Adiabatic theory of strong-field ionization of molecules with nuclear motion
Jens Svensmark
March 14, 2021
Intro
Strong-field ionization
Adiabatic approximation
- Assume field varies slowly
- Laser field momentarily constant
- Sequence of stationary Siegert states
Sequence of stationary states
Objective
Develop adiabatic theory for molecules including nuclear motion
Adiabatic theory
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}
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*}
Mapping time to momentum
Convergence of PEMD
Isotope effect
Channel ionization probabilities
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*}
Channel ionization probabilities
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
Deleted scenes
Introduction
Breakdown of BOA
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 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 \(R\)
Different grids
Adiabatic theory
Electronic hamiltonian
\(H_x(F) = -\frac{1}{2}\frac{\partial^2}{\partial x^2} + Fx + V\left(x; R\right)\)
Half cycle pulse
Time is mapped to momentum
PEMD
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*}