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Attosecond coupled electron-nuclear dynamics of H\(_2\) molecule under intense laser fields

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Abstract

Sequential double ionization and fragmentation dynamics of the H\(_2\) molecule exposed to an 750 nm, 4.5 fs elliptically polarized laser pulse is investigated by employing a quasi-classical model. In the model, momentum-dependent auxiliary potentials are added to the Hamiltonian to account for non-classical effects. Through theoretical exploitation of the molecular clock technique, the evolution of the vibrational wave packet of H\(_2^+\) formed by over-the-barrier ionization of the H\(_2\) molecule is tracked between the first and second ionization events with the temporal resolution of 140 attoseconds. The role of electron correlation in strong field ionization is captured. Our results show that the quasi-classical model is quite capable of describing and predicting light-induced multi-electron processes in the molecules. Our study provides a simple path of explaining and understanding the physical mechanism of the strong field multi-electron processes.

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Data Availability Statement

This manuscript has associated data in a data repository. [Authors comment:Data is available from the corresponding author on reasonable request. ].

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Acknowledgements

We acknowledge Dr. Diptesh Dey for the helpful discussion. This work is financially supported by Science and Engineering Research Board (SERB) New Delhi, India, through Project No. CRG/2020/000040.

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  1. Department of Chemical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur, 741246, India

    Gaurav Pandey & Ashwani K. Tiwari

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Correspondence to Ashwani K. Tiwari.

Appendix A: equilibrium configuration and sampling of initial conditions

Appendix A: equilibrium configuration and sampling of initial conditions

The equilibrium configuration of the H\(_2\) molecule is obtained by minimizing the quasi-classical Hamiltonian of H\(_2\) given in Eq. 1 of the main text. Minimization is performed by using the downhill simplex algorithm. The minimization procedure is repeated several times with various random initial positions and momenta to ensure the attainment of the global energy minimum.

The calculated ground-state energy of the molecule, \(E_\mathrm{initial} = -1.17283\) a.u. is in good agreement with the experimental value \(-1.16698\) a.u. Taking the obtained equilibrium configuration as an initial condition, a field trajectory is run for a sufficiently large time (more than 3 times of the simulation time in the presence of laser field) to check the stability of the molecule. The energy of the field-free molecule has been found to be constant with time which confirms that in the absence of the laser field, our molecule is stable and remains in the ground state. Fig. 6 shows the temporal evolution of the total energy of the molecule. A constant value of energy with time indicates that the molecule is stable when the laser field is not turned on.

Once the initial configuration of the molecule is obtained, since the system is stationary, vibrational energy is imparted to both the nucleus. By optimizing the position and momentum of the electrons for this new configuration of both the nucleus, the total energy of the system \((E_0)\) is computed. The new optimized configuration is taken as the initial condition for a field-free trajectory if \(0.006 \le (E_\mathrm{initial} - E_0) \ge 0.1 \). A field-free trajectory is then evolved in time with this initial condition. On this field-free trajectory, random points are taken for the position and momentum of the particles for the subsequent trajectories in the presence of a laser field. Such 200 field-free trajectories are run, and on each trajectory, 500 random points are chosen to generate the ensemble of \(10^5\) molecules.

One important feature of the quasi-classical Hamiltonian is that the Hamiltonian is invariant to the overall rotation of the position and momentum of all the particles, \(H({\varvec{r}}_1,{\varvec{p}}_1,{\varvec{r}}_2,{\varvec{p}}_2, {\varvec{r}}_{a},{\varvec{p}}_{a},{\varvec{r}}_{b},{\varvec{p}}_{b}) = H(\varvec{\Omega }_1 {\varvec{r}}_1,\varvec{\Omega }_1 {\varvec{r}}_2,\varvec{\Omega }_1 {\varvec{r}}_a,\varvec{\Omega }_1 {\varvec{r}}_b,\varvec{\Omega }_2 {\varvec{p}}_1,\varvec{\Omega }_2 {\varvec{p}}_2,\varvec{\Omega }_2 p_a,\varvec{\Omega }_2 {\varvec{p}}_b)\), where \(\Omega _1\) and \(\Omega _2\) are two sets of rotation matrices given by \(\begin{pmatrix} \cos \theta &{} \sin \theta \\ -\sin \theta &{} \cos \theta \\ \end{pmatrix}\), where \(\theta \) is chosen randomly from 0 to \(360^{\circ }\) for each trajectory. A random rotation matrix is applied to each trajectory before the run to generate the ensemble of randomly oriented molecules. Following this procedure, we obtained the initial ensemble of the randomly oriented molecules

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Pandey, G., Tiwari, A.K. Attosecond coupled electron-nuclear dynamics of H\(_2\) molecule under intense laser fields. Eur. Phys. J. D 76, 75 (2022). https://doi.org/10.1140/epjd/s10053-022-00401-x

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