1. Quantum-orbit approach for an elliptically polarized laser field Wilhelm Becker Max-Born-Institut, Berlin, Germany Workshop „Attoscience: Exploring and Controlling Matter on its Natural Time Scale“, KITPC, Beijing, May 12, 2011

2. Collaborators: C. Figueira de Morisson Faria, University College, London S. P. Goreslavski, MEPhI, Moscow R. Kopold, Siemens, Regensburg X. Liu, CAS, Wuhan D. B. Milosevic, U. Sarajevo G. G. Paulus, U. Jena S. V. Popruzhenko, MEPhI, Moscow N. I. Shvetsov- Shilovski, U. Jena

5. Motivation NSDI knee experimentally measured for circular polarization NSDI knee observed in completely classical (CC) and semiclassical (tunneling-classical; TC) simulations for circular polarization Dependence of a process on ellipticity is indicative of the mechanism

6. Nonsequential double ionization exists for circular polarization

7. NSDI for a circularly polarized laser field linearcircular G. D. Gillen, M. A. Walker, L. D. Van Woerkom, PRA 64, 043413 (2001) magnesium, I p1 = 7.6 eV, I p2 = 15.0 eV, I p3 = 80 eV, 120 fs, 800 nm

8. Double-ionization yield from completely classical (CC) simulations X. Wang, J. H. Eberly, NJP 12, 093047 (2010) I p = 1.3 a.u.

9. Electron trajectories from completely classical double-ionization simulations X. Wang, J. H. Eberly, NJP 12, 093047 (2010) doubly-ionizing orbits tend to be „long orbits“

10. CC simulation: escape over the Stark saddle depends on parameters helium, I p = 2.24 a.u. a = 1 b = 1 no knee magnesium, I p = 0.83 a.u. a = 3 b = 1 a knee F. Mauger, C. Chandre, T. Uzer, PRL 105, 083002 (2010)

11. Elliptical polarization helps revealing the mechanism

12. Ellipticity dependence reveals the mechanism P. Dietrich, N. H. Burnett, M. Yu. Ivanov, P. B. Corkum, PRA 50, R3589 (1994) HH 21 in argon, measured and simulated NSDI of argon, measured and simulated ellipticity

13. An example of ellipticity as a diagnostic tool NSDI of neon as a function of wavelength for various ellipticities calculated by the tunneling-classical-trajectory model transition to the standard rescattering mechanism at about 200 nm constant intensity I = 1.0 x 10 15 Wcm -2 X. Hao, G. Wang, X. Jia, W. Li, J. Liu, J. Chen, PRA 80, 023408 (2009)

14. Recollision and elliptical (linear --> circular) polarization Simplest simple-man argument: for sufficiently large ellipticity, especially for circular polarization, an electron released with zero velocity will not return to its place of birth no recollision-induced processes However, electrons are released with nonzero distribution of transverse velocities recollision is possible for suitable transverse momentum (But, no HHG for circular polarization, QM dipole selection rule)

15. Quantum-orbit formalism

16. Formal description of recollision processes HATI into a state with final (drift) momentum p: V f = continuum scattering potential), <  f | = <  p Volkov | = „direct“ + rescattered 1st-order Born approximation

17. Formal description of recollision processes HATI into a state with final (drift) momentum p: V f = continuum scattering potential), <  f | = <  p Volkov | = „direct“ + rescattered Low-frequency approximation (LFA)

18. Evaluation by stationary phase (steepest descent) with respect to the integration variables t, t‘, k

19. Saddle-point equations for high-order ATI the (complex) solutions t s ‘, t s, and q s (s=1,2,...) determine electron orbits in the laser field („quantum orbits“)

20. Saddle-point equations elastic rescattering return to the ion tunneling at constant energy

21. Many returns: for given final state, there are many solutions of the saddle-point equations „Long orbits“

22. Building up the ATI spectrum from quantum orbits shortest two orbits 1+2 shortest six orbits 1 +...+ 6 shortest 14 orbits Magnitude of the contributions of the various pairs of orbits Significance of longer orbits decreases due to spreading

23. x(t=t s ‘) = 0, but Re [x(Re t s ‘)] = „tunnel exit“ different from 0

24. Quantum orbits (real parts) for elliptical polarization Re y (a.u.) position of the ion tunnel exit x = semimajor axis y = semiminor axis Note: the shortest orbits require the largest transverse momenta to return semimajor polarization axis

25. Why longer orbits require lower transverse momenta to return short orbit: transverse drift is significant

26. Why longer orbits require lower transverse momenta to return longer orbit: transverse drift is much reduced

27. The contribution of an orbit is weighted exponentially prop. to exp(-p drift 2 /  p 2 ) short orbits have large p drift and are suppressed

28. What is the difference between the saddle points for linear and for elliptical polarization? linear pol.: for I p = 0 and q T = 0, the solution t‘ is real simple-man model elliptical pol.: even for Ip = 0 and q T = 0, the solution t‘ is complex (cannot have both q x - eA x (t‘) = 0 and q y - eA y (t‘) = 0) can only say that q - eA(t‘) is a complex null vector

29. Examples: HHG and HATI

30. Above-threshold ionization by an elliptically polarized laser field  = 0.5  = 1.59 eV I = 5 x 10 14 Wcm -2 R. Kopold, D. B. Milosevic, WB, PRL 84, 3831 (2000) The plateau becomes a stair The shortest orbits make the smallest contributions, but with the highest cutoff

31. Quantum orbits for elliptical polarization: Experiment vs. theory The plateau becomes a staircase  = 0.36 xenon at 0.77 x 10 14 Wcm -2 Salieres, Carre, Le Deroff, Grasbon, Paulus, Walther, Kopold, Becker, Milosevic, Sanpera, Lewenstein, Science 292, 902 (2001) The shortest orbits are not always the dominant orbits

32. Alternative description: quasienergy formalism (zero-range potential or effective-range theory) B. Borca, M. V. Frolov, N. L. Manakov, A. F. Starace, PRL 87, 133001 (2001) N. L. Manakov, M. V. Frolov, B. Borca, A. F. Starace, JPB 36, R49 (2003) A. V. Flegel, M. V. Frolov, N. L. Manakov, A. F. Starace, JPB 38, L27 (2005)

33. Staircase for HATI  = 0.5 I p = 0.9 eV  = 1.59 eV I = 5 x 10 14 Wcm -2 R. Kopold, D. B. Milosevic, WB, PRL 84, 3831 (2000)

34. Staircase for HHG I p = 0.9 eV  = 1.59 eV I = 5 x 10 14 Wcm -2  = 0.5

35. Quantum orbits in the complex t 0 and t 1 plane Im  t 0 t1t1 Re  t i t0t0 orbits 1,2 orbits 3,4 orbits 5,6 HATI: * (asterisk) HHG: (diamond)  = 0.5, 780 nm, He 5 x 10 14 Wcm -2

36. HATI for various ellipticities I p = 0.9 eV  = 1.59 eV I = 5 x 10 14 Wcm -2 strong drop for  > 0.3

37. HHG for various ellipticities D. B. Milosevic, JPB 33, 2479 (2000) I p = 13.6 eV  = 1 eV I = 1.4 x 10 14 Wcm -2 dramatic drop for  > 0.2

38. Cutoffs for HHG orbits pair of orbits 1 c 1 = 3.17 c 2 = 1.32 (Lewenstein, Ivanov) 2 c 1 = 1.54 c 2 = 0.88 3 c 1 = 2.40 c 2 = 1.10 HATI cutoff D. B. Milosevic, JPB 33, 2479 (2000) pair 1 E max = 10.01 U p + 0.54 I p M. Busuladzic, A. Gazibegovic-Busuladzic, D. B. Milosevic, Laser Phys. 16, 289 (2006)

39. Interference of direct and rescattered electrons G. G. Paulus, F. Grasbon, A. Dreischuh, H. Walther, R. Kopold, WB, PRL 84, 3791 (2000)

41. experiment: 7.7 x 10 13 Wcm -2 Xe 800 nm  = 0.36 theory: 5.7 x 10 13 Wcm -2 „Xe (I p = 0.436)“  = 0.48 Mechanism of the second plateau

42. rescattered direct The contributions of just the rescattered and just the direct electrons individually are only smoothly dependent on the angle, only the superposition is structured Interference of direct and rescattered electrons

43. Conditions for interference between direct and rescattered electrons energy yield energy yield direct rescattered direct rescattered for elliptical polarization, the yields of direct and rescattered electrons are comparable over a larger energy range linear elliptical See, however, Huismans et al., Science (2011)

44. Example: NSDI for elliptical polarization

45. NSDI from a simple semiclassical model R(t) = ADK tunneling rate t = start time, t‘(t) = recollision time E(t‘) = kinetic energy of the recolliding electron (t‘ - t) -3 = effect of spreading V p1p2 = form factor (to be ignored)  (...) = energy conservation in rescattering C. Figueira de Morisson Faria, H. Schomerus, X. Liu, WB, PRA 69, 043405 (2004)

46. NSDI by an elliptically polarized field: the bad news  = 0 --> 0.4 8 o.o.m.! N. I. Shvetsov-Shilovski, S. P. Goreslavski, S. V. Popruzhenko, WB, PRA 77, 063405 (2008) Ti:Sa neon I = 8 x 10 14 Wcm -2

47. NSDI for elliptical polarization: ion-momentum distribution Ti:Sa neon I=8 x10 14 Wcm -2 first six returns first return only N. I. Shvetsov-Shilovski, S. P. Goreslavski, S. V. Popruzhenko, WB, PRA 77, 063405 (2008) this case to be realized by a single-cycle pulse

48. N. I. Shvetsov-Shilovski, S. P. Goreslavski, S. V. Popruzhenko, WB, PRA 77, 063405 (2008) NSDI for elliptical polarization: electron-electron-momentum correlation W(p 1x,p 2x |p 1y >0,p 2y >0) first six returns first return only single-cycle pulse case!

49. Asymmetry of the momentum-momentum correlation between the first and the third quadrant 8 x 10 14 Wcm -2 4 x 10 14 Wcm -2 asymmetry is strongly intensity-dependent depending upon which orbits are dominant  = 10% for  = 0.1 yield is down by 3 should be measurable

50. Try some ellipticity Coulomb focusing is desirable to increase the effects ATI spectra for elliptical polarization are coming up from X. Y. Lai and X. Liu