1. Recollision, Time Delay, and Double Ionization studied with 3-D Classical Ensembles S.L. Haan, A. Karim, and Z. SmithJ.H. Eberly Calvin CollegeUniversity of Rochester Grand Rapids MI USARochester NY USA Also acknowledging contributions of C. Cully, A. Vache, D. Tannor, and L. Breen of Calvin College R. Panfili of U. Rochester in helping develop the 3-D ensemble program and technique & Phay Ho (UR) for numerous discussions regarding double ionization Work supported by National Science Foundation Grant PHY-0355035 and DOE Grant DE-FG02-05ER15713

2. We set up an ensemble of classical two-electron atoms. –Each atom has slightly different initial conditions –Ensemble sizes 400,000 We evolve each two-electron atom in time through a laser pulse, using Newton’s laws of motion. After each run, we can sort the trajectories and –Study statistical behavior; –Backtrack individual trajectories to learn their history. Overview of Method

3. Some Details –To prevent self-ionization of our starting state, we shield the electron-nucleus interaction RE Starting Distribution: –Gaussian, spherically symmetric, radial motion only, and total energy = He Ground State Energy; available KE is randomly distributed between electrons –We allow system to propagate without any laser field for time of 1 laser cycle (~100 a.u.). Resulting distribution is basically independent of details of initial radial distribution We use a 10-cycle trapezoidal pulse (2+6+2), polarized in z direction; =780 nm (16-photon single ionization, 50 double)

4. Final momenta parallel to laser polarization for ionized electron pairs Population in quadrants 2 and 4 indicates emission into opposite momentum hemispheres Having population in all 4 quadrants is consistent with experiment (e.g., V.L.B. de Jesus, et al., Journal of Electron Spectroscopy 141, 127 (2004)). I=.2 PW/cm 2 I=.4 PW/cm 2 I=.6 PW/cm 2 I=.8 PW/cm 2

5. Cause of opposite hemisphere emissions? We can backtrack doubly ionizing trajectories to learn cause Trajectories show recollision typically followed by a short time delay before final ionization. Careful sorting… Recollision time -- time of closest approach of two electrons after first electron achieves E>0. Double ionization time -- time at which both electrons achieve E>0 or escape nuclear well.

6. Delay time between recollision & double ionization Most DI trajectories show a part-cycle phase delay between recollision and double ionization

7. Final momenta sorted by: delay times from recollision to ionization and by final direction relative to recollision direction delay<1/25 cycle delay<1/4 cycle delay<1/2 cycledelay≥1/2 cycle RE directions--adjust signs of momenta so all collisions occur with returning electron traveling in the +z direction. For small delay times, almost all final z- momenta are opposite from the recollision direction. With increased delay times, there is increased spillover into the 2nd and 4th quadrants. I=6x10 14 W/cm 2

8. So… Q: When in the laser cycle do the recollisions and ionizations typically occur? –Recollision model: The most energetic recollision events occur just before a laser zero –[e.g. Corkum 71, 1994 (1993)] –Ionization: The confining potential-energy barrier is most suppressed when the field is maximal a quarter cycle later

9. When in laser cycle do recollisions and ionizations occur? Background curve shows laser cycle. Red--double ionization within 1/2 cycle of recollision and emergence in same momentum hemisphere Green--similar, but emerge in opposite momentum hemispheres Blue--remaining DI trajectories (i.e., delay time > 1/2 cycle). Collisions peak just before a zero of the laser. But Ionizations peak just before the laser reaches full strength.

10. Classical description of the DI process Up to about 15% of the time (depending on intensity), recollision leads nearly immediately to double ionization. Recollisions most often occur as laser field passes through zero; both electrons have small momentum immediately after collision and are pushed back opposite from the recollision direction Direction change after collision  the maximum drift momentum for either electron is (2Up) 1/2 Sample has I =4x10 14 W/cm 2

11. Classical description of the DI process Up to about 15% of the time (depending on intensity), recollision leads nearly immediately to double ionization. Recollisions most often occur as laser field passes through zero; both electrons have small momentum immediately after collision and are pushed back opposite from the recollision direction Direction change after collision  the maximum drift momentum for either electron is (2Up) 1/2 Sample has I =4x10 14 W/cm 2

12. In most cases there is a time lag between recollision and the ionization of the second electron If second electron ionizes before laser peaks then (to first approximation) it can follow the other electron out in the negative direction (opposite from the recollision direction) Energy (au) time (cycles) time lag for this trajectory is 0.18cycle

13. In most cases there is a time lag between recollision and the ionization of the second electron time lag for this trajectory is 0.18cycle Energy diagram (shows z only)

14. Here’s an example with a slightly longer time lag time lag for this trajectory is 0.22 cycle

15. If, to first approximation, second electron ionizes after the field peaks, the electrons can have drift velocities in opposite momentum hemispheres.

16. And, finally, sometimes the phase delay between recollision and ionization is > half a cycle. In that case the field ionization of the second electron is basically uncorrelated with the drift direction of the first. Other notes *Electron exchange occurs in about 1/3 the recollisions *Recolliding electron often misses on first return

17. Conclusions so far: The 3D ensemble method predicts population distributions in semi-quantitative agreement with experiment; The method indicates that there is typically a phase delay between recollision and double ionization and this phase delay is crucial in determining final electron correlations. Because of the direction change after recollision, maximum momentum is about (4U p ) 1/2, maximum energy about 2U p

18. Preliminary results for =390 nm Parker et al (PRL 96, 133001 (2006)) considered =390 nm –Total electron pair energy to about 5.3 U p –Experiment and 3-d quantum theory in agreement (4U p ) 1/2 Our results (I=1.1x10 15 W/cm 2, =390 nm ): 5.3U p 4U p I=1.1x10 15 W/cm 2 I=0.8x10 15 W/cm 2 Our classical ensemble also gives high-energy (E>2U p ; |p|>(4U p ) 1/2) electrons Electron momentum distribution, from their paper: Our result for =390 nm:

19. Minimum delay of at least 0.2 cycles between recollision and ionization We can back analyze the trajectories

20. Energy (au) prob density Final Energies of the two electrons- recolliding (blue) & “struck” (red) (for trajectories with a high energy electron) 2U p The high-energy electron is usually the struck electron

21. For high-energy electrons Recollision times: red--final momenta in same hemisphere w/in half cycle green--opposite hemisphere w/in half cycle blue--time delay of > 1/2 cycle Ionization Times:

22. The production of a high-energy (E>2U p ) electron

23. Conclusions for =390 nm Ensemble method gives electrons of energy >2U p The higher energy electron is most often the struck electron In our ensemble the high-energy electrons result from ionizations that feature the right phase match between motion of the electron in the nuclear well and the laser field