## Simulations using Supercomputers at CU Boulder

Theoretical research in strong-field and ultrafast physics is related to the solution of the time-dependent Schrödinger equation (TDSE) and Maxwell equations. For the problems of interest in our research there exist no analytical solutions of the corresponding equations. We therefore develop and apply methods to numerically solve the TDSE and Maxwell equations. To this end, we frequently use in our work the Summit (previously: Janus) supercomputer at CU Boulder. Besides a few start-up projects, allocations for the following projects were funded:

*Simulation of Ultrafast Atomic and Molecular Processes*
(Group project, since 06/2017 - 05/2017, 4.055M CPU hours)

*Calculating the time-dependent susceptibility of noble gases*
(Project leader: Andrew Spott, 10/2015 - 05/2017, 0.95M CPU hours)

*Attosecond time resolution of electron dynamics in physical and chemical processes*
(Project leader: Cory Goldsmith, 05/2015 - 05/2017, 1.0M CPU hours)

*Coherent attosecond soft X-rays with controlled polarization*
(Project leader: Carlos Hernandez-Garcia, 04/2015 - 05/2017, 3.0M CPU hours)

*Simulations of laser-induced electron dynamics in atoms and molecules*
(Project leader: Michelle Miller, 07/2015 - 06/2016, 1.0M CPU hours)

*Numerical simulations of the interaction of atoms and molecules with ultrashort intense laser pulses*
(Group allocation, 12/2013 - 05/2015, 5.7M CPU hours)

*Numerical simulations of the interaction of atoms and molecules with ultrashort intense laser pulses*
(Group allocation, 12/2012 - 12/2013, 2.7M CPU hours)

Results from numerical simulations using the Janus supercomputer are published in 23 peer-reviewed articles, various conference presentations, and contributed to the completion of 4 PhD thesis at CU Boulder (as of June 2017).

## Time-Dependent Density Functional Theory

In order to consider multielectron effects in strong-field ionization and high harmonic generation we make use of the time-dependent density functional theory (TDDFT) approach. The theory relies on the solution of the Kohn-Sham equations for the density of the noninteracting particles; correlation and exchange interactions are considered via different approximation methods. We have applied the TDDFT approach to analyze the importance of multielectron contributions in high-order harmonic generation in molecules.

Y. Xia and A Jaron-Becker, *Opt. Lett.* **39**, 1461 (2014)

## Numerical Basis State Method

In this method we numerically obtain the energy eigenstates of the field-free Hamiltonian in a box on the grid. The solution of the full time-dependent Schrödinger equation, including the interaction of the system with the external field, is then expanded in this numerical basis and propagated in time. Some observables, such as transition probabilities, can be easily obtained with this method and even monitored at certain times during the interaction. The method has been applied to recently observed phenomena related to strong-field excitation of atoms and calculations of the nonlinear susceptibilities - in collaboration with X. Gao (Beijing Computational Science Research Center, China) and J. Li (Shanghai Jiao Tong University and Tsinghua University, China).

S.H. Chen et al., *Phys. Rev. A* **86**, 013410 (2012)

## A Virtual AMO Lab

In the nonperturbative intensity regime exact solutions of the time-dependent Schrödinger equation of a few-body system interacting with an ultrashort intense laser pulse can be obtained by direct numerical integration. Such simulations distinguish themselves just in the respective Hamiltonian of the Schrödinger equation. This is the basis of our program package, which provides an unified basis for a number of strong-field problems. It consists of routines for the propagation of the wavefunction on the grid and the post-processing of the data. The result is a virtual lab for the analysis and visualization of few-body processes on an attosecond time scale. Applications for single-active-electron as well as correlated electron dynamics have been realized.

##
Intense-Field Many-Body *S*-Matrix Theory

The IM*S*T provides a systematic *ab-initio* approach to
investigate the dynamics of atoms and molecules interacting with
intense laser radiation. Structurally, the usual *S*-matrix
expansions, as the time-dependent perturbation theory, are based
on a single partition of the total Hamiltonian of the system into
an unperturbed reference Hamiltonian and the interaction potential.
Such an 'one-potential' scheme is not very useful for the analysis
of strong-field processes, in which the internal Coulomb interaction
between the charged particles in the atom or molecule and the
external laser-electron interaction energy are of comparable strength.
Thus, one requires to be able to account simultaneously of
different reference Hamiltonians in the initial, intermediate and
final states. The IM*S*T is such a more general *S*-matrix
expansion scheme. It provides an effective method for analyses of
direct and rearrangement processes that can occur in the presence
of intense laser fields.

**Review:** A. Becker and F.H.M. Faisal, *J. Phys. B* **38**, R1 (2005)