Non-Resonant Two-Photon Transitions in Length and Velocity Gauges

TL;DR

This paper investigates the gauge invariance of two-photon transition matrix elements, revealing that true invariance only occurs at exact resonance, supported by analytic calculations involving hypergeometric functions and the Schroedinger propagator.

Contribution

It demonstrates that gauge invariance in two-photon transitions is limited to resonance conditions, clarifying the subtle role of asymptotic states and damping parameters in gauge transformations.

Findings

Gauge invariance holds only at exact resonance.

Analytic expressions involve hypergeometric functions.

Connections between asymptotic states and gauge transformations are clarified.

Abstract

We reexamine the invariance of two-photon transition matrix elements and corresponding two-photon Rabi frequencies under the "gauge" transformation from the length to the velocity gauge. It is shown that gauge invariance, in the most general sense, only holds at exact resonance, for both one-color as well as two-color absorption. The arguments leading to this conclusion are supported by analytic calculations which express the matrix elements in terms of hypergeometric functions, and ramified by a "master identity" which is fulfilled by off-diagonal matrix elements of the Schroedinger propagator under a the transformation from the velocity to the length gauge. The study of the gauge dependence of atomic processes highlights subtle connections between the concept of asymptotic states, the gauge transformation of the wave function, and infinitesimal damping parameters for perturbations and interaction Hamiltonians that switch off the terms in the infinite past and future [of the form exp(-epsilon * | t |)]. We include a pertinent discussion.