1. Workshop finale dei Progetti Grid del PON
Study of the effects of nuclear motion on High Harmonic Generation in simple molecules
P. P. Corso1, G. Castiglia1, R. Daniele1, E. Fiordilino1, F. Morales2, G. Orlando1 , F. Persico1
1 Dipartimento di Scienze Fisiche ed Astronomiche - Università di Palermo and CNISM.
2 Dipartimento di Fisica e Tecnologie Relative - Università di Palermo and CNISM.
Workshop finale dei Progetti Grid del PON
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Catania,

2. Outline Introduction Theory and model Numerical approach
Physical Results
MATLAB on the COMETA Grid
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3. Introduction
The physics of laser–molecule interaction in the presence of intense laser pulses has recently undergone considerable advances particularly in highly nonlinear phenomena. Among these:
high-order harmonic generation (HHG);
molecular ionization, molecular dissociation, Coulomb explosion;
charge resonance enhanced ionization (CREI);
attosecond pulse generation.
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4. Introduction HHG from atoms
The radiation spectrum scattered by atoms is characterized by a wide plateau of odd harmonics of the laser frequency, followed by a sudden cutoff of the emitted radiation
plateau
cut-off
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5. Introduction HHG from atoms the position of the plateau is given by
where
IP is the ionization energy of the active electron;
Up is the laser ponderomotive energy;
wM can be as large as 300 wL, wL being the laser frequency
These features are essentially independent of the atomic species.
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6. Introduction HHG from molecules and molecular ions
In contrast, the radiation scattered from molecules lacks much of the above universality, as the larger number of molecular degrees of freedom leads to a strong dependence on the molecular species.
In particular, the spectra emitted by molecules are strongly affected by the positions and the mass of the nuclei and (for simple molecules) by the alignment angle of the molecules with respect to the laser polarization direction.
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7. Introduction
We have studied the dynamics of homonuclear diatomic molecules and molecular ions driven by a laser pulse of frequency wL ( nm) and peak electric field amplitude EL ( a.u.).
All the results presented have been obtained numerically solving the Time-Dependent Schroedinger Equation (TDSE) of the molecular systems.
The solution of molecular TDSE is obtained by means of parallel codes based on split-operator algorithms implemented on the COMETA grid.
The nuclear dynamics is investigated either in a semiclassical approximation, i.e. considering the nuclei as moving point-like classical particles, or in a full quantum framework.
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8. Introduction In particular we show that:
The laser field induces slow nuclear oscillations at frequency ωN;
Satellite and evenly spaced peaks of the odd harmonics of the laser frequency appear in the emitted spectra – this effect can be explained in terms of both classical and quantum model of the nuclei dynamics;
The height of the emitted plateau’s are strongly affected by the mass of the nuclei and by the duration of the laser pulse – this effect can only be explained in terms of a full-quantum approach;
HHG provides a valuable tool for observing of the internal degrees of freedom of the molecules.
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9. THEORY AND MODEL
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10. Theory and model
The TDSE for a diatomic molecule driven by an intense, linearly polarized, laser field is given by
where
coulomb interaction energy
interaction energy with external laser field in the dipole approximation
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11. Theory and model The analytical approach to the problem is very hard
In order to solve the TDSE, it is necessary to use a numerical approach
Basic approximations:
reduced dimensionality (1D model for both electron and nuclei coordinates)
soft-core potential
Screen parameter
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12. Theory and model The Schroedinger equation become:
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13. Theory and model One more approximation
Since the nuclear mass M is much larger than the mass m of the electron, for many purposes the nuclei can be considered as classical particles whose dynamics can be described by Newton equations.
With this assumption, the time evolution of the system is calculated by solving both the TDSE for the electron:
(in this case the Ri coordinates are treated as external parameters)
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14. Theory and model and the classical Newton equation for nuclear motion:
Nucleus–nucleus interaction force
Force exerted by the electron-cloud upon the n-th nucleus
Force acting upon the nuclei by the laser.
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15. Theory and model Emitted radiation
The laser field induces a strong acceleration on the electrons, which are stimulated to emit electromagnetic radiation; the radiated power is given by the Larmor formula
where ai(t) is the quantum-averaged acceleration of the i-th electron and can be obtained by using the Ehrenfest theorem. The spectrum of the emitted radiation is
with ãi(ω) the Fourier transform of the electron acceleration ai(t)
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16. Theory and model Physical idea
The acceleration is a local dynamical quantity, affected by the physical properties of the space where and when the charge is moving.
The emitted radiation should carry information on the behavior of the charge itself.
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17. NUMERICAL APPROACH
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18. Numerical approach
In order to solve the TDSE of a one-dimensional diatomic molecule, we have implemented a numerical parallel code based on the technique of the Split-Operator Method (SOM).
Let’s consider the following Schroedinger equation:
The SOM consists on approximating the time evolution of the wave equation by means of the following scheme:
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19. Numerical approach
This is accomplished by applying consecutively the following operators on the wave function:
Multiplication by a phase factor in coordinate space
Multiplication by a phase factor in momentum space
Multiplication by a phase factor in coordinate space
IFFT
FFT
Inverse Fast Fourier Transform
Fast Fourier Transform
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20. Numerical approach Ground state evaluation
GROUND STATE Φ0(R, x1, x2)
EIGENENERGY E0(R)
 depend on R
The equilibrium internuclear distance in the absence of the laser is given by the value of R, which minimizes the ground state energy of the molecule.
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21. IMAGINARY-TIME PROPAGATION
Numerical approach
Ground state evaluation
The generic eigenenergy value is obtained by solving the time-independent Schrödinger equation (TISE)
We have used the
IMAGINARY-TIME PROPAGATION
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22. Physical Results
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23. Physical Results: Satellite Peaks
1D semiclassical model of H2+ molecular ion
It is evident the presence of regularly spaced sidebands around the odd harmonic peaks
The satellite peak spacing is exactly given by the nuclei vibration frequency
Corso etal, J. Phys. B 40, 1383 (2007)
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24. Physical Results: Satellite Peaks
1D semiclassical model for various molecular isotopes
Different molecular isotopes are characterized by different satellite peak spacing
Each satellite peak spacing is directly related to the corresponding nuclear vibration frequency
Corso etal, J. Phys. B 40, 1383 (2007)
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25. Physical Results: Satellite Peaks
3D semiclassical model for various molecular isotopes
Corso etal, J. Phys. B 40, 1383 (2007)
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26. Physical Results: Satellite Peaks
1D semiclassical model for various H2 molecular isotopes
Morales etal, Laser Physics 18, 592 (2008)
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27. Physical Results: Isotopic Effect
1D full-quantum model for various H2+ molecular isotopes
T = 4 o.c.
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28. Physical Results: Isotopic Effect
1D full-quantum model for various H2+ molecular isotopes
T = 8 o.c.
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29. Physical Results: Isotopic Effect
1D full-quantum model for various H2+ molecular isotopes
T = 16 o.c.
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30. Physical Results: Isotopic Effect
1D full-quantum model for various H2+ molecular isotopes
T = 32 o.c.
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31. Physical Results: Isotopic Effect
1D full-quantum model for various H2+ molecular isotopes
T = 64 o.c.
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32. MATLAB on the COMETA Grid
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33. MATLAB on the COMETA Grid
On October 2008 the Physics Department of the University of Palermo has bought a composite MATLAB license consisting of:
5 concurrent licenses of MATLAB
5 concurrent licenses of MATLAB Parallel Computing Toolbox
128 licenses of MATLAB Distributed Computing Engine
These licenses have been installed on the COMETA Grid
In few days they will be available, for only Academic purposes to the whole COMETA community
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34. MATLAB on the COMETA Grid
In the following slide we show some very preliminary results about the scalability of a first porting in MATLAB of the semiclassical code for the H2+ molecular ion (moderate core-core communication over the Infiniband network)
These preliminary results clearly show how well MATLAB scales with the number of core involved in the computation
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35. MATLAB on the COMETA Grid
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36. MATLAB on the COMETA Grid
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37. MATLAB on the COMETA Grid
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38. MATLAB on the COMETA Grid
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39. Thank you for your kind attention !
Any questions ?
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