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 "Ricerca" 2000-2006 - Avviso 1575 Catania, 10-12.02.2009

Outline Introduction Theory and model Numerical approach Physical Results MATLAB on the COMETA Grid Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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. Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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 Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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. Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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. Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

Introduction We have studied the dynamics of homonuclear diatomic molecules and molecular ions driven by a laser pulse of frequency wL (390-780 nm) and peak electric field amplitude EL (10-2-1 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. Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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. Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

THEORY AND MODEL Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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 Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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 Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

Theory and model The Schroedinger equation become: Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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) Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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. Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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) Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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. Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

NUMERICAL APPROACH Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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: Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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 Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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. Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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 Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

Physical Results Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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) Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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) Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

Physical Results: Satellite Peaks 3D semiclassical model for various molecular isotopes Corso etal, J. Phys. B 40, 1383 (2007) Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

Physical Results: Satellite Peaks 1D semiclassical model for various H2 molecular isotopes Morales etal, Laser Physics 18, 592 (2008) Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

Physical Results: Isotopic Effect 1D full-quantum model for various H2+ molecular isotopes T = 4 o.c. Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

Physical Results: Isotopic Effect 1D full-quantum model for various H2+ molecular isotopes T = 8 o.c. Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

Physical Results: Isotopic Effect 1D full-quantum model for various H2+ molecular isotopes T = 16 o.c. Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

Physical Results: Isotopic Effect 1D full-quantum model for various H2+ molecular isotopes T = 32 o.c. Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

Physical Results: Isotopic Effect 1D full-quantum model for various H2+ molecular isotopes T = 64 o.c. Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

MATLAB on the COMETA Grid Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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 Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

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 Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

MATLAB on the COMETA Grid Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

MATLAB on the COMETA Grid Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

MATLAB on the COMETA Grid Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

MATLAB on the COMETA Grid Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009

Thank you for your kind attention ! Any questions ? Catania, Workshop finale dei Progetti Grid del PON "Ricerca" 2000-2006 - Avviso 1575 - 10-12.02.2009