Analytic sources of inequivalence of the velocity gauge and length gauge

Abstract

It has been known for many years now that the descriptions of electromagnetic couplings in velocity gauge and length gauge can yield different results for atoms and molecules in strong fields. We point out that it is mathematically consistent to mix velocity gauge for some components of a material with length gauge for other components, although this should not be possible for a bona fide gauge transformation. For many-particle systems in a Hartree approximation, it is even possible to mix velocity gauge and length gauge for different particles of the same kind. Four main sources of analytic differences between velocity gauge and length gauge are then identified, and it is pointed out that these sources imply differences between velocity gauge and length gauge in particular for subfemtosecond spectroscopy, for experiments involving strong fields, and for chiral materials. Finally, it is emphasized that the transformation from velocity gauge to length gauge is just a particular example of a picture-changing unitary transformation. However, all these transformations lead to nonunitary shifts of the Hamiltonian, irrespective of whether the transformation can be described as a gauge transformation. Therefore, all these descriptions of quantum optics in dipole approximation are formally equivalent if agreement is achieved which particular formulation of the time-dependent interaction terms perturbs the ``true'' energy eigenstates of a system. However, this is where the actual discrepancies between velocity gauge, length gauge, and also other formulations such as acceleration gauge originate. This implies a generalization of the results of Galstyan et al. [Phys. Rev. A 93, 023422 (2016)] from two different classes of theoretical formulations to many different classes of theoretical formulations.