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

Jens Svensmark

The University of Electro-Communications, Tokyo

June 3, 2021

Intro

Strong-field ionization

ionization_process_rotated_1.svg

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Adiabatic approximation

  • Assume field varies slowly
  • Laser field momentarily constant
  • Sequence of stationary Siegert 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*} \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)

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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*}

ionization_process_rotated_v_and_k_rot.svg

\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

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Isotope effect

Channel ionization probabilities

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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

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Recap and outlook

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

Rescattering

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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?

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Introduction

Breakdown of BOA

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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}

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

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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

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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

compare_TDSE_BO_total_PEMDs-T50_F0.2.svg

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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*}

Full wavefunction evolution