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

Jens Svensmark

June 25, 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

• First half of talk: adiabatic part [PRA 101, 053422 (2020)]
• Second half of talk: rescattering

### 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*} \small{\left[-\frac{1}{2}\frac{\partial^2}{\partial x^2} + V(x;R) + F x - E_e(R,F)\right] \psi_e(x;R,F) = 0} \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*} \small{\left[-\frac{1}{2}\frac{\partial^2}{\partial x^2} + V(x;R) + F x - 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

## Rescattering

### One-cycle pulse

\begin{align*} F(t) & = - F_0 \sqrt{2e}\tau \exp[-\tau ^2] \\ \tau &=\frac{2t}{T} \end{align*}
\begin{align*} v_0 & = \tfrac{\sqrt{2e}}{4} F_0T \\ r_0 & = T v_0 \end{align*}

### Rescattering contributions

\begin{align*} I_v(k) = I_v^{a}(k) + %I_v^r(k) I_v^{r,I}(k) + I_v^{r,S}(k) \end{align*}
\begin{align*} I_v^{r,S}(k) = \left. \sum_{\pm,i,r} p_{\pm}^{r,S}(t) \phi_{v, i}(t) \exp\left[i \mathcal{S}^{\text{elec}}(t;k) + i\varepsilon_v t\right] \right|_{t=t_{r, \pm}} \end{align*}
\begin{align*} \phi_{v, i}(t) = \sum_{v'} e^{- i\varepsilon_{v'} (t - t_i) } {\color{#ec6800}{S_{v v'}^{ \pm \text{sgn}(u_f(t))}(u_f(t))}} %{\color{#1e50a2}{S_{v v'}^{ \pm \text{sgn}(u_f(t))}(u_f(t))}} g_{v'}(t_i) \end{align*}

## Wave packet reconstruction

But before that, BO for scattering states

### Wanted

Has anybody here ever used BO for scattering states?

\begin{align} S_{v' v}^{\pm,\text{BO}}(k) = \int_0^{\infty} \chi_{v'}(R) S^{\pm}(k; R) \chi_{v}(R) dR \end{align}

### Nuclear wave packet

\begin{align*} I_v^{r,S}(k) = \left. \sum_{\pm,i,r} p_{\pm}^{r,S}(t) \phi_{v, i}(t) \exp\left[i \mathcal{S}^{\text{elec}}(t;k) + i\varepsilon_v t\right] \right|_{t=t_{r, \pm}} \end{align*}
\begin{align*} \phi_{v,i}(t) = \int_{0}^\infty \chi_{v}(R) S^{\pm \text{sgn}(u_f(t))}(u_f(t); R) g^{\text{wp}}(R, t) dR \end{align*}
\begin{align*} g^{\text{wp}}(R, t) = e^{- i H_R (t-t_i)} [ f(R,F(t_i)) \Psi(R, t_i) ]. \end{align*}

### Sum over trajectories

\begin{align*} I_v^{r,S}(k) = \left. {\color{#ec6800}{\sum_{\pm,i,r}}} p_{\pm}^{r,S}(t) \phi_{v, i}(t) \exp\left[i \mathcal{S}^{\text{elec}}(t;k) + i\varepsilon_v t\right] \right|_{t=t_{r, \pm}} \end{align*}

### PEMD factorization

\begin{align*} P_v^{r, c}(k) & = |I_v^{r, c}(k)|^2 \\ & = p_c^2 \text{Ai}^2(z(k)) |\phi_{v,i}(\tilde{t}_{r,c}, t_{r,c})|^2 \end{align*}

## Recap and outlook

• 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

Any questions?

## BO for scattering states

### Usual BO

\begin{align*} \Phi_0^{\text{BO}} (x,R)=\Phi_0 (R)\phi_e (x;R), \end{align*}
\begin{align*} \left[- \frac{1}{2} \frac{\partial^2}{\partial x^2} + V\left(x; R\right) - E_e(R)\right] \phi_e(x;R) = 0 \end{align*}
\begin{align*} \left[ - \frac{1}{2\mu} \frac{\partial^2}{\partial R^2} + U_\text{mol}(R) - E^\text{BO}_0\right] \Phi_0(R) = 0 \end{align*}
\begin{align*} U_\text{mol}(R) = U_\text{ion}(R) + E_e(R) \end{align*}

### BO for scattering

\begin{align*} \left[- \frac{1}{2} \frac{\partial^2}{\partial x^2} + V\left(x; R\right) - E_e(R)\right] \phi_e(x;R) = 0 \end{align*}

No discretization, nothing to fix $$E_e(R)$$

### BO for scattering

H. Gao and C. H. Greene, PRA 42, 6946 (1990)

\begin{align*} \tilde{U}_\text{mol}(R) = U_\text{ion}(R) + E_e \end{align*}
\begin{align*} \left[ - \frac{1}{2\mu} \frac{\partial^2}{\partial R^2} + \tilde{U}_\text{mol}(R) - E^\text{BO} \right] \phi_v(R) = 0 \end{align*}
\begin{align*} \left[- \frac{1}{2} \frac{\partial^2}{\partial x^2} + V\left(x; R\right) - E_{e, v}\right] \phi_{e, v}(x;R,k) = 0 \end{align*}

## 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*}