Strong optical magnetic fields. Part 2: Poincaré gauge

We continue our last post on the interaction of nanoparticles with twisted light having strong magnetic fields.

In Part 1 we showed an example of an antiparallel beam (opposite spin and orbital angular momenta) that has a strong magnetic field. And we argued that one has to include the optical-magnetic interaction.

Figure 1: Twisted light for $\ell=2$ in parallel and antiparallel configuration.

Here, we take a gauge-invariance approach, and look for a Hamiltonian that is explicitly written in terms of electric and magnetic field [1]. We transform to a Poincaré gauge [2] with the formulas
\begin{eqnarray}
    \mathbf A(\mathbf r,t)
&=&
    -\int_0^1 \, du \, u \,\mathbf r \times \mathbf B(u\mathbf r,t)
\\
    U(\mathbf r,t)
&=&
    -\int_0^1 \, du \,\mathbf r \cdot \mathbf E(u\mathbf r,t)
\,.
\end{eqnarray}
relating the magnetic and electric field to the potentials. If we consider the interaction with small particles, we can simplify the electric and magnetic fields of TL, and reduce the above Poincaré expressions to a local version, where the potentials depend on the electric and magnetic fields on a single point. Then, we insert these formulas in the minimal coupling:
\begin{equation}
H =\frac{1}{2m}[\mathbf p - q \mathbf A(\mathbf r,t)]^2 + V(\mathbf r) + qU(\mathbf r,t),
\end{equation}
and obtain the gauge-invariant Hamiltonian
\begin{eqnarray} \label{Eq:H_final}
    H &=& \frac{\mathbf p^2}{2m}  + V(\mathbf r) - \mathbf E^{\mathrm {eff}}(\mathbf r, t) \cdot \mathbf d  - \mathbf B^{\mathrm {eff}}(\mathbf r, t) \cdot \mathbf m_B \,,
\end{eqnarray}
that clearly separate electric from magnetic interactions. These terms resemble the well-known electric and magnetic dipole formulas for homogeneous fields, but retain the spatial dependence that makes TL so interesting. They also contain in a compact form electric and magnetic 2$^n$-poles of the multipolar expansion. Finally, we note that all three components of the field are present.

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).