Adiabatic theory of strong-field ionization of molecules with nuclear motion
Jens Svensmark
June 25, 2021
Intro
Strong-field ionization
Adiabatic approximation
- 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
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*}
\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*}
Siegert states (tunneling)
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*}
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
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*}
Trajectories
Trajectories
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*}
Trajectories and momentum
PEMD
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}
BO for S-matrix
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*}
Nuclear wave packet
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*}
Trajectories and momentum
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*}
PEMD at the caustic
Wave packet reconstruction
PEMD modulation
PEMD modulation
Trajectories
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?
Deleted scenes
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
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*}