Dynamic polarizability of Rydberg atoms: Applicability of near-free electron approximation, gauge invariance and the Dirac sea

TL;DR

This paper investigates the quantum mechanical basis of ponderomotive energy shifts in Rydberg atoms, confirming the free-electron approximation's validity and gauge invariance, especially for high principal quantum numbers.

Contribution

It provides a detailed quantum analysis of ponderomotive shifts, deriving corrections, confirming gauge invariance, and demonstrating the approximation's accuracy for high Rydberg states.

Findings

Free-electron AC Stark shift is an expectation value of a vector potential squared term.

Corrections to the free-electron value decrease with higher principal quantum numbers.

The free-electron approximation is accurate within 1% for 60s Rb and Sr states away from resonances.

Abstract

Ponderomotive energy shifts experienced by Rydberg atoms in optical fields are known to be well approximated by the classical quiver energy of a free electron. We examine such energy shifts quantum mechanically and elucidate how they relate to the ponderomotive shift of a free electron in off-resonant fields. We derive and evaluate corrections to the ponderomotive free electron polarizability in the length and velocity (transverse or Coulomb) gauges, which agree exactly as mandated by the gauge invariance. We also show how the free electron value emerges from the Dirac equation through summation over the Dirac sea states. We find that the free-electron AC Stark shift comes as an expectation value of a term proportional to the square of the vector potential in the velocity gauge. On the other hand, the same dominant contribution can be obtained to first order via a series expansion of the exact energy shift from the second order perturbation theory in the length gauge. Finally, we numerically examine the validity of the free-electron approximation. The correction to the free-electron value becomes smaller with increasing principal quantum number, and it is well below a per cent for 60s states of Rb and Sr away from the resonances.