1. Imaging molecules with strong laser fields: successes and surprises Misha Ivanov

2. Olga Smirnova, MBI Berlin Ryan Murray, University of Waterloo Serguei Patchkovskii, NRC, Ottawa Michael Spanner, NRC Ottawa

3. A la carte Ionization in strong IR fields & electronic structure – the naïve view Tunnel ionization of small molecules revisited

4. Ionization in strong IR fields CO 2 Naïve view: Tunnel ionization should map out the orbital Agrees with experiment for HCl, N 2, O 2 – but not CO 2 -eF Dyson orbital for the CO 2 + ground state Fix the axes, Rotate F X2gX2g ~

5. Ionization in strong IR fields: Reality check Dyson orbital X2gX2g ~

6. Ionization in strong IR fields: Reality check X2gX2g ~ Dyson orbital Tunneling theory – dotted. Max @ 28 o Experiment: Max @ 45 o

7. Ionization in strong IR fields: Reality check Standard tunneling theory – blue dotted. Max @ 28 o TDSE at 800 nm – solid: Max @ 45 o : Same as experiment Is tunneling idea fundamentally flawed? Let us revisit the theory X2gX2g ~ Dyson orbital

8. The Talk Trailer* X2gX2g ~ Abu-samha & Madsen, PRA 2009 *: approved for all audiences Our results

9. A la carte Ionization in strong IR fields & electronic structure – the naïve view Tunnel ionization of small molecules revisited The basics Reminder: 1D WKB and the role of polarization 3D WKB: fitting a round peg into a square hole Results

10. A la carte Ionization in strong IR fields & electronic structure – the naïve view Tunnel ionization of small molecules revisited The basics Reminder: 1D WKB and the role of polarization 3D WKB: fitting a round peg into a square hole Results

11. 1. Match polarized bound state at some plane z=z 0 near the core 2. Outgoing wave at z=+ infinity E b = -  2 /2 Z Z0Z0 Outgoing wave Polarized bound state Tunnel Ionization as a boundary value problem

12. Tunnel Ionization Rate Ionization rate: current across a plane z=z* E b = -  2 /2 Z Z* The role of the bound wavefunction is taken by the Dyson orbital

13. Our approach to single channel ionization matching point Quantum Chemistry, polarized 3D WKB tunneling How to approach 3D tunneling in a simple way?

14. A la carte Ionization in strong IR fields & electronic structure – the naïve view Tunnel ionization of small molecules revisited The basics Reminder: 1D WKB and the role of polarization 3D WKB: fitting a round peg into a square hole Results

15. Simple Example: 1D tunneling, I E b = -  2 /2 Z Z0Z0 Outgoing wave We need to solve subject to the boundary conditions at z 0 and at +infinity WKB

16. Simple Example: 1D tunneling, II The WKB solution that matches the boundary conditions is E b = -  2 /2 Z Z0Z0 Outgoing wave where s(z,z 0 ) has to satisfy the HJ equation and the condition at z 0 WKB

17. Simple Example: 1D tunneling, III The solution of the HJ equation with this condition is E b = -  2 /2 Z Z0Z0 Outgoing wave where is velocity WKB

18. Result in 1D E b = -  2 /2 Z Z0Z0 Z ex z> z ex Need to know polarized bound state!

19. Role of polarization, I E b = -  2 /2 Z Z0Z0 Z ex Options: Correct: use polarized state – quantum chemistry for real systems Cheap: neglect polarization & use field-free bound state (??) Need to know polarized bound state!

20. E b = -  2 /2 Z Z0Z0 Z ex Polarization appears in the exponent! Field included Field NOT included Integrands are different, z 0 does not cancel Role of polarization, II Errors here

21. E b = -  2 /2 Z Z0Z0 Z ex The trick: Expand exponent in Fz 0 /  2 & keep only linear term. Neglecting polarization & getting away with it Effect scales with z 0 2 … can only be done analytically Example: short-range potential

22. If U(z>z 0 ) is small compared to I p =  2 /2 Long-range potentials If one does not include polarization, then one should use short-range part long-range correction

23. A la carte Ionization in strong IR fields & electronic structure – the naïve view Tunnel ionization of small molecules revisited The basics Reminder: 1D WKB and the role of polarization 3D WKB: fitting a round peg into a square hole Results

24. WKB in 3D? WKB in 3D is always a problem. What can we do here? P  ≠0 puts exponential penalty on the rate,  TUN is small Experiment: W(P  ) ~ exp[-P  2 /P 0 2 ] But the orbital is not For z>>1 U(r)-Fz is almost separable in z and  But the orbital is not  TUN z, F p≠p≠ How does one force separability onto an arbitrary  b ?

25. Example: the short-range potential The equation we need to solve … is separable in x,y,z. But our boundary condition  b is not. How does one force separability onto an arbitrary  b ? E b = -  2 /2 Z Z0Z0 Z ex

26. Forcing separability: Partial Fourier transform NB: The SE is linear Use 2D Fourier in x,y to re-write  b as sum of separable functions Step 1. Solve the SE for each with the boundary condition

27. Forcing separability: Partial Fourier transform Step 2: Full solution is The ionization rate can be calculated directly in the mixed space

28. Using Partial Fourier transform, I The Schrödinger Equation for yields effectively 1D equation for The only difference from before is higher Ip for tunneling with px, py≠0

29. Using Partial Fourier transform, II Then the answer is already known: where the tunneling amplitude a T depends on p x,p y via effective Ip tunneling time

30. Using Partial Fourier transform, III The wavefunction just after the barrier is And the ionization rate becomes Long-range correction: modify v(z’) to include the core potential

31. Results and Analysis Both coordinate and momentum matter! In weak fields exp(-P 2 perp  T ) filtering is severe:  T =[2Ip] 1/2 /F Tunneling is along the field, The rate is coordinate-domain dominated Z0Z0

32. Analysis Both coordinate and momentum matter! 2. For strong fields, filtering is not as severe – momentum features will start to show up

33. Predictions momentum coordinate Weak fields: rate follows coordinate Strong fields: rate mixes up coordinate and momentum

34. Example: CO 2 X2gX2g ~ Abu-samha & Madsen, PRA 2009

35. Results for C0 2 TDSE, numerics for 1600 nm 3D WKB, analytics

36. HOMO (X-channel) Results for N2 HOMO-1 (A-channel) Angle  deg Ionization probability I=8.10 13 W/cm 2 HOMO (X-channel) HOMO-1 (A-channel) I=1.7.10 14 W/cm 2 Angle  deg

37. The matching point and polarized wavefunction matching point Quantum Chemistry, polarized 3D WKB tunneling If we implement our approach numerically, polarized wf must be used. Then z 0 should drop out

38. Test with H2 It works! Ratio of 0 to 90 deg is 1.4

39. Simple analytical results Let Then  (  L )=  s,Atom R(  L ) TT

40. Limiting cases  (  L )=  s,Atom R(  L ) Neglect the deviation angle  T The orbital density is imaged directly TT

41. Limiting cases  (  L )=  s,Atom R(  L ) What about nodal planes? F 0 =0!! TT

42. Limiting cases  (  L )=  s,Atom R(  L ) What about nodal planes? Gives the PPT / Smirnov-Chibisov result for atoms Is equivalent to MO-ADK for molecules F 0 is the orbital, F 1 its derivative vs  TT

43. Full expression, strong fields  (  L )=  s,Atom R(  L ) R=

44. Conclusions Physically transparent theory for single – channel ionization in molecules Ionization images Dyson molecular orbitals in coordinate space if the fields are not too high. Close to barrier suppression intensities the momentum space features of the orbital become very important. CO2 experiments can be explained within the tunneling picture and a single-channel approximation Need to extend beyond barrier suppression and to oscillating fields

45. Tunnel Ionization as a boundary value problem with boundary conditions: Stationary SE (SSE) 1. Match to a polarized bound state at some plane z=z 0 near the core 2. Outgoing wave at z=+ infinity

46. Simple Example: 1D tunneling, II Using WKB approximation, we can write where s(z) is the reduced action, given by Full action: S (z,t)=-Et+s(z) E b = -  2 /2 Z Z0Z0 Outgoing wave and v(z)=∂s/∂z is velocity