Instantaneous ionization rate as a functional derivative

TL;DR

This paper introduces a gauge-invariant, physically measurable definition of the instantaneous ionization rate (IIR) as a functional derivative of ionization probability, and applies it to hydrogen in strong laser fields, revealing a Coulomb tail effect.

Contribution

It proposes a new unambiguous, gauge-invariant method to define and compute IIR based on measurable quantities, advancing the understanding of ionization dynamics.

Findings

IIR lags behind the electric field due to Coulomb tail effects.

The IIR does not show measurable delay in strong field tunnel ionization.

The approach is validated through numerical solutions of the Schrödinger equation.

Abstract

We describe an approach defining instantaneous ionization rate (IIR) as a functional derivative of the total ionization probability. The definition is based on physical quantities which are directly measurable, such as the total ionization probability and the waveform of the pulse. The definition is, therefore, unambiguous and does not suffer from gauge non-invariance. We compute IIR by solving numerically the time-dependent Schrodinger equation for the hydrogen atom in a strong laser field. We find that the IIR lags behind the electric field, but this lag is entirely due to the long tail effect of the Coulomb field. In agreement with the previous results using attoclock methodology, therefore, the IIR we define does not show measurable delay in strong field tunnel ionization.