Intrinsic multipole moments of non-Gaussian wave packets

Charged wave packets with non-Gaussian spatial profiles are shown to possess intrinsic multipole moments. The magnetic dipole moment and the electric quadrupole moment are found for a wide class of packets, including the vortex electrons with orbital angular momentum $\ell$, the Airy beams, the so-called Schrödinger-cat states, and their generalizations. For the packets with no phase vortices, the electric quadrupole moment is shown to grow quadratically with the packet's width $|Q_{\alpha \beta }|\sim e{\sigma }_{\perp }^2$, while it is also $\left|\ell \right|$ times enhanced for the vortex beams. For available beams of electron microscopes, these multipole moments are relatively easily adjusted and can be quite large, which affects the packets' electromagnetic fields and also allows one to develop new diagnostic tools for materials science, atomic and molecular physics, nuclear physics, and so forth.

Figure 1

Unlike the vortex packet, the Airy beam has no azimuthal symmetry. This results in an azimuthal dependence of the radial field $E_{\xi ,\rho }$ [Eq. (24)] of its quadrupole moment (16). The parameters are $\eta =0, {\xi }_y=0$, and $\theta =\pi /2$.