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

Jens Svensmark

January 21, 2020

PhD

Postdoc

Postdoc

## Intro

• Previously been used for atoms
• Here we look at molecules

### Time scales

• Laser
• Motion inside molecule

• Laser field momentarily constant
• Sequence of stationary Siegert states

### Previous work

• Looked at atoms/fixed nuclei
• Electrons are fast
• Laser is slow

PRA 86, 043417 (2012)

PRA 92, 043402 (2015)

PRL 116, 173001 (2016)

### What we want to do

Extend adiabatic theory to molecules including nuclear motion

## Numerical calculations

### Time-dependent Schrödinger 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}

### Numerical methods

Split step fourier method

Scattering states found using R-matrix propagation

## Fixed nuclei example

$$\epsilon = \frac{T_\text{e}}{T_\text{f}} \to 0$$

Photo electron momentum distribution (PEMD)

\begin{align} P(k) & = \frac{2\pi}{\left| F(t_i) \right|} \left| f(F(t_i)) \right|^2 e^{- \int_{-\infty}^{t_i} \Gamma(F(t)) dt} \end{align}

$$k:$$ Momentum of outgoing electron

$$t_i$$ is time at which electron with momentum $$k$$ ionized, $$k = -\int_{t_i}^{\infty} F(t')dt'$$

### Pulse

$$\epsilon = \frac{T_\text{e}}{T_\text{f}} \to 0$$

Photo electron momentum distribution (PEMD)

\begin{align} P(k) & = \frac{2\pi}{\left| F(t_i) \right|} \left| f(F(t_i)) \right|^2 e^{- \int_{-\infty}^{t_i} \Gamma(F(t)) dt} \end{align}

$$k:$$ Momentum of outgoing electron

$$t_i$$ is time at which electron with momentum $$k$$ ionized, $$k = -\int_{t_i}^{\infty} F(t')dt'$$

$$f(F), \Gamma(F):$$ Properties of Siegert state

### Siegert state

Solution of time independent Schrödinger equation

\begin{align} [H_0 + Fx] \Phi(x, R; F) & = E(F) \Phi(x, R; F) \end{align}

with outgoing-wave boundary conditions.

### Outgoing wave

Photo electron momentum distribution (PEMD)

\begin{align} P(k) & = \frac{2\pi}{\left| F(t_i) \right|} \left| f(F(t_i)) \right|^2 e^{- \int_{-\infty}^{t_i} \Gamma(F(t)) dt} \end{align}

$$k:$$ Momentum of outgoing electron

$$t_i$$ is time at which electron with momentum $$k$$ ionized, $$k = -\int_{t_i}^{\infty} F(t')dt'$$

$$f(F):$$ Asymptotic wave function coefficient

$$\Gamma(F):$$ Total rate

### Weak field limit

For $$F\to 0$$ adiabatic theory reduces to Keldysh theory

### Nuclear-electron interaction

\begin{align} V(x; R) = V\left(x+\frac{1}{2}R\right) + V\left(x-\frac{1}{2}R\right) \end{align}

Finite range potential

\begin{align} V(x)&=-\frac{a}{\cosh^2(bx)}, \end{align}

with $$a = 0.62772$$ and $$b = 0.857$$.

### Gaussian half-cycle pulse

\begin{align} F(t) & = F_0\exp\left(-\left[\frac{2t}{T}\right]^2\right) \end{align}

## Including nuclear motion

### Nuclear-nuclear interaction

H$$_2^+$$ like interaction

\begin{align} U_\text{ion}(R) & = \frac{A}{R^2} + B + C R^2 \end{align}

with $$A=0.26, B =-0.732635$$ and $$C =0.01625$$

$$R$$ is the internuclear seperation

### Vibrational states, molecular ion

\begin{align*} \left[-\frac{1}{2\mu} \frac{d^2}{dR^2} + U_\text{ion}(R)\right] \chi_v(R) = \varepsilon_v \chi_v(R) \end{align*}

$$\epsilon_\text{f} = \max\left(\frac{T_\text{e}}{T_\text{f}}, \frac{T_\text{n}}{T_\text{f}}\right) \to 0$$

PEMD (for half-cycle pulse)

\begin{align*} P_v(k) & = \frac{2\pi}{m^2} \frac{|f_{v}(F(t_i))|^2}{ |F(t_i)|} e^{-\int_{-\infty}^{t_i} \Gamma(F(t')) dt'} \end{align*}

Total PEMD

\begin{align*} P(k) = \sum_v P_v(k) \end{align*}

## Slow nuclei and field (AAnf)

### AAnf

$$\epsilon_\text{nf} = \max\left(\frac{T_\text{e}}{T_\text{f}}, \frac{T_\text{e}}{T_\text{n}}\right) \to 0$$

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 $$\psi_e(x;R,F(t))$$

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

where

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

### AAnf

Born-Oppenheimer ansatz

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

### PEMD

\begin{align*} P_v^{\text{AAnf}}(k) & = \frac{2\pi }{|F(t_i)|} \left| g_v(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, \end{align*}
\begin{align*} \psi_e(x;R,F)|_{|x|\to \infty} = f(R,F) f(x,E_e(R,F),F) \end{align*}

### Regions of applicability

AAf

\begin{align*} \epsilon_\text{f} = \max\left(\frac{T_\text{e}}{T_\text{f}}, \frac{T_\text{n}}{T_\text{f}}\right) \to 0 \end{align*}

AAnf

\begin{align*} \epsilon_\text{nf} = \max\left(\frac{T_\text{e}}{T_\text{f}}, \frac{T_\text{e}}{T_\text{n}}\right) \to 0 \end{align*}

### Small nuclear masses

$$T=30$$ a.u.

### Small nuclear masses

$$T=150$$ a.u.

## Few-cycle pulse

### Pulse

\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)} g_v(t_i) \right|^2 \end{align*}

## Many-cycle pulse

### Many-cycle model

\begin{align} F(t) = F_0 \cos \omega t \end{align}
\begin{align} P_{v}^{\text{AAnf}}(k) \propto % 2\pi \frac{\left|\tilde{g}_{v}(t_{(0, 1)})\right|^2}{\left|F(t_{(0, 1)})\right|} \left|p^{\text{inter}}(k)\right|^2 \left|p^{\text{intra}}(k)\right|^2 \end{align}

Similar to SFA based model in PRA 93, 031401

### Inter-cycle factor

\begin{align} \left|p^{\text{inter}}(k)\right|^2 = e^{-c(k)} \left|\frac{\sin^2 [N d(k)]}{\sin^2 d(k)}\right| \end{align}
\begin{align} d(k) & = \pi \omega^{-1} \Delta E(k) \color{gray} { + \frac{1}{2}s_0^{\text{cycle}}}\\ \Delta E(k) & = E_0^{\text{BO}}-\varepsilon_v - \frac{1}{2 m} k^2 - U_p \end{align}

Ponderomotive energy: $$U_p = \frac{F_0^2}{4m\omega^2}$$

### Intra-cycle factor

\begin{align} p^{\text{intra}}(k) = 2 \cos \frac{\Delta\mathcal{S}_a(k)}{2} \end{align}
\begin{align} \Delta \mathcal{S}_a(k) = & - \omega^{-1} \Delta E(k) \left[-2\sin^{-1} \frac{k\omega}{F_0}+\text{sgn}(k) \pi\right] \\ & - \frac{3}{2}k \frac{F_0}{m\omega^2} \sqrt{1 - \left(\frac{k\omega}{F_0}\right)^2}- \Delta s_0 \end{align}

## Recap

• AAf only works for small nuclear masses in slow fields
• AAnf works for large nuclear masses in slow fields
• Possible next: include rescattering
• Possible next: include dissociation

## Deleted scenes

### 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

#### Time-independent Schrödinger 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 $$R$$

## 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*}