Bogolyubov’s averaging applied to the Kramers-Henneberger Hamiltonian

• KHBogolyubov.mlx: Matlab livescript for the manuscript Bogolyubov’s averaging theorem applied to the Kramers-Henneberger Hamiltonian by E. Floriani, J. Dubois, C. Chandre

Reference: E. Floriani, J. Dubois, C. Chandre, Bogolyubov's averaging theorem applied to the Kramers-Henneberger Hamiltonian, Physica D 431, 133124 (2022); arxiv:2107.01946

@article{floriani2021,
         title = {Bogolyubov's averaging theorem applied to the Kramers-Henneberger Hamiltonian}, 
         author = {Floriani, E. and Dubois, J. and Chandre, C.},
         journal = {Physica D},
         volume = {431},
         pages = {133124},
         year = {2022},
         doi = {10.1016/j.physd.2021.133124},
         URL = {https://doi.org/10.1016/j.physd.2021.133124}
}

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Time-dependent Schrödinger equation in the dipole approximation (TDSE)

Numerical integration of the following Schrödinger equation (in the dipole approximation and in atomic units)

$$i \frac{\partial \psi}{\partial t} = \left( -\frac{\Delta}{2} + V(x) + x E(t) \right) \psi(x,t),$$

where $E(t)=E_0 f(t) \Phi(\omega t)$ with $f(t)$ the laser envelope, and $\Phi$ a $2\pi$-periodic function. The frequency $\omega$ is defined by the laser wavelength, and the amplitude of the electric field $E_0$ is defined by the laser intensity. Here $V$ is the ionic potential.

• TDSE_params.py: to be edited to change the parameters of the TDSE computation (see below for a list of parameters)

• TDSE_classes.py: contains the TDSE class and main functions

• TDSE.py: contains the methods to execute TDSE

Once TDSE_params.py has been edited with the relevant parameters, run the file as

python3 TDSE.py

or

nohup python3 -u TDSE.py &>TDSE.out < /dev/null &

The list of Python packages and their version are specified in requirements.txt

Parameters

The file TDSE_params.py should contain the following parameters:

• Method: string; 'eigenstates', 'wavefunction', 'HHG', 'ionization'; choice of method
  • 'eigenstates': plot the first k eigenstates and eigenvalues of the potential specified in InitialState[1], where k is equal to InitialState[0]+1
  • 'wavefunction': displays the wavefunction as a function of time obtained by integrating the TDSE equation
  • 'HHG': compute the high-harmonic generation (HHG) spectrum as a function of time
  • 'ionization': computes the ionization probability as well as displaying the wavefunction as a function of time
  • 'Husimi': computes the Husimi representation of the wavefunction as a function of time; 'p_husimi' and 'sigma_husimi' need to be defined
• laser_intensity: float; intensity of the laser field in W cm^-2
• laser_wavelength: float; wavelength of the laser field in nm
• laser_E: lambda function returning an array of n floats; n components (where n is the dimension of configuration space) of the electric field (dipole approximation); the electric field is then given by E0 * laser_envelope(t) * laser_E(ω t) where E0 = sqrt(laser_intensity)
• te: array of 3 floats; duration of the ramp-up, plateau and ramp-down in laser cycles
• V: lambda function; ionic potential
• InitialState: integer or array [integer or tuple of integers or lambda function, string]; integer = index of the initial eigenstate (0 corresponds to the ground state, 1 is the first excited state...); string = potential with which the initial state is computed ('V', 'VKH2' or 'VKH3'); in case a tuple of integers is entered, the initial state is a linear combination of the various states in the tuple with the coefficients given in InitialCoeffs; if InitialState or InitialState[0] is a lambda function, the initial state is computed on the grid using this function
• L: array of n floats; size of the box in each direction
• N: array of n integers; number of points in each direction

Some additional (optional) parameters could be defined in TDSE_params.py:

• laser_envelope: string; 'trapez', 'sinus', 'const'; envelope of the laser field during ramp-up and ramp-down
• InitialCoeffs: array of floats; the initial state is a linear combination of eigenstates $\Psi_k(x)$ of the potential defined in InitialState[1], i.e., $\psi(x,0)=\sum_k c_k \Psi_k(x)$ where $k$ belongs to InitialState[0]; if not specified, the coefficients are equal to 1
• DisplayCoord: string; 'lab', 'KH2' or 'KH3'; if KH (Kramers-Henneberger), the wave function is moved to the KH frame (for display and for saving) of order 2 or 3; if not specified, 'lab' is the default
• delta: float or array of n floats; size of the absorbing boundary in each direction (if float, the size is taken equal in all dimensions)
• Lg: float or array of n floats; size of the box for the initial computation of the initial state along each dimension; if float, [-Lg, Lg] in each dimension; if not specified, Lg=L
• nsteps_per_period: integer; number of steps per laser period for the integration; the time-step is then defined as 2π /ω / nsteps_per_period
• scale: string; 'linear' or 'log'; the axis scale type to apply for the representation of the wavefunction (if Method='wavefunction')
• legend: string; location of the legend; for more details, see matplotlib legend
• xlim: tuple of floats; x-axis view limits (in atomic units)
• ylim: tuple of floats or string; y-axis view limits (in atomic units); if 'auto', let the y-axis scale automatically
• figsize: tuple of floats; width and height in inches of the figure
• SaveWaveFunction: boolean; if True, saves the animation of the wavefunction as an animated .gif image
• PlotData: boolean; if True, displays the wavefunction on the screen as time increases (only for 1D and 2D)
• SaveData: boolean; if True, the time evolution of the wave function are saved in a .mat file
• dpi: integer; number of dots per inch for the movie frames (if SaveWaveFunction is True)
• refresh: integer; the wavefunction is displayed every refresh time steps
• darkmode: boolean; if True, plots are done in dark mode
• tol: relative accuracy for eigenvalues (stopping criterion) (default=10^-10, 0 implies machine precision); see eigsh
• maxiter: maximum number of Arnoldi update iterations allowed (default=1000); see eigsh
• ncv: number of Lanczos vectors generated (default=100); see eigsh
• Nkh: integer; number of points in one period to compute the Kramers-Henneberger potentiel V_KH(x) (default=2¹²)
• ode_solver: string; choice of splitting symplectic integrator; for a list see pyHamSys (default='BM4')

Reference: E. Floriani, J. Dubois, C. Chandre, Scars of Kramers-Henneberger atoms, arxiv:2407.18575

@misc{floriani2024,
      title={Scars of Kramers-Henneberger atoms}, 
      author={Elena Floriani and Jonathan Dubois and Cristel Chandre},
      year={2024},
      eprint={2407.18575},
      archivePrefix={arXiv},
      primaryClass={nlin.CD},
      url={https://arxiv.org/abs/2407.18575}, 
}

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For more information: cristel.chandre@cnrs.fr