1. Electron wavefunction in strong and ultra-strong laser field One- and two-dimensional ab initio simulations and models Jacek Matulewski Division of Collision Theory and Nonlinear Systems Faculty of Physics, Astronomy and Informatics Nicolaus Copernicus University in Toruń, Poland Marburg, 20 I 2004

2. 2/38 Outline 1. Stabilisation and related phenomena in 1D strong field ionisation 2. Beyond the dipole approximation - magnetic drift in 2D ultra-strong field 3. Brief report on other projects (control of wavefunction by pulses) Authors: Andrzej Raczyński, Jarosław Zaremba and Jacek Matulewski

3. 3/38 Outline 1. Stabilisation and related phenomena in 1D strong field ionisation 2. Beyond the dipole approximation - magnetic drift in 2D ultra-strong field 3. Brief report on other projects (control of wavefunction by pulses)

4. 4/38 Language Space built with : Time depended quantities we’re interested in Initial state = ground bound state of quantum system

5. 5/38 Simulation in 1D - typical result for strong laser field Evolution of wavefunction (fast oscillations with frequency of laser field was removed) Potential well: 2a = 0.244 a.u. = 1.3 ·10 -11 m V 0 = 2.049 a.u. = 55.7 eV Electric field:  = 1 a.u. = 6.6 ·10 15 s -1  = 1 a.u. = 5.1 ·10 11 V/m (I =  2 = 10 16 W/cm 2 ) Phase and pulse shape must be chosen carefully because of possibility of electron escape due to fast drift (Newton’s equations)

6. 6/38 Simulation in 1D - typical result for strong laser field Population of initial state: Photoelectron spectra: ATI phenomena fast oscillations

7. 7/38 Simulation in 1D - slow drift and stabilisation Evolution of wavefunction in  = 5 a.u. (precisely: of its part located near potential well) Wavefunction properties slowly changes position remains its shape

8. 8/38 Simulation in 1D - slow drift and stabilisation Some photos of wavefunction in  = 5 a.u. (precisely: of its part located near potential well) Wavefunction properties slowly changes position remains its shape The idea is to find the potential which keeps shape of the wavefunction unchanged and model its movement due to perturbation of free oscillating electron by well potential

9. 9/38 Simulation in 1D - Generalised KH model Phys Rev A 61, 043402 (2000) Transformation to Kramers frame, where and is putted together fast oscillations and slow drift Spreading of oscillating potential around (x 0 is average value of  0 (t)) One dimensional Schrödinger equation for one electron in laser field (dipole approximation):

10. 10/38 Simulation in 1D - Generalised KH model Phys Rev A 61, 043402 (2000) Approximation: Function describing slow drift is well fitted by solution of Ehrenfest equation The reason of slow drift is interaction of wavefunction with the well edges Replacing time depended potential with HK well which is zeroth term of its Fourier expansion: In our parameters HK well has only one bound state

11. 11/38 One period of oscillations - marks of the interaction with the potential well Simulation in 1D - Generalised KH model (results) Phys Rev A 61, 043402 (2000)

12. 12/38 Simulation in 1D - Generalised KH model (results) Phys Rev A 61, 043402 (2000) Position of the electron

13. 13/38 Simulation in 1D - Generalised KH model (results) Phys Rev A 61, 043402 (2000) Occupation of the potential well ground state (initial state)

14. 14/38 Simulation in 1D - Generalised KH model (results) Phys Rev A 61, 043402 (2000) What else influence the final population of the initial state? The way of switching on the pulse Time of live in HK-well (Volkova et all, Zh. Eksp. Teor. Fiz. 106, 1360 (1994)) Transformed back to laboratory frame wavefunction of HK eigenstate Slow drift and fast oscillations Stabilisation is not permanent because of HK state is ionised (higher terms of Fourier expansion)

15. 15/38 1. Stabilisation in Kramers- Henneberger well 2. Slow drift of almost free electrons (interaction with well) Simulation in 1D - Summary Phys Rev A, 61, 043402 (2000)

16. 16/38 Outline 1. Stabilisation and related phenomena in 1D strong field ionisation 2. Beyond the dipole approximation - magnetic drift in 2D ultra-strong field 3. Brief report on other projects (control of wavefunction by pulses)

17. 17/38 Typical simulation set in 3D: wave propagation direction: y electric field polarisation direction: x magnetic field polarisation direction: z

18. 18/38 Atom + laser field in 3D without dipole approximation: classical description of free electron (Lorentz force) quantum description of atom in classical external field dipole approximation Atom + pole lasera w 1D: stabilisation - HK well slow drift Atom + laser field in 2D: stabilisation - HK well in 2D magnetic drift Omitting free evolution in z direction Simplifying the y dependence of hamiltonian Possible description approaches

19. 19/38 Classical description in 3D (de facto 2D) Electromagnetic field:Lorentz force acting on free electron: Equations of motion: and trivial equation for z(t)

20. 20/38 Classical description in 3D (de facto 2D) Solution: x(t) and y(t)y(t) Equations of motion:

21. 21/38 Quantum description in 3D in Coulomb gauge Schrödinger equation (omitting evolution in z direction): Dipole approximation + unitary transformation One-dimensional calculations dE (oscillations, stabilisation, slow drift) Two-dimensional calculation pA (no dipole approximation!!!)

22. 22/38 Solutions of two-dimensional Schrödinger equations in pA gauge, no dipole approximation, ultra-strong laser field Stabilisation (HK well) Magnetic drift in propagation direction atom: radial well a = 1 a.u., V 0 = 2 a.u. field:  = 1 a.u., E 0 = 15 a.u. Reiss, Phys. Rev 63 013409 (2000)

23. 23/38 Wavefunction - phenomenology of stabilisation Phys Rev A 68, 013408 ( 2003 ) atom: radial well a = 1 a.u., V 0 = 2 a.u. field:  = 1 a.u. two scattering centres in x = 0 and x = 30 a.u. E 0 = 15 a.u. E 0 = 20 a.u.4T, 7T, 10T,

24. 24/38 Characteristics of wavefunction 1. Regardless the phase of laser field magnetic drift is always directed to +y 2. Drift is linear in time + 2  oscillations 3. Constant drift velocity depends on E 0 2 4. Wave of electron finding probability has double frequency in y direction Wavefunction - phenomenology of stabilisation Phys Rev A 68, 013408 ( 2003 ) E 0 = 20 a.u.

25. 25/38 (1) Schrödinger equation with no dipole approximation (pA gauge) Quantum model of stabilisation and magnetic drift in 2D Simplifying the hamiltonian spreading of vector potential in series Patel et al., Phys.Rev.A 64 013411 (2) Lowest order approximation of nondipol Schrödinger equation (additional term proportional to y in hamiltonian describing magnetic field) (3) Equation in Kramers frame with additional magnetic field transformation to frame of oscillating electron transformation removing A 2 term averaging of oscillating potential (4) Electron motion equation in HK well and coupling with continuum by magnetic field

26. 26/38 Quantum model of stabilisation and magnetic drift in 2D Looking for hamiltonian with simplier dependence on y (1) In high frequency regime the distribution of wavepacket in momentum space is concentrated around the zero (2) Term describing magnetic field

27. 27/38 (2) Quantum model of stabilisation and magnetic drift in 2D Transformation to the Kramers frame Transformation removing the A 2 term Transformation to frame moving with an electron In Kramers frame potential oscillates (3)

28. 28/38 Quantum model of stabilisation and magnetic drift in 2D Fourier expansion of the time depended potential of the oscillating well In high frequency regime one can replace fast oscillating potential with time averaged potential of Kramers-Henneberger well: (4) „typical” time independed part of hamiltonian with bounding potential Term similar to dE (but here all in pA) where and

29. 29/38 Quantum model of stabilisation and magnetic drift in 2D Bounded states of Kramers-Henneberger well V HK (x, y) in 2D Potential and ground state (E 1 = – 0.0192 a.u.). Only excited state (E 2 = – 0.0158 a.u.) E 0 = 15 a.u. Eigenvalue problem:

30. 30/38 Quantum model of stabilisation and magnetic drift in 2D HK well bound states coupling with continuum (4) Approximated hamiltonian for linear polarised laser field propagating in y direction: Characteristics of wavefunction 1. Regardless the phase of laser field magnetic drift is always directed to +y 2. Drift is linear in time + 2  oscillations 3. Constant drift velocity depends on E 0 2 4. Wave of electron finding probability has double frequency in y direction Classical model in 2D/3D Dipole approxination (regarding y dependence in E x (y,t) and B z (y,t))

31. 31/38 Stabilisation and magnetic drift in ultra-strong laser field Summary of 2D simulations Probability of finding electron inside the o box with size 100 a.u. around the well Field intensity with dipole app. without dipole app. Ryabikin, Sergeev, Optics Express 417, 7 12 (2000)

32. 32/38 Saving the stabilisation using constant magnetic field for quantum model and details see Phys Rev A 68, 045401 ( 2003 ) Wavefunction without and with constant magnetic field

33. 33/38 Outline 1. Stabilisation and related phenomena in 1D strong field ionisation 2. Beyond the dipole approximation - magnetic drift in 2D ultra-strong field 3. Brief report on other projects (control of wavefunction by pulses)

34. 34/38 Outline 1. Stabilisation and related phenomena in 1D strong field ionisation 2. Beyond the dipole approximation - magnetic drift in 2D ultra-strong field 3. Brief report on other projects (control of wavefunction by pulses)

35. 35/38 Other projects (quantum engineering)

36. 36/38 Control of ionisation of hydrogen atom by two-colour pulse Physics Letters A 25 205-211 ( 1999) detuning

37. 37/38 Control of Rydberg atom by chirped pulse Phys Rev A 57, 4561 (1998) Chirped pulse: By changing the depth of modulation one can control the coupling amplitudes depending on  0 =  13 = 0.518 a.u.  1 =  34 = 0.0587 a.u.  = 0.05 a.u. Rochester atom:

38. 38/38 Control of Rydberg atom by chirped pulse Phys Rev A 57, 4561 (1998) Chirped pulse: (no modulation) Photoelectron spectra

39. 39/38 Control of Rydberg atom by chirped pulse Phys Rev A 57, 4561 (1998) Chirped pulse: Bound states population

40. 40/38 Outline 1. Stabilisation and related phenomena in 1D strong field ionisation 2. Beyond the dipole approximation - magnetic drift in 2D ultra-strong field 3. Brief report on other projects (control of wavefunction by pulses)