Simple proof of gauge invariance for the S-matrix element of strong-field photoionization

TL;DR

This paper provides a straightforward proof that the length gauge form of the photoionization probability amplitude is gauge-invariant within the Keldysh-Faisal-Reiss theory, clarifying conditions for the velocity gauge form.

Contribution

It offers a simple proof confirming gauge invariance of the length gauge amplitude and clarifies the conditions needed for the velocity gauge form to be gauge-invariant.

Findings

Length gauge form is gauge-invariant under standard assumptions.

Velocity gauge form requires neglecting the exponential factor or specific conditions on the vector potential.

Clarifies the theoretical foundation of strong-field photoionization models.

Abstract

The relationship between the length gauge (LG) and the velocity gauge (VG) exact forms of the photoionization probability amplitude is considered. Our motivation for this paper comes from applications of the Keldysh-Faisal-Reiss (KFR) theory, which describes atoms (or ions) in a strong laser field (in the nonrelativistic approach, in the dipole approximation). On the faith of a certain widely-accepted assumption, we present a simple proof that the well-known LG form of the exact photoionization (or photodetachment) probability amplitude is indeed the gauge-invariant result. In contrast, to obtain the VG form of this probability amplitude, one has to either (i) neglect the well-known Goeppert-Mayer exponential factor (which assures gauge invariance) during all the time evolution of the ionized electron or (ii) put some conditions on the vector potential of the laser field.