Strong optical magnetic fields. Part 1: Introduction

Antiparallel beams of twisted light exhibit a remarkable property: Their magnetic field may overcome their electric field.

A good example of a twisted light beam for which the magnetic field dominates is the antiparallel $\large \ell=2, \sigma=-1$ [1]. Let's compare the $x$ components close to $r=0$

(the tilde means the part of the field without the time and $z$ dependence). We are assuming that $q_r r \ll 1$, which is true for an optical field exciting a QD, for example. Furthermore, the magnetic field is actually a constant different from zero at $r=0$, so close enough the magnetic field is stronger than the electric field.

How do we treat the interaction with matter in this atypical case?

Or in other words, what is the form of the light-matter Hamiltonian? Please, take a minute to answer this question to yourself before proceeding ... A common answer would be:

the dipole-moment interaction: $\large \mathbf d \cdot \mathbf E_0$

or maybe you thought about the minimal coupling: $\large [\mathbf p - q \mathbf A(\mathbf r)]^2/2m$

The second answer makes sense, but did you consider the magnetic effects of the vector potential $\mathbf A$? The point is: we are in used to thinking of the interaction of light with the orbital degree of freedom as purely electric.

Clearly, the dipole-moment interaction is useless, but the minimal coupling Hamiltonian contains both electric and magnetic contributions. However, expressed in terms of the vector potential, it isn't clear what is the magnetic part. Of course one can resort to the multipolar expansion [2].

But we will take another approach that yields gauge-invariant expression (in terms of electric and magnetic fields) that are both intuitive and local, that correctly treat the interaction with nanoparticles. They are intuitive because they resemble the well-know dipole-moment interactions, and are local for they depend on a single point. Stay tuned!

References

[1] Quinteiro, G. F., D. E. Reiter, and T. Kuhn. "Formulation of the twisted-light–matter interaction at the phase singularity: The twisted-light gauge." Physical Review A 91, no. 3 (2015): 033808.
[2] Tannoudji, C. Cohen, J. Dupont Roc, and G. Grynberg. "Photons and Atoms: Introduction to Quantum Electrodynamics." (1989).