1. Classical and quantum electrodynamics e®ects in intense laser pulses Antonino Di Piazza Workshop on Petawatt Lasers at Hard X-Ray Sources Dresden, September 9 th 2011

2. OUTLINE Introduction –Classical vs quantum vacuum –Strong-field CED and QED in intense laser fields Light-by-light diffraction Radiation reaction in CED and QED –Entering the quantum radiation dominated regime Strong-field QED in intense laser fields –Photon splitting in a strong laser field Conclusion

3. Classical vs. quantum vacuum In quantum field theory the vacuum state is the state in which no real particles are present (electrons, positrons, photons etc...) –Virtual particles are present –They live for a very short time and cover a very short distance (   =}/  2 and   =}/ , respectively). For electrons and positrons:   »10 -11 cm and  »10 -21 s. Quantum Field TheoryQuantum MechanicsSpecial Relativity Virtual Particles i. e. Quantum Vacuum Fluctuations Time-Energy Uncertainty Principle  &} Mass-Energy Equivalence  =  2 Can we somehow detect the presence of the virtual particles?

4. Critical ¯elds of QED Physical meaning of the critical fields: Since virtual particles live for a very short time, then very strong fields are needed to make apparent the effects of their presence A strength scale is given by the critical fields (here ®=1/137) Ze n=1

5. A particle (e {, e + or °) with energy E (}! for a photon) collides head on with a plane wave with amplitude E L and angular frequency ! L (wavelength ¸ L ) Strong-field QED in intense laser pulses Lorentz- and gauge-invariant parameters (Ritus 1985) E L, ! L E (}!) Numerical values (with an e - beam of E=1 GeV (Leemans 2006)) Name Mathematical definition It controls… SF-QED regime Classical nonlinearity parameter Relativistic and multiphoton effects » & 1 Quantum nonlinearity parameter Quantum effects (photon recoil etc…) Â & 1 Laser facility Energy (J) Pulse duration (fs) Spot radius (¹m) Intensity (W/cm 2 ) PENELOPE (HZDR, Rossendorf) 150 5 (1) 10 21 (3£10 22 ) »=18, Â=0.14 (»=180, Â=1.4)

6. Light-by-light diffraction A. Di Piazza, K. Z. Hatsagortsyan and C. H. Keitel, Phys. Rev. Lett. 97, 083603 (2006); B. King, A. Di Piazza and C. H. Keitel, Phys. Rev. A 82, 032114 (2010) Strong beam: I 0 =10 23 W/cm 2, w 0 =¸ 0 =0.8 ¹m. Probe beam: ¸ p =0.4 nm, w p =8 ¹m The inclusion of the spatial focusing of the strong beam significantly affects the results à and " » 10 -9 rad are measurable in the soft x-ray regime (Marx et al. 2011) In CED electromagnetic ¯elds do not interact in vacuum. Instead, in QED virtual charged particles can mediate a pure quantum interaction between electromagnetic ¯elds also in vacuum

7. Radiation reaction in classical electrodynamics The Lorentz equation does not take into account that while being accelerated the electron generates an electromagnetic radiation ¯eld and it loses energy and momentum One has to solve self consistently the coupled Lorentz and Maxwell equations where now m 0 is the electron’s bare mass and F T,¹º =@ ¹ A T,º {@ º A T,¹ is the total electromagnetic ¯eld (Dirac 1938) : What is the equation of motion of an electron in an external, given electromagnetic ¯eld F ¹º (x)? 1.Find the solution of the wave equation and substitute it into the Lorentz equation 2.Renormalize the electron mass and obtain the Lorentz- Abraham-Dirac equation

8. The Lorentz-Abraham-Dirac equation is plagued by serious inconsistencies: runaway solutions, preacceleration In the realm of classical electrodynamics, i.e. if quantum e®ects are negligible, the Lorentz-Abraham-Dirac equation can be approximated by the so-called Landau-Lifshitz equation (Landau and Lifshitz 1947) The analytical solution of the Landau-Lifshitz equation in a plane wave (A. Di Piazza, Lett. Math. Phys. 83, 305 (2008)) indicates that 1.radiation-reaction e®ects scale with the parameter R C ©, where © is the total phase of the laser pulse and R C =®Â» 2.if R C t1 the e®ects of radiation reaction on the electron dynamics are dominant: it implies that the energy emitted by the electron in one laser period is of the order of the initial energy (classical radiation dominated regime) 3.if 2° 0 t», the electron may be temporarily reflected by the laser ¯eld only if radiation reaction is taken into account and this regime turns out to be very sensitive to radiation reaction Recall that an ultrarelativistic electron mainly emits along its velocity within a cone of aperture » 1/° (Landau and Lifshitz 1947)

9. Numerical parameters: electron energy 40 MeV (2° 0 =156), laser wavelength 0.8 ¹m, laser intensity 5  10 22 W/cm 2 (»=150), pulse duration 30 fs, focused to 2.5 ¹m (note that R C =0.08!) The red parts of the trajectory are those where the longitudinal velocity of the electron is positive The black lines indicate the cut- o® position from the formula ! c =3! 0 ° 3 Electron trajectories and emission spectra without and with radiation reaction A. Di Piazza, K. Z. Hatsagortsyan and C. H. Keitel, Phys. Rev. Lett. 102, 254802 (2009)

10. Radiation reaction in quantum electrodynamics One could say that radiation reaction is automatically taken into account in QED already in the “basic” emission diagram because energy-momentum conservation is automatically ful¯lled. However: Quantum radiation dominated regime: the electron emits many photons incoherently already in one laser period (emission probability P »®» =R Q & 1) all in general with a large recoil ( & 1) At  & 1 we have identified the quantum origin of radiation reaction in the incoherent multiple emissions by the electron No multiple coherent emission Classical limit: the electron emits a large number of photons but all with a small recoil 1.photon recoil has no classical analogue 2.the corresponding spectrum reduces in the classical limit ¿1 to the classical spectrum calculated via the Lorentz equation, i.e. without radiation reaction

11. We calculated the average energy emitted per unit of electron energy (emission spectrum) in the realm of strong- ¯eld QED, by taking into account the incoherent emission of N>1 photons (quantum radiation reaction) Numerical parameters: electron energy 1 GeV, laser wavelength 0.8 ¹m, laser intensity 5  10 22 W/cm 2 (»=150) corresponding to R Q =®»=1.1 and Â=1.8, laser pulse duration 5 fs E®ects of radiation reaction: 1.increase of the spectrum yield at low energies 2.shift to lower energies of the maximum of the spectrum yield 3.decrease of the spectrum yield at high energies Classical radiation reaction arti¯cially ampli¯es all the above e®ects Classical spectra both without and with radiation reaction give unphysical results at high photon energies A. Di Piazza, K. Z. Hatsagortsyan and C. H. Keitel, Phys. Rev. Lett. 105, 220403 (2010)

12. A. Di Piazza, A. I. Milstein and C. H. Keitel, Phys. Rev. A 76, 032103 (2007) Photon splitting in a laser field Split a Petawatt laser beam and with a fraction of it back-scatter 10 10 electrons with 5 GeV energy: 10 8 photons with energy of 400 MeV Laser ¯eld: intensity 10 22 W/cm 2, laser photon energy 1 eV and pulse duration of 100 fs For these parameters it is: »=100 and Â=0.34 A photon-splitting rate of 1.4£10 4 s -1 is obtained, i.e. about 1.4 photon-splitting events per second at 10 Hz

13. CONCLUSION Classical and quantum electrodynamics are well established theories but there are still areas to be investigated both theoretically and experimentally The combination of a PW laser with the European X-FEL can be a unique tool –to measure vacuum polarization (X-FEL photon beam+PW laser) –to test the classical equations of motion including radiation reaction and understand the quantum origin of radiation reaction (electron bunch+PW laser) –to detect photon splitting (back-scattered photon beam+PW laser)