# Adiabatic theory of strong-field ionization of molecules with nuclear motion

Jens Svensmark

March 14, 2021

## Intro

### Strong-field ionization

• Assume field varies slowly
• Laser field momentarily constant
• Sequence of stationary Siegert states

### Objective

Develop adiabatic theory for molecules including nuclear motion

### Time-dependent SchrÃ¶dinger equation

\begin{align} i\frac{\partial }{\partial t} \Psi(x,R,t) & = [H_0+F(t) x] \Psi(x,R,t) \end{align}
\begin{align} H_0 = -\frac{1}{2m}\frac{\partial^2}{\partial x^2}-\frac{1}{2\mu} \frac{\partial^2}{\partial R^2} + V(x;R)+U_\text{ion}(R) \end{align}

### AAnf

Assuming $$\ T_\text{electron} \ll T_\text{field}, T_\text{nuclei}$$

Born-Oppenheimer ansatz

\begin{align*} \Psi_{\text{BO}}(x,R,t) = \Psi(R, t)\psi_e(x;R,F(t)) \end{align*}

Electronic Siegert state

\begin{align*} \left[H_x(F) - E_e(R,F)\right] \psi_e(x;R,F) = 0 \end{align*}

Nuclear wavefunction fulfills

\begin{align*} i\frac{\partial}{\partial t}\Psi(R, t) & = \left[-\frac{1}{2\mu} \frac{\partial^2}{\partial R^2} + E_e(R,F(t))\right] \Psi(R, t) \end{align*}

### Photo-electron momentum distribution (PEMD)

\begin{align*} P_v^{\text{AAnf}}(k) & = 2\pi \left|\sum_i \frac{e^{i\mathcal{S}_a(t_i,v; k)} g_v(t_i)}{\sqrt{|F(t_i)|}} \right|^2 \end{align*}
\begin{align*} g_v(t) = \int_0^\infty \chi_v (R) f(R,F(t)) \Psi(R,t) dR,\\ \psi_e(x;R,F)|_{|x|\to \infty} = f(R,F) f(x,E_e(R,F),F) \end{align*}

## Isotope effect

### Reflection approximation

Classical turning points

\begin{align*} U_\text{ion}(R_v)=\varepsilon_v \end{align*}

Channel ionization probabilities

\begin{align*} P_v^\text{ion} \propto |\Psi_0(R_v)|^2 \end{align*}

## Recap

• Extended adiabatic theory to molecules
Phys. Rev. A 101, 053422 (2020)
• Work in progress: include rescattering
• Possible next: include dissociation

### Acknowledgment

• Prof. T. Morishita
The University of Electro-Communications
• Prof. O. Tolstikhin
Moscow Institute of Physics and Technology
• Funding from JSPS

## Introduction

### What we want to look at

• BOA breaks down in weak-field limit for total rate
• WFAT gives total rate
• Adiabatic theory gives differential quantities
• Differential quantities might show other BO-breakdowns

## Numerical calculations

### Numerical methods

Split step fourier method

Scattering states found using R-matrix propagation

### Time-independent SchrÃ¶dinger equation

Scattering boundary conditions

\begin{align} H_0 \Psi(x,R) & = E \Psi(x,R) \end{align}

### Boundary conditions

\begin{align} \Psi_-^{\text{out}}(x,k) & = \begin{cases} e^{-ikx} - r^* e^{ikx} & x\leq x_-\\ t^* e^{-ikx} & x\geq x_+ \end{cases} \end{align}

$$\Psi_+^{\text{out}}(x,k)$$

### Scattering solution method

• R-matrix propagation in $$x$$
• Adiabatic basis in $$R$$ for fixed $$x$$
• Sectorized legendre DVR in $$x$$
• Sine-DVR in $$R$$

### Different grids

Electronic hamiltonian

$$H_x(F) = -\frac{1}{2}\frac{\partial^2}{\partial x^2} + Fx + V\left(x; R\right)$$

## One cycle pulse

### Pulse

\begin{align*} P_v^{\text{AAnf}}(k) & = \left|\sum_i \frac{\sqrt{2\pi} e^{i\pi/4} }{\sqrt{s_i qF(t_i)}} e^{i\mathcal{S}_a(t_i,v; k)} R_v(t_i) \right|^2 \end{align*}

### Build the PEMD

\begin{align*} P_v^{\text{AAnf}}(k) & = \left|\sum_i \frac{\sqrt{2\pi} e^{i\pi/4} }{\sqrt{s_i F(t_i)}} e^{i\mathcal{S}_a(t_i,v; k)} R_v(t_i) \right|^2 \end{align*}