# Nonlinear photoionization of transparent solids: a nonperturbative theory obeying selection rules

## TL;DR

This paper develops a nonperturbative, analytical theory for photoionization in transparent solids that incorporates selection rules and interference effects, improving upon Keldysh's original model and aligning well with experimental data.

## Contribution

It introduces a novel nonperturbative approach that accounts for interference and selection rules in photoionization, enhancing the accuracy of theoretical predictions.

## Key findings

- Analytical expressions for photoionization rate including selection rules.
- Interference of transition amplitudes significantly affects ionization.
- The improved theory matches experimental measurements of nonlinear optical properties.

## Abstract

We provide a nonperturbative theory for photoionization of transparent solids. By applying a particular steepest-descent method, we derive analytical expressions for the photoionization rate within the two-band structure model, which consistently account for the $selection$ $rules$ related to the parity of the number of absorbed photons ($odd$ or $even$). We demonstrate the crucial role of the interference of the transition amplitudes (saddle-points), which in the semi-classical limit, can be interpreted in terms of interfering quantum trajectories. Keldysh's foundational work of laser physics [Sov. Phys. JETP 20, 1307 (1965)] disregarded this interference, resulting in the violation of $selection$ $rules$. We provide an improved Keldysh photoionization theory and show its excellent agreement with measurements for the frequency dependence of the two-photon absorption and nonlinear refractive index coefficients in dielectrics.