceeded in various laboratories. For example, theTexas Petawatt Laser (TPL) at the University of Texas atAustin (Texas, USA) has exceeded the Petawatt thresh-old thanks to the OPCPA technique, by compressing anenergy of 186 J in a pulse lasting only 167 fs (TPL, 2011).The TPL has been employed for investigating laser-plasma interactions at extreme conditions, particularlyrelevant for astrophysics. Also, the two laser systemsVulcan (Vulcan, 2011) and Astra Gemini (Astra Gemini,2011) at the Central Laser Facility (CLF) in the UnitedKingdom provide powers of the order of 1 PW. TheVulcan facility can deliver an energy of 500 J in a pulselasting 500 fs. It is a Nd:YAG laser system amplifiedvia CPA and can provide intensities up to 10 W/cm .Whereas, the Astra Gemini laser consists of two inde-pendent Ti:Sa laser beams of 0 . W/cm . The particular layoutof the Astra Gemini laser renders this system especiallyversatile for unique applications in strong-field physics,where two ultrastrong beams are required. Two lasersystems are likely to be updated to the Petawatt levelin Germany. The first one is the Petawatt High-EnergyLaser for heavy Ion eXperiments (PHELIX) Nd:YAGlaser at the Gesellschaft f¨ur Schwerionenforschung (GSI)in Darmstadt, capable now of delivering an energy of120 J in about 500 fs (PHELIX, 2011). The 1-PWthreshold should be reached by increasing the pulse en-ergy to 500 J. Combined with the highly-charged ionbeams at GSI, the PHELIX facility can be attractive forexperimental investigations in strong-field quantum elec-trodynamics (QED). The second system to be updatedto 1 PW is the Petawatt Optical Laser Amplifier for Ra-diation Intensive Experiments (POLARIS) laser in Jena(Hein et al. , 2010). At the moment, a power of about100 TW (energy of 10 J for a pulse duration of 100 fs)has been reached and the goal of 1 PW power shouldbe achieved by compressing 120 J in about 120 fs. TheScottish Centre for the Application of Plasma-based Ac-celerators (SCAPA) research center is one of the maininitiatives within the Scottish Universities Physics Al-liance (SUPA) project dedicated to the high-power laserinteraction with plasmas. A laser system will be devel-oped, which will generate pulses of 5-7 J energy and of25-30 fs duration at a repetition rate of 5 Hz, correspond-ing to a peak power of 200-250 TW, with potential forfuture upgrades to the petawatt level (SCAPA, 2012).The 1-PW threshold has been also exceeded in Ti:Salaser systems like those described in Sung et al. , 2010(energy of 34 J for a pulse duration of 30 fs) and inWang et al. , 2011 (energy of 32 . . . et al. , 2010). The Ti:Sa Petawatt Field Synthe-sizer (PFS) system under development in Garching (Ger-many) aims to be the first high-repetition rate petawattlaser system with an envisaged repetition rate of 10 Hz(PFS, 2011). By adopting the OPCPA technique thePFS should reach the petawatt level by compressing anenergy of about 5 J in 5 fs (Major et al. , 2010). Fora recent review on petawatt-class laser systems, see Ko-rzhimanov et al. , 2011.Finally, we also want to mention other high-powerlasers, mainly devoted to fast ignition and characterizedby relatively long pulses of the order of 1 ps-1 ns. Amongothers we mention the OMEGA EP system at Rochester(New York, USA) (energy of 1 kJ for a pulse durationof 1 ps) (OMEGA EP, 2011) and the National Igni-tion Facility (NIF) at the Lawrence Livermore NationalLaboratory (LLNL) at Livermore (California, USA) (en-ergy of 2 MJ distributed in 192 beams with a pulse du-ration of about 3-10 ns) (NIF, 2011). Another high-power laser facility is the PETawatt Aquitaine Laser(PETAL) in Le Barp close to Bordeaux (France), whichis a multi-petawatt laser, generating pulses with energyup to 3.5 kJ and with a duration of 0.5 to 5 ps (PETAL,2011). The PETAL facility is planned to be coupled tothe Laser M´egaJoule (LMJ) under construction in Bor-deaux (France). In the LMJ a total energy of 1 . .
1. Next-generation PW optical laser systems
The possibility of building a 10 PW laser system isunder consideration in various laboratories. At the CLFin the United Kingdom the 10 PW upgrade of the Vul- can laser has already started (Vulcan 10PW, 2011). Thenew laser will provide beams with an energy of 300 Jin only 30 fs via the OPCPA. A 10 PW laser system isin principle capable of unprecedented intensities largerthan 10 W/cm if the beam is focused to about 1 µ m.The front-end stage of the Vulcan 10 PW is already com-pleted and it delivers pulses with about 1 J of energy at acentral wavelength of 0 . µ m, with sufficient bandwidthto support a pulse with duration less than 30 fs.Another 10 PW laser project is the ILE APOLLONto be realized at the Institut de Lumi´ere Extreme (ILE)in France (Chambaret et al. , 2009). The laser pulsesare expected to deliver an energy of 150 J in 15 fs atthe last stage of amplification after the front end (en-ergy of 100 mJ in less than 10 fs), with a repetition rateof one shot per minute. Laser intensities of the orderof 10 W/cm are envisaged at the ILE APOLLON sys-tem, entering the so-called ultrarelativistic regime, wherealso ions (rest energy of the order of 1 GeV) become rel-ativistic within one laser period of such an intense laserfield.We mention here also the PEtawatt pARametric Laser(PEARL-10) project at the Institute of Applied Physicsof the Russian Academy of Sciences in Nizhny Novgorod(Russia), which is an upgrade of the present 0.56-PWlaser employing the OPCPA technique, to 10 PW (200 Jof energy compressed in 20 fs) (Korzhimanov et al. ,2011).
2. Multi-Petawatt and Exawatt optical laser systems
The Extreme Light Infrastructure (ELI) (ELI, 2011)(see Fig. 1), the Exawatt Center for Extreme LightStudies (XCELS) (XCELS, 2012) and the High Powerlaser Energy Research (HiPER) facility at the CLF inthe United Kingdom (HiPER, 2011) are envisaged lasersystems with a power exceeding the 100 PW level.ELI is a large-scale laser facility consisting of four “pil-lars” (see Fig. 1): one devoted to nuclear physics, oneto attosecond physics, one to secondary beams (photonbeams, ultrarelativistic electron and ion beams) and oneto high-intensity physics. This last one is of relevancehere and it is supposed to comprise ten beams each witha power of 10-20 PW that, when combined in phase,should deliver a single beam of about 100-200 PW at arepetition rate of one shot per minute. The relativelyhigh repetition rate is obtained by compressing in eachbeam alone 0 . . W/cm are envisaged, which are well above the ultrarelativisticregime. At such intensities, it will be possible to test dif-ferent aspects of fundamental physics for the first time. FIG. 1 (Color online) Summary of the four pillars of ELI. Apower value of 10( ×
2) PW indicates the availability of twolaser systems each with 10 PW power. Reprinted with per-mission from Feder, 2010. Copyright 2010, American Instituteof Physics.
The XCELS infrastructure is planned to be built inNizhny Novgorod (Russia) and it will consist of 12 beamseach with energy of 300-400 J and with duration of20-30 fs. The pulse resulting from the superposition ofthese beams is expected to have a power of 200 PW, apulse duration of about 25 fs, and divergence less than3 diffraction limits (at a central wavelength of 0 . µ m).Apart from aiming to overcome the 100 PW threshold,the main priorities of XCELS are the creation of sourcesof attosecond and subattosecond, X-ray and γ -ray pulses,the development of laser based electron and ion acceler-ators with electron and ion energies exceeding 100 GeVand up to 10 GeV, respectively, the realization in labo-ratory of astrophysical and early-cosmological conditionsand the investigation of the structure of the quantumvacuum.The main goal of the other large-scale facility HiPERis the first demonstration of laser-driven fusion, or fastignition. To this end HiPER will deliver: 1) an energy ofabout 200 kJ distributed in 40 beams with a pulse dura-tion of several nanoseconds and a photon energy of 3 eVin the compression side; 2) an energy of about 70 kJ dis-tributed in 24 beams with a pulse duration of 15 ps anda photon energy of 2 eV in the ignition side. EmployingHiPER for high-intensity physics would imply a feasiblereconfiguration of the ignition side to deliver 10 kJ inonly 10 fs via the OPCPA technique. This would renderHiPER a laser facility with Exawatt (10 W) power andwith a potential intensity of 10 W/cm .Finally, we briefly mention the GEKKO EXA facilityconceptually under design in Osaka (Japan) (GEKKOEXA, 2011). This facility is expected to deliver pulsesof 2 kJ energy and of 10 fs duration corresponding to200 PW and with an intensity up to 10 W/cm . Photon energy [eV]1 10 10 P e a k b r illi a n ce [ ph o t o n s / ( s m r a d mm . % b a nd w i d t h ] FLASH European XFELLCLS
FIG. 2 (Color online) Comparison among the peak brilliancesof the three facilities FLASH, LCLS and European XFEL asa function of the laser photon energy. An envisaged peakbrilliance of 5 × photons/(s mrad mm . B. Brilliant x-ray laser sources
Strong optical laser systems are sources of coherentradiation at wavelengths of the order of 1 µ m, corre-sponding to photon energies of the order of 1 eV. Con-siderable efforts have been devoted in the past few yearsto develop coherent radiation sources at photon energieslarger than 100 eV. The discovery of the Self-AmplifiedSpontaneous Emission (SASE) regime (Bonifacio et al. ,1984) has opened the possibility of employing Free Elec-tron Lasers (FELs) to generate coherent light at suchshort wavelengths. In a FEL relativistic bunches of elec-trons pass through a spatially-periodic magnetic field(undulator) and emit high-energy photons. In the SASEregime the interaction of the electron bunch with its ownelectromagnetic field “structures” the bunch itself intoslices (microbunches) each one emitting coherently evenat wavelengths below 1 nm (FELs at such small wave-lengths are dubbed X-ray Free Electron Lasers (XFELs)).The Free-Electron Laser in Hamburg (FLASH) facil-ity at the Deutsches Elektronen-SYnchrotron (DESY)in Hamburg (Germany) (FLASH, 2011) is one of themost brilliant operating FEL’s. It delivers short pulses(duration of about 10-100 fs) of coherent radiation inthe extreme ultraviolet-soft x-ray regime (fundamentalwavelength from 60 nm down to 6 . . -10 photons/(s mrad mm et al. , 2010). Since the electron beam en-ergy can be varied from 4 . . . .
15 nm (corresponding to photon energies from 0 . -10 photons/(s mrad mm .
08 nm wavelength (corresponding to a photon en-ergy of 15 . × photons/(s mrad mm . .
05 nm, which corresponds to aphoton energy of 24 . . . . . et al. , 2009; San-sone et al. , 2006), though with intensities several orderssmaller than XFELs. Less stable sources of coherent softx-rays are the so-called x-ray lasers, which are based onthe amplification of spontaneous emission by multiplyionized atoms in dense plasmas created by intense laserpulses (Suckewer and Jaegle, 2009; Wang et al. , 2008;Zeitoun et al. , 2004). III. FREE ELECTRON DYNAMICS IN A LASER FIELD
In this Section we review, for the benefit of the reader,some important basic results on the dynamics of a freeelectron in a laser field (see also the review Eberly, 1969)and link them to recent investigations on the subject.Results in the realm of classical and quantum electro-dynamics are considered separately. Radiation-reactionand electron self-interaction effects are not included hereand their discussion is developed in Sec. VI.
A. Classical dynamics
The motion of a charged particle in a laser field is usu-ally associated to an oscillation along the laser polariza-tion direction. This is pertinent to the non-relativisticregime, while the charge dynamics in the relativistic do-main is enriched by new features like the drift along thelaser propagation direction and other non-dipole effects(like the well-known figure-8 trajectory), as well as bythe sharpening of the trajectory at those instants wherethe velocity along the polarization direction reverses. Asa consequence, laser-driven relativistic free electrons alsoemit high harmonics of the laser frequency (see Sec. V).The classical motion of an electron (electric charge e < m ) in an arbitrary external electromag-netic field F µν ( x ) is determined by the Lorentz equation mdu µ /ds = eF µν u ν , where u µ = dx µ /ds is the elec-tron four-velocity and s its proper time (Landau andLifshitz, 1975). If the external field is a plane wave,the field tensor F µν ( x ) depends only on the dimensionalphase φ = ( n x ), where n µ = (1 , n ), with n being theunit vector along the propagation direction of the wave.In this case, for an arbitrary four-vector v µ = ( v , v )it is convenient to introduce the notation v k = n · v , v ⊥ = v − v k n and v − = ( n v ) = v − v k . The four-vector potential of the wave can be chosen in the Lorentzgauge as A µ ( φ ) = (0 , A ( φ )), with A − ( φ ) = − A k ( φ ) = 0.We indicate as p µ = ( ε, p ) = mu µ the (kinetic) four-momentum of the electron. Since a plane-wave field de-pends only on φ , the canonical momenta p ⊥ ( φ ) + e A ( φ )and p − ( φ ) are conserved as they are the conjugated mo-menta to the cyclic coordinates x ⊥ and t + x k , respec-tively. For p µ ( φ ) = p µ = ( ε , p ) = mγ (1 , β ) beingthe initial condition for the electron’s four-momentum ata given phase φ , the above-mentioned conservation lawsalready allow to write the electron’s four-momentum atan arbitrary phase φ as (Landau and Lifshitz, 1975) ε ( φ ) = ε − e p , ⊥ · [ A ( φ ) − A ( φ )] p , − + e A ( φ ) − A ( φ )] p , − , (1) p ⊥ ( φ ) = p , ⊥ − e [ A ( φ ) − A ( φ )] , (2) p k ( φ ) = p , k − e p , ⊥ · [ A ( φ ) − A ( φ )] p , − + e A ( φ ) − A ( φ )] p , − , (3)where the on-shell condition ε ( φ ) + p k ( φ ) = [ p ⊥ ( φ ) + m ] /p , − was employed. For the paradigmatic case ofa linearly polarized monochromatic plane wave, it is A µ ( φ ) = A µ cos( ω φ ), with A µ = (0 , E u /ω ), where E is the electric field amplitude, ω the angular frequencyand u the polarization direction (perpendicular to n ).The above analytical solution indicates that even if anelectron is initially at rest, it becomes relativistic withinone laser period T = 2 π/ω if the parameter ξ = | e | p − A m = | e | E mω (4)is of the order of or larger than unity. In the rela-tivistic regime the magnetic component of the Lorentzforce, which depends on the electron’s velocity, becomescomparable to the electric one and the electron’s dy-namics becomes highly nonlinear in the laser-field am-plitude. Thus, the parameter ξ is known as classicalnonlinearity parameter. An heuristic interpretation ofthe parameter ξ is as the work performed by the laserfield on the electron in one laser wavelength λ = T inunits of the electron mass, which clearly explains whyrelativistic effects become important at ξ &
1. Alter-natively, Eqs. (1)-(3) indicate that the figure-8 trajec-tory has a longitudinal (transverse) extension of the or-der of λ ξ ( λ ξ ), implying that the electron trajectorydeviates from the unidirectional oscillating one and be-comes nonlinear in the field amplitude at ξ &
1. Notethat numerically ξ = 6 . p I [10 W/cm ] λ [ µ m] =7 . p I [10 W/cm ] /ω [eV], where I = E / π is thewave’s peak intensity, and that ξ is gauge- and Lorentz-invariant: the gauge invariance has to be intended withrespect to gauge transformations which do not alter thedependence of the four-vector potential on φ (see Heinzland Ilderton, 2009 for a thorough analysis of this issue).The solution in Eqs. (1)-(3) also indicates that in theultrarelativistic regime at ξ ≫
1, the electron acquiresa drift momentum along the propagation direction of thelaser field which is proportional to ξ , in contrast to thetransverse momentum which is proportional to ξ . Inthe case of an electron initially at rest, for example, themomentum p ( ∞ ) of the electron after the laser pulse has been switched off ( A ( ∞ ) = ) has the components p ⊥ ( ∞ ) = e A ( φ ) and p k ( ∞ ) = e A ( φ ) / m .Realistic laser pulses, as those produced in laborato-ries, have a more complicated structure than a planewave, essentially because they are spatially focused onthe transverse planes and the area of the focusing spotchanges along the laser’s propagation axis. Generallyspeaking, if the radius of the minimal focusing area (spotradius) is much larger than the central wavelength of thelaser pulse, then the pulse can be reasonably approxi-mated by a plane wave. A Gaussian beam in the paraxialapproximation offers a more accurate analytical descrip-tion of a realistic laser pulse, which shows a Gaussian pro-file in the transverse planes (Salamin and Keitel, 2002).The dynamics of an electron in such a field cannot be de-rived analytically and a numerical solution of the Lorentzequation is required (Salamin et al. , 2002). B. Quantum dynamics
In the realm of relativistic quantum mechanics, i.e.,when e + - e − pair production is negligible (see also Sec.VIII) and the single-particle quantum theory is appli-cable, the dynamics of an electron in an external elec-tromagnetic field with four-vector potential A µ ( x ) is de-scribed by the Dirac equation { γ µ [ i∂ µ − eA µ ( x )] − m } Ψ = 0 , (5)where γ µ are the Dirac matrices and where Ψ( x ) is thefour-component electron bi-spinor (Berestetskii et al. ,1982). Analogously to the classical case, if the exter-nal field is a plane wave, the Dirac equation can besolved exactly. If p µ = ( ε , p ) and σ / ± / φ → −∞ andif A µ ( −∞ ) = 0, the positive-energy ( ε >
0) solutionΨ p ,σ ( x ) of Eq. (5) reads (Berestetskii et al. , 1982;Volkov, 1935)Ψ p ,σ ( x ) = (cid:20) e p , − ˆ n ˆ A ( φ ) (cid:21) u p ,σ √ V ε e iS p , (6)where in general ˆ v = γ µ v µ for a generic four-vector v µ ,where u p ,σ is a positive-energy free bi-spinor (Berestet-skii et al. , 1982), V is the quantization volume, and where S p ( x ) = − ( p x ) − Z φ −∞ dφ ′ (cid:20) e ( p A ( φ ′ )) p , − − e A ( φ ′ )2 p , − (cid:21) (7)is the classical action of an electron in the planewave (Landau and Lifshitz, 1975). The above electronstates are known as positive-energy Volkov states. Thenegative-energy states Ψ − p , − σ ( x ) can be formally ob-tained by the replacements p µ → − p µ and σ → − σ inEq. (6) except for the energy in the square root (the re-sulting bi-spinor u − p , − σ is the corresponding negative- FIG. 3 (Color) Free wave packet evolution in a plane wavefield. The solid gray line indicates the center of mass tra-jectory, coinciding essentially with the classical trajectory,and the laser pulse travels from left to right. The blue re-gions indicate the copropagating self-adaptive numerical grid.Time and space coordinates are given in “atomic units”, with1 a.u. = 24 as and 1 a.u. = 0 .
05 nm, respectively. FromBauke and Keitel, 2011. energy free bi-spinor (Berestetskii et al. , 1982)). Al-though it has been shown long ago that positive- andnegative-energy Volkov states form a complete set of or-thogonal states on the hypersurfaces φ = const (Ritus,1985), the corresponding property on the hypersurfaces t = const is not straightforward and it has been provedonly recently (see Ritus, 1985 and Zakowicz, 2005, andBoca and Florescu, 2010 for a proof of the orthogonal-ity and of the completeness of the Volkov states, respec-tively).Since the Volkov states form a basis of the space ofthe solutions of Dirac equation in a plane wave, they canbe employed to build electron wave packets and studytheir evolution. A pedagogical example of laser-inducedDirac dynamics is displayed in Fig. 3 for a plane wavewith peak intensity of 6 . × W/cm and centralwavelength of 2 nm. The figure shows the drift of thewave packet in the propagation direction of the wave, itsspreading and its shearing due to non-dipole effects. InFillion-Gourdeau et al. , 2012 an alternative method ofsolving the time-dependent Dirac equation in coordinatespace is presented, which explicitly avoids the fermiondoubling, i.e., the appearance of unphysical modes whenthe Dirac equation is discretized.As in the classical case, we shortly mention here theparadigmatic case of a monochromatic, linearly polar-ized plane-wave field A µ ( φ ) = A µ cos( ω φ ). In this casethe action S p ( x ) can be written in the form S p ( x ) = We point out that the discussed Volkov states Ψ ± p , ± σ ( x ) arethe so-called Volkov in-states, as they transform into free-statesin the limit t → −∞ (Fradkin et al. , 1991). Volkov out-states,which transform into free-states in the limit t → ∞ , can bederived analogously and differ from the Volkov in-states only byan inconsequential constant phase factor (recall that A ( ∞ ) = ). − ( q x ) + “oscillating terms”, with (Ritus, 1985) q µ = p µ + m ξ p , − n µ . (8)The four-vector q µ plays the role of an “effective” four-momentum of the electron in the laser field and it is indi-cated as electron “quasimomentum”. The correspondingelectron “mass” p q = m ∗ = m p ξ / q µ and the dressed mass m ∗ in the case of acircularly polarized laser field with the same amplitudeand frequency is obtained from the above ones with thereplacement ξ → ξ . The quasimomentum coincidesclassically with the average momentum of the electronin the plane wave. Correspondingly, the mass dressingdepends only on the classical nonlinearity parameter ξ and it is an effect of the quivering motion of the electronin the monochromatic wave (see also the recent reviewEhlotzky et al. , 2009). As we will see in Sec. V.A, itis important that conservation laws in QED processesin the presence of a monochromatic plane-wave field in-volve the quasimomentum q µ for the incoming electronsrather than the four-momentum p µ . The question of theelectron dressed mass in pulsed laser fields has been in-vestigated in Heinzl et al. , 2010a and Mackenroth andDi Piazza, 2011.In the realm of QED the parameter ξ can also beheuristically interpreted as the work performed by thelaser field on the electron in the typical QED length λ C = 1 /m ≈ . × − cm (Compton wavelength) inunits of the laser photon energy ω (see Eq. (4)). Thisqualitatively explains why multiphoton effects in a laserfield become important at ξ &
1, such that the laser fieldhas to be taken into account exactly in the calculations(Ritus, 1985). In the framework of QED this is achievedby working in the so-called Furry picture (Furry, 1951),where the e + - e − field Ψ( x ) is quantized in the presenceof the plane-wave field. This amounts essentially in em-ploying the Volkov (dressed) states and the correspond-ing Volkov (dressed) propagators (Ritus, 1985) insteadof free particle states and free propagators to computethe amplitudes of QED processes. In the Furry picturethe effects of the plane wave are accounted for exactlyand only the interaction between the e + - e − field Ψ( x )and the radiation field F µν ( x ) ≡ ∂ µ A ν ( x ) − ∂ ν A µ ( x )is accounted for by means of perturbation theory. Thecomplete evolution of the system “ e + - e − field+radiationfield” is obtained by means of the S -matrix S = T (cid:20) exp (cid:18) − ie Z d x ¯Ψ γ µ Ψ A µ (cid:19)(cid:21) , (9)where T is the time-ordering operator and ¯Ψ( x ) =Ψ † ( x ) γ . For an initial state containing only a single elec-tron with four-momentum p µ , the quantitative descrip-tion of the interaction between the electron, the laser fieldand the radiation field involves in particular the gauge-and Lorentz-invariant quantum parameter χ = | e | p − ( F ,µν p ν ) m = p , − m E F cr , (10)where F cr = m / | e | = 1 . × V/cm = 4 . × Gbeing the critical electromagnetic field of QED (Ritus,1985). The definition of F cr indicates that a constantand uniform electric field with strength of the order of F cr , provides an e + - e − pair with an energy of the or-der of its rest energy 2 m in a distance of the order ofthe Compton wavelength λ C , implying the instabilityof the vacuum under e + - e − pair creation in the pres-ence of such a strong field (Heisenberg and Euler, 1936;Sauter, 1931; Schwinger, 1951). In Eq. (10) we con-sidered the case of a linearly polarized plane wave ofthe form A µ ( φ ) = A µ ψ ( φ ), with ψ ( φ ) being an arbi-trary function with max | dψ ( φ ) /dφ | = 1 and we intro-duced the tensor amplitude F µν = k µ A ν − k ν A µ , with k µ = ω n µ . For an ultrarelativistic electron initiallycounterpropagating with respect to the plane wave it is χ = 5 . × − ε [GeV] p I [10 W/cm ]. The parame-ter χ can be interpreted as the amplitude of the electricfield of the plane wave in the initial rest-frame of the elec-tron in units of the critical field of QED and it controlsthe magnitude of pure quantum effects like the photonrecoil in multiphoton Compton scattering and spin ef-fects. This is why it is known as “nonlinear quantumparameter”.Since the probability dP e /dV dt per unit volume andunit time of a quantum process is a gauge- and Lorentzinvariant quantity, for those processes in a plane-wavefield involving an incoming electron, as, e.g., multiphotonCompton scattering, it can depend only on the two pa-rameters ξ and χ (Ritus, 1985). For an electromagneticfield F µν ( x ) = ( E ( x ) , B ( x )) either constant or slowly-varying, the quantity dP e /dV dt , calculated in the lat-ter case in the leading order with respect to the fields’derivatives, can in principle also depend on the two fieldinvariants F ( x ) = 14 F µν ( x ) F µν ( x ) = −
12 [ E ( x ) − B ( x )] , (11) G ( x ) = 14 F µν ( x ) ˜ F µν ( x ) = − E ( x ) · B ( x ) (12)which identically vanish for a plane wave. In the sec-ond equation ˜ F µν ( x ) = ǫ µναβ F αβ ( x ) / F µν ( x ) and ǫ µναβ is the four-dimensional com-pletely anti-symmetric tensor with ǫ = +1 (since G ( x ) is actually a pseudo-scalar function, the probabil-ity dP e /dV dt can only depend on G ( x )). Note, how-ever, that if | F ( x ) | , | G ( x ) | ≪ min(1 , χ ( x )) F , with χ ( x ) = | e | p | ( F µν ( x ) p ν ) | /m , then the dependence of dP e /dV dt on F ( x ) and G ( x ) can be neglected. In thiscase the probability dP e /dV dt essentially coincides withthe analogous quantity calculated for a constant crossed field F µν , with the replacement F µν → F µν ( x ) (Ritus,1985). For a monochromatic plane wave with angular fre-quency ω this occurs if ξ ≫
1. As will be seen in Sec.V.A, this condition corresponds, e.g., to the formationtime of multiphoton Compton scattering ( ∼ m/ | e | E )being much shorter than the laser period T .As has been mentioned, the S -matrix in Eq. (9)describes all possible electrodynamical processes amongelectrons, positrons and photons. The above consid-erations can be easily adapted for discussing processesinvolving an initial positron. Whereas, the probability dP γ /dV dt of a quantum process in a plane-wave field in-volving an incoming photon, as, e.g., multiphoton e + - e − pair production, depends on the parameters ξ and κ = | e | p − ( F ,µν k ν ) m = k − m E F cr , (13)where k µ = ( ω, k ) is the four-momentum of the incomingphoton (see Ritus, 1985 and Secs. VII-IX). For a pho-ton counterpropagating with respect to the plane waveit is κ = 5 . × − ω [GeV] p I [10 W/cm ]. In thecase of multiphoton e + - e − pair production, the param-eter κ can be interpreted as the amplitude of the elec-tric field of the plane wave in units of the critical field F cr in the center-of-mass system of the created electronand positron (Ritus, 1985). The above remarks on pro-cesses occurring in a constant or slowly-varying back-ground field F µν ( x ) and involving an incoming electron,also apply to the case of an incoming photon once onereplaces χ ( x ) with κ ( x ) = | e | p | ( F µν ( x ) k ν ) | /m . IV. RELATIVISTIC ATOMIC DYNAMICS IN STRONGLASER FIELDS
When super-intense infrared laser pulses, as those de-scribed in Sec. II.A, impinge on an atom, the latter isimmediately partly or fully ionized (Becker et al. , 2002;Keitel, 2001; Protopapas et al. , 1997). The ejected elec-trons experience the typical “zig-zag” motion of a freeelectron in both laser polarization and propagation direc-tions (see Eqs. (2)-(3) and Fig. 3) and will not, in gen-eral, return to the ionic core. With an enhanced bindingforce on the remaining electrons, the ionization dynamicsbecomes increasingly complex and may experience sub-tle relativistic and correlation effects. When the bind-ing force of the ionic core and that of the applied laserfield eventually become comparable, the electrons mayin special cases return to and interact with the parention (rescattering (Corkum, 1993; Kuchiev, 1987; Schafer et al. , 1993)). This interaction leads, for example, to theejection of other electrons, to the absorption of energy ina scattering process or to the emission of high-harmonicphotons in case of recombination.0
A. Ionization
Previously, atomic or molecular ionization was stud-ied with laser pulses of intensity below 10 W/cm , andthe relativistic laser-matter interaction was dominatedby the plasma community. The pioneering experimentreported in Moore et al. , 1999 on the ionization behav-ior of atoms and ions in interaction with a laser withintensity of 3 × W/cm has thus attracted consid-erable interest. The laser magnetic field component wasshown to alter the direction of ionization characteristi-cally (see Sec. III.A). This is because in the relativis-tic regime the ionized electron in a laser field acquiresa large momentum along the laser propagation direction(see Eq. (3)) and photoelectrons are emitted mostly inthat direction within a characteristic opening angle θ :tan θ ∼ p ⊥ ( ∞ ) /p k ( ∞ ) ∼ /ξ (see the discussion belowEq. (4)). With highly-charged ions becoming more easilyavailable in a wide range of charges, e.g., via super-stronglaser fields or by passing the ion beams through metallicfoils, relativistic laser-induced ionization has been fur-ther studied (Chowdhury et al. , 2001; Dammasch et al. ,2001; DiChiara et al. , 2008, 2010; Gubbini et al. , 2005;Palaniyappan et al. , 2008; Yamakawa et al. , 2003, 2004).In this situation rescattering is generally suppressedand multiple ionization of atoms and ions takes placemostly via direct ionization, especially including tunnel-ing. On the theoretical side attention was then focusedon the relativistic generalization (Milosevic et al. , 2002;Popov et al. , 1997; Popov, 2004; Popov et al. , 2006) ofthe so-called Perelomov-Popov-Terent’ev (PPT) theoryor Ammosov-Delone-Krainov (ADK) model (Ammosov et al. , 1986; Perelomov, 1967), which describes atomicionization in the quasistatic tunneling regime. While thecommon intuitive interpretation of the laser induced tun-neling fails in the relativistic regime (Reiss, 2008), a re-vised picture has been proposed in Klaiber et al. , 2012.The Strong Field Approximation (SFA) (Faisal, 1973;Keldysh, 1965; Reiss, 1980), which treats in a universalway both the multiphoton and the tunneling regimes ofstrong-field ionization, has also been extended to the rel-ativistic regime (Reiss, 1990a,b). Both the PPT theoryand the SFA assume that the direct ionization process oc-curs as a single-electron phenomenon and thus neglectsatomic structure effects.When the tunneling process proceeds very fast, multi-electron correlation effects can occur due to the so-calledshake-up processes. Thus, the detachment of one elec-tron from the atom or ion via tunneling modifies theself-consistent potential sensed by the remaining elec-trons and may result, consequently, in the excitation ofthe atomic core (inelastic tunneling). A strong excitationmay also trigger the simultaneous escape of several elec-trons from the bound state through the potential barrier(collective tunneling). These effects were known to oc-cur also in the nonrelativistic regime (Zon, 1999, 2000) Log I [W/cm ] T o t a l p o pu l a t i o n FIG. 4 (Color) The total Rb (black lines) and Rb (redlines) ion populations in a gaseous target as a function ofthe peak intensity of a linearly-polarized laser field with awavelength of 0 . µ m and with a pulse duration of 5 fs. Thesolid lines display two-electron inelastic tunneling, the dashedlines one-electron inelastic tunneling and the dashed-dottedlines the results via the PPT theory. Adapted from Kornev et al. , 2009. but were concealed by competing rescattering effects. InKornev et al. , 2009, it was shown that, in the relativisticregime, the role of inelastic and collective tunneling cansignificantly increase and the relativistic PPT rate hasbeen generalized in this respect. In a linearly polarizedfield, the rate R ( N )coll of inelastic collective tunneling of N equivalent electrons from the outer shell of an ion is de-scribed by the following universal formula (Kornev et al. ,2009): R ( N )coll = m √ α N − C Nκl M !( l + 1 / N M N M +1 √ π N Y j =1 ( l + m j )!( m j !) ( l − m j )! × I κ N − (cid:18) E a E (cid:19) ν − N − M +1 / e − NE a / E , (14)where α = e ≈ /
137 is the fine-structure constant, m , . . . , m N are the magnetic quantum numbers of thebound electrons, M = P Nj =1 m j , l is the orbital quan-tum number of the electrons, κ = q I ( N ) p /mN , I ( N ) p = P Nj =1 ( I (0) p,j − ∆ j ), I (0) p,j is the j th ionization potential ofa parent ion, ∆ j is the energy of the core excitation, E a = κ F cr is the atomic field, Z | e | is the charge ofthe residual ion, I is the adimensional overlap integral(see Kornev et al. , 2009 for its precise definition) and C κl ≈ (2 /ν ) ν / √ πν , with ν = Zα/κ . According to thecalculations in Kornev et al. , 2009, inelastic and collec-tive tunneling effects contribute significantly to the rel-ativistic ionization dynamics at intensities larger than10 W/cm , thus changing the ionization probability bymore than one order of magnitude (see Fig. 4).Spin effects of bound systems in strong laser fields wereshown to moderately alter the quantum dynamics andits associated radiation via spin-orbit coupling in highly-1 FIG. 5 (Color) (a) Experimental photoelectron spectra forargon at I = 1 . × W/cm and at an angle of 62 ◦ fromthe laser propagation direction. Analytical results are shownfor all photoelectrons (continuous line) and for the L-shell(dashed line). The angular distributions are at an electronenergy of (b) 60 keV, (c) 400 keV, and (d) 770 keV. FromDiChiara et al. , 2008. charged ions already at an intensity of ∼ W/cm (Hu and Keitel, 1999; Walser et al. , 2002). More recentlya nonperturbative relativistic SFA theory has been de-veloped, describing circular dichroism and spin effects inthe ionization of helium in an intense circularly polarizedlaser field (Bhattacharyya et al. , 2011). Here, two-photonionization has been studied in the nonrelativistic inten-sity range 10 -10 W/cm with a photon energy of 45eV, yielding small relative spin-induced corrections of theorder of 10 − .A series of experiments has been devoted to the mea-surement of atomic multi-electron effects in relativisti-cally strong laser fields. In DiChiara et al. , 2008, the en-ergy distribution of the ejected electrons and the angle-resolved photoelectron spectra for atomic photoioniza-tion of argon at I ∼ W/cm have been inves-tigated experimentally. Here, isolation of the single-atom response in the multicharged environment has beenachieved by measuring photoelectron yields, energies,and angular distributions as functions of the sample den-sity. Ionization of the entire valence shell along with sev-eral inner-shell electrons was shown at I ∼ -10 W/cm . A typical spectrum in the case of linear polar-ization is displayed in Fig. 5. An extended plateau-likestructure appears in the spectrum due to the electronsoriginating from the L-shell and the longitudinal com-ponent of the focused laser field. A surprising featureis observed in the energy-resolved angular distribution.In contrast to the nonrelativistic case with increasingrescatterings and, thus, angular-distribution widths athigh energies, here azimuthally isotropic angular distri-butions are observed at low energies ( ∼
60 keV in Fig. 5),which become narrower for high-energy photoelectrons.The authors attribute the anomalous broad angular dis- tribution for low-energy electrons to electron-correlationeffects. A similar experiment on the energy- and angle-resolved photoionization was later reported for xenon at alaser intensity of 10 W/cm (DiChiara et al. , 2010). Forenergies below 0.5 MeV, the yield and the angular distri-bution were shown not to be described by a one-electronstrong-field model, but rather involve most likely mul-tielectron and high-energy atomic excitation processes.A further experiment on relativistic ionization of themethane molecule at I ∼ -10 W/cm (Palaniyap-pan et al. , 2008) indicated that molecular mechanisms ofionization play no role, and that C ions are producedat these intensities mostly via the cross-shell rescatteringatomic ionization mechanism. All these experimental re-sults still await an accurate theoretical description.On the computational side, various numerical meth-ods have been developed to describe the laser-driven rel-ativistic quantum dynamics in highly-charged ions. AFast-Fourier-Transform split-operator code was imple-mented in Mocken and Keitel, 2008 for solving the Diracequation in 2+1 dimensions by employing adaptive gridand parallel computing algorithms. Another method hasbeen developed in Selstø et al. , 2009 to solve the 3DDirac equation by expanding the angular part of thewave-function in spherical harmonics. The latter wasapplied to hydrogenlike ions in intense high-frequencylaser pulses with emphasis on investigating the role ofnegative-energy states. In Bauke et al. , 2011, the clas-sical relativistic phase-space averaging method, general-ized to arbitrary central potentials, and the enhancedtime-dependent Dirac and Klein-Gordon numerical treat-ments are employed to investigate the relativistic ioniza-tion of highly-charged hydrogenlike ions in short intenselaser pulses. For ionization dynamics beyond the tunnel-ing regime, quantum mechanical and classical methodsgive similar results, for laser wavelengths from the near-infrared region to the soft x-ray regime. Furthermorea useful procedure has been developed, which employsthe over-the-barrier ionization yields for highly-chargedions, to determine the peak laser field strength of shortultrastrong pulses in the range I ∼ -10 W/cm (Hetzheim and Keitel, 2009). In addition, in this arti-cle the ionization angle of the ejected electrons is inves-tigated by the full quantum mechanical solution of theDirac equation and the laser field strength is shown to bealso linked to the electron emission angle. The magneticfield-induced tilt in the lobes of the angular distributionsof photoelectrons in laser-induced relativistic ionizationhas also been discussed in Klaiber et al. , 2007.There are also several new theoretical results for theionic quantum dynamics in strong high-frequency laserfields, in the so-called stabilization regime, where theionization rate decreases or remains constant also withincreasing laser intensity. An unexpected nondipole ef-fect has been reported in Førre et al. , 2006 via numeri-cally solving the Schr¨odinger equation for a hydrogenic2 FIG. 6 (Color online) Dipole (left) and nondipole (right)probability densities of the Kramers-Henneberger wave-function in the x - z plane for a x -polarized, 10-cycle sin-likepulse propagating in the positive z direction (upward), with E = 1 . × V/cm and ω = 54 eV. The snapshots aretaken at t = 0, t = T /
2, and t = 1 . T from top to bot-tom. The length of the horizontal line corresponds to about50 a B ≈ . a B = λ C /α ≈ . × − cm beingthe Bohr radius . Note that the scale is logarithmic with fourcontours per decade. From Førre et al. , 2006. atom beyond the dipole approximation. For this purposethe Kramers-Henneberger transformation (Henneberger,1968; Kramers, 1956) has been employed, i.e., the trans-formation to the instantaneous rest-frame of a classicalfree electron in the laser field, and the terms ∼ ξ havebeen neglected in the Hamiltonian (the value of ξ con-sidered was approximately 0 . z axisbut this velocity tends to zero at the end of the pulse.Thus, the electromagnetic forces alone do not change theelectron momentum along the propagation direction atthe end of the pulse. The net effect of the Coulomb forceson the electron wave packet is consequently a momentumcomponent along the negative z axis: the electron, whichis most probably situated in the upper hemisphere overthe pulse, undergoes a momentum kick in the negative z direction each time it passes close to the nucleus. A sim-ilar effect has been reported for molecules (Førre et al. ,2007).Radiative recombination, being the time-reversed pro-cess of photoionization, of a relativistic electron with ahighly-charged ion in the presence of a very intense laserfield has been considered in M¨uller et al. , 2009. It wasshown that the strong coupling of the electron to the laser field may lead to a very broad energy spectrum of emit-ted recombination photons, with pronounced side wings,and to characteristic modifications of the photon angulardistribution.Specific features of nondipole quantum dynamics instrong and ultrashort laser pulses have also been inves-tigated employing the so-called Magnus approximation(Dimitrovski et al. , 2009). The dominant nondipole ef-fect is found to be a shift of the entire wave-function to-wards the propagation direction, inducing a substantialpopulation transfer into states with similar geometry.The recent experiment reported in Smeenk et al. , 2011addresses the question of how the photon momenta areshared between the electron and ion during laser-inducedmultiphoton ionization. Theoretically, this problem re-quires a nondipole treatment, even in the nonrelativis-tic case, to take into account explicitly the laser photonmomentum. Energy-conservation of ℓ -photon ionizationhere means that ℓω = I p + U p + K , where I p is the ion-ization energy of the atom, U p = e E / mω is the pon-deromotive energy and K is the electron’s kinetic energy.The experimental results in Smeenk et al. , 2011, obtainedusing laser fields with wavelength of 0 . µ m and 1 . µ m inthe intensity range of 10 -10 W/cm has show that thefraction of the momentum, corresponding to the numberof observed photons needed to overcome the ionizationenergy I p , is transferred to the created ion rather thanto the photoelectron. The electron carries only the mo-mentum corresponding to the kinetic energy K , whilethe ponderomotive energy and the corresponding por-tion of the momentum are transferred back to the laserfield. This experiment shows that the tunneling conceptfor the ionization dynamics is only an approximation. Infact, the quasistatic tunneling provides no mechanism totransfer linear momentum to the ion, a conclusion thatagrees with recent concerns in Reiss, 2008. B. Recollisions and high-order harmonic generation
Tunneling in the nonrelativistic regime is generally fol-lowed by recollisions with the parent ion along with var-ious subsequent effects (Corkum, 1993; Kuchiev, 1987;Schafer et al. , 1993). A characteristic feature of strong-field processes in the relativistic regime is the suppressionof recollisions due to the magnetically induced relativisticdrift of the ionized electron in the laser propagation direc-tion (see Sec. III.A). Although relativistic effects becomesignificant when the parameter ξ exceeds unity, signa-tures of the drift in the laser propagation direction can beobserved already in the weakly relativistic regime ξ . d k in the laser propagation direction islarger than the electron’s wave packet size a wp, k in thatdirection (Palaniyappan et al. , 2006). The drift distance3is approximately given by d k ∼ λ ξ / a wp, k can be estimatedfrom a wp, k ∼ v k ∆ t , where v k is a typical electron velocityalong the laser propagation direction and ∆ t is the excur-sion time of the electron in the continuum. The velocity v k can be related to the tunneling time τ tun via the time-energy uncertainty: mv k / ∼ /τ tun . In turn, one canestimate the tunneling time τ tun as τ tun ∼ l tun /v b , where l tun ∼ I p / | e | E is the tunneling length and v b ∼ p I p /m the velocity of the bound electron. In the above esti-mate, it was assumed that the work carried out by thelaser field along the tunneling length equals I p . Thus, atthe rescattering moment ∆ t ∼ T , the wave packet size a wp, k is of the order of λ q | e | E / p m I p and the role ofthe drift can be characterized by means of the parameter r = ( d k /a wp, k ) as estimated by r ∼ ξ p mI p ω . (15)The condition r & p d ( ∼ mξ /
4, see Eq. (3)), opposite to thelaser propagation direction. The probability P i ( p d ) ofthis process is exponentially damped, though, due to thenonzero momentum p d (see, e.g., Salamin et al. , 2006): P i ( p d ) ∼ exp (cid:20) −
23 (2 mI p ) / m | e | E (cid:18) p d mI p (cid:19)(cid:21) . (16)The drift term in the exponent proportional to p d will beimportant if p mI p p d /m | e | E &
1, which is equivalentto the condition r &
3. At near-infrared wavelengths( ω ≈ I p = 13 . I approximately above 3 × W/cm . Then, HHGand other recollision phenomena are suppressed.The attainability of relativistic recollisions would, how-ever, be very attractive for ultrahigh HHG (Kohler et al. ,2012) as well as for the realization of laser controlledhigh-energy (Hatsagortsyan et al. , 2006) and nuclear pro-cesses (Chelkowski et al. , 2004; Milosevic et al. , 2004).Various methods for counteracting the relativistic drifthave been proposed, such as by utilizing highly-chargedions (Hu and Keitel, 2001; Keitel and Hu, 2002) whichmove relativistically against the laser propagation di-rection (Chiril˘a et al. , 2004; Mocken and Keitel, 2004),by employing Positronium (Ps) atoms (Henrich et al. ,2004), or through preparing antisymmetric atomic (Fis-cher et al. , 2007) and molecular (Fischer et al. , 2006) or-bitals. Here the impact of the drift of the ionized electron FIG. 7 (Color) The HHG setup with two counterpropagatingAPTs. After ionization by the laser pulse 1, the ejected elec-tron is driven in the same pulse (light blue), propagates freelyafter the pulse 1 has left (gray dashed) and is driven back tothe ion by the laser pulse 2 (dark blue). From Kohler et al. ,2011. is reduced by the increase of the laser frequency in thesystem’s center of mass, an equally strong drift via twoconstituents with equal mass or via appropriate initialmomenta from antisymmetric orbitals, respectively.On the other hand, the laser field can also be modi-fied to suppress the relativistic drift by employing tightlyfocused laser beams (Lin et al. , 2006), two counter-propagating laser beams with linear polarization (Kei-tel et al. , 1993; Kylstra et al. , 2000; Taranukhin, 2000;Taranukhin and Shubin, 2001, 2002) or equal-handed cir-cular polarization (Milosevic et al. , 2004). In the firsttwo cases, the longitudinal component in the tightly fo-cused laser beam may counteract the drift, or the Lorentzforce may be eliminated in a small area near the antin-odes of the resulting standing wave, respectively. In thethird case involving circularly polarized light, the rela-tivistic drift is eliminated because the electron velocityis oriented in the same direction as the magnetic field.This setup is well suited for imaging attosecond dynam-ics of nuclear processes but not for HHG because of thephase-matching problem (Liu et al. , 2009). In the weaklyrelativistic regime the Lorentz force may also be com-pensated by a second weak laser beam polarized alongthe direction of propagation of the strong beam (Chiril˘a et al. , 2002). Furthermore, the relativistic drift can besignificantly reduced by means of special tailoring of thedriving laser pulse, which strongly reduces the time whenthe electron’s motion is relativistic with respect to a si-nusoidal laser pulse (Klaiber et al. , 2006, 2007). Twoconsecutive laser pulses (Verschl and Keitel, 2007a) ora single laser field assisted by a strong magnetic fieldcan also be used to reverse the drift (Verschl and Kei-tel, 2007b). In addition two strong Attosecond PulseTrains (APTs) (Hatsagortsyan et al. , 2008) or an infraredlaser pulse assisted by an APT (Klaiber et al. , 2008)have been employed to enhance relativistic recollisions.In fact, due to the presence of the APT the ionizationcan be accomplished by one XUV photon absorption andthe relatively large energy ω X of the XUV-photon with ω X = I p + p d / m can compensate the subsequent mo-mentum drift p d ∼ mξ / et al. ,1994) has been developed as a reliable source of coher-ent XUV radiation and attosecond pulses (Agostini andDiMauro, 2004) opening the door for attosecond time-resolved spectroscopy (Krausz and Ivanov, 2009). Non-relativistic HHG in an atomic gas medium allows alreadyto generate coherent x-ray photons up to keV energies(Sansone et al. , 2006) and to produce XUV pulses shorterthan 100 as (Goulielmakis et al. , 2008). The most favor-able conversion efficiency for nonrelativistic keV harmon-ics is anticipated with mid-infrared driving laser fields(Chen et al. , 2010; Popmintchev et al. , 2009). However,progress in this field has slowed down, especially becauseof the inhibition, alluded to above, of recollisions dueto optical driving-field intensity above 3 × W/cm .This indicates the limit on the cut-off frequency ω c ofnonrelativistic HHG to ω c ≈ . U p ∼
10 keV.Another factor hindering HHG at high intensities isthe less favorable phase-matching. In strong laser fields,outer-shell electrons are rapidly ionized and produce alarge free electron background causing a phase mismatchbetween the driving laser wave and the emitted x-rays.The feasibility of phase-matched relativistic HHG in amacroscopic ensemble was first investigated in Kohler et al. , 2011. Here, the driving fields are two counterprop-agating APTs consisting of 100 as pulses with a peakintensity of the order of 10 W/cm (see Fig. 7). Theelectron is driven to the continuum by the laser pulse 1in Fig. 7, followed by the usual relativistic drift. There-after, the laser pulse 2 overtakes the electron, reversesthe drift and imposes the rescattering, yielding a muchhigher HHG signal than for a conventional laser field atthe same cut-off energy. Here phase-matching can be ful-filled due to an additional intrinsic phase specific to thissetup, depending on the time delay between the pulsesand on the pulse intensity. The latter, being unique forthis laser setup, mainly affects the electron excursiontime and varies along the propagation direction. Thephase-matching is achieved by modifying the laser in-tensity along the propagation direction and by balanc-ing the phase slip due to dispersion with the indicatedintrinsic phase. Note, however, that HHG in the rela-tivistic regime has been observed experimentally ratherefficiently in laser plasma interactions (Dromey et al. ,2006). V. MULTIPHOTON THOMSON AND COMPTONSCATTERING
In this section we discuss one of the most fundamentalprocesses in QED in a strong laser field: the emission ofradiation by an accelerated electron. After reporting on recent theoretical investigations on this process, we dis-cuss its possible applications for producing high-energyphoton beams.
A. Fundamental considerations
When an electron is wiggled by an intense laser wave,it emits electromagnetic radiation. This process occurswith absorption of energy and momentum by the electronfrom the laser field and it is named as multiphoton Thom-son scattering or multiphoton Compton scattering, de-pending on whether quantum effects, like photon recoil,are negligible or not. Multiphoton Thomson and Comp-ton scattering in a strong laser field have been studiedtheoretically since a long time (see Salamin and Faisal,1998; Sarachik and Schappert, 1970; and Sengupta, 1949for multiphoton Thomson scattering and Brown and Kib-ble, 1964; Goldman, 1964; and Nikishov and Ritus, 1964afor multiphoton Compton scattering). The classical cal-culation of the emitted spectrum is based on the analyt-ical solution in Eqs. (1)-(3) of the Lorentz equation ina plane wave and the substitution of the correspondingelectron trajectory in the Li´enard-Wiechert fields (Jack-son, 1975; Landau and Lifshitz, 1975). Whereas, as wehave discussed in Sec. III.B, the quantum calculation ofthe amplitude of the process is performed in the Furrypicture of QED. As a result, the total emission probabil-ity depends only on the two Lorentz- and gauge-invariantparameters ξ (see Eq. (4)) and χ (see Eq. (10)).The parameter ξ has already been discussed in Sec.III.A. In the contest of multiphoton Compton scatteringthis parameter controls in particular the effective order ℓ eff of the emitted harmonics, which, for an ultrarela-tivistic electron, can be estimated in the following way.In order to effectively emit a frequency ω ′ , the formationlength l f of the process must not exceed the coherencelength l coh , because, otherwise, interference effects wouldhinder the emission. Since an electron with instanta-neous velocity β and energy ε = mγ = m/ p − β ≫ m mainly emits along the direction of β , within a cone withapex angle ϑ ∼ /γ ≪
1, the formation length l f can beestimated from l f ∼ ̺/γ , with ̺ being the instantaneousradius of curvature of the electron trajectory (Jackson,1975). On the other hand, l coh = π/ω ′ (1 − β cos ϑ ) ∼ γ /ω ′ (Baier et al. , 1998; Jackson, 1975). By requir-ing that l f . l coh , we obtain the following estimatefor the largest-emitted frequency (cut-off frequency) ω ′ c : ω ′ c ∼ γ /̺ . Now, in the average rest-frame of the elec-tron, i.e., in the reference frame where the average elec-tron velocity along the propagation direction of the laservanishes (see Eq. (3)), it is γ ⋆ ∼ ξ (corresponding to theenergy ε ⋆ ∼ mξ ) and l ⋆f ∼ λ ⋆ /ξ , where the upper in-dex ⋆ indicates the variable in this frame. Consequently, ω ′ ⋆c ∼ ξ ω ⋆ and the effective order of the emitted har-monics is ℓ eff ∼ ξ (note that ℓ eff is a Lorentz scalar). As5the order of the emitted harmonics corresponds quantummechanically to the number of laser photons absorbed bythe electron during the emission process, the parameter ξ is also said to determine the “multiphoton” characterof the process.On the other hand, the nonlinear quantum parameter χ (see Eq. (10)) in the contest of multiphoton Comptonscattering controls the importance of quantum effects asthe recoil of the emitted photon. In fact, we can esti-mate classically the importance of the emitted photonrecoil from the ratio ω ′ c /ε and our considerations aboveexactly indicate that ω ′ c /ε ∼ ξ ω ⋆ /m ∼ χ . Thus, multi-photon Thomson scattering is characterized by the con-dition χ ≪
1, while multiphoton Compton scatteringby χ &
1. This result can also be obtained in the caseof a monochromatic laser wave starting from the energy-momentum conservation relation q µ + ℓk µ = q ′ µ + k ′ µ (17)in the case in which ℓ laser photons are absorbed in theprocess (Ritus, 1985). Here q µ and q ′ µ are the quasimo-menta of the initial and final electron (see Eq. (8)) and k ′ µ = ω ′ n ′ µ is the four-momentum of the produced pho-ton ( n ′ = 0). From this expression it is easy to obtainthe energy ω ′ of the emitted photon as ω ′ = ℓω p , − ( n ′ p ) + (cid:16) ℓω + m ξ p , − (cid:17) n ′− . (18)By reminding that ℓ eff ∼ ξ and by estimating the typicalemission angle of the photon (Mackenroth and Di Piazza,2011), it is possible to show that ω ′ ∼ χ ε at ξ ≫
1. Asit has also been throughroughly investigated analyticallyand numerically in Boca and Florescu, 2011 and Seiptand K¨ampfer, 2011a, multiphoton Compton and Thom-son spectra coincide in the limit χ →
0, although inSeipt and K¨ampfer, 2011a differences have been observednumerically in the detailed structure of the classical andquantum spectra also for χ ≪
1. The most importantdifference between classical and quantum spectra is cer-tainly the presence of a sharp cut-off in the latter asan effect of the photon recoil: the energy of the photonemitted in a plane wave is limited by the initial energyof the electron . This does not occur classically, as therethe frequency of the emitted radiation does not have thephysical meaning of photon energy. The dependence ofthe energy cut-off on the laser intensity has been recently In the case of a plane-wave background field, this limitationrather concerns the quantity k ′− of the emitted photon, as k ′− = p , − − p ′− < p , − . However, for an ultrarelativistic electronwith p , − ≫ mξ and initially counterpropagating with respectto the laser field, it is p , − ≈ ε , k ′− ≈ ω ′ and p ′− ≈ ε ′ (Baier et al. , 1998). recognized as a possible experimental signature of multi-photon Compton scattering (Harvey et al. , 2009).First calculations on multiphoton Thomson and Comp-ton scattering have mainly focused on the easiest case of amonochromatic background plane wave, either with cir-cular or linear polarization. The main results of theseinvestigations, like the dependence of the emitted fre-quencies on the laser intensities have been recently re-viewed in Ehlotzky et al. , 2009. The complete descriptionof the multiphoton Compton scattering process with re-spect to the polarization properties of the incoming andthe outgoing electrons, and of the emitted photon in amonochromatic laser wave has been presented in Ivanov et al. , 2004. Recently significant attention has been de-voted to the investigation of multiphoton Thomson andCompton scattering in the presence of short and evenultrashort plane-wave pulses (we recall that such pulseshave still an infinite extension in the directions perpen-dicular to the propagation direction). In Boca and Flo-rescu, 2009 multiphoton Compton scattering has beenconsidered in the presence of a pulsed plane wave. Theangular-resolved spectra are practically insensitive to theprecise form of the laser pulse for ω τ ≥
20, with τ being the pulse duration. The main differences with re-spect to the monochromatic case are: 1) a broadening ofthe lines corresponding to the emitted frequencies; 2) theappearance of sub-peaks, which are due to interferenceeffects in the emission at the beginning and at the end ofthe laser pulse. On the one hand, the continuous natureof the emission spectrum in a finite pulse in contrast tothe discrete one in the monochromatic case has a clearmathematical counterpart. In both cases, in fact, thetotal transverse momenta P ⊥ with respect to the laserpropagation direction and the total quantity P − are con-served in the emission process (see Sec. III). However, inthe monochromatic case the following additional conser-vation law holds (for a linearly polarized plane wave, seeEq. (17)) ε + p , k + m ξ p , − + 2 ℓω = ω ′ + k ′k + ε ′ + p ′k + m ξ p ′− , (19)so that the resulting four-dimensional energy-momentumconservation law allows only for the emission of the dis-crete frequencies in Eq. (18). On the other hand, theappearance of sub-peaks has been in particular investi-gated in Heinzl et al. , 2010c, where it has been foundthat the number N s-p of sub-peaks within the first har-monic scales linearly with the pulse duration τ and with ξ : N s-p = 0 . ξ τ [fs]. In this paper the effects of spa-tial focusing of the driving laser pulse are also discussed.The authors investigate in particular the dependence ofthe deflection angle α out undergone by the electron af-ter colliding head-on with a Gaussian focused beam as afunction of the impact parameter b (see Fig. 8).By further decreasing the laser pulse duration, it hasbeen argued that effects of the relative phase between the6 FIG. 8 (Color online) Deflection α out of an electron initiallycounterpropagating with respect to a laser field with an en-ergy of 3 J and a pulse duration of 20 fs, as a function of theimpact parameter b for different laser waist radii w . FromHeinzl et al. , 2010c. pulse profile and the carrier wave (the so-called CarrierEnvelope Phase (CEP)) should become visible in multi-photon Thomson and Compton scattering. In Boca andFlorescu, 2009 the case of ultrashort pulses with ω τ & et al. , 2010 the dependence of theangular distribution of the emitted radiation in multi-photon Thomson and Compton scattering on the CEP offew-cycles pulses has been exploited to propose a schemeto measure the CEP of ultrarelativistic laser pulses (in-tensities larger than 10 W/cm ). The method is es-sentially based on the high directionality of the photonemission by an ultrarelativistic electron, because the tra-jectory of the electron, in turn, also depends on the laser’sCEP. Accuracies in the measurement of the CEP of theorder of a few degrees are theoretically envisaged. Multi-photon Compton scattering in one-cycle laser pulses hasbeen considered in Mackenroth and Di Piazza, 2011 anda substantial broadening of the emission lines with re-spect to the monochromatic case has been observed. Thehigh-directionality of radiation emitted via multiphotonThomson scattering has also been employed as a diag-nostic tool in Har-Shemesh and Di Piazza, 2012, wherea new rather precise method has been proposed to mea-sure the peak intensity of strong laser fields (intensitiesbetween 10 W/cm and 10 W/cm ) from the angularaperture of the photon spectrum.The study of multiphoton Thomson and Comptonscattering in short laser pulses has also stimulated the in-vestigation of scaling laws for the photon spectral density(Boca and Oprea, 2011; Heinzl et al. , 2010c; Seipt andK¨ampfer, 2011a,b). For example, in Heinzl et al. , 2010c ascaling law has been found for backscattered radiation inthe case of head-on laser-electron collisions, which sim-plifies the averaging over the electron-beam phase space. A more general scaling law has been determined in Seiptand K¨ampfer, 2011b, which relaxes the previous assump-tions on head-on collision and on backscattered radiationemployed in Heinzl et al. , 2010c. Moreover, in Seipt andK¨ampfer, 2011a a simple relation is determined betweenthe classical and the quantum spectral densities. Finally,in Boca and Oprea, 2011 it is found that in the ultra-relativistic case γ ≫ ξ /γ and not on theindependent values of ξ and γ (see also Mackenroth et al. , 2010).In the above-mentioned publications the spectral prop-erties of the emitted radiation in the classical and quan-tum regimes have been considered. In Kim et al. , 2009and Zhang et al. , 2008, instead, the temporal propertiesof the emitted radiation in multiphoton Thomson scatter-ing have been investigated. In both papers the feasibilityof generating single attosecond pulses is discussed.Photoemission by a single-electron wave packet viaThomson scattering in a strong laser field has been dis-cussed in Peatross et al. , 2008. It was shown that thepartial emissions from the individual electron momentumcomponents do not interfere when the driving field is aplane wave. In other words, the size of the electron wavepacket, even when it spreads to the scale of the wave-length of the driving field, does not affect the Thomsonemission.Finally, we shortly mention that multiphoton effectsin Thomson and Compton scattering have been mea-sured in various laboratories. The second-harmonic ra-diation was first observed in the collision of a 1 keV elec-tron beam with a Q-switched Nd:YAG laser, althoughthe laser intensity was such that ξ ≈ .
01 (Englertand Rinehart, 1983), and then in the interaction of amode-locked Nd:YAG laser ( ξ = 2) with plasma elec-trons (Chen et al. , 1998). Multiphoton Thomson scat-tering of laser radiation in the x-ray domain has beenreported in Babzien et al. , 2006 (see Pogorelsky et al. ,2000 for a similar proof-of-principle experiment). Single-shot measurements of the angular distribution of the sec-ond harmonic (photon energy 6 . laser beamwith ξ = 0 .
35. In the prominent SLAC experiment(Bula et al. , 1996) multiphoton Compton emission wasdetected for the first time. In this experiment an ultra-relativistic electron beam with energy of about 46 . W/cm ( ξ ≈ . χ ≈ .
3) and four-photonCompton scattering has been observed indirectly via anonlinear energy shift in the spectrum of the outcomingelectrons.7
B. Thomson- and Compton-based sources of high-energyphoton beams
The single-particle theoretical analysis presented aboveindicates that high-energy photons can be emitted viamultiphoton Thomson and Compton scattering of an ul-trarelativistic electron. For example, an electron withinitial energy ε ≫ m colliding head-on with an opticallaser field ( ω ≈ ξ .
1) isbarely deflected by the laser field ( ̺ ∼ λ γ /ξ ≫ λ )and potentially emits photons with energies ω [keV] . . × − ε [MeV]. This feature has boosted the ideaof so-called Thomson- and Compton-based sources ofhigh-energy photons as a valid alternative to conven-tional synchrotron sources, the main advantages of theformer being the compactness, the wide tunability, theshortness of the photon beams in the femtosecond scaleand the potential for high brightness. Unlike the exper-iments on multiphoton Thomson and Compton scatter-ing where laser systems with ξ & ξ .
1, such that multipho-ton effects are suppressed and shorter bandwidths of thephoton beam are achieved. On the other hand, the elec-tron beam quality is crucial for Thomson- and Compton-based radiation sources. In particular, the brightness ofthe photon beam scales inversely quadratically with theelectron beam emittance, and linearly with the electronbunch current density.Proof-of-principle experiments have demonstratedThomson- and Compton-based photon sources by cross-ing a high-energy laser pulse with a picosecond relativis-tic electron beam from a conventional linear electron ac-celerator (Chouffani et al. , 2002; Leemans et al. , 1996;Pogorelsky et al. , 2000; Sakai et al. , 2003; Schoenlein et al. , 1996; Ting et al. , 1995, 1996). We also mentionthe benchmark experiment carried out at LLNL, wherephotons with an energy of 78 keV have been producedwith a total flux of 1 . × photons/shot, by collid-ing an electron beam with an energy of 57 MeV with aTi:Sa laser beam with an intensity of about 10 W/cm ( ξ ≈ .
5) (Gibson et al. , 2004).Another achievement in the development of Thomson-and Compton-based photon sources has been the exper-imental realization of a compact all-optical setup, wherethe electrons are accelerated by an intense laser. Inthe first experiment with an all-optical setup (Schwoerer et al. , 2006), x-ray photons in the range of 0.4 keV to 2keV have been generated. In this experiment the elec-tron beam was produced by a high-intensity Ti:Sa laserbeam ( I ≈ × W/cm ) focused into a pulsed he-lium gas jet. The characteristic feature of the all-opticalsetup is that the electron bunches and, consequently, thegenerated x-ray photon beams have an ultrashort dura-tion ( ∼
100 fs) and a linear size of the order of 10 µ m.Another advantage is that the electrons can be precisely synchronized with the driving laser field. In order tofurther improve the all-optical setup, design parametersfor a proof-of-concept experiment have been analyzed inHartemann et al. , 2007. For the calculation of the Comp-ton scattering parameters, a 3D Compton scattering codehas been used, which was extensively tested for Comp-ton scattering experiments performed at LLNL (Brown et al. , 2004; Brown and Hartemann, 2004; Hartemann et al. , 2004, 2005) (see Sun and Wu, 2011 for an alter-native numerical simulation scheme). It is shown thatx-ray fluxes exceeding 10 s − and a peak brightnesslarger than 10 photons/(s mrad mm et al. , 2010). Production of gamma-rays ranging from 75 keV to 0.9 MeV has been demon-strated with a peak spectral brightness of 1 . × pho-tons/(s mrad mm . × photons/shot. An experimental setup for high-flux gamma-ray generation has been constructed in theSaga Light-Source facility in Tosu (Japan), by collid-ing a 1.4 GeV electron beam with a CO laser (wave-length 10 . µ m) (Kaneyasu et al. , 2011). A flux of about3 . × photons/s gamma photons with energy largerthan 0.5 MeV has been obtained.In the basic setups of Thomson- and Compton-basedphoton sources the electrons experience the intense laserfield for a time-interval much shorter than that neededto cross the whole laser beam, the former being of the or-der of the laser’s Rayleigh length divided by the speed oflight. Thus, the quest for a more intense laser pulse at agiven power in order to increase the photon yield impliesa tighter focusing and therefore a shorter effective inter-action time, which in turn causes a broadening of thephoton spectrum. In Debus et al. , 2010 the Traveling-Wave Thomson Scattering (TWTS) setup is proposed,which allows the electrons to stay in the focal regionof the laser beam during the whole crossing time (seeFig. 9). This is achieved by employing cylindrical op-tics to focus the laser field only along one direction (redlines in Fig. 9) and, depending on the angle betweenthe initial electron velocity and the laser wave-vector, bytilting the laser pulse front. As a result, an interactionlength ∼ × photons/shotat 20 keV). An alternative way of reaching longer effec-tive laser-electron interaction times has been proposed inKaragodsky et al. , 2010, where a planar Bragg structureis employed to guide the laser pulse and realize Thom-son/Compton scattering in a waveguide. In this way,the yield of x-rays can be enhanced by about two ordersof magnitude with respect to the conventional free-spaceGaussian-beam configuration at given electron beam andinjected laser power in both configurations. However,there are two constraints specific to this setup. On the8 FIG. 9 (Color) Schematic setup of TWTS with the red linesindicating the laser focal lines. In the notation of Debus et al. ,2010 φ is the angle between the initial electrons’ velocity andthe laser’s wave vector and α is the angle between the laserpulse front and the laser propagation direction. Adapted fromDebus et al. , 2010. one hand, the electron beam has to have a small angularspread in order to be injected into the planar Bragg struc-ture without causing wall damage. On the other hand,the laser field strength has to be such that ξ . × − to avoid surface damage.Finally, in Hartemann et al. , 2008 a setup has beenproposed to obtain bright GeV gamma-rays via Comp-ton scattering of electrons by a thermonuclear plasma.In fact, a thermonuclear deuterium-tritium plasma pro-duces intense blackbody radiation with a temperature ∼
20 keV and a photon density ∼ cm − (Tabak et al. , 1994). When a thermal photon with energy ω ∼ γ ∼ ), a Doppler-shifted high-energy pho-ton ω ′ ∼ γ ω ∼ ω ′ has to be smaller than the initial electron energy ε , akinematical photon pileup is induced in the emitted pho-ton spectrum at ε (Zeldovich and Sunyaev, 1969) (seeFig. 10). This results in a quasimonochromatic GeVgamma-ray beam with a peak brightness & pho-tons/(s mrad mm in situ iso-tope detection (Albert et al. , 2010). In this respect, suchphoton sources will represent the main experimental toolfor nuclear-physics investigations at the Romanian pil-lar of ELI (see Fig. 1). Other possible applications,at photon energies beyond the MeV threshold, includethe production of positron beams (Hugenschmidt et al. ,2012; Omori et al. , 2003) as well as the investigations ofhigh-energy processes occurring in gamma-gamma andgamma-lepton collisions (Telnov, 1990). ω ′ /ε N o r m a li ze d o n - a x i s b r i g h t n e ss FIG. 10 (Color online) Normalized on-axis brightness for dif-ferent values of the rapidity ρ = cosh − γ at a plasma tem-perature of 20 keV. See Hartemann et al. , 2008 for the mean-ing of the blue circles at ρ = 6. Adapted from Hartemann et al. , 2008. VI. RADIATION REACTION
The issue of “radiation reaction” (RR) is one of theoldest and most fundamental problems in electrodynam-ics. Classically it corresponds to the determination ofthe equation of motion of a charged particle, an electronfor definiteness, in a given electromagnetic field F µν ( x ).In fact, the Lorentz equation mdu µ /ds = eF µν u ν (seeSec. III.A) does not take into account that the electron,while being accelerated, emits electromagnetic radiationand loses energy and momentum in this way. The firstattempt of taking into account the reaction of the radia-tion emitted by the electron on the motion of the electronitself (from here comes the expression “radiation reac-tion”), was accomplished by H. A. Lorentz in the nonrel-ativistic regime (Lorentz, 1909). Starting from the knownLarmor formula P L = (2 / e a for the power emittedby an electron with instantaneous acceleration a , Lorentzargued that this energy-loss corresponds to a “damping”force F R = (2 / e d a /dt acting on the electron. Theexpression of the damping force was generalized to therelativistic case by M. Abraham in the form (Abraham,1905) F µR = 23 e (cid:18) d u µ ds + du ν ds du ν ds u µ (cid:19) . (20)In order to solve the problem of a radiating electron self-consistently, P. A. M. Dirac suggested in Dirac, 1938 tostart from the coupled system of Maxwell and Lorentz9equations ∂ µ F µνT = 4 πj ν ∂ λ F T,µν + ∂ µ F T,νλ + ∂ ν F T,λµ = 0 m du µ ds = eF µνT u ν , (21)where F µνT ( x ) = F µν ( x ) + F µνS ( x ), with F µνS ( x ) be-ing the “self” electromagnetic field generated by theelectron four-current j µ ( x ) = e R dsδ ( x − x ( s )) u µ andwhere the meaning of the symbol m for the electronmass will be clarified below. In order to write an “ef-fective” equation of motion for the electron which in-cludes RR, one “removes” the degrees of freedom ofthe electromagnetic field (Teitelboim, 1971). This isachieved in Landau and Lifshitz, 1975 at the level ofthe Lagrangian of the system electron+electromagneticfield and an interesting connection of the RR problemwith the derivation of the so-called Darwin Lagrangianis indicated. By working at the level of the equationsof motion (21), one first employs the Green’s functionmethod and formally determines the retarded solution F µνT, ret ( x ) of the (inhomogeneous) Maxwell’s equations: F µνT, ret ( x ) = F µν ( x ) + F µνS, ret ( x ) (Teitelboim, 1971). Sub-stitution of F µνT, ret ( x ) in the Lorentz equation eliminatesthe electromagnetic field’s degrees of freedom, but it isnot straightforward because F µνT, ret ( x ) has to be calcu-lated at the electron’s position, where the electron cur-rent diverges. This difficulty is circumvented by model-ing the electron as a uniformly charged sphere of radius a tending to zero. After performing the substitution, andby neglecting terms which vanish in the limit a →
0, oneobtains the equation ( m + δm ) du µ /ds = eF µν u ν + F µR with δm = (4 / e /a being formally diverging. How-ever, it is important to note that the only diverging termin the limit a → m is the overall coefficient of the four-acceleration du µ /ds .Therefore, one sets m = m + δm and obtains the so-called Lorentz-Abraham-Dirac (LAD) equation: m du µ ds = eF µν u ν + 23 e (cid:18) d u µ ds + du ν ds du ν ds u µ (cid:19) . (22)We point out that renormalization in quantum field the-ory is based on the fact that the bare quantities, likecharge and mass, appear in the Lagrangian density of thetheory, which is not an observable physical quantity. Onthe other hand, the bare electron mass m , which is for-mally negatively diverging for m to be finite, appears herein the system of equations (21), which should “directly”provide classical physical observables, like the electrontrajectory. On the other hand, it is also known that theLAD equation is plagued with physical inconsistencieslike, for example, the existence of the so-called “runaway” solutions with an exponentially-diverging electron accel-eration, even in the absence of an external field (see thebooks Hartemann, 2001 and Rohrlich, 2007 for reviewson these issues).It was first shown in Landau and Lifshitz, 1975 that inthe nonrelativistic limit the RR force given by Eq. (20)is much smaller than the Lorentz force, if the typicalwavelength λ and the typical field-amplitude F of theexternal electromagnetic field fulfill the two conditions λ ≫ αλ C , F ≪ F cr α , (23)where F cr is the critical electromagnetic field of QED (seeSec. III.B). This allows for the reduction of order in theLAD equation, i.e., for the substitution of the electronacceleration in the RR force via the Lorentz force dividedby the electron mass. In order to perform the analogousreduction of order in the relativistic case, the conditions(23) have to be fulfilled in the instantaneous rest-frameof the electron (Landau and Lifshitz, 1975). The resultis the so-called Landau-Lifshitz (LL) equation m du µ ds = eF µν u ν + 23 e h em ( ∂ α F µν ) u α u ν − e m F µν F αν u α + e m ( F αν u ν )( F αλ u λ ) u µ (cid:21) . (24)The LL equation is not affected by the shortcomings ofthe LAD equation: for example, it is evident that if theexternal field vanishes, so does the electron acceleration.Most importantly, the conditions (23) in the instanta-neous rest-frame of the electron have always to be fulfilledin the realm of classical electrodynamics, i.e., if quantumeffects are neglected. In order for this to be true, in fact,the two weaker conditions λ ≫ λ C and F ≪ F cr haveto be fulfilled in the instantaneous rest-frame of the elec-tron: the first guarantees that the electron’s wave func-tion is well localized and the second ensures that purequantum effects, like photon recoil or spin effects arenegligible (Baier et al. , 1998; Berestetskii et al. , 1982;Ritus, 1985) (see also Sec. III.B). This observation ledF. Rohrlich to state recently that the LL equation is the“physically correct” classical relativistic equation of mo-tion of a charged particle (Rohrlich, 2008). Rohrlich’sstatement is also supported by the findings in Spohn,2000, where it is shown that the physical solutions ofthe LAD equation, i.e., those which are not runaway-like, are on the critical manifold of the LAD equationitself and are governed there exactly by the LL equa-tion. On the other hand, since the LL equation is derivedfrom the LAD equation, one may still doubt on its rigor-ous validity, due to the application in the latter equationof the “suspicious” classical mass-renormalization proce-dure. However, this procedure is avoided in Gralla et al. ,2009 by employing a more sophisticated zero-size limit-ing procedure, where also the charge and the mass of the0particle are sent to zero but in such a way that their ra-tio remains constant. The authors conclude that at theleading-order level the LL equation represents the self-consistent perturbative equation of motion for a chargewithout electric and magnetic moment. The motion ofa continuous charge distribution interacting with an ex-ternal electromagnetic field is also investigated by a self-consistent model and at a more formal level in Burton et al. , 2007.From the original derivation of the LL equation fromthe LAD equation in Landau and Lifshitz, 1975, it isexpected that the two equations should predict the sameelectron trajectory, possibly with differences smaller thanthe quantum effects. This conclusion has been recentlyconfirmed by analytical and numerical investigations inHadad et al. , 2010 for an external plane-wave field withlinear and circular polarization and in Bulanov et al. ,2011 for different time-dependent external electromag-netic field configurations. An effective numerical methodto calculate the trajectory of an electron via the LL equa-tion, which explicitly maintains the relativistic covari-ance and the mass-shell condition u = 1, has been ad-vanced in Harvey et al. , 2011a. An alternative numericalmethod for determining the dynamics of an electron in-cluding RR effects has been proposed in Mao et al. , 2010.We should emphasize that the LL equation is not theonly equation which has been suggested to overcome theinconsistencies of the LAD equation. A list of alternativeequations can be found in the recent review (Hammond,2010) (see also Seto et al. , 2011). A phenomenologicalequation of motion, including RR and quantum effectsrelated to photon recoil, has been suggested in Sokolov et al. , 2010a, 2009 (see also Sec. VI.B). The authors writethe differential variation of the electron momentum asdue to two contributions: one arising from the externalfield and one corresponding to the recoil of the emittedphoton. The resulting equation can be written as thesystem m dx µ dτ = p µ + 23 e I QED I L eF µν p ν m dp µ dτ = eF µν dx ν dτ − I QED p µ m , (25)where τ is the time in the “momentarily comovingLorentz frame” of the electron where the spatial compo-nents of p µ instantaneously vanish, I QED is the quantumradiation intensity (Ritus, 1985) and I L = (2 / αω ξ .The expression of I QED in the case of a plane wave isemployed, which is valid only for an ultrarelativistic elec-tron in the presence of a slowly-varying and undercriticalotherwise arbitrary external field (see Sec. III.B).It has also to be stressed that the original LAD equa-tion is still the subject of extensive investigation (thefirst study of the LAD equation in a plane-wave field wasperformed in Hartemann and Kerman, 1996). In Ferrisand Gratus, 2011, for example, the origin of the Schott term in the RR force, i.e., the term proportional to thederivative of the electron acceleration (see Eq. (20)),is thoroughly investigated and in Kazinski and Shipulya,2011 the asymptotics of the physical solutions of the LADequation at large proper times are obtained. Whereas, inNoble et al. , 2011 a kinetic theory of RR is proposed,based on the LAD equation and applicable to study sys-tems of many particles including RR (this last aspectis also considered in Rohrlich, 2007). Enhancement ofRR effects due to the coherent emission of radiation by alarge number of charges is discussed in Smorenburg et al. ,2010. In this respect, we mention here that, in order toinvestigate strong laser-plasma interactions at intensitiesexceeding 10 W/cm , RR effects have been also imple-mented in Particle-In-Cell (PIC) codes (Tamburini et al. ,2012, 2010; Zhidkov et al. , 2002) by modifying the Vlasovequation for the electron distribution function accordingto the LL equation. Specifically, in Zhidkov et al. , 2002it is shown that in the collision of a laser beam withintensity I = 10 W/cm with an overdense plasmaslab, about 35% of the absorbed laser energy is con-verted into radiation and that the effect of RR amountsto about 20%. One-dimensional (Tamburini et al. , 2010)and three-dimensional (Tamburini et al. , 2012) PIC sim-ulations have shown that RR effects strongly depend onthe polarization of the driving field: while for circular po-larization they are negligible even at I ∼ W/cm ,at those intensities they are important for linear polariza-tion. The simulations also show the beneficial effects ofRR in reducing the energy spread of ion beams generatedvia laser-plasma interactions (see also Sec. XII.A). A dif-ferent beneficial effect of RR on ion acceleration has beenfound in Chen et al. , 2011 for the case of a transparentplasma: RR strongly suppresses the backward motion ofthe electrons, cools them down and increases the num-ber of ions to be bunched and accelerated. Finally, thesystem in Eq. (25) has been implemented in a 3D PICcode in Sokolov et al. , 2009 showing that a laser pulsewith intensity 10 W/cm loses about 27% of its energyin the collision with a plasma slab. The same system ofequations has been employed to study the penetrationof ultra-intense laser beams into a plasma in the hole-boring regime (Naumova et al. , 2009) and to investigatethe process of ponderomotive ion acceleration at ultra-high laser intensities in overcritical bulk targets (Schlegel et al. , 2009). A. The classical radiation dominated regime
As it was already observed in Landau and Lifshitz,1975, the fact that the RR force in the LL equation hasto be much smaller than the Lorentz force in the instan-taneous rest-frame of the electron does not exclude thatsome components of the two forces can be of the sameorder of magnitude in the laboratory system. This occurs1if the condition αγ F/F cr ∼ γ being the relativistic Lorentz factor of the elec-tron and F the amplitude of the external electromagneticfield. For an ultrarelativistic electron this condition canbe fulfilled also in the realm of classical electrodynam-ics (quantum recoil effects are negligible if γF/F cr ≪ et al. , 2005 an equivalentdefinition of the CRDR in the presence of a backgroundlaser field has been formulated, as the regime where theaverage energy radiated by the electron in one laser pe-riod is comparable with the initial electron energy. Byestimating the radiated power P L from the relativisticLarmor formula P L = − (2 / e ( du µ /ds )( du µ /ds ) (Jack-son, 1975) with du µ /ds → ( e/m ) F µν u ν , one obtains that P L ∼ αχ ξ ε . Therefore, the conditions of being in theCRDR are R C = αχ ξ ≈ , χ ≪ , (26)where, as has been seen in Sec. III.B, the second condi-tion ensures in particular that the quantum effects likephoton recoil are negligible. The same condition R C ≈ u − ( φ ) is constant if the equation of motion isthat due to Lorentz, it decreases here with respect to φ as u − ( φ ) = u , − /h ( φ ), where u µ is the four-velocity atan initial φ and h ( φ ) = 1 + 23 R C ω Z φφ dϕ (cid:18) dψ ( ϕ ) dϕ (cid:19) , (27)where the four-potential of the wave has been assumedto have the form A µ ( φ ) = A µ ψ ( φ ) (see Sec. III.B, be-low Eq. 10). This effect has been recently suggested inHarvey et al. , 2011b as a possible signature to measureRR (see also Lehmann and Spatschek, 2011). The twoconditions in Eq. (26) are, in principle, compatible forsufficiently large values of ξ . For example, for an opti-cal ( ω = 1 eV) laser field with an average intensity of10 W/cm and for an electron initially counterpropa-gating with respect to the laser field with an energy of20 MeV, it is χ = 0 .
16 and R C = 1 .
3. This exampleshows that, in general, it is not experimentally easy toenter the CRDR at least with presently-available lasersystems. In Di Piazza et al. , 2009a a different regime hasbeen investigated, which is parametrically less demand-ing than the CRDR but in which the effects of RR arestill large. In this regime the change in the longitudi-nal (with respect to the laser field propagation direction)momentum of the electron due to RR in one laser period is of the order of the electron’s longitudinal momentumitself in the laser field. As a result, it is found that inthe ultrarelativistic case and for a few-cycle pulse, if theconditions R C & γ − ξ ξ > γ ≶ ξ including RR effects). This can have measurable effectsif one exploits the high directionality of the radiationemitted by an ultrarelativistic electron (see Sec. V.A).The results in Di Piazza et al. , 2009a show in fact thatthe apex angle of the angular distribution of the emittedradiation, with and without RR effects included, maydiffer by more than 10 ◦ already at an average opticallaser intensity of 5 × W/cm ( ξ ≈ γ ≈ R C ≈ .
08. Small RR effects on photon spectra emittedby initially bound electrons had already been predictedvia numerical integration of the LL equation in Keitel et al. , 1998 well below the CRDR.
B. Quantum radiation reaction
The shortcomings of the classical approaches to theproblem of RR suggest that it can be fully understoodonly at the quantum level. In the seminal paper Monizand Sharp, 1977 the origin of the classical inconsisten-cies, like the existence of runaway solutions of the LADequation, were clarified in the nonrelativistic case. Theauthors first show that such inconsistencies are also ab-sent in classical electrodynamics if one considers chargedistributions with a typical radius larger than the classi-cal electron radius r = αλ C ≈ . × − cm. Goingto the nonrelativistic quantum theory and by analyzingthe Heisenberg equations of motion of the electron in anexternal time-dependent field, the authors conclude thatthe quantum theory of a pointlike particle does not ad-mit any runaway solutions, provided that the externalfield varies slowly along a length of the order of λ C (thisis an obvious assumption in the realm of nonrelativis-tic theory, as time-dependent fields with typical wave-lengths of the order of λ C would in principle allow for e + - e − pair production, see Sec. VIII). From this pointof view a classical theory of RR has only physical mean-ing as the classical limit ( ~ →
0) of the correspondingquantum theory and the authors indicate that the result-ing equation of motion is the nonrelativistic LL equationwith the bare mass m (it is shown that the electrostaticself-energy of a point charge vanishes in nonrelativisticquantum electrodynamics). On the other hand, if oneconsiders the quantum equations of motion of a charge2distribution and performs the classical limit before thepointlike limit, then the classical equations of motion ofthe charge distribution are, of course, recovered and, oncethe point-like limit is then performed, runaway solutionsappear again. The nonrelativistic form of the LL equa-tion has been also recovered from quantum mechanicsin Krivitskii and Tsytovich, 1991 by including radiativecorrections to the time-dependent electron momentumoperator in the Heisenberg representation, and by calcu-lating the time-derivative of the average momentum in asemiclassical state.The situation in the relativistic theory is less straight-forward because relativistic quantum electrodynamics,i.e., QED, is a field theory fundamentally different fromclassical electrodynamics. The first theory of relativis-tic quantum RR goes back to W. Heitler and his group(Heitler, 1984; Jauch and Rohrlich, 1976). However, theevaluations of the QED amplitudes in Heitler’s theory in-volve the solution of complicated integral equations andit has given a practical result only in the calculation ofthe total energy emitted by a nonrelativistic quantumoscillator, with and without RR.At first sight one would say that RR effects are auto-matically taken into account in QED, because the elec-tromagnetic field is treated as a collection of photonsthat take away energy and momentum, when they areemitted by charged particles. However, photon recoil isalways proportional to ~ , making it a purely quantumquantity with no classical counterpart. Moreover, if onecalculates the spectrum of multiphoton Compton scat-tering in an external plane-wave field, for example, andthen performs the classical limit χ →
0, one obtainsthe corresponding multiphoton Thomson spectrum cal-culated via the Lorentz equation and not via the LADor the LL equation (see also Sec. V.A). Finally, it hasalso been seen that in classical electrodynamics the RReffects may not be a small perturbation on the Lorentzdynamics and they cannot be obtained as the result of asingle limiting procedure. Otherwise they would alwaysappear as a small correction.In order to understand what RR is in QED, it is moreconvenient to go back to Eq. (21) and to notice thatthe LAD, namely the LL, equation is equivalent to thecoupled system of Maxwell and Lorentz equations. If onedetermines the trajectory of the electron via the LL equa-tion and then calculates the total electromagnetic field F µνT ( x ) via the Li´enard-Wiechert four-potential (Landauand Lifshitz, 1975), one has solved completely the classi-cal problem of the radiating electron in the given electro-magnetic field. As has been discussed in Sec. III.B, thesolution of the analogous problem in strong-field QEDwould correspond to completely determine the S -matrixin Eq. (9), as well as the asymptotic state | t → + ∞i for the given initial state | t → −∞i = | e − i , which rep-resents a single electron. The first-order term in theperturbative expansion of the S -matrix corresponds to the process of multiphoton Compton scattering and then,classically, to the Lorentz dynamics. Whereas, all high-order terms give rise to radiative corrections and to high-order coherent and incoherent (cascade) processes, anddetermine what we call “quantum RR”. Here, by high-order coherent processes is meant those involving morethan one basic QED process (photon emission by anelectron/positron or e + - e − photoproduction) but all oc-curring in the same formation region. Analogously, inhigher-order incoherent or cascade processes each basicQED process occurs in a different formation region. Now,in the case of a background plane wave at ξ ≫ χ .
1, the quantum effects are certainly important butthe radiative corrections and higher-order coherent pro-cesses scale with α and can be neglected (Ritus, 1972).Also, if χ does not exceed unity then the photons emit-ted by the electron are mainly unable, by interactingagain with the laser field, to create e + - e − pairs, as thepair production probability is exponentially suppressed(see also Secs. VIII and IX). Therefore, it can be con-cluded that at ξ ≫ χ .
1, RR in QED corre-sponds to the overall photon recoil experienced by theelectron when it emits many photons consecutively andincoherently (Di Piazza et al. , 2010a).A qualitative understanding of the above conclusioncan be attained by assuming that χ ≪ N γ of photons emitted byan electron in one laser period at ξ ≫
1. Since theprobability of emitting one photon in a formation lengthis of the order of α and since one laser period containsabout ξ formation lengths (Ritus, 1985) then N γ ∼ αξ .Also, the typical energy ω ′ of a photon emitted by anelectron is of the order ω ′ ∼ χ ε , then the average en-ergy E emitted by the electron is E ∼ αξ χ ε = R C ε .This estimate is in agreement with the classical result ob-tained from the LL equation. In other words, the classicallimit of RR in this regime corresponds to the emission ofa higher and higher number of photons all with an en-ergy much smaller than the electron energy, in such away that even though the recoil at each emission is al-most negligible, the cumulative effect of all photon emis-sions may have a finite nonnegligible effect. Note that ω ′ and N γ are both pure quantum quantities and only theirproduct E has a classical analogue in the limit χ → et al. , 2010a of the Quantum Radiation-Dominated Regime (QRDR), which is characterized bymultiple emission of photons already in one laser period.This regime is then characterized by the conditions R Q = αξ ≈ , χ & . (29)Quantum photon spectra have been calculated numeri-cally in Di Piazza et al. , 2010a without RR, i.e., by in-cluding only the emission of one photon (four-momentum k ′ µ ), and with RR, i.e., by including multiple-photonemissions (and by integrating with respect to all the four-3 ̟ ′ d S RR Q /d̟ ′ d S no RR Q /d̟ ′ d S RR C /d̟ ′ d S no RR C /d̟ ′ FIG. 11 (Color online) Quantum photon spectra as a functionof ̟ ′ = k ′− /p , − calculated with (solid, black line) and with-out (long dashed, red line) RR and the corresponding classicalones with (short dashed, blue line) and without (dotted, ma-genta line) RR. The error bars in the quantum spectrum withRR stem from numerical uncertainties in multidimensionalintegrations. The numerical parameters in our notation are: ε = 1 GeV, ω = 1 .
55 eV and I = 10 W/cm ( R Q = 1 . χ = 1 . et al. , 2010a. momenta of the emitted photons except one indicated as k ′ µ ). The results show that in the QRDR the effects ofRR are essentially three (see Fig. 11): 1) increase of thephoton yield at low photon energies; 2) decrease of thephoton yield at high photon energies; 3) shift of the max-imum of the photon spectrum towards low photon ener-gies. Figure 11 also shows that the classical treatment ofRR (via the LL equation) artificially overestimates theabove effects, the reason being that quantum correctionsdecrease the average energy emitted by the electron withrespect to the classical value (Ritus, 1985). However, at χ ≪
1, i.e., when the recoil of each emitted photon ismuch smaller than the electron energy, then the quantumspectra converge into the corresponding classical ones.As mentioned in Sec. VI, a semiclassical phenomenolog-ical approach to RR in the quantum regime has beenproposed in Sokolov et al. , 2009, 2010b.Finally, the quantum modifications induced by theelectron’s self-field onto the Volkov states (see Eq. (6))have been recently investigated in Meuren and Di Pi-azza, 2011. It is found that the classical expression ofthe electron quasimomentum q µ in a linearly polarizedplane wave (see Eq. (8)) admits a correction depend-ing on the quantum parameter χ and also that self-fieldeffects induce a peculiar dynamics of the electron spin. VII. VACUUM-POLARIZATION EFFECTS
QED predicts that photons interact with each otheralso in vacuum (Berestetskii et al. , 1982). Effects aris-ing from this purely quantum interaction are referred toas vacuum polarization effects. This is in contrast toclassical electrodynamics where the linearity of Maxwell’sequations in vacuum forbids self-interaction of the elec- (cid:1)
FIG. 12 Vacuum polarization diagram in an external back-ground electromagnetic field. The thick electron lines indicateelectron propagators calculated in the Furry picture, account-ing exactly for the presence of the background field. tromagnetic field in the vacuum itself. The possibility ofphoton-photon interaction in vacuum, within the frame-work of QED, can be understood qualitatively by observ-ing that a photon may locally “materialize” into an e + - e − pair which, in turn, interacts with other photons. For thesame reason a background electromagnetic field can in-fluence photon propagation (see Fig. 12 and Berestetskii et al. , 1982). In the latter case the extension l f of theregion where this transformation occurs, i.e., its forma-tion length, depends, in principle, on the structure ofthe background field (Baier and Katkov, 2005). How-ever, in some cases it can be estimated qualitatively viathe Heisenberg uncertainty principle from the typical mo-mentum p flowing in the e + - e − loop in Fig. 12. We con-sider, for example, a constant background electromag-netic field (or a slowly-varying one, at leading order inthe space-time derivatives of the field itself). In this case,if the energy ω of the incoming photon (see Fig. 12) is atmost of the order of m , then the momentum p flowing inthe e + - e − loop is of the order of m and l f ∼ /p ∼ λ C .If ω ≫ m the analysis is more complicated and the for-mation length strongly depends on the structure of thebackground field.From the theoretical point of view it is convenient todistinguish between low-energy vacuum-polarization ef-fects if ω ≪ m and high-energy ones if ω & m . A. Low-energy vacuum-polarization effects
The scattering in vacuum of a real photon by anotherreal photon is possibly the most fundamental vacuum-polarization process (Berestetskii et al. , 1982) and ithas not yet been observed experimentally. The totalcross section of the process depends only on the Lorentz-invariant parameter η = ( k k ) /m , with k µ and k µ be-ing the four-momenta of the colliding photons or, equiv-alently, on the energy ω ∗ of the two colliding photons intheir center-of-momentum system ( η = 2 ω ∗ /m ). Thisprocess has been investigated in Euler, 1936 in the low-energy limit η ≪ η ≫
1. The complete expression of thecross section σ γγ → γγ was calculated in Karplus and Neu-man, 1950 and can also be found in Berestetskii et al. ,1982 (see also Fig. 13). In the low-energy limit η ≪ ω ∗ /m − σ γγ → γγ [ − c m ] − − FIG. 13 Cross section of real photon-photon scattering as afunction of the energy of the colliding photons in their center-of-momentum system in units of the electron mass. Adaptedfrom Berestetskii et al. , 1982. Copyright Elsevier (1982). the cross section σ γγ → γγ is given by (Berestetskii et al. ,1982) σ γγ → γγ = 97381000 π α λ C η η ≪ η ≫ et al. , 1974;Berestetskii et al. , 1982) σ γγ → γγ = 1 π (cid:20) π − π ζ (3) + 148225 π − ζ (5) i α λ C η , η ≫ ζ ( x ) is the Riemann zeta function (Olver et al. ,2010). In terms of the center-of-momentum energy ω ∗ , the cross section becomes σ γγ → γγ [cm ] = 7 . × − ( ω ∗ [eV]) at ω ∗ ≪ m and σ γγ → γγ [cm ] = 5 . × − / ( ω ∗ [GeV]) at ω ∗ ≫ m . The steep dependenceof σ γγ → γγ on η for η ≪ et al. , 2006 for experiments and ex-perimental proposals until 2005 aiming to observe realphoton-photon scattering in vacuum).However, various proposals have been put forward re-cently in order to observe this process by colliding stronglaser beams which contain a large number of photons.A common theoretical starting point of all these pro-posals is the “effective-Lagrangian” approach (Dittrichand Gies, 2000; Dittrich and Reuter, 1985). In this ap-proach the interaction among photons in vacuum is de-scribed via an effective Lagrangian density of the elec-tromagnetic field. By starting from the total Lagrangiandensity of the classical electromagnetic field and of thequantum e + - e − Dirac field, one integrates out the de-grees of freedom of the latter field and is left with aLagrangian density depending only on the electromag-netic field. As has been seen above, the formation re-gion of photon-photon interaction at low energies is of the order of λ C , therefore if the classical electromag-netic field F µν ( x ) = ( E ( x ) , B ( x )) comprises only wave-lengths much larger than λ C , the interaction is approxi-mately pointlike and the effective Lagrangian density isaccordingly a local quantity. Also, since the effective La-grangian density is a Lorentz invariant, it can dependonly on the electromagnetic field invariants F ( x ) and G ( x ) already introduced in Sec. III.B. The complete ex-pression of the effective Lagrangian density was reportedfor the first time in Heisenberg and Euler, 1936 and Weis-skopf, 1936 (see also Schwinger, 1951) and it is known asthe Euler-Heisenberg Lagrangian density. Here we are in-terested only in the experimentally-relevant low-intensitylimit | F ( x ) | , | G ( x ) | ≪ F and the leading-order Euler-Heisenberg Lagrangian density L EH ( x ) reads (Dittrichand Gies, 2000; Dittrich and Reuter, 1985) L EH ( x ) = − π F ( x ) + α π F ( x ) + 7 G ( x ) F cr . (32)Different experimental observables have been suggestedto detect low-energy vacuum-polarization effects. Thosewill be reviewed next.
1. Experimental suggestions for direct detection ofphoton-photon scattering
The most direct way to search for photon-photon scat-tering events in vacuum by means of laser fields is to lettwo laser beams collide and to look for scattered photons.However, by employing a third “assisting” laser beam, ifone of the final photons is kinematically allowed to beemitted along this beam with the same frequency and po-larization, then the number of photon-photon scatteringevents can be coherently enhanced (Varfolomeev, 1966).In this laser-assisted setup the “signal” of photon-photonscattering is, of course, the remaining outgoing photon.In Lundin et al. , 2007 and Lundstr¨om et al. , 2006 anexperiment has been suggested to observe laser-assistedphoton-photon scattering with the Astra-Gemini lasersystem (see Sec. II.A). The authors found a particular“three-dimensional” setup, which turns out to be espe-cially favorable for the observation of the process (seeFig. 14). The number N γ of photons scattered in oneshot for an optimal choice of the geometrical factors andof the polarization angles between the incoming and theassisting beams is found to be N γ ≈ . P [PW] P [PW] P [PW]( λ [ µ m]) . (33)Here, P and P are the powers of the incoming beams, P is the power of the assisting beam and λ is the wave-length of the scattered wave to be measured. By plug-ging in the feasible values for Astra-Gemini P = P =0 . P = 0 . N γ ≈ .
07, i.e.,5
FIG. 14 (Color) Schematic “three-dimensional” setup forlaser-assisted photon-photon scattering involving two incom-ing beams (in blue), an assisting one (in red) and a scatteredone (in violet). From Lundstr¨om et al. , 2006. roughly one photon scattered every 15 shots (for the pa-rameters of Astra Gemini the wavelength of both incom-ing beams is chosen as 0 . µ m, that of the assisting beamas 0 . µ m, so that the wavelength λ = 0 . µ m of thescattered photon is different from those of the incomingand assisting beams).The quantum interaction among photons in vacuumhas been exploited in King et al. , 2010a to propose, forthe first time, a double-slit setup comprised only of light(see also Marklund, 2010). In this setup two strong par-allel beams collide head-on with a counterpropagatingprobe pulse. The photons of the probe have the choiceto interact either with one or with the other strong beam,and, when scattered, they are predicted to build an in-terference pattern with alternating minima and maximatypical of double-slit experiments (see Fig. 15). Also,if one of the slits is closed, i.e., if the probe collideswith only one strong beam, the interference fringes dis-appear. The key idea behind this setup is that thevacuum-scattered beam (intensity I d ), although propa-gating along the same direction as the probe, has a muchwider angular distribution than the latter, offering thepossibility of detecting vacuum-scattered photons out-side the focus of the undiffracted probe beam. For astrong-field intensity of I ≈ × W/cm that maybe in the near future be available at ELI or at HiPERand for a probe beam with wavelength λ p = 0 . µ mand intensity I p = 4 × W/cm , it is predicted thatabout four photons per shot would contribute to build upthe interference pattern in the observable region (it is theregion outside the circle in Fig. 15, where I d > I p ).The diffraction of a probe beam in vacuum by a single fo-cused strong laser pulse is also investigated in Tommasiniand Michinel, 2010 in the case of almost counterpropa- FIG. 15 (Color) Intensity I d of the vacuum-scattered wavefor a probe beam propagating along the positive y directionand colliding with two strong beams aligned along the x axis.The crosses correspond to coordinates x n according to theclassical prediction x n = ( n + 1 / λ p d/D , where n in an inte-ger number, λ p is the wavelength of the probe field, d is thedistance between the interaction region and the observationscreen and D is the distance between the centers of the twostrong beams. The numerical values of the parameters can befound in King et al. , 2010a. Adapted from King et al. , 2010a. gating beams. For optimal lasers parameters and at astrong-laser power of 100 PW the diffracted vacuum sig-nal is predicted to be measurable in a single shot. Theeffects on photon-photon scattering of the temporal pro-file of the laser pulses have been recently investigatedin King and Keitel, 2012, showing a suppression of thenumber of vacuum scattered photons with respect to theinfinite-pulse (monochromatic) case.The concept of Bragg scattering has been exploited inKryuchkyan and Hatsagortsyan, 2011 to observe the scat-tering of photons by a modulated electromagnetic-fieldstructure in vacuum. If a probe wave passes througha series of parallel strong laser pulses and if the Braggcondition on the impinging angle is fulfilled, the num-ber of diffracted photons can be strongly enhanced. Ata fixed intensity for each strong beam the enhancementfactor with respect to laser-assisted photon-photon scat-tering is equal to the number of beams in the periodicstructure. However, in experiments usually the total en-ergy of the laser beams is fixed and an enhancement bya factor of two is predicted. By considering N G equalGaussian pulses propagating along the x direction andtheir centers separated by a distance D > w z from eachother, the resulting photon-photon scattering probabil-ity will be proportional to the phase-matching factor P ,with P = sin ( δk z N G D/ ( δk z D/ , (34)where the vector δ k = k − k is the difference betweenthe wave vectors of the reflected and incident waves. The6Bragg condition is satisfied for δk z = 2 πl/D , with l be-ing an integer. By employing ten optical laser beamswith a wavelength of 1 µ m and each with an intensityof 2 . × W/cm , about five vacuum-scattered pho-tons are predicted per shot. Finally, an enhancementof vacuum-polarization effects in laser-laser collision hasbeen predicted in Monden and Kodama, 2011 by em-ploying strong laser beams with large angular aperture.For example, it is predicted that the number of vacuum-radiated photons will be enhanced by two orders of mag-nitude, if the angular aperture of the colliding beams isincreased from 53 ◦ to 103 ◦ .Other experimental suggestions to measure photon-photon scattering in vacuum can be found in Eriksson et al. , 2004 and Tommasini et al. , 2008.
2. Polarimetry-based experimental suggestions
The expression of the Euler-Heiseberg Lagrangian den-sity in Eq. (32), suggests to interpret a region where onlyan electromagnetic field is present as a material mediumcharacterized by a polarization P EH = ∂ L EH /∂ E − E / π and a magnetization M EH = ∂ L EH /∂ B + B / π (Jackson, 1975) given by P EH = α π F (cid:2) E − B ) E + 7( E · B ) B (cid:3) , (35) M EH = α π F (cid:2) B − E ) B + 7( E · B ) E (cid:3) . (36)Note that an arbitrary single plane wave cannot “polar-ize” the vacuum, as in this case P EH and M EH identi-cally vanish. Equations (35) and (36) indicate that thepresence of an electromagnetic field in the vacuum altersthe vacuum’s refractive index. The situation is even morecomplicated because of the vectorial nature of the back-ground electromagnetic field which polarizes the vacuumand introduces a privileged direction in it. As a result,the vacuum’s refractive index is altered in a way thatdepends, in general, on the mutual polarizations of theprobe electromagnetic field and of the background field:the polarized vacuum behaves as a birefringent medium.For example, in the case of an arbitrary constant electro-magnetic field ( E , B ), the refractive indices n EH, / ofa wave propagating along the direction n and polarizedalong one of the two independent directions u = E / | E | and u = B / | B | , with E = E − ( n · E ) E + n × B and B = B − ( n · B ) B − n × E are given by (Dittrich andGies, 2000) n EH, = 1 + 4 α π ( n × E ) + ( n × B ) − n · ( E × B ) F , (37) n EH, = 1 + 7 α π ( n × E ) + ( n × B ) − n · ( E × B ) F , (38) respectively.The birefringence of the polarized vacuum is exploitedin Heinzl et al. , 2006 to show that if a linearly-polarizedprobe x-ray beam (wavelength λ p ) propagates along astrong optical standing wave, then it emerges from theinteraction elliptically polarized with ellipticity ǫ givenby ǫ = 2 α κ l ,R λ p I I cr , (39)where κ ∼ l ,R is theRayleigh length of the intense laser beam. If this beamis generated by a laser like ELI ( I ∼ W/cm ), val-ues of the ellipticities of the order of 10 − are predictedat λ p = 0 . − at a wavelength of 0 . et al. , 2011). InFerrando et al. , 2007 a phase-shift has been found the-oretically to be induced by vacuum-polarization effectswhen two laser beams cross in the vacuum, which is pre-dicted to be measurable at laser intensities available atELI or at HiPER.When an electromagnetic wave with wavelength λ im-pinges upon a material body, the features of the scatteredradiation depend on the so-called diffraction parameter D = l ⊥ /λd (Jackson, 1975). Here, l ⊥ is the spatial di-mension of the body perpendicular to the propagationdirection of the incident wave and d is the distance ofthe screen, where the radiation is detected, from the in-teraction region. The near region, D ≫
1, is known asthe “refractive-index limit”, because the effects of thepresence of the body can be described as if the wavepropagates through a medium with a given refractiveindex. However, if D . et al. , 2006 within the context of light-lightinteraction in vacuum (see also Di Piazza et al. , 2007a).Tight focusing required to reach high intensities usuallyrenders the interaction region so small that diffractioneffects may become substantial at typical experimentalconditions. In some cases diffractive effects reduce by anorder of magnitude the values of the ellipticity calculatedvia the refractive-index approach and also induce a ro-tation of the main axis of the polarization ellipse withrespect to the initial polarization direction of the probefield (Di Piazza et al. , 2006). Tight focusing of the strongpolarizing beam requires quite a detailed mathematicaldescription employing a realistic focused Gaussian beam,while a simpler description was employed for the usuallyweakly-focused probe beam. This prevented the appli-cability of the results in the so-called far region where D ≪ et al. , 2010b, where it was also pointed7out that by considering the diffraction of a probe beamby two separated beams instead of that by a single stand-ing wave, an increase in the ellipticity and in the rotationof polarization angle by a factor 1.5 is expected.A different method based on the phase-contrast Fourierimaging technique has been suggested in Homma et al. ,2011 to detect vacuum birefringence. This technique pro-vides a very sensitive tool to measure the absolute phaseshift of a probe beam when it crosses an intense laserfield. Numerical simulations demonstrate the feasibilityof measuring vacuum birefringence also by employing anoptical probe field and a 100-PW strong laser beam.Photon “acceleration” in vacuum due to vacuum polar-ization has been studied in Mendon¸ca et al. , 2006. Thiseffect corresponds to a shift of the photon frequency whenit passes through a strong electromagnetic wave. If k µp is the four-momentum of a probe photon with energy ω p when it enters a region where a strong laser beamis present, then, due to vacuum-polarization effects, ω p becomes sensitive to the gradient of the intensity of thestrong beam. As a result, a frequency up-shift (down-shift) is predicted at the rear (front) of the strong beam.According to Eqs. (37) and (38) the phase velocityof light in vacuum is smaller than unity. This circum-stance has been exploited in Marklund et al. , 2005, whereCherenkov radiation by ultrarelativistic particles mov-ing with constant velocity in a photon gas has been pre-dicted, if the speed of the particle exceeds the phase ve-locity of light. Finally, in Zimmer et al. , 2012 an inducedelectric dipole moment of the neutron has been proposedas a signature of the polarization of the QED vacuum.
3. Low-energy vacuum-polarization effects in a plasma
Equations (37) and (38) indicate that vacuum-polarization effects elicited by a plane wave with intensity I would alter the vacuum refractive index by an amountof the order of ( α/ π )( I /I cr ). It was first realized inDi Piazza et al. , 2007b that this aspect can be in prin-ciple significantly improved in a plasma. For the sakeof simplicity the case of a cold plasma was consideredand the vacuum-polarization effects were implemented inthe inhomogeneous Maxwell’s equations as an additional“vacuum four-current” (Di Piazza et al. , 2007b). Now,unlike in the vacuum, the field invariant F ( x ) for a singlemonochromatic circularly polarized plane wave does notvanish in a plasma. Thus, vacuum-polarization effects ina plasma already arise in the presence of a single travel-ing plane wave. In Di Piazza et al. , 2007b the vacuum-corrected refractive index n of a two-fluid electron-ionplasma (ion mass, density and charge number given by m i , n i and Z , respectively) in the presence of a circularlypolarized plane wave has been found as n = r n + 2 α π I I cr (1 − n ) , (40) where n = vuut − πe n i mω p ξ + Z p ( m i /m ) + Z ξ ! , (41)is the refractive index of the plasma without vacuum-polarization effects (Mulser and Bauer, 2010). Equation(40) already indicates the possibility of enhancing the ef-fects of vacuum polarization by working at laser frequen-cies ω such that n ≪
1, i.e., close to the effective plasmacritical frequency. This region of parameters is in generalcomplex to investigate, due to the arising of different in-stabilities. However, the idealized situation investigatedin Di Piazza et al. , 2007b shows, at least in principle, thepossibility of enhancing the vacuum-polarization effectsby an order of magnitude at a given intensity I withrespect, for example, to the results in Di Piazza et al. ,2006. The effects of the presence of an additional strongconstant magnetic field have been analyzed in Lundin et al. , 2007. The general theory presented in this pa-per covers different waves propagating in a plasma asAlfv´en modes, whistler modes, and large-amplitude lasermodes. We also mention the recent paper Bu and Ji,2010, in which the photon acceleration process has beeninvestigated in a cold plasma and the reference Brodin et al. , 2007, where vacuum-induced photon splitting ina plasma is studied. Finally, we mention the possibilityof testing nonlinear vacuum QED effects in waveguides.In Brodin et al. , 2001 the generation of new modes inwaveguides due to vacuum-polarization effects has beenpredicted. More recently signatures of nonlinear QEDeffects in the transmitted power along a waveguide havebeen analyzed in Ferraro, 2010. B. High-energy vacuum-polarization effects
Generally speaking the treatment of vacuum-polarization effects for an incoming photon with energy ω in the presence of a background electromagnetic fieldwith a typical angular frequency ω b cannot be performedin an effective Lagrangian approach if ωω b /m &
1: theincoming photon “sees” the nonlocality of its interactionwith the background electromagnetic field through itslocal “transformation” into an e + - e − pair (see Fig. 12).The technical difficulty in treating vacuum-polarizationeffects at high energies arises from the fact that theinteraction between the virtual e + - e − pair and thebackground field has to be accounted for exactly. Thishas been accomplished for the background field of anucleus with charge number Z such that Zα ∼ et al. , 2003 and Milstein and Schumacher, 1994).As we have recalled in Sec. III.B, the Dirac equa-tion in a background plane wave described by the four-8vector potential A µ ( φ ) can be solved exactly and ana-lytically. Accordingly, the exact electron (Volkov) prop-agator G ( x, y | A ) in the same background field has alsobeen determined (see, e.g., Ritus, 1985). The so-called“operator technique”, developed in Baier et al. , 1976a,b,turns out to be very convenient for investigating vacuum-polarization effects at high energies (the operator tech-nique for a constant background field was developedin Baier et al. , 1975a,b; and Schwinger, 1951). Inthis technique a generic electron state Ψ( x ) in a plane-wave field and the propagator G ( x, y | A ) are intendedas the configuration representation of an abstract state | Ψ i and of an operator G ( A ) such that Ψ( x ) = h x | Ψ i and G ( x, y | A ) = h x | G ( A ) | y i . In particular, since thepropagator G ( x, y | A ) is the solution of the equation { γ µ [ i∂ µ − eA µ ( φ )] − m } G ( x, y | A ) = δ ( x − y ), then theabstract operator G ( A ) is simply G ( A ) = 1 γ µ [ P µ − eA µ ( φ )] − m , (42)with P µ being the four-momentum operator. Evaluationvia the operator technique of the matrix element corre-sponding to a generic vacuum-polarization process is thencarried out by manipulating abstract operators, which iseasier than by working with the corresponding quantitiesin configuration space.In Di Piazza et al. , 2007c the operator technique hasbeen employed to calculate the rate of photon splittingin a strong laser field for an incoming photon with four-momentum k µ . This was the first investigation of a QEDprocess involving three Volkov propagators. The calcu-lated rate is valid for an arbitrary plane-wave field, pro-vided that radiative corrections can be neglected, i.e., at α κ / ≪
1, with κ = ( k − /m )( E /F cr ), in the most un-favorable regime ξ , κ ≫ ξ and κ . In Di Piazza et al. , 2007c it turnedout to be more convenient to perform a parametric studyof the photon-splitting rate by varying the two parame-ters ξ and η = ω k − /m (note that κ = η ξ ). Byemploying the Furry theorem (Berestetskii et al. , 1982),it is shown that photon splitting in a laser field only oc-curs with absorption of an odd number of laser photons.In particular, if the strong field is circularly polarized andif it counterpropagates with respect to the incoming pho-ton, conservation of the total angular momentum alongthe propagation direction of the beams implies that for η ≪ et al. , 2008a a physical scenario has beenadvanced in which nonperturbative vacuum-polarizationeffects can be in principle observed (here “nonperturba-tive” means “high-order” in the quantum nonlinearityparameter, see below). In this scenario a high-energy proton collides head-on with a strong laser field. Thequantum interaction of the Coulomb field of the protonwith the laser field allows for a merging of laser photonsinto a single high-energy photon. The use of a proton,instead of an electron, for example, is required in order tosuppress the background process of multiphoton Thom-son/Compton scattering, where again many photons ofthe laser can be directly absorbed by the proton and con-verted into a single high-energy photon (see Sec. V.A).In fact, the kinematics of the two processes, vacuum-mediated laser photon merging and multiphoton Thom-son/Compton scattering is the same, except that in thetreatment in Di Piazza et al. , 2008a only an even num-ber of laser photons can merge in the vacuum-mediatedprocess. The probability of ℓ -photon Thomson/Comptonscattering of a particle with charge Q and mass M de-pends on the parameter ξ ,c = | Q | E /M ω and scalesas ξ ℓ ,c at ξ ,c ≪
1. Thus, the use of a “heavy” particlelike a proton (mass m p = 1 . × m = 938 MeV) isessential to suppress this background process. Note thatin order to have ξ ,p = | e | E /m p ω ≈ W/cm is required. It is found that for an optical backgroundfield such that ξ ≫
1, the amplitude of the (2 ℓ )-photonmerging process depends only on the nonlinear quantumparameter χ (2 ℓ )0 ,p = E F cr ℓ (1 + v p ) ω m − cos ϑ v p cos ϑ , (43)where v p is the proton velocity and ϑ is the angle betweenthe direction of the emitted photon and the propagationdirection of the plane wave. By colliding a proton beam,of energy available at the Large Hadron Collider (LHC)of the order of 7 TeV (LHC, 2011) with an optical laserbeam of intensity of 3 × W/cm , it is demonstratedthat the vacuum-mediated merging of two laser photonsand the analogous two-photon Thomson/Compton scat-tering have comparable rates, implying that the inclu-sive signal should be twice the one expected withoutvacuum-polarization effects. By accounting for the de-tails of the proton beams available at LHC and of thelaser system PFS (see Sec. II.A), about 670 two-photonmerging events and about 5 four-photon merging eventsare expected per hour. In the discussed setup most of thephotons are emitted almost in the direction of the protonvelocity ( ϑ ≈ π ) in which χ (2)0 ,p ∼
1. It is also indicated inDi Piazza et al. , 2008a that the use of the perturbativeexpression of the laser-photon merging rate at leadingorder in χ (2)0 ,p ≪ et al. , 2008b, involving, for example, XFEL or single in-tense XUV pulses. Finally, the process of Delbr¨uck scat-tering in a combined Coulomb and laser field has beenstudied in Di Piazza and Milstein, 2008. Here an incom-ing photon is scattered by the Coulomb field of a nucleus9 (cid:1) (cid:1) (cid:1) x a) b) c) FIG. 16 Feynman diagrams corresponding to processes (i)(part a)), (ii) (part b)) and (iii) (part c)), respectively. Thethick continuous lines in parts a) and b) indicate Volkovpositive- and negative-energy states. The crossed vertex inpart b) stands for the Coulomb electromagnetic field. Thediagram in part c) is related to the vacuum current j µ vac ( x ),that one has to determine in order to calculate the e + - e − pairyield (Dittrich and Gies, 2000). The double line indicates theelectron propagator calculated in the Furry picture includingexactly the background standing wave. and by a strong laser field. While the presence of thelaser field is taken into account exactly in the calcula-tions, only leading-order effects in the nuclear parameter Zα are accounted for. Analogously to Di Piazza et al. ,2008a, it is found that high-order nonlinear correctionsin the parameter κ to the cross section of the processalready become important at κ ≈ .
2. For example,these corrections amount to about 50% at κ = 0 . VIII. ELECTRON-POSITRON PAIR PRODUCTION
One of the most important predictions of QED hasbeen the possibility of transforming light into matter(Dirac, 1928). If two photons with four-momenta k µ and k µ collide at an angle such that the parameter η =( k k ) /m exceeds two, the creation of an e + - e − pair be-comes kinematically allowed (Breit-Wheeler e + - e − pairproduction (Breit and Wheeler, 1934)). Shortly after therealization of the laser in 1960, theoreticians started tostudy possibilities for the creation of e + - e − pairs fromvacuum by very strong laser fields (Nikishov and Ritus,1964a; Reiss, 1962; Yakovlev, 1965). Because of con-straints from energy-momentum conservation, a singleplane-wave laser field cannot create pairs from vacuum,no matter how intense it is. For a single plane wave, infact, all the photons propagate along the same directionand the parameter η vanishes identically for any pair ofphotons in the plane wave. Thus, an additional source ofenergy is therefore required to trigger the process of pairproduction in a plane wave. There are essentially threedifferent possibilities (see also Fig. 16):(i) pair production by a high-energy photon propa-gating in a strong laser field (multiphoton Breit-Wheeler pair production);(ii) pair production by a Coulomb field in the presenceof a strong laser field;(iii) pair production by two counterpropagating stronglaser beams forming a standing light wave. These processes share the common feature to possessdifferent interaction regimes which are mainly character-ized by the value of the parameter ξ . When ξ ≪ R of the form R e + - e − ∼ mξ ℓ m , where ℓ m is the minimuminteger number which kinematically allows the process.For the process (i) it is ℓ m ( k k ) > m , with k µ and k µ being the four-momentum of the laser photon and theincoming photon, respectively. For the process (ii) it is ℓ m ω ⋆ > m , where ω ⋆ = ω u c, − is the laser angular fre-quency in the rest-frame of the charge which produces theCoulomb field and which has four-velocity u µc . Finally,for the process (iii) it is ℓ m ω > m . Due to the specificdependence of the pair production rate on ξ , this regimeof pair production is called multiphoton regime. In con-trast, when ξ ≫ ξ ≫ κ introduced in Sec. III.Band it scales as ∼ m κ / exp( − / κ ) if κ ≪ ∼ m κ / if κ ≫ Z . In the first case the pair production rate dependson the parameter χ , already introduced in Sec. III.B,and the recoil due to the pair creation on the electronhas to be taken into account (see also Sec. VIII.A). Inthe second case the motion of the nucleus is usually as-sumed not to be altered by the pair creation process andthe nucleus itself is described as a background Coulombfield (see also Sec. VIII.B). The pair-production ratedepends on the parameter χ ,n = u n, − ( E /F cr ), with u µn being the four-velocity of the nucleus, and on thenuclear parameter Zα . Specifically, the pair-productionrate scales as m ( Zα ) exp( − √ /χ ,n ) if χ ,n ≪ m ( Zα ) χ ,n ln χ ,n if χ ,n ≫ et al. , 2006;Yakovlev, 1965). Finally, for process (iii), the rate R e + - e − has been derived mainly by approximating the stand-ing wave as an oscillating electric field (see also Sec.VIII.C). It is found that R e + - e − depends on the ratioΥ = E /F cr , with E being the amplitude of the stand-ing wave in the (fixed) laboratory frame, and that itscales as m Υ exp( − π/ Υ ) if Υ ≪ m Υ ifΥ ≫ E (Schwinger, 1951).The physical meaning of the three parameters κ , χ ,n and Υ can be qualitatively understood in the follow-0ing way. For process (i) the dressing of the electronand positron mass (see Sec. III.B) modifies the thresh-old of e + - e − pair production at ξ ≫ ℓ m ( k k ) & m ξ . Now, analogously to multiphotonThomson and Compton scattering (see Sec. V.A), thetypical number of laser photons absorbed in pair pro-duction via photon-laser collision is of the order of ξ (Nikishov and Ritus, 1964a) and the threshold conditionbecomes κ &
1. Concerning the process (ii), the ap-pearance of the parameter χ ,n in the quasistatic limit ξ ≫ u n, − E is the amplitude of the laser field in the rest-frameof the nucleus and that a constant and uniform electricfield with strength of the order of F cr = m / | e | suppliesa e + - e − pair with its rest energy 2 m along the typicallength scale of QED λ C = 1 /m (see also Sec. III.B).This last observation also demonstrates the presence ofthe parameter Υ for process (iii). The typical exponen-tial scaling of the pair production rate for ξ ≫
1, andat κ ≪ χ ,n ≪ ≪ et al. , 2012; Reiss, 2008)).In all processes discussed above, the laser field is al-ways participating directly in the pair creation step andfundamental properties of the quantum vacuum underextreme high-field conditions are probed. However, aswill be seen shortly, lying at the border of experimentalfeasibility, the expected pair yields are generally rathersmall. It is worth mentioning here that lasers can alsobe applied for abundant generation of e + - e − pairs (Chen et al. , 2009, 2010). When a solid target is irradiated by anintense laser pulse, a plasma is formed and electrons areaccelerated to high energies. They may emit radiationby bremsstrahlung which efficiently converts into e + - e − pairs through the Bethe-Heitler process. The laser fieldplays an indirect role in the pair production here by serv-ing solely as a particle accelerator. The prolific amountof antimatter generated this way may lead to interestingapplications in various fields of science (M¨uller and Kei-tel, 2009). Abundant production of e + - e − pairs and ofhigh-energy photons in the collision of a multipetawattlaser beam and a solid target has been recently investi-gated in Nakamura et al. , 2011 and Ridgers et al. , 2012.In particular, in Nakamura et al. , 2011 it has been shownthat almost all the laser pulse energy is converted afterthe collision into a well collimated high-power gamma-ray flash. Whereas, the numerical simulations in Ridgers et al. , 2012 indicate that about 35 % of the energy ofa 10 PW laser pulse after the laser-target interaction isconverted into a gamma-ray burst and that simultane-ously a pure e + - e − plasma is produced with a maximumpositron density of 10 m − . A. Pair production in photon-laser and electron-lasercollisions
Among the pair-production processes mentionedabove, only laser-induced pair production for ξ < et al. , 1999; Burke et al. , 1997) (seeReiss, 1971 for a corresponding theoretical proposal).The experiment relied on collisions of the 46.6 GeV elec-tron beam from SLAC’s linear accelerator with a coun-terpropagating intense laser pulse of photon energy of ω = 2 . . × W/cm ( ξ ≈ . et al. , 2010). Good agreement with the experimental re-sults has been obtained. Moreover, it was shown that theSLAC study observed the onset of nonperturbative paircreation dynamics, which adds even further significanceto this benchmarking experiment (see also Reiss, 2009).A formal treatment of the process has also been given inIlderton, 2011, where special emphasis is put on effectsstemming from the finite duration of the laser pulse.Figure 17 shows a survey of various combinations ofincoming electron energies and optical laser intensitieswhich give rise to an observable pair yield. It covers therange from the perturbative few-photon regime ( ξ ≈ . I ≈ W/cm ) to the highly nonperturbative do-main ( ξ ≈
10 at I ≈ W/cm ), where the con-tributions from thousands of photon absorption chan-nels need to be included. We note that few-GeV elec-tron beams can be produced today using compact laser-plasma accelerators (Leemans et al. , 2006) (see also Sec.XII). Future pair creation studies may therefore rely onall-optical setups, where a laser-generated electron beamcollides with a counterpropagating laser pulse. Anotherall-optical setup for pair creation by a seed electron ex-posed to two counterpropagating laser pulses was putforward in Bell and Kirk, 2008, which will be discussedin Sec. IX.Pair creation studies could also be conducted as a non-standard application of the 17.5 GeV electron beam atthe upcoming European XFEL beamline at DESY (Euro-pean XFEL, 2011), which will normally serve to generatecoherent x-ray pulses. However, in combination with a1 à à à à à àà à à à à àà Rate ~ s - tunnelingtransition totunnelingnonperturbativemultiphotonSLACexperimentperturbativefew - photon @ W (cid:144) cm D E l ec t r o n E n er gy @ G e V D FIG. 17 (Color online) Transition from the perturbative tothe fully nonperturbative regimes of e + - e − pair creation inelectron-laser collisions. The laser photon energy is 2 . et al. , 2010. table-top 10-TW optical laser system, it would also bevery suitable to probe the various regimes of pair pro-duction. In particular, the production channel (ii) couldbe investigated by a suitable choice of beam parameters(Hu et al. , 2010).Other aspects of pair creation by a high-energy photonand a strong laser field have been investigated in recentyears. In Heinzl et al. , 2010a, process (i) was consideredin the case where the laser pulse has finite duration. Itwas found that the finite pulse duration is imprinted onthe spectra of created particles. Pair production by ahigh-energy photon and an ultrashort laser pulse was alsoconsidered in Tuchin, 2010. Quantum interference effectscan arise in photon-induced pair creation in a two-modelaser field of commensurate frequencies (Narozhny andFofanov, 2000).In addition, the fundamental process (i) may allow forapplications as a novel tool in ultrashort pulse spectrom-etry. A corresponding detection scheme for the char-acterization of short gamma-ray pulses of GeV photonsdown to the zeptosecond scale, called Streaking at HighEnergies with Electrons and Positrons (SHEEP), hasbeen proposed in Ipp et al. , 2011. The basic conceptof SHEEP is based on e + - e − pair production in vacuumby a photon of the test pulse, assisted by an auxiliarycounter-propagating intense laser pulse. In contrast toconventional streak imaging, two particles with oppositecharges, electron and positron, are created in the samerelative phase within the third streaking pulse that co-propagates with the test pulse. By measuring simulta-neously the energy and momentum of the electrons andthe positrons originating from different positions withinthe test pulse, its length and, in principle, even its shapecan be reconstructed. The time resolution of SHEEP fordifferent classes of tests, streaking and strong pulses canrange from femtosecond to zeptosecond duration. B. Pair production in nucleus-laser collisions
While in electron-laser collisions the contribution ofreaction (ii) to pair production is in general small, it be-comes accessible to experimental observation when theprojectile electrons are replaced by heavier particles suchas protons or other nuclei. The two-step production pro-cess via multiphoton Compton scattering will then bestrongly suppressed by the large projectile mass. The re-cent commissioning of the LHC at CERN has stimulatedsubstantial activities on pair production in combinedlaser and nuclear Coulomb fields, which may be viewed asa generalization of the well-known Bethe-Heitler processto strong fields (multiphoton Bethe-Heitler pair produc-tion). The large Lorentz factors γ n of the ultrarelativisticnuclear beams lead to efficient enhancement of the laserparameters in the projectile rest-frame.Indeed, when a proton beam with Lorentz factor γ p ≈ I ≈ W/cm ,the Lorentz-boosted laser field strength approaches thecritical value F cr . This circumstance motivated the firstcalculations of nonperturbative pair production in colli-sions of a relativistic nucleus with a superintense near-optical laser beam (M¨uller et al. , 2003a,b). Smaller pro-jectile Lorentz factors may be sufficient, when ultrastrongXFEL pulses are employed (Avetissian et al. , 2003). Thecalculations were based on an S-matrix treatment and as-sumed laser fields of circular polarization. Later on, alsothe case of linear field polarization was studied (Kami´nski et al. , 2006; Krajewska et al. , 2006; M¨uller et al. , 2004;Sieczka et al. , 2006). This case is rendered more involveddue to the appearance of generalized Bessel functions,which are of very high order when ξ ≫
1. The underly-ing S-matrix element is generally of the form S p + ,σ + ,p − ,σ − = − ie Z d x Ψ † p − ,σ − ( x ) V n ( r )Ψ − p + , − σ + ( x ) . (44)It describes the transition of an electron from thenegative-energy Volkov state Ψ − p + , − σ + ( x ) to a positive-energy Volkov state Ψ † p − ,σ − ( x ), which is mediated by theCoulomb potential V n ( r ) = Z | e | /r of the projectile nu-cleus. An alternative approach to the problem based onthe polarization operator in a plane electromagnetic wavehas been developed in Milstein et al. , 2006. It allowsto obtain total production rates analytically. Both ap-proaches rely on the strong-field approximation and in-clude the laser field exactly to all orders, whereas thenuclear field is treated at leading order in Zα .Since the high-intensity Bethe-Heitler process has notbeen observed in experiment yet, in recent years physi-cists have proposed scenarios which may allow to realizethe various interaction regimes of the process by present-day technology. Few-photon Bethe-Heitler pair produc-tion in the perturbative domain could be realized in colli-2sions of the LHC proton beam with an XUV pulse of an-gular frequency ω X ≈
100 eV and of moderate intensity I X ∼ W/cm (M¨uller, 2009). Corresponding radi-ation sources of table-top dimension are available nowa-days in many laboratories. They are based on HHG fromatomic gas jets or solid surfaces (see Sec. IV.B). The rate R e + - e − of pair creation by two-photon absorption close tothe energetic threshold (i.e., ω ⋆X = u n, − ω X & m , for theangular frequency ω ⋆X of the XUV pulse in the rest-frameof the nucleus) is given by (Milstein et al. , 2006) R e + - e − = 14 − j ( Zα ) ξ ω ⋆X (cid:18) ω ⋆X m − (cid:19) j +2 , (45)with j = 0 for linear polarization and j = 2 for circularpolarization.In the quasistatic regime of the process sizable pairyields require superintense laser fields from a petawattsource in conjunction with an LHC proton beam (M¨uller et al. , 2003b; Sieczka et al. , 2006). Such experimentswill become feasible when petawatt laser pulses are madeavailable by high-power devices of table-top size, ratherthan by immobile large-scale facilities as they exist atpresent. A method to enable tunneling pair productionwith more compact multiterawatt laser systems has beenproposed in Di Piazza et al. , 2009b, 2010b. It relies onthe application of an additional weak XUV field, whichis superimposed on a powerful optical laser wave. In thistwo-color setup, the energy threshold for pair creationcan be overcome by the absorption of one photon fromthe high-frequency field and several additional photonsfrom the low-frequency field. As a result, by choosing theXUV frequency ω ⋆X such that the parameter δ = (2 m − ω ⋆X ) /m fulfills the conditions 0 < δ ≪
1, the tunnelingbarrier can be substantially lowered and even controlled.The pair production rate in the quasistatic regime for0 < δ ≪ δ and χ ,n , and on the classical nonlinearity parameter ξ X of the XUV field. For a circularly polarized strong laserfield it becomes (Di Piazza et al. , 2009b) R e + - e − = 164 √ π m ( Zα ) ξ X χ ,n p ζ exp (cid:18) −
23 1 ζ (cid:19) (46)for ζ = χ ,n / δ / ≪
1, to be compared with the usualscaling ∼ ( − √ /χ ,n ) in the absence of the XUV field.A related process is laser-assisted Bethe-Heitler paircreation, where the high-frequency photon energy sat-isfies ω ⋆X > m . A pronounced channeling of the e + - e − pair due to the forces exerted by the laser field after theircreation was found (L¨otstedt et al. , 2008, 2009). Multi-photon Bethe-Heitler pair creation in a two-color laserwave was investigated in Roshchupkin, 2001.Analytical formulas for positron energy spectra andangular distributions in the tunneling regime of the pro-cess were obtained in Kuchiev and Robinson, 2007. Forpair production at ξ ∼ m ( Zα ) exp( − . /χ ,n ) was obtained inM¨uller et al. , 2009b, which closely resembles the tunnel-ing exponential behaviour m ( Zα ) exp( − √ /χ ,n ).While in Eq. (44) the influence of the projectile is de-scribed by an external Coulomb field, the projectile canalso be treated as a quantum particle which allows tostudy nuclear recoil effects (Krajewska and Kami´nski,2010, 2011; M¨uller and M¨uller, 2009). Besides, in laser-nucleus collisions, bound-free pair creation can occurwhere the electron is created in a deeply bound atomicstate of the nucleus. The process was studied first forcircular laser polarization (Matveev et al. , 2005; M¨uller et al. , 2003c) and later on also for linear polarization(Deneke and M¨uller, 2008), including contributions fromthe various atomic subshells. C. Pair production in a standing laser wave
Purely light-induced pair production can occur whentwo noncopropagating laser waves are superimposed.The simplest field configuration consists of two coun-terpropagating laser pulses of equal frequency and in-tensity. The resulting field is a standing wave which isinhomogeneous both in space and time and a theoreti-cal treatment of the process is very challenging. In or-der to render the problem tractable and since the pro-duction process mainly occurs where the electric fieldcomponent of the background field is stronger than themagnetic one (Dittrich and Gies, 2000), in the standardapproach the resulting standing light wave is approxi-mated by a purely electric field oscillating in time. Thisapproximation is expected to be justified for a strong( I > W/cm ), optical laser field where the typ-ical spatial scale of the field variation λ ∼ µ m ismuch larger than the pair formation length m/ | e | E = λ C F cr /E ≈ . × − µ m / p I [10 W/cm ] (Ritus,1985). Note the analogy between the formation length m/ | e | E for pair production and the tunneling length l tun ∼ I p / | e | E in atomic ionization (see Sec. IV.B),with I p being the ionization potential energy.Pair production in an oscillating electric field is ageneralization of the Schwinger mechanism (Schwinger,1951) to time-dependent fields and it has been consideredby many theoreticians, starting from the seminal worksBrezin and Itzykson, 1970 and Popov, 1971, 1972 (for acomprehensive list of references until 2005, see Salamin et al. , 2006).While in the laser-electron and laser-nucleus collisionsof the previous subsections the Doppler boost of the laserparameters due to a highly relativistic Lorentz factorcould be exploited, in laser-laser collisions this is not pos-sible so that high field strengths E or high frequencies3 E /F cr FIG. 18 (Color) Number of pairs produced by the x-ray as-sisted Schwinger mechanism for two different values of the x-ray angular frequency ω X , indicated as ω in the figure, (blueand black solid lines) and the ratio of these catalyzed pairs tothose produced by the standard Schwinger mechanism (blueand black dashed lines), both as functions of the optical fieldstrength in units of F cr . Adapted from Dunne et al. , 2009. ω are required in the laboratory frame. Theoreticiansare therefore aiming to find ways for enhancing the pairproduction probability in order to render the process ob-servable in the foreseeable future.A first possibility to facilitate the observability of paircreation in a standing optical laser wave is to superim-pose an x-ray photon (or any other high-frequency com-ponent) onto the high-field region (Dunne et al. , 2009;Monin and Voloshin, 2010; Sch¨utzhold et al. , 2008). Inthis way, the Schwinger mechanism is catalyzed so thatthe usual exponential suppression ∼ exp( − π/ Υ ) is sig-nificantly lowered. For example, in the limit when thex-ray energy approaches the threshold value 2 m , the pairproduction rate R e + - e − becomes R e + - e − ∼ m exp (cid:18) − π − (cid:19) (47)assuming that the x-ray propagation direction is perpen-dicular to the electric field vector of the strong opticalfield. An overview of the pair production enhancementeffect due to the x-ray assistance is shown in Fig. 18.Another proposal to enhance the pair yield is the ap-plication of multiple colliding laser pulses instead of onlytwo (Bulanov et al. , 2010a). It has been demonstratedthat the threshold laser energy necessary to produce asingle pair, decreases when the number of colliding pulsesis increased. The results are summarized in Table I. Pairproduction exceeds the threshold when eight laser pulses,with a total energy of 10 kJ, are simultaneously focusedon one spot. Doubling (tripling) the number of pulsesleads to an enhancement by two (six) orders of magni-tude. The threshold energy drops from 40 kJ for twopulses to 5.1 kJ for 24 pulses, clearly indicating that themultiple-pulse geometry is strongly favorable. Besides, itwas noticed that the pre-exponential volume factor in the TABLE I Number N e + - e − of e + - e − pairs produced by differ-ent numbers n of laser pulses, with a total energy W of 10 kJ.The threshold value total energy W th needed to produce one e + - e − pair is shown in the third column. The precise collisiongeometry and the pulse parameters can be found in Bulanov et al. , 2010a. Adapted from Bulanov et al. , 2010a. n N e + - e − at W = 10 kJ W th [kJ]2 < − < −
208 4.0 1016 1 . ×
824 4 . × pair creation probability can be very large and partiallycompensate for the exponential suppression (Narozhny et al. , 2006).Fine details of pair production in a time-dependent os-cillating electric field are being studied nowadays becausethey might serve as characteristic signatures to discrim-inate the process of interest from potentially strongerbackground processes. For example, it was found thatthe momentum spectrum of the created particles is highlysensitive to a subcycle structure of the field (Hebenstreit et al. , 2009) and that in the presence of an alternating-sign time-dependent electric field, coherent interferenceeffects are observed in the Schwinger mechanism (Akker-mans and Dunne, 2012). The observation in Hebenstreit et al. , 2009 found an elegant mathematical explanationvia the Stokes phenomenon (Dumlu and Dunne, 2010).Further effects stemming from the precise shape of theexternal field were analyzed in Dumlu, 2010 and Dumluand Dunne, 2011. Also, the oscillating dynamics of the e + - e − plasma created by a uniform electric field, in-cluding backreaction effects, was investigated (Apostol,2011; Benedetti et al. , 2011; Han et al. , 2010) (see alsoBialynicki-Birula and Rudnicki, 2011 and Kim and Schu-bert, 2011).In addition to pair creation in superstrong laser pulsesof low frequency, the process is also extensively discussedin connection with the upcoming XFEL facilities (see,e.g., Alkofer et al. , 2001 and Ringwald, 2001). Here thequestion arises as to what extent the spatial field depen-dence may influence the pair creation process, both interms of total probabilities and particle momentum dis-tributions. According to Noether’s theorem, pair produc-tion in a time-dependent oscillating electric field occurswith conservation of the total momentum, as well as ofthe total spin. The problem therefore reduces effectivelyto a two-level system since the field couples negative andpositive-energy electron states of same momentum andspin only. The production process exhibits resonancewhen the energy gap is an integer multiple of the laser fre-quency, leading to a characteristic Rabi flopping between4 ω /m M a x i m a l p r o du c t i o np r o b a b ili t y FIG. 19 (Color online) Probability spectrum of e + - e − pairproduction in two counterpropagating laser pulses, with thelaser magnetic field included (black triangles) and neglected(red crosses). In the first case, the labeling ( ℓ r - ℓ l ) signifies thenumber of absorbed photons from the right-left propagatingwave; in the second case, the peak labels denote the total pho-ton number ( ℓ ). A vanishing initial momentum (i.e., positronmomentum) and ξ = 1 have been assumed. Adapted fromRuf et al. , 2009. the negative and positive-energy Dirac continua (Popov,1971). Due to the electron dressing by the oscillatingfield, the resonant laser frequencies are determined bythe equation ℓω = 2 h ε i , where h ε i is the time-averagedelectron energy in the time-dependent oscillating electricfield. Accordingly, when the particle momentum is var-ied, several resonances occur corresponding to differentphoton numbers ℓ . This gives rise to a characteristic ringstructure in the momentum distribution (Mocken et al. ,2010).Modifications of these well-established properties ofthe pair creation process, when the spatial field depen-dence and, thus, the laser magnetic-field component areaccounted for, have been revealed in Ruf et al. , 2009.Utilizing an advanced computer code for solving the cor-responding Dirac equation numerically, it was shown thatthe positions of the resonances are shifted, several newresonances occur, and the resonance lines are split due tothe influence of the spatial field dependence (see Fig. 19).The basic reason for these effects is that, in contrast toa uniform oscillating electric field, the photons in thecounterpropagating laser pulses carry momentum alongthe beam axis. Therefore not only the total number ℓ of absorbed laser photons matters, but also how manyof them have originated from the laser pulse travellingto the right and left, respectively. For example, for themultiphoton order ℓ = 5 two different resonance frequen-cies exist now, corresponding to ℓ r = 3, ℓ l = 2 on theone hand, and to ℓ r = 4, ℓ l = 1, on the other. Dueto the photon momentum, the former two-level schemeis also broken into a V -type three-level scheme. Thiscauses a splitting of the resonance lines, in analogy withthe Autler-Townes effect known from atomic physics. For another numerical approach to space-time depen-dent problems in quantum field theory, we refer to thereview Cheng et al. , 2010. Moreover, Schwinger pair pro-duction in a space-time dependent electric field pulse hasbeen treated very recently within the Wigner formalism(Hebenstreit et al. , 2011). Here, a self-bunching effectof the created particles in phase space, due to the spa-tiotemporal structure of the pulse, was found.Finally, we mention that, unlike a plane-wave field, aspatially focused laser beam is capable to produce e + - e − pairs from vacuum and this process has been investigatedin Narozhny et al. , 2004 for different field polarizations.Spontaneous pair production may, in principle, also occurin a nuclear field for charge numbers Z exceeding a crit-ical value Z c , which depends on the nuclear model. Forexample, Z c = 173 for a uniformly charged sphere withradius 1 . × − cm (Berestetskii et al. , 1982). Seethe reviews Baur et al. , 2007 and Zeldovich and Popov,1972 for more detailed information also on e + - e − pairproduction in heavy ion collisions. D. Spin effects and other fundamental aspects oflaser-induced pair creation
A particularly interesting aspect of tunneling pair cre-ation is the electron and positron spin-polarization. Ingeneral, pronounced spin signatures in a field-inducedprocess may be expected when the background fieldstrength approaches the critical value F cr (Kirsebom et al. , 2001; Walser et al. , 2002). This indeed coincideswith the condition for a sizable yield of tunneling pairproduction. Studies of spin effects in pair production bya high-energy photon and a strong laser field were per-formed in Tsai, 1993 and Ivanov et al. , 2005, based onconsiderations on helicity amplitudes and on the spin-polarization vector, respectively. Characteristic differ-ences between fermionic and bosonic particles have beenrevealed with respect to pair creation in an oscillatingelectric field (Popov, 1972) and in recent studies of theKlein paradox (Cheng et al. , 2009b; Krekora et al. , 2004;Wagner et al. , 2010a). In the latter case it was shownthat the existence of a fermionic (bosonic) particle in theinitial state leads to suppression (enhancement) of thepair production probability due to the different quan-tum statistics. The enhancement in the bosonic case maybe even exponential due to an avalanche process (Wag-ner et al. , 2010b). Concerning tunneling pair creation incombined laser and nuclear Coulomb fields, it has beenshown that the internal spin-polarization vector is pro-portional in magnitude to χ ,n and, to leading order,directed along the transverse momentum component ofthe electron (Di Piazza et al. , 2010c). A helicity analy-sis of pair production in laser-proton collisions revealedthat: 1) right-handed leptons are emitted in the labora-tory frame under slightly smaller angles with respect to5the proton beam than left-handed ones; 2) the rate ofpair creation of spin-1 / e + - e − correlations (Fedorov et al. ,2006; Krajewska and Kami´nski, 2008; Krekora et al. ,2005), multiple pair creation (Cheng et al. , 2009a), ques-tions of locality (Cheng et al. , 2008) and vacuum decaytimes (Labun and Rafelski, 2009), and consistency re-strictions on the maximum laser field strength to guar-antee the validity of the external-field approximation(Gavrilov and Gitman, 2008). IX. QED CASCADES
As it was discussed in the previous section, the E-144experiment at SLAC is the only one, so far, where laser-driven multiphoton e + - e − pair production has been ob-served. Considering that about 100 positrons have beendetected in 22000 shots, each comprising the collision ofabout 10 electrons with the laser beam, the process re-sults to be rather inefficient. One could attribute thisto the relatively low intensity I of the laser system of1 . × W/cm ( ξ ≈ . ω = 2 . ε of the electron beam (about 46 . χ wasabout unity ( χ ≈ . et al. , 2010a) has pointed out in general that in thementioned setup, i.e., an electron beam colliding witha strong laser pulse, RR effects prevent the developmentof a cascade or avalanche process with an efficient, pro-lific production of e + - e − pairs even at much larger laserintensities such that ξ ≫
1. By an avalanche or cas-cade process we mean here a process in which the in-coming electrons emit high-energy photons in the laserfield, which can interact with the field itself generating e + - e − pairs, which, in turn, emit photons again and soon (of course a cascade process may also be initiated bya photon beam rather than by an electron beam). Theabove result has been obtained by numerically integrat-ing the kinetic equations, which describe the evolution ofthe electron, the positron and the photon distributions ina plane-wave background field from a given initial elec-tron distribution, and by accounting for the two basicprocesses that couple these distributions, i.e., multipho-ton Compton scattering and multiphoton Breit-Wheelerpair production. The physical reason why an avalancheprocess cannot develop in a single plane-wave field canbe understood in the following way. In the ultrarelativis-tic case ξ ≫
1, the above-mentioned basic processes ina plane-wave field are essentially controlled by the pa-rameters χ and κ , respectively (see also Secs. III.B, V.A and VIII). Now, at the j th step in which an elec-tron/positron emits a photon or a photon transforms intoan e + - e − pair, the initial quantity p ( j )0 , − or k ( j )0 , − is con-served and it is distributed over the two final particles (anelectron/positron and a photon in multiphoton Comp-ton scattering and an e + - e − pair in multiphoton Breit-Wheeler pair production). Thus, both resulting particlesat each step will have a value of their own parameter χ ′ ( j )0 = ( p ′ ( j )0 , − /m )( E /F cr ) or κ ′ ( j )0 = ( k ′ ( j )0 , − /m )( E /F cr )smaller than that of the incoming particle. Moreover,due to the special symmetry of the plane-wave field, thequantities p ( j )0 , − or k ( j )0 , − are also rigorously conserved be-tween two steps (see also Sec. III.A). Then, the avalancheends when the parameters χ ( k )0 ,i and κ ( k )0 ,i at a certain step k are smaller than unity for i ∈ [1 , . . . , N k ], with N k be-ing the number of particles at that step.The question arises as to whether other field configura-tions exist, where an avalanche process can be efficientlytriggered (see the review Aharonian and Plyasheshnikov,2003 for the development of QED cascades in matter,photon gas and magnetic field). A positive answer tothis question has first been given in Bell and Kirk, 2008:even the presence of a single electron initially at restin a standing wave generated by two identical counter-propagating circularly polarized laser fields can prime anavalanche process already at field intensities of the orderof 10 W/cm . We note that in the presence of a singleplane wave the same process would require an intensityof the order of I cr , because for an electron initially at rest χ = E /F cr . From this point of view, the authors of Belland Kirk, 2008 explain qualitatively the advantage of em-ploying two counterpropagating laser beams by means ofan analogy taken from accelerator physics: a collision be-tween two particles in their center-of-momenta is muchmore efficient than if one of the particles is initially atrest, because much more of the initial energy can betransferred, for example, to create new particles. Theauthors approximated the standing wave by a rotatingelectric field (see Sec. VIII.C). In such a field and foran ultrarelativistic electron the controlling parameter is˜ χ = ( p ⊥ /m )( E /F cr ), where p ⊥ is the component of theelectron momentum perpendicular to the electric field.By estimating p ⊥ ∼ mξ (see also Eq. (2)), one ob-tains ˜ χ ≈ I [10 W/cm ] /ω [eV] (here ω , E and I are the standing-wave’s angular frequency, electric fieldamplitude and intensity). The investigation in Bell andKirk, 2008 is based on the analysis of the trajectory ofthe electron in the rotating electric field including RReffects via the LL equation. Since the momentum of theelectron oscillates around a value of the order of mξ , theelectron emits high-energy photons efficiently that can inturn trigger the cascade (see Fig. 20). The possibility ofdescribing the evolution of the electron via its classicaltrajectory, can be justified as follows. When an elec-tron interacts with a background electromagnetic field6 FIG. 20 (Color online) The number of e + - e − pairs ( N ± ) andthe number of photons ( N γ ) created by an initial single elec-tron in a rotating electric field as a function of the field inten-sity. The other plotted quantities are described in Bell andKirk, 2008. From Bell and Kirk, 2008. like that of a laser, quantum effects are essentially of twokinds (Baier et al. , 1998): the first one is associated withthe quantum nature of the electron motion and the sec-ond one with the recoil undergone by the electron when itemits a photon. For an ultrarelativistic electron it can beshown that, while the first kind of quantum effects is neg-ligible, the second kind is large and has to be taken intoaccount (Baier et al. , 1998). Thus, the basic assumptionis that, since the background laser field is strong, the elec-tron is promptly accelerated to ultrarelativistic energies,the motion between two photon emissions is essentiallyclassical and, if necessary, only the emissions have to betreated quantum mechanically by including the photonrecoil. On the other hand, photons are assumed to prop-agate in the field along straight lines.The model employed in Bell and Kirk, 2008 was im-proved in Kirk et al. , 2009 by considering colliding pulsedfields with finite time duration and a realistic represen-tation for the synchrotron spectrum emitted by a rela-tivistic electron. The results in Bell and Kirk, 2008 wereessentially confirmed and numerical simulations with lin-early polarized beams have shown a general insensitiv-ity of the cascade development to the polarization ofthe beams. Another interesting finding in Kirk et al. ,2009 is that the electrons in the standing wave tend tomigrate to regions where the electric field vanishes andthen they do not contribute to the pair-production pro-cess anymore. In both papers Bell and Kirk, 2008 andKirk et al. , 2009 the emission of radiation by the electronwas treated classically, i.e., the electron was supposedto loose energy and momentum continuously althoughin Kirk et al. , 2009 the damping force-term in the LL equation was evaluated by employing the total emittedpower calculated quantum mechanically. The stochasticnature of the emission of a photon has been taken intoaccount in Duclous et al. , 2011. Analogously, the energyof the emitted photon is chosen randomly following thesynchrotron spectral distribution and the momentum ofthe photon is always chosen to be parallel to that of theemitting electron. By contrast, in the pair productionprocess by a photon, since the photon is not deflectedby the laser field, it is assumed that after it has propa-gated one wavelength in the field, it decays into an e + - e − pair. The main result in Duclous et al. , 2011 is that atrelatively low intensities of the order of 10 W/cm thepair production rate is increased if the quantum natureof the photon emission is taken into account. The reasonis that, due to the stochastic nature of the emission pro-cess, the electron can propagate for an unusually largedistance before emitting. In this way it may gain an un-usually large amount of energy and consequently emit ahigh-energy photon, that can be more easily convertedinto an e + - e − pair. Moreover, the discontinuous natureof the (curvature of the) electron trajectory is shown toslow down the tendency of the electrons to migrate toregions where the electric field vanishes.The intensity threshold of the avalanche process in arotating electric field has also been investigated in Fedo-tov et al. , 2010. Denoting by t acc the time an electronneeds to reach an energy corresponding to χ = 1 start-ing from rest in the given field, by t e ( t γ ) the electron(photon) lifetime under photon emission ( e + - e − pair pro-duction) and by t esc the time after which the electron es-capes from the laser field, the authors give the followingconditions for the occurrence of the avalanche process: t acc . t e , t γ ≪ t esc . Estimates based on the classicalelectron trajectory without including RR effects, lead tothe simple condition E & αF cr for the avalanche to beprimed in an optical field. The above estimate corre-sponds to an intensity of about 2 . × W/cm , i.e.,one order of magnitude larger than what was found inBell and Kirk, 2008. However, the main result of Fedo-tov et al. , 2010 concerns the limitation, brought abouton the maximal laser intensity that can be producedin the discussed field configuration by the starting ofthe avalanche process. In fact, the energy to acceler-ate the electrons and the positrons participating in thecascade has to come from the background electromag-netic field. By assuming an exponential increase of thenumber of electrons and positrons, it is found that al-ready at laser intensities of the order of 10 W/cm thecreated electrons and positrons have an energy which ex-ceeds the initial total energy of the laser beams. Thishints at the fact that at such intensities the collidinglaser beams are completely depleted due to the avalancheprocess. The results obtained from qualitative estimatesin Fedotov et al. , 2010 have been scrutinized in Elkina et al. , 2011 by means of more realistic numerical meth-7ods based on kinetic or cascade equations. In general,if f ∓ ( r , p , t ) ( f γ ( r , k , t )) is the electron/positron (pho-ton) distribution function (upper and lower sign for elec-tron and positron, respectively) in the phase-space ( r , p )(( r , k )) and ε = p m + p ( ω = | k | ), their evolutionin the presence of a given classical electromagnetic field( E ( r , t ) , B ( r , t )) is described by the kinetic equations (cid:20) ∂∂t + p ε · ∂∂ r ± F L ( r , p , t ) · ∂∂ p (cid:21) f ∓ ( r , p , t )= Z d k w rad ( r , p + k → k , t ) f ∓ ( r , p + k , t ) − f ∓ ( r , p , t ) Z d k w rad ( r , p → k , t )+ Z d k w cre ( r , k → p , t ) f γ ( r , k , t ) , (48) (cid:18) ∂∂t + k ω · ∂∂ r (cid:19) f γ ( r , p , t )= Z d p w rad ( r , p → k , t )[ f + ( r , p , t ) + f − ( r , p , t )] − f γ ( r , k , t ) Z d p w cre ( r , k → p , t ) , (49)where F L ( r , p , t ) = e [ E ( r , t )+( p /ε ) × B ( r , t )] and where w rad ( r , p → k , t ) ( w cre ( r , k → p , t )) is the local prob-ability per unit time and unit momentum that an elec-tron/positron (photon) with momentum p ( k ) emits (cre-ates) at the space-time point ( t, r ) a photon with momen-tum k (an e + - e − pair with the electron/positron havinga momentum p ). It is worth pointing out here a con-nection between the development of a QED cascade andthe quantum description of RR. In fact, as has been dis-cussed in Sec. VI, from a quantum point of view, RR cor-responds to the multiple recoils experienced by the elec-tron in the incoherent emission of many photons. Thus,in the kinetic approach RR is described by those terms inEqs. (48) and (49), which do not involve e + - e − pair pro-duction. In fact, in Elkina et al. , 2011 it has been shownin the ultrarelativistic case that the equation of motionfor the average momentum of the electron distribution,as derived from Eq. (48), coincides with the LL equationin the classical regime χ ≪ χ ≪ et al. , 2011 the background field is approx-imated as a uniform, rotating electric field E ( t ) and f ± ( r , p , t ) → f ± ( p , t ) ( f γ ( r , k , t ) → f γ ( k , t )). Analo-gously to Duclous et al. , 2011, it is assumed that themomentum of the photon emitted in multiphoton Comp-ton scattering is parallel to that of the emitting electron(positron); in the same way, the momenta of the elec-tron and positron created in multiphoton Breit-Wheelerpair production are assumed to be parallel to that of thecreating photon. The evolution of the electron, positronand photon distributions has been investigated by nu-merically integrating the resulting cascade equations via a Monte Carlo method. Whereas, the instants of radia-tion and pair production have been randomly generated.In particular, the exponential increase of the number of e + - e − pairs and the qualitative estimate, for example, ofthe typical energy of the electron at the moment of thephoton emission carried out in Fedotov et al. , 2010 havebeen confirmed (apart from discrepancies within one or-der of magnitude).In both papers Elkina et al. , 2011 and Fedotov et al. ,2010 only the case of a rotating electric field was con-sidered. In Bulanov et al. , 2010b it is pointed out thatthe limitation on the maximal laser intensity reachablebefore the cascade is triggered, strongly depends on thepolarization of the laser beams which create the stand-ing wave. The paradigmatic cases of a rotating electricfield and of an oscillating electric field are compared.The estimates presented in Bulanov et al. , 2010b for thecase of a rotating electric field essentially confirm thatthe avalanche starts at laser intensities of the order of10 W/cm . The main physical reason why the cascadeprocess in a circularly polarized standing wave starts atsuch an intensity is that in a rotating electric field theelectron emits photons with typical energies of the orderof 0 . ω γ , i.e., proportional to the cube of the Lorentzfactor of the emitting electron γ (Bulanov et al. , 2010b).Whereas, in an oscillating electric field the typical emit-ted energy scales as γ , such that in order to radiate ahard photon with a given energy, a much more energeticelectron is needed. Hence, the authors of Bulanov et al. ,2010b conclude that in an oscillating electric field RR andquantum effects do not play a fundamental role at laserintensities smaller than I cr and that avalanche processesdo not constitute a limitation. It is crucial however, forthe conclusion in Bulanov et al. , 2010b that the collisionof the laser beams occurs in vacuum, i.e., the seed elec-trons and positrons which would trigger the cascade aresupposed to be created in the collision itself.The question of the occurrence of the avalanche for twocolliding linearly polarized pulses has also been addressedin Nerush et al. , 2011b, where a detailed description ofthe system under investigation has been provided. Infact, previous models had assumed the background elec-tromagnetic field as given, neglecting in this way the fieldgenerated by the electrons and positrons. The approachfollowed in Nerush et al. , 2011b exploits the existence oftwo energy scales for the photons: one is that of the ex-ternal laser field and of the plasma fields which is muchsmaller than m , and the other is that of the photonsproduced by the high-energy electrons which is, by con-trast, much larger than m . The evolution of the low-energy photons is described by means of Maxwell’s equa-tions which are solved with a PIC code, i.e., the photonsare treated as a classical electromagnetic field. Whereas,the production of hard photons as well as the creationof e + - e − pairs is described as a stochastic process em-ploying a Monte Carlo method. Unless low-energy ones,8 x/λ y / λ y / λ y / λ FIG. 21 (Color online) Snapshot of the normalized electrondensity ρ e (part a)), of the normalized photon density ρ γ (partb)), and of the normalized laser intensity ρ l (part c)) 25 . I = 3 × W/cm and the common wave-length is λ = 0 . µ m. Adapted from Nerush et al. , 2011b. hard photons are treated as particles and their evolutionis described via a distribution function. It is surpris-ing that by considering a single seed electron initiallyat rest at a node of the magnetic field of a linearly po-larized standing wave, an avalanche process is observedin the numerical simulation already for a laser intensity I = 3 × W/cm and a laser wavelength λ = 0 . µ m(see Fig. 21). The figure clearly shows the formation ofan overdense e + - e − plasma in the central region | x | . λ .In the same numerical simulations it is found that afterabout 20 laser periods almost half of the initial energy ofthe laser field has been transferred to the plasma. Dis-agreement with the predictions in Bulanov et al. , 2010bis stated to be due to the formation of the avalanche inregions between the nodes of the electric and magneticfield, where the simplified analysis of the electron motioncarried out in Bulanov et al. , 2010b is not valid. Fur-ther analytical insight into the formation of the cascadehas been reported in Nerush et al. , 2011a by analyzingapproximate solutions of the cascade equations in thepresence of a rotating electric field. X. MUON-ANTIMUON AND PION-ANTIPION PAIRPRODUCTION
The production of e + - e − pairs in strong laser fieldshas been discussed in Sec. VIII. In view of the ongo-ing technical progress the question arises as to whetheralso heavier particles such as muon-antimuon ( µ + - µ − )or pion-antipion ( π + - π − ) pairs can be produced with theemerging near-future laser sources. The production of µ + - µ − pairs from vacuum in the tunneling regime ap-pears rather hopeless, though, since the required fieldneeds to be close to F cr ,µ = ̺ µ F cr = 5 . × V / cm,with the ratio ̺ µ = m µ /m ≈
207 between the muonmass m µ and the electron mass m . Even by boostingthe effective laser fields with the Lorentz factors ( ∼ )of the most energetic electron beams available (Bamber et al. , 1999), the value of F cr ,µ seems out of reach. Thetunneling production of π + - π − pairs is even more diffi-cult as ̺ π = m π /m ≈ µ + - µ − and π + - π − production can occur in microscopic collision processesin laser-generated or laser-driven plasmas, as well as byfew-photon absorption from a high-frequency laser wave. A. Muon-antimuon and pion-antipion pair production inlaser-driven collisions in plasmas
Energetic particle collisions in a plasma can in prin-ciple drive µ + - µ − and π + - π − production. The plasmamay consist either of electrons and ions or of electronsand positrons. Both kinds of plasmas can be produced byintense laser beams interacting with a solid target. Withrespect to e + - e − plasmas, this has been predicted byLiang et al. , 1998. As has been mentioned in Sec. VIII,abundant amounts of e + - e − pairs have been recently pro-duced in this manner at LLNL with pair densities ofthe order of 10 cm − (Chen et al. , 2009, 2010) andmuch higher densities of the order of 10 cm − have beenalso predicted (Shen and Meyer-ter-Vehn, 2001). Theo-reticians have therefore started to investigate the prop-erties and time evolution of relativistic e + - e − plasmas(Aksenov et al. , 2007; Hu and M¨uller, 2011; Kuznetsova et al. , 2008; Kuznetsova and Rafelski, 2012; Mustafa andK¨ampfer, 2009; Thoma, 2009a,b). In particular, it hasbeen shown (Kuznetsova et al. , 2008; Thoma, 2009a,b)that in an e + - e − plasma of 10 MeV temperature, µ + - µ − pairs, π + - π − pairs as well as neutral π can be createdin e + - e − collisions. The required energy stems from thehigh-energy tails of the thermal distributions.Also cold e + - e − plasmas of high density can be gener-ated nowadays due to dedicated positron accumulationand trapping techniques (Cassidy et al. , 2005). Whensuch a nonrelativistic low-energy plasma interacts with asuperintense laser field, µ + - µ − pair production can oc-cur as well (M¨uller et al. , 2006, 2008b). In this case,the plasma particles acquire the necessary energy by9strong coupling to the external field which drives the elec-trons and positrons into violent collisions. The minimumlaser peak intensity to ignite the reaction e + e − → µ + µ − amounts to about 7 × W/cm at a typical opticallaser photon energy of ω = 1 eV, corresponding to ξ , min = ̺ µ ≈ R e + e − → µ + µ − of the pro-cess in the presence of a linearly polarized field reads R e + e − → µ + µ − ≈ π α m ξ s − ξ , min ξ N + N − V , (50)with the number N ± of electrons/positrons and the in-teraction volume V , which is determined by the laser fo-cal spot size. Equation (50) may be made intuitivelymeaningful by introducing the invariant cross section σ e + e − → µ + µ − of µ + - µ − production in an e + - e − collisionin vacuum (Peskin and Schroeder, 1995): σ e + e − → µ + µ − = 4 π α ε ∗ r − m µ ε ∗ m µ ε ∗ ! , (51)where the upper index ∗ indicates quantities in thecenter-of-mass system of the colliding electron andpositron, as, e.g., the common electron and positron en-ergy ε ∗ . Now, by exploiting the fact that the quan-tity R e + - e − /V is a Lorentz invariant and that in thepresent physical scenario ε ∗ can be estimated as mξ ,Eq. (50) implies the usual relation R ∗ e + e − → µ + µ − /V ∗ ∼ σ e + e − → µ + µ − n ∗ + n ∗− between the number of events per unitvolume and per unit time R ∗ e + e − → µ + µ − /V ∗ and the crosssection σ e + e − → µ + µ − ; n ∗± = N ± /V ∗ denote here the par-ticle densities. The process e + e − → µ + µ − in the pres-ence of an intense laser wave has also been considered inNedoreshta et al. , 2009.In laser-produced electron-ion plasmas resulting fromintense laser-solid interactions, µ + - µ − and π + - π − pairscan be generated by the cascade mechanism via energeticbremsstrahlung, like in the case of e + - e − pair produc-tion mentioned in Sec. VIII. Assuming a laser-generatedfew-GeV electron beam, several hundreds to thousandsof µ + - µ − pairs arise from bremsstrahlung conversion ina high- Z target material (Titov et al. , 2009). The pro-duction of π + - π − pairs by laser-accelerated protons wasconsidered in Bychenkov et al. , 2001, where a thresholdlaser intensity of 10 W/cm for the process to occurwas determined. B. Muon-antimuon and pion-antipion pair production inhigh-energy XFEL-nucleus collisions
In this subsection another mechanism of µ + - µ − and π + - π − pair creation by laser fields will be pursued, whichis based on the collision of an x-ray laser beam with anultrarelativistic nuclear beam. This setup is similar tothe one of Sec. VIII.B. In the case of µ + - µ − pair creation, by considering anx-ray photon energy of ω = 12 keV and a nuclear rela-tivistic Lorentz factor of γ n = 7000, the photon energy inthe rest-frame of the nucleus amounts to ω ⋆ ≈ γ n ω =168 MeV. The energy gap of 2 m µ for µ + - µ − pair pro-duction can thus be overcome by two-photon absorption(M¨uller et al. , 2008a, 2009a). Note that because of pro-nounced recoil effects, the Lorentz factor which would berequired for two-photon µ + - µ − production by a projec-tile electron is much larger: γ & , corresponding to acurrently unavailable electron energy in the TeV range.At first sight, e + - e − and µ + - µ − pair production incombined laser and Coulomb fields seem to be very sim-ilar processes since the electron and muon only differ bytheir mass (and lifetime). In this picture, the corre-sponding production probabilities would coincide whenthe laser field strength and frequency are scaled in accor-dance with the mass ratio ̺ µ , i.e., P µ + - µ − ( E ,µ , ω ,µ ) = P e + - e − ( E , ω ) for E ,µ = ̺ µ E and ω ,µ = ̺ µ ω . Thissimple scaling argument does not apply, however, as thelarge muon mass is connected with a correspondinglysmall Compton wavelength λ C,µ = λ C /̺ µ ≈ .
86 fm(1 fm = 10 − cm), which is smaller than the radius ofmost nuclei. As a result, while the nucleus can be ap-proximately taken as pointlike in e + - e − pair production( λ C ≈
386 fm), its finite extension must be taken intoaccount in µ + - µ − pair production. Pronounced nuclearsize effects have also been found for µ + - µ − production inrelativistic heavy-ion collisions (Baur et al. , 2007).Muon pair creation in XFEL-nucleus collisions can becalculated via the amplitude in Eq. (44), with the nu-clear potential V n ( r ) arising from an extended nucleus.It leads to the appearance of a nuclear form factor F ( q )which depends on the recoil momentum q . For example, F ( q ) = exp( − q a /
6) for a Gaussian nuclear charge dis-tribution of root-mean-square radius a . Since the typi-cal recoil momentum is q ∼ m µ , the form factor leads tosubstantial suppression of the process. The fully differen-tial production rate dR µ + - µ − = dR (el) µ + - µ − + dR (inel) µ + - µ − maybe split into an elastic and an inelastic part, dependingon whether the nucleus remains in its ground state orgets excited during the process. They read dR (el) µ + - µ − = dR (0) µ + - µ − Z F ( q ) and dR (inel) µ + - µ − ≈ dR (0) µ + - µ − Z [1 − F ( q )],respectively, with dR (0) µ + - µ − being the production rate fora pointlike proton.Figure 22 shows total µ + - µ − production rates R ⋆µ + - µ − in the rest-frame of the nucleus for several nuclei col-liding with an intense XFEL beam. For an extendednucleus, the elastic rate increases with its charge but de-creases with its size. This interplay leads to the emer-gence of maximum elastic rates for medium-heavy ions.Figure 22 also implies that the total rate R µ + - µ − = R (el) µ + - µ − + R (inel) µ + - µ − in the laboratory frame saturates athigh Z values since R (inel) µ + - µ − increases with nuclear charge.0 Z R ⋆ µ + - µ − [ s ] FIG. 22 Total rates for µ + - µ − pair creation by two-photonabsorption from an intense XFEL beam ( ω = 12 keV, I =2 . × W/cm ) colliding with various ultrarelativistic nu-clei ( γ n = 7000). The triangles show elastic rates, whereasthe squares indicate total (“elastic + inelastic”) rates. Thenumerical data are connected by fit curves. The dotted lineholds for a pointlike nucleus. The production rates are calcu-lated in the rest-frame of the nucleus. Adapted from M¨uller et al. , 2008a. For highly-charged nuclei the main contribution stemsfrom the inelastic channel where the protons inside thenucleus act incoherently ( R (inel) µ + - µ − ∝ Z ). This impliesthat despite the high charges, high-order Coulomb cor-rections in Zα are of minor importance. In the collision,also e + - e − pairs are produced by single-photon absorp-tion in the nuclear field. However, this rather strongbackground process does not deplete the x-ray beam.In XFEL-proton collisions, π + - π − pairs can be gener-ated as well. A corresponding calculation has been re-ported in Dadi and M¨uller, 2011, which includes boththe electromagnetic and hadronic pion-proton interac-tions. The latter was described approximately by a phe-nomenological Yukawa potential. It was shown that,despite the larger pion mass, π + - π − pair productionby two-photon absorption from the XFEL field largelydominates over the corresponding process of µ + - µ − pairproduction in the Doppler-boosted frequency range of ω ⋆ ≈ µ + - µ − pairs are predominantly producedindirectly via two-photon π + - π − production and subse-quent pion decay, π + → µ + + ν µ and π − → µ − + ¯ ν µ .In relativistic laser-nucleus collisions, µ + - µ − or π + - π − pairs can also be produced indirectly within a two-stepprocess (Kuchiev, 2007). First, upon the collision, an e + - e − pair is created via tunneling pair production. Af-terwards, the pair, being still subject to the electromag-netic forces exerted by the laser field, is driven by thefield into an energetic e + - e − collision. If the collision en-ergy is large enough, the reaction e + e − → µ + µ − may be triggered. This two-step mechanism thus represents acombination of the processes considered in Sec. VIII.Band X.A. Besides, it may be considered as a generaliza-tion of the well-established analogy between strong-fieldionization and pair production (see Sec. VIII) to includealso the recollision step. XI. NUCLEAR PHYSICS
Influencing atomic nuclei with optical laser radiationis, in general, a difficult task because of the large nuclearlevel spacing ∆ E of the order of 1 keV-1 MeV, whichexceeds typical laser photon energies ω ∼ r n ∼ | e | E r n ∼ ∆ E can only be satisfied for atlaser-field amplitudes at least close to F cr . Direct laser-nucleus interactions have therefore mostly been dismissedin the past.On the other hand, laser-induced secondary reactionsin nuclei have been explored especially in the late 1990s.Via laser-heated clusters and laser-produced plasmas,various nuclear reactions have been ignited, such as fis-sion, fusion and neutron production. In all these cases,the interaction of the laser field with the target firstproduces secondary particles such as photo-electrons orbremsstrahlung photons which, in a subsequent step,trigger the nuclear reaction. For a recent review on thissubject we refer to Ledingham and Galster, 2010 andLedingham et al. , 2003.In recent years, however, the interest in direct laser-nucleus coupling has been revived by the ongoing techno-logical progress towards laser sources of increasingly highintensities as well as frequencies. Indeed, when suitablenuclear isotopes are considered, intense high-frequencyfields or superstrong near-optical fields may be capableof affecting the nuclear structure and dynamics. A. Direct laser-nucleus interaction
1. Resonant laser-nucleus coupling
There are several low-lying nuclear transitions in thekeV range, and even a few in the eV range. Examples ofthe latter are
Th (∆
E ≈ . U (∆
E ≈ et al. , 2007). These isotopes can be excitedby the 5th harmonic of a Ti:Sa laser ( ω = 1 .
55 eV)and by pulses envisaged at the ELI attosecond source(ELI, 2011), respectively. Even higher frequencies canbe attained by laser pulse reflection from relativistic fly-ing mirrors of electrons extracted from an underdenseplasma (Bulanov et al. , 1994) or possibly also from anoverdense plasma (Habs et al. , 2008). Otherwise keV-energy photons are generated by XFELs, which, as we1have seen in Sec. II.B, are presently emerging as large-scale facilities, e.g., at SLAC (LCLS, 2011) and DESY(European XFEL, 2011), and which could be employedwith focusing (Mimura et al. , 2010) and reflection devices(Shvyd’ko et al. , 2011). In addition, also XFEL facili-ties of table-top size (Gr¨uner et al. , 2007; Kneip et al. ,2010) and even fully coherent XFEL sources are envis-aged such as the future XFEL Oscillator (XFELO) (Kim et al. , 2008) or the Seeded XFEL (SXFEL) (Feldhaus et al. , 1997; LCLS II, 2011). Brilliant gamma-ray beamswith spectra peaked between 20 KeV and 150 KeV havebeen recently produced from resonant betatron motion ofelectrons in a plasma wake (Cipiccia et al. , 2011). A newmaterial research center, the Matter-Radiation Interac-tions in Extremes (MaRIE) (MaRIE, 2011) is planned,allowing for both fully coherent XFEL light with photonenergy up to 100 keV and accelerated ion beams. Thephotonuclear pillar of ELI to be set up near Bucharest(Romania) is planned to provide a compact XFEL alongwith an ion accelerator aiming for energies of 4-5 GeV(ELI, 2011). In addition, coherent gamma-rays reach-ing few MeV energies via electron laser interaction areenvisaged at this facility (ELI, 2011; Habs et al. , 2009).With these sources of coherent high-frequency pulses,driving electric dipole (E1) transitions in nuclei is becom-ing feasible (B¨urvenich et al. , 2006a). Table II displaysa list of nuclei with suitable E1 transitions. Along withan appropriate moderate nucleus acceleration, resonancemay be induced due to the Doppler shift via the factor(1 + v n ) γ n in the counterpropagating setup, with v n and γ n being the nucleus velocity and its Lorentz factor, re-spectively (B¨urvenich et al. , 2006a). For example, with Ra and an XFEL frequency of 12 . v n ) γ n = 4 would be sufficient. In general such mod-erate pre-accelerations of the nuclei would be of greatassistance since they increase the number of possible nu-clear transitions for the limited number of available lightfrequencies. Note that the electric field strength of thelaser pulse transforms analogously in the rest-frame ofthe nucleus, such that the applied laser field in the labo-ratory frame may correspondingly be weaker for a coun-terpropagating setup.For optimal coherence properties of the envisaged high-frequency facilities, subsequent pulse applications wereshown to yield notable excitation of nuclei (B¨urvenich et al. , 2006a). In addition, many low-energy electricquadrupole (E2) and magnetic dipole (M1) transitionsare available. Here it is interesting to note that certain E2or M1 transitions can indeed be competitive in strengthwith E1 transitions (P´alffy, 2008; P´alffy et al. , 2008).While the majority of transitions is available for highfrequencies in the MeV domain (requiring substantial nu-cleus accelerations), Fig. 23 displays also numerous suit-able nuclear transitions below 12.4 keV along with theexcitation efficiencies from realistic laser pulses. Whileindeed experimental challenges are high, resonant direct TABLE II The transition energy ∆ E , the dipole moment µ ,the life-time τ g ( τ e ) of the ground (excited) state of few rel-evant nuclear systems and E1 transitions (Aas et al. , 1999;NNDC, 2011). Adapted from B¨urvenich et al. , 2006a.Nucleus Transition ∆ E [keV] µ [ e fm] τ g τ e [ps] Sm 3 / − → / + > . a
47 h < Ta 9 / − → / + a stable 6 × Ac 3 / + → / − a Ra 3 / − → / + Th 3 / − → / + b b231 Th 5 / − → / +
186 0.017 25.52 h 1030 a Estimated via the Einstein A coefficient from τ e and ∆ E b Not listed in the National Nuclear Data Center (NNDC)(NNDC, 2011) Z N γ FIG. 23 (Color online) Number N γ of signal photons per nu-cleus per laser pulse for several isotopes with first excitedstates below 12.4 keV (green squares) and above 12.4 keV(red crosses). The results are plotted versus the atomicnumber Z . The considered European XFEL has a pulseduration of 100 fs and an average brilliance of 1.6 × photons/(s mrad mm et al. , 2008. interactions of laser radiation with nuclei is expected topave the way for nuclear quantum optics. Especially con-trol in exciting and deexciting certain long-living nuclearstates would have dramatic implications for nuclear iso-mer research (Aprahamian and Sun, 2005; P´alffy et al. ,2007; Walker and Dracoulis, 1999). As an obvious ap-plication this would be of relevance for nuclear batter-ies (Aprahamian and Sun, 2005; Walker and Dracoulis,1999), i.e., for controlled pumping and release of energystored in long-lived nuclear states. In atomic physics, theSTimulated Raman Adiabatic Passage (STIRAP) tech-nique has proven to be highly efficient in controlling pop-ulations robustly with high precision (Bergmann et al. ,1998). On the basis of currently envisaged acceleratorsand coherent high-frequency laser facilities, it has beenrecently shown that such an efficient coherent populationtransfer will also be feasible in nuclei (Liao et al. , 2011).2Most recently, a nuclear control scheme with optimizedpulse shapes and sequences has been developed in Wong et al. , 2011.Serious challenges are certainly imposed by the nuclearlinewidths that may either be too narrow to allow for suf-ficient interaction with the applied laser pulses or inhibitexcitations and coherences due to large spontaneous de-cay. Decades of research in atomic physics allowing nowfor shaping atomic spectra via quantum interference (Ev-ers and Keitel, 2002; Kiffner et al. , 2010; Postavaru et al. ,2011) raise hopes that such obstacles may be overcomein the near future as well.Direct photoexcitation of giant dipole resonanceswith few-MeV photons via laser-electron interaction wasshown to be feasible (Weidenm¨uller, 2011) based on en-visaged experimental facilities such as ELI. Finally, carehas to be taken to compare the laser-induced nuclearchannels with competing nuclear processes via, for ex-ample, bound electron transitions or electron captures inthe atomic shells (P´alffy, 2010; P´alffy et al. , 2007).
2. Nonresonant laser-nucleus interactions
Already decades ago a lively debate was started onwhether nuclear β -decay may be significantly affectedby the presence of a strong laser pulse or not (Akhme-dov, 1983; Becker et al. , 1984; Nikishov and Ritus, 1964b;Reiss, 1983) and a conclusive experimental answer to thisissue is still to come. Most recently the notion of affect-ing nuclear α -decay with strong laser pulses has beendiscussed showing that moderate changes of such nuclearreactions with the strongest envisaged laser pulses areindeed feasible (Casta˜neda Cort´es et al. , 2012, 2011).When the laser intensity is high enough ( I > W/cm ) low-frequency laser fields are able to influencethe nuclear structure without necessarily inducing nu-clear reactions. In such ultrastrong fields, low-lying nu-clear levels get modified by the dynamic (AC-) Stark shift(B¨urvenich et al. , 2006b). These AC-Stark shifts are ofthe same order as in typical atomic quantum optical sys-tems relative to the respective transition frequencies. Ateven higher, supercritical intensities ( I > W/cm )the laser field induces modifications to the proton root-mean-square radius and to the proton density distribu-tion (B¨urvenich et al. , 2006b). B. Nuclear signatures in laser-driven atomic and moleculardynamics
Muonic atoms represent traditional tools for nuclearspectroscopy by employing atomic physics techniques.Due to the large muon mass compared to that of theelectron, m µ ≈ m , and because of its correspond-ingly small Bohr radius a B,µ = λ C,µ /α ≈
255 fm, the muonic wave-function has a large overlap with the nu-cleus. Precise measurements of x-ray transitions be-tween stationary muonic states are therefore sensitive tonuclear-structure features such as finite size, deforma-tion, surface thickness, or polarization.When a muonic atom is subjected to a strong laserfield, the muon becomes a dynamic nuclear probe whichis periodically driven across the nucleus by the field. Thiscan be inferred, for example, from the high-harmonic ra-diation emitted by such systems (Shahbaz et al. , 2010,2007). Figure 24 compares the HHG spectra from muonichydrogen versus muonic deuterium subject to a verystrong XUV laser field. Such fields are envisaged at theELI attosecond source (see Fig. 1). Due to the differ-ent masses M n of the respective nuclei ( M n = m p for ahydrogen nucleus and M n ≈ m p + m n for a deuterium nu-cleus by neglecting the binding energy, with m p/n beingthe proton/neutron mass), muonic hydrogen gives rise toa significantly larger harmonic cut-off energy. The reasoncan be understood by inspection of the ponderomotiveenergy U p = e E ω M r = e E ω (cid:18) m µ + 1 M n (cid:19) , (52)which depends in the present case on the reduced mass M r = m µ M n / ( m µ + M n ) of the muon-nucleus system.The reduced mass of muonic hydrogen ( ≈
93 MeV) issmaller than that of muonic deuterium ( ≈
98 MeV) andthis implies a larger ponderomotive energy and an en-larged plateau extension. The influence of the nuclearmass can also be explained by the separated motions ofthe atomic binding partners. The muon and the nucleusare driven by the laser field into opposite directions alongthe laser’s polarization axis. Upon recombination theirkinetic energies sum up as indicated on the right-handside of Eq. (52). Within this picture, the larger cut-offenergy for muonic hydrogen results from the fact that,due to its smaller mass, the proton is more strongly ac-celerated by the laser field than the deuteron.Due to the large muon mass, very high harmonic cut-off energies can be achieved via muonic atoms with chargenumber Z in the nonrelativistic regime of interaction.Since the harmonic-conversion efficiency as well as thedensity of muonic atom samples are rather low, it is im-portant to maximize the radiative signal strength. A siz-able HHG signal requires efficient ionization on the onehand, and efficient recombination on the other. The for-mer is guaranteed if the laser’s electric field amplitude E lies just below the border of over-barrier ionization, E . M r Q eff ( Zα ) . (53)Here, Q eff = | e | ( Z/M n + 1 /m µ ) M r represents an effec-tive charge (Reiss, 1979; Shahbaz et al. , 2010). Efficientrecollision is guaranteed if the magnetic drift along the3 -18 -15 -12 -9 H a r m on i c S i gn a l [ a r b . u . ] FIG. 24 (Color) HHG spectra emitted from muonic hydrogen(black) and muonic deuterium (red) in a laser field of intensity I ≈ W/cm and photon energy ω ≈
60 eV. “Arb. u.”stands for “arbitrary units”. From Shahbaz et al. , 2007. laser propagation direction can be neglected. Equation(15) indicates that this is the case here, provided (cid:18) Q eff E M r ω (cid:19) . ω p M r I p . (54)The two above inequalities define a maximum laser in-tensity and a minimum laser angular frequency which arestill in accordance with the conditions imposed. At theselaser parameters, the maximum harmonic cut-off ener-gies are attained and an efficient ionization-recollisionprocess is guaranteed. For muonic hydrogen the corre-sponding lowest frequency lies in the Vacuum Ultraviolet(VUV) range ( ω ≈
27 eV) and the maximum field in-tensity amounts to 1 . × W/cm . At these values,the harmonic spectrum extends to a maximum energy ofapproximately 0 .
55 MeV. For light muonic atoms withnuclear charge number
Z >
1, the achievable cut-offenergies are even higher, reaching several MeVs. Thisholds prospects for the production of coherent ultrashortgamma-ray pulses (see also Xiang et al. , 2010).In principle, also nuclear size effects arise in the HHGsignal from muonic atoms. This has been shown qualita-tively in 1D numerical simulations, where a 50% enhance-ment of the harmonic plateau height has been obtainedfor muonic hydrogen compared with muonic deuterium(Shahbaz et al. , 2010, 2007). This has been attributed tothe enhanced final muon acceleration towards the hydro-genic core. For more precise predictions, 3D calculationsare desirable.Existing results also indicate that muonic atoms inhigh-intensity, high-frequency laser fields can be utilizedto dynamically gain structure information on nuclearground states via their high-harmonic response. Besides,the laser-driven muonic charge cloud, causing a time-dependent Coulomb field, may lead to nuclear excita-tion (Shahbaz et al. , 2009). The excitation probabilities are quite small, however, because of the large differencebetween the laser photon energy and the nuclear tran-sition energy. Nuclear excitation can also be triggeredby intense laser-induced recollisions of field-ionized high-energy electrons (Kornev and Zon, 2007; Milosevic et al. ,2004; Mocken and Keitel, 2004).Finally, muonic molecules are of particular interest fornuclear fusion studies. Modifications of muon-catalyzedfusion in strongly laser-driven muonic D +2 molecules havebeen investigated (Chelkowski et al. , 2004; Paramonov,2007). It was found that applied field intensities of theorder of 10 W/cm can control the molecular recol-lision dynamics by triggering the nuclear reaction ona femtosecond time scale. Similar theoretical studieshave recently been carried out on aligned (electronic)HT molecules, i.e., involving a Tritium atom (Zhi andSokolov, 2009). XII. LASER COLLIDERS
The fast advancement in laser technology is opening upthe possibility of employing intense laser beams to effi-ciently accelerate charged particles and to make them col-lide for eventually even initiating high-energy reactions.
A. Laser acceleration
Strong laser fields provide new mechanisms for par-ticle acceleration alternative to conventional acceleratortechnology (Tajima and Dawson, 1979). With presently-available laser systems an enormous electron accelerationgradient ∼ et al. , 2010; Faure et al. , 2004; Geddes et al. ,2004; Hafz et al. , 2008; Mangles et al. , 2004). Differ-ent schemes of laser-electron acceleration have been pro-posed. These include the laser wakefield accelerator, theplasma beat wave accelerator, the self-modulated laserwakefield accelerator and plasma waves driven by mul-tiple laser pulses (see the recent reviews Esarey et al. ,2009 and Malka, 2011). High-gradient plasma wakefieldscan also be generated with an ultrashort bunch of pro-tons (Caldwell et al. , 2009), allowing electron acceler-ation to TeV energies in a single stage. The achiev-able current and emittance of presently-available laser-accelerated electron beams is sufficient to build syn-chrotron radiation sources or even to aim at compactXFEL lasers (Schlenvoigt et al. , 2008).Laser acceleration of ions provides quasimonoenergeticbeams with energy of several MeVs per nucleon (Fuchs et al. , 2007; Haberberger et al. , 2012; Hegelich et al. ,2006; Schwoerer et al. , 2006; Toncian et al. , 2006). Itmostly employs the interaction of high-intensity lasers4with solid targets. One of the main goals of laser-ionacceleration is to create low-cost devices for medical ap-plications, such as for hadron cancer therapy (Combs et al. , 2009). Several regimes have been identified forlaser-ion acceleration (see also the forthcoming reviewMacchi et al. , 2012). For laser intensities in the range10 -10 W/cm and for solid targets with a thicknessranging from a few to tens of micrometers, the so-calledtarget-normal-sheath acceleration is the main mecha-nism (Fuchs et al. , 2006). A further laser-ion interac-tion process is the skin-layer ponderomotive acceleration(Badziak, 2007). By contrast, the radiation-pressure ac-celeration regime operates when the target thickness isdecreased (see Esirkepov et al. , 2004 and Macchi et al. ,2009 for the so-called “laser piston” and “light sail”regimes, respectively). In Galow et al. , 2011 a chirpedultrastrong laser pulse is applied to proton accelerationin a plasma. Chirping of the laser pulse ensures optimalphase synchronization of the protons with the laser fieldand leads to efficient proton energy gain from the field.In this way, a dense proton beam (with about 10 protonsper bunch) of high energy (250 MeV) and good quality(energy spread ∼ et al. , 2003). Especially efficient accelerations(Bochkarev et al. , 2011; Gupta et al. , 2007; Salamin,2006) can be achieved in a radially polarized axicon laserbeam (Dorn et al. , 2003). For example, the generationof mono-energetic GeV electrons from ionization in a ra-dially polarized laser beam is theoretically demonstratedin Salamin, 2007, 2010. A setup for direct laser acceler-ation of protons and bare carbon nuclei is considered inSalamin et al. , 2008. It has been shown that laser pulsesof 0 . et al. , 2010 and further optimization studies inHarman et al. , 2011 indicate that protons stemming fromlaser-plasma processes can be efficiently post-acceleratedemploying single and crossed pulsed laser beams, focusedto spot radii of the order of the laser wavelength. Theprotons in the resulting beam have kinetic energies ex-ceeding 200 MeV and small energy spreads of about 1%.The direct-acceleration method has proved to be efficientalso for other applications. In Salamin, 2011 it is shownthat 10 keV helium and carbon ions, injected into 1 TW-power crossed laser beams of radial polarization, can beaccelerated in vacuum to energies of hundreds of keVnecessary for ion lithography. B. Laser-plasma linear collider
Laser-electron accelerators have already entered theGeV energy domain where the realm of particle physics starts (Leemans et al. , 2006). In fact, the strong interac-tion comes into play at distances d of the order of d ∼ ε ∼ /d ∼ L at least as highas 10 -10 cm − s − . Meanwhile, for the ultimate goalof being competitive with the next International LinearCollider (ILC, 2011), energies on the order of 1 TeV andluminosities of the order of 10 cm − s − are required(Ellis and Wilson, 2001).The potential of the Laser-Plasma Accelerator (LPA)scheme to develop a laser-plasma linear collider is dis-cussed in Schroeder et al. , 2010. Two LPA regimes areanalyzed which are distinguished by the relationship be-tween the laser beam waist size w and the plasma fre-quency ω p = p πn p e /m , with n p being the plasmadensity: 1) the quasilinear regime at large radius of thelaser beam ω p w > ξ / p ξ / ξ ∼ ω p w . √ ξ ( ξ > et al. , 2010.In the standard LPA scheme, the electron plasma waveis driven by an intense laser pulse with duration τ ofthe order of the plasma wavelength λ p = 2 π/ω p , whichaccelerates the electrons injected in the plasma wave bywave breaking (Esarey et al. , 2009; Leemans and Esarey,2009; Malka et al. , 2008). The accelerating field E p ofthe plasma wave can be estimated from E p ∼ mω p | e | ∝ n / p . (55)In fact, in the plasma wave, the charge separation occurson a length scale of the order of λ p , producing a sur-face charge density σ p ∼ | e | n p λ p and a field E p ∼ πσ p ,which corresponds to Eq. (55). The number of elec-trons N e that can be accelerated in a plasma wave isgiven approximately by the number of charged particlesrequired to compensate for the laser-excited wakefield,having a longitudinal component E k . From the relation E k ∼ πN e | e | /πw , it follows that N e ∼ πn p ω p ∝ n − / p , (56)because w ω p ∼ L d . We can estimate the latter by equat-ing the energy spent for accelerating the N e electronsalong L d ( ∼ N e | e | E p L d ) to the energy of the laser pulse( ∼ E πw τ / π ). Recalling that w ω p ∼
1, this yields L d ∼ ω ω p λ p ∝ n − / p . (57)A staging of LPA is required to achieve high current den-sities along with high energies. The electron energy gain∆ ε s in a single-stage LPA is∆ ε s ∼ | e | E p L d ∼ m ω ω p ∝ n − p . (58)Therefore, the number of stages N s to achieve atotal acceleration energy ε is N s = ε / ∆ ε s ∼ ( ε /m )( ω p /ω ) ∝ n p . It corresponds to a total colliderlength L c of L c ∼ N s L d ∼ ε m λ p ∝ n − / p . (59)When two identical beams each with N particles andwith horizontal (vertical) transverse beam size σ x ( σ y )collide with a frequency f , the luminosity L is defined as L = N f / πσ x σ y . In LPA N = N e and f is the laserrepetition rate, then L ∼ π ω p σ x σ y fr . (60)The laser energy W s required in a single stage inthe LPA collider is approximately given by W s ∼ m ( λ p /r )( ω /ω p ) at ξ ∼ P T amounts to P T ∼ N s W s f ∼ ε f ( λ p /r ).The above estimates show that, although the number N e of electrons in the bunch as well as the single-stageenergy gain ∆ ε s increase at low plasma densities, theaccelerating gradient ∆ ε s /L d nevertheless decreases be-cause the laser depletion length L d and the overall col-lider length L c increase as well. Limiting the total lengthof each LPA in a collider to about 100 m will requirea plasma density n p ∼ cm − to provide a center-of mass energy of ∼ ∼ et al. , 2010). Fora number of electrons per bunch of N e ∼ , a laserrepetition rate of 15 kHz and a transverse beam size ofabout 10 nm would be required to reach the goal-valueof 10 cm − s − for the accelerator luminosity. At theusual condition w ∼ /ω p ∼ µ m, instead, the lu-minosity amounts to L ∼ cm − s − . In the above conditions each acceleration stage would be powered bya laser pulse with an energy of 30 J corresponding toan average power of about 0 . C. Laser micro-collider
We turn now to another scheme for a laser collider,which is based on principles quite different than those ofthe LPA scheme discussed above. In the LPA scheme,the electron is accelerated due to its synchronous motionwith the propagating field. In this way the symmetryin the energy exchange process between the electron andthe oscillating field is broken, as required by the Lawson-Woodward theorem (Lawson, 1979; Woodward, 1947).Another way to exploit the energy gain of the electronin the oscillating laser field is to initiate high-energy pro-cesses in situ , i.e., inside the laser beam (McDonald andShmakov, 1999). In this case the temporary energy gainof the electron during interaction with a half cycle of thelaser wave is used to trigger some processes during whichthe electron state may change (in particular, the electronmay annihilate with a positron) and the desired asymme-try in the energy exchange can be achieved. In fact, thisapproach is widely employed in the nonrelativistic regimevia laser-driven recollisions of an ionized electron with itsparent ion (see Sec. IV.B).The question arises also as to whether the temporaryenergy gain of the electron in the laser beam can also beemployed in the relativistic regime at ultrahigh energies.As pointed out in Sec. IV, an extension of the establishedrecollision scheme with normal atoms into the relativisticregime is hindered by the relativistic drift. However, thedrift will not cause any problem when positronium (Ps)atoms are used because its constituent particles, electronand positron, have the same absolute value of the charge-to-mass ratio (see Sec. III.A).A corresponding realization of high-energy e + - e − rec-ollisions in the GeV domain aiming at particle reactionshas been proposed in Hatsagortsyan et al. , 2006. It re-lies on (initially nonrelativistic) Ps atoms exposed tosuper-intense laser pulses. After almost instantaneousionization of Ps in the strong laser field, the free electronand positron oscillate in opposite directions along thelaser electric field and experience the same ponderomo-tive drift motion along the laser propagation direction. Inthis way, the particles acquire energy from the field andare driven into periodic e + - e − collisions (Henrich et al. ,6 FIG. 25 (a) In conventional e + - e − colliders bunches of ac-celerated electrons and positrons are focused to collide head-on-head incoherently , i.e., the bunches collide head-on-headbut electrons and positrons in the bunch do not. (b) In therecollision-based collider, the electron and positron originat-ing from the same Ps atom may collide head-on-head coher-ently (Henrich et al. , 2004). From Hatsagortsyan et al. , 2006. ε ∗ of the electron and thepositron at the recollision time arises mainly from thetransversal momentum of the particles and it scales as ε ∗ ∼ mξ . A basic particle reaction which could be trig-gered in a laser-driven collider is e + - e − annihilation withproduction of a µ + - µ − pair, i.e., e + e − → µ + µ − . Theenergy threshold for this process in the center-of-masssystem is 2 m µ ≈
210 MeV. It can be reached with a laserfield such that ξ ∼ W/cm , currently within reach.In addition, the proposed recollision-based laser col-lider can yield high luminosities compared to conven-tional laser accelerators. In the latter, bunches of elec-trons and positrons are accelerated and brought intohead-on-head collision. However, the particles in thebunch are distributed randomly such that each micro-scopic e + - e − collision is not head-on-head but has a meanimpact parameter b i ∼ a b determined by the beam ra-dius a b , characterizing the collision as incoherent (seeFig. 25). Instead, in the recollision-based collider theelectron and the positron stem from the same Ps atomwith initial coordinates being confined within the rangeof one Bohr radius a B ≈ . × − cm. Since they aredriven coherently by the laser field, they can recollidewith a mean impact parameter b c ∼ a wp of the order ofthe electron wave packet size a wp (see Fig. 25). Con-sequently, the luminosity contains a coherent component(Hatsagortsyan et al. , 2006): L = (cid:20) N p ( N p − b i + N p b c (cid:21) f, (61)where N p is the number of particles in the bunch, and f is the bunch repetition frequency. The coherent com-ponent ( N p /b c ) f can lead to a substantial luminosityenhancement in the case when the particle number islow and the particle’s wave packet spreading is small, N p a wp < a b . Note that the reaction Ps → µ + µ − arising in a strongly laser-driven e + - e − plasma may be consid-ered as the coherent counterpart of the incoherent process e + e − → µ + µ − discussed in Sec. X.A.Rigorous quantum-electrodynamical calculations for µ + - µ − pair production in a laser field have been per-formed in M¨uller et al. , 2006, 2008b,c. In agreementwith Eq. (61), they enabled the development of a simple-man’s model in which the rate of the laser-driven pro-cess can be expressed via a convolution of the rescat-tering electron wave packet with the field-free cross sec-tion σ e + e − → µ + µ − (see Eq. 51). The latter attains themaximal value σ (max) e + e − → µ + µ − ∼ α λ C,µ ∼ − cm at ε ∗ ≈
260 MeV (Peskin and Schroeder, 1995). However,when the field driving the Ps atoms is a single laser wave,the e + - e − recollision times are long and the µ + - µ − pro-duction process is substantially suppressed by extensivewave packet spreading. This obstacle can be overcomewhen two counterpropagating laser beams are employed.The role of the spreading of the electron wave packet incounterpropagating focused laser beams of circular andlinear polarization has been investigated in detail in Liu et al. , 2009. The advantage of the circular-polarizationsetup is the focusing of the recolliding electron wavepacket. However, this advantage is reduced by a spa-tial offset in the e + - e − collision when the initial coordi-nate of the Ps atom deviates from the symmetric positionbetween the laser pulses. The latter imposes a severe re-striction on the Ps gas size along the laser propagationdirection. Thus, the linear-polarization setup is prefer-able when the offset at the recollision is very small andthe wave packet size at the recollision is within accept-able limits. Results from a Monte Carlo simulation ofthe e + - e − wave-packet dynamics in counterpropagatinglinearly polarized laser pulses are shown in Fig. 26.The luminosity L and the number of reactionevents N for the recollision-based collider with counter-propagating laser pulses can be estimated as: L ∼ N Ps a wp τ r f, (62) N ∼ σ (max) e + e − → µ + µ − a wp τ r N Ps N L , (63)respectively, where N Ps is the number of Ps atoms, τ r the recollision time of the order of the lasers period, N L the number of laser pulses, and f the laser repetitionrate. Taking N Ps ≈ (Cassidy and Mills, 2005), f = 1Hz and the spatial extension of the e + - e − pair from Fig.26, one estimates a luminosity of L ∼ cm − s − andabout one µ + - µ − pair production event every 10 lasershots at a laser intensity of 4 . × W/cm .In conclusion, the scheme of the recollision-based lasercollider allows to realize high-energy and high-luminositycollisions in a microscopic setup. However, it is not easilyscalable to the parameters of the ILC, namely, to TeVenergies and luminosities of the order of 10 cm − s − .7 -3 -2 -1 0 1 2 3-3-2-10123 x [a.u.] (a) -10 -5 0 5 10-3-2-10123 z [a.u.] y [ a . u .] (c) y [ a . u .] -3 -2 -1 0 1 2 3 x [a.u.]Positron (b) Electron -10 -5 0 5 10 z [a.u.] (d)
FIG. 26 (Color) The coordinate-space distributions of theelectron and the positron wave-packets at the recollision timein focused counterpropagating pulses along the z directionwith w = 10 µ m, λ = 0 . µ m and with I = 4 . × W/cm (parts (a) and (c)) and I = 1 . × W/cm (parts (b) and (d)). The Ps atom is initially located at theorigin. Spatial coordinates are given in “atomic units”, with1 a.u. = 0 .
05 nm. Adapted from Liu et al. , 2009.
XIII. PARTICLE PHYSICS WITHIN AND BEYOND THESTANDARD MODEL
The sustained progress in laser technology towardshigher and higher field intensities raises the question asto what extent ultrastrong laser fields may develop intoa useful tool for particle physics beyond QED. Below wereview theoretical predictions regarding the influence ofsuper-intense laser waves on electroweak processes andtheir potential for probing new physics beyond the Stan-dard Model.
A. Electroweak sector of the Standard Model
The energy scale of weak interactions is set by themasses of the W ± and Z exchange bosons, m W ≈ m Z ∼
100 GeV. Therefore, the influence of external laser fields,even if strong on the scale of QED, is generally rathersmall. An overview of weak interaction processes in thepresence of intense electromagnetic fields has been givenin Kurilin, 1999.Various weak decay processes in the presence of in-tense laser fields have been considered. They can be di-vided into two classes: 1) laser-assisted processes whichalso exist in the absence of the field but may be modi-fied due to its presence; 2) field-induced processes whichcan only proceed when a background field is present,providing an additional energy reservoir. With respectto processes from the first category, π → µ + ν and µ − → e − + ¯ ν e + ν µ have already been examined (Ritus,1985). Laser-assisted muon decay has also been revis-ited recently (Dicus et al. , 2009; Farzinnia et al. , 2009;Narozhny and Fedotov, 2008). W ± and Z boson decay into a fermion-antifermion pair was calculated in Kurilin,2004, 2009. In all cases, the effect of the laser field wasfound to be small. As a general result, the presence of thelaser field modifies the field-free decay rate R M, of a par-ticle with mass M to R M = R ,M (1+∆), with the correc-tion ∆ being of the order of χ ,M in the range of param-eters ξ ,M = ξ m/M > χ ,M = χ ( m/M ) ≪ et al. , 2012 and multiphoton effects in the crosssection are predicted.External fields can also induce decay processes, whichare energetically forbidden otherwise. In Kurilin, 1999,the field-induced lepton decay l − → W − + ν l wasconsidered. Since the mass of the initial-state parti-cle is smaller than the mass of the decay products,the process is clearly impossible in vacuum. The pres-ence of the field does allow for such an exotic decay,but the probability P l − → W − + ν l remains exponentiallysuppressed, i.e., P l − → W − + ν l ∼ exp( − /χ ,W ), where χ ,W = χ ( m/m W ) .Finally, the production of an e + - e − pair by high-energyneutrino impact on a strong laser pulse has been cal-culated in Tinsley, 2005. The setup is similar to the e + - e − pair production processes in QED discussed inSecs. VIII.A and VIII.B. However, as it was shown inTinsley, 2005, the laser-induced process ν → ν + e + + e − is extremely unlikely. At a field intensity of about I = 3 × W/cm , the production length is on theorder of a light year, even for a neutrino energy of 1 PeV. B. Particle physics beyond the Standard Model
Recently, a lot of attention has been devoted to thepossibility of employing intense laser sources to test as-pects of physical theories which go even beyond the Stan-dard Model. In Heinzl et al. , 2010b, for example, it isenvisaged that effects of the noncommutativity of space-time modify the kinematics of multiphoton Comptonscattering by inducing a nonzero photon mass. We recallthat in noncommutative quantum field theories operators X µ are associated to spacetime coordinates x µ , which donot commute, but rather satisfy the commutation rela-tions [ X µ , X ν ] = i Θ µν , with Θ µν being an antisymmetricconstant tensor (Douglas and Nekrasov, 2001).On a different side, one of the still open problems ofthe Standard Model is the so-called strong CP problem(Kim and Carosi, 2010). The nontrivial structure ofthe vacuum, as predicted by Quantum Chromodynam-ics (QCD), allows for the violation within QCD of thecombined symmetry of charge conjugation (C) and par-ity (P). This implies a value for the neutron’s electricdipole moment which, however, is already many ordersof magnitude larger than experimental upper limits. Oneway of solving this problem was suggested in Peccei and8Quinn, 1977 which required the existence of a massivepseudoscalar boson, called axion. The axion has neverbeen observed experimentally although some of its prop-erties can be predicted on theoretical grounds: it shouldbe electrically neutral and its mass should not exceed1 eV in order of magnitude. Although being electri-cally neutral, the axion is predicted to couple to the elec-tromagnetic field F µν ( x ) through a Lagrangian-densityterm L aγ ( x ) = g a ( x ) F µν ( x ) ˜ F µν ( x ) , (64)with g being the photon-axion coupling constant and a ( x )the axion field.The photon-axion Lagrangian density in Eq. (64) hasmainly two implications: 1) the existence of axions in-duces a change in the polarization of a light beam pass-ing through a background electromagnetic field; 2) aphoton can transform into an axion (and vice versa)in the presence of a background electromagnetic field.The first prediction has been tested in experimentslike the Brookhaven-Fermilab-Rochester-Trieste (BFRT)(Cameron et al. , 1993) and the Polarizzazione del Vuotocon LASer (PVLAS) (PVLAS, 2011), where a linearlypolarized probe laser field crossed a region in which astrong magnetic field was present of 3 .
25 T and 5 Tat BFRT and at PVLAS, respectively. Testing the sec-ond prediction is the aim of the so-called “light shiningthrough a wall” experiments like the Any Light ParticleSearch (ALPS) (ALPS, 2010), the CERN Axion SolarTelescope (CAST) (CAST, 2008) and Gamma to milli-eV particle search (GammeV) (GammeV, 2011) (see alsothe detailed theoretical analysis in Adler et al. , 2008).In the GammeV experiment, for example, the light of aNd:YAG laser passes through a region in which a 5 Tmagnetic field is present. A mirror is positioned behindthat region in order to reflect the laser light. The axionswhich would eventually be created in the magnetic-fieldregion pass through the mirror undisturbed and can bereconverted to photons by means of a second magneticfield, activated after the mirror itself. So far these exper-iments have given negative results. An interesting exper-imental proposal has been put forward in Rabad´an et al. ,2006, where the high-energy photon beam delivered byan XFEL facility has been suggested as a probe beam totest regions of parameters (like the axion mass m a or thephoton-axion coupling constant g ) which are inaccessiblevia optical laser light.The perspective for reaching ultrahigh intensities atfuture laser facilities has stimulated new proposals foremploying such fields to elicit the photon-axion interac-tion (Gies, 2009). In fact, an advantage of using stronguniform magnetic fields is that they can be kept strongfor a macroscopically long time (of the order of hours)and on a macroscopic spatial region (of the order of 1 m)(for QED processes occurring in a strong magnetic field, see the standard review Erber, 1966 and the very currentoverview paper Dunne, 2012 for recent progresses in thefield). On the other hand, laser beams deliver fields muchstronger than those employed in the mentioned experi-ments (the magnetic field strength of a laser beam withthe available intensity of 10 W/cm amounts to about6 . × T) but in a microscopic space-time region. How-ever, it has been first realized in Mendon¸ca, 2007 that en-visaged ultrahigh intensities at future laser facilities maycompensate for the tiny space-time extension of the laserspot region. In Mendon¸ca, 2007 the coupled equations ofthe electromagnetic field F µν ( x ) and the axion field a ( x ) ∂ µ ∂ µ a + m a a = 14 g F µν ˜ F µν ∂ µ F µν = g ( ∂ µ a ) ˜ F µν (65)are solved approximately. It is shown that if a probelaser field propagates through a strong plane-wave field,the axion field “grows” at the expense mainly of theprobe field itself, whose intensity should be observed todecrease. Laser powers of the order of 1 PW have al-ready been shown to provide stronger hints for the pres-ence of axions than magnetic-field-based experiments likethe PVLAS. More realistic Gaussian laser beams areconsidered in D¨obrich and Gies, 2010 where the start-ing point is also represented by Eq. (65). The sug-gested experimental setup assumes a probe electromag-netic beam with angular frequency ω p, in passing througha strong counterpropagating Gaussian beam with angu-lar frequency ω , k and another strong Gaussian beampropagating perpendicularly and with angular frequency ω , ⊥ . By choosing ω , ⊥ = 2 ω , k , it is found that aftera photon-axion-photon double conversion, photons aregenerated with angular frequencies ω p, out = ω p, in ± ω , k .The amplitudes of these processes are shown to bepeaked at specific values of the axion mass m a, ± =2 q ω p, in ω , k + ω , k (1 ± /
2. Since the optical photonenergies are of the order of 1 eV, this setup allows for theinvestigation of values of the axion mass in this regime.This is very important because such a region of the ax-ion mass is inaccessible to experiments based on strongmagnetic fields, which can probe regions at most in themeV range.In addition to electrically neutral new particles suchas axions, yet unobserved particles with nonzero chargemay also exist. The fact that they have so far escapeddetection implies that they are either very heavy (ren-dering them a target for large-scale accelerator experi-ments), or that they are light but very weakly charged.In the latter case, these so-called minicharged particles,i.e., particles with absolute value of the electric chargemuch smaller than | e | , are suitable candidates for laser-based searches (Gies, 2009). Let m ǫ and Q ǫ = ǫe , with0 < ǫ ≪
1, denote the minicharged particle mass andcharge, respectively. Then, the corresponding critical9field scale F cr ,ǫ = m ǫ / | Q ǫ | can be much lower than F cr . As a consequence, vacuum nonlinearities associ-ated with minicharged particles may be very pronouncedin an external laser field with intensity much less than I cr ∼ W/cm . Moreover, even at optical photonenergies ∼ m ǫ . et al. , 2006, vac-uum dichroism and birefringence effects due to the ex-istence of minicharged particles were analyzed when aprobe laser beam with ω p > m ǫ traverses a magneticfield. It was shown that polarization measurements inthis setup would provide much stronger constraints onminicharged particles in the mass range below 0.1 eVthan in previous laboratory searches. XIV. CONCLUSION AND OUTLOOK
The fast development of laser technology has beenpaving the way to employ laser sources for investigatingrelativistic, quantum electrodynamical, nuclear and high-energy processes. Starting with the lowest required in-tensities, relativistic atomic processes are already withinthe reach of available laser systems, while the proposedmethods to compensate the deteriorating effects of therelativistic drift still have to be tested experimentally.Moreover, a fully consistent theoretical interpretation ofrecent experimental results on correlation effects in rela-tivistic multielectron tunneling is still missing.Concerning the interaction of free electrons with in-tense laser beams, we have seen that experiments havebeen performed to explore the classical regime. Onlythe E-144 experiment at SLAC has so far been realizedon multiphoton Compton scattering, although presentlyavailable lasers and electron beams would allow for prob-ing this regime in full detail. We have also pointed outthat at laser intensities of the order of 10 -10 W/cm ,RR effects come into play at electron energies of the orderof a few GeV. It is envisaged that the quantum radiation-dominated regime, where quantum and RR effects sub-stantially alter the electron dynamics, could be one ofthe first extreme regimes of light-matter interaction tobe probed with upcoming petawatt laser facilities. Onthe theoretical side, most of the calculations have beenperformed by approximating the laser field as a planewave, as the Dirac equation in the presence of a focusedbackground field cannot be solved analytically. Certainly,new methods have to be developed to calculate photonspectra including quantum effects and spatio-temporalfocusing of the laser field in order to be able to quanti-tatively interpret upcoming experimental results.Nonlinear quantum electrodynamical effects have beenshown to become observable at future multipetawattlaser facilities, as well as at ELI and HiPER. Here themain challenges concern the measurability of tiny effects on the polarization of probe beams and on the detectionof a typically very low number of signal photons out oflarge backgrounds. Similar challenges are envisaged todetect the presence of light and weakly-interacting hypo-thetical particles like axions and minicharged particles.The physical properties (mass, coupling constants, etc.)of such hypothetical particles are, of course, unknown.In this respect intense laser fields may be employed hereto set bounds on physical quantities like the axion massand, in particular, to scan regions of physical parameters,which are inaccessible to conventional methods based, forexample, on astrophysical observations.Different schemes have been proposed to observe e + - e − pair production at intensities below the Schwinger limit,which seems now to be feasible in the near future, at leastfrom a theoretical point of view. Corresponding studieswould complement the results of the pioneering E-144experiment and deepen our understanding of the QEDvacuum in the presence of extreme electromagnetic fields.This is also connected with the recent investigations onthe development of QED cascades in laser-laser collisions.In addition to being intrinsically interesting, the devel-opment of QED cascades is expected to set a limit on themaximal attainable laser intensity. However, the study ofquantum cascades in intense laser fields has started rela-tively recently and is still under vivid development. Moreadvanced analytical and numerical methods are requiredin order to describe realistically and quantitatively sucha complex system as an electron-positron-photon plasmain the presence of a strong driving laser field.Nuclear quantum optics is also a new exciting andpromising field. Since the energy difference between nu-clear levels is typically in the multi-keV and MeV range,high-frequency laser pulses, especially in combinationwith accelerators, are preferable in controlling nucleardynamics. As pointed out, especially table-top highly co-herent x-ray light beams, envisaged for the future, openup perspectives for exciting applications including nu-clear state preparation and nuclear batteries.Finally, we want to point out that most of the consid-ered processes have not yet been observed or tested ex-perimentally. This is in our opinion one of the most chal-lenging aspects of upcoming laser physics, not only froman experimental point of view, but also from the point ofview of theoretical methods. Experimentally, the mainreason is that in order to test, for example, nonlinearquantum electrodynamics or to investigate nuclear quan-tum optics, high-energy particle beams (including photonbeams) are required to be available in the same labora-tory as the strong laser. On the one hand, the combinedexpertise from different experimental physical communi-ties is required to perform such complex but fundamentalexperiments. On the other hand, the fast technologicaldevelopment of laser-plasma accelerators is very promis-ing and exciting, as this seems the most feasible waytowards the realization of stable, table-top high-energy0particle accelerators. Combining such high-energy probebeams with an ultrarelativistic laser beam in a single all-optical setup will certainly result in a unique tool foradvancing our understanding of intense laser-matter in-teractions. ACKNOWLEDGMENTS
We are grateful to many students, colleagues and col-laborators for inspiring discussions, ideas, and sugges-tions, in particular in joint publications during the heremost relevant last six years, to H. Bauke, T. J. B¨urvenich,A. Dadi, C. Deneke, J. Evers, B. Galow, R. Grobe, Z.Harman, H. Hu, A. Ipp, U. D. Jentschura, B. King, M.Klaiber, M. Kohler, G. Yu. Kryuchkyan, W.-T. Liao, C.Liu, E. L¨otstedt, A. Macchi, F. Mackenroth, S. Meuren,A. I. Milstein, G. Mocken, S. J. M¨uller, T.-O. M¨uller, A.P´alffy, F. Pegoraro, M. Ruf, Y. I. Salamin, M. Tamburini,M. Verschl, and A. B. Voitkiv.
LIST OF FREQUENTLY-USED SYMBOLS E laser field amplitude F cr = m / | e | critical electromagnetic field( ≈ . × V/cm)( ≈ . × G) I = E / π laser peak intensity I cr = F / π critical laser intensity( ≈ . × W/cm ) Z nuclear charge number e electron charge( ≈ − / √ ≈ − . k µ = ( ω, k ) = ωn µ initial or incoming photonfour-momentum k ′ µ = ( ω ′ , k ′ ) = ω ′ n ′ µ final or outgoing photonfour-momentum k µ = ω n µ = ω (1 , n ) laser photon four-momentum m electron mass( ≈ .
511 MeV) p µ = ( ε , p ) = mγ (1 , β ) initial or incoming electronfour-momentum α = e fine-structure constant( ≈ / ≈ . × − ) χ = ( p , − /m )( E /F cr ) nonlinear electron quantumparameter κ = ( k − /m )( E /F cr ) nonlinear photon quantumparameter λ = T = 2 π/ω laser wavelength and period λ C = 1 /m Compton wavelength( ≈ . × − cm) ξ = | e | E /mω classical relativisticparameter ω laser angular frequency REFERENCES
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