Relativistic Anyon Beam: Construction and Properties

Motivated by recent interest in photon and electron vortex beams, we propose the construction of a relativistic anyon beam. Following Jackiw and Nair [R. Jackiw and V. P. Nair, Phys. Rev. D 43, 1933 (1991).] we derive an explicit form of the relativistic plane wave solution of a single anyon. Subsequently we construct the planar anyon beam by superposing these solutions. Explicit expressions for the conserved anyon current are derived. Finally, we provide expressions for the anyon beam current using the superposed waves and discuss its properties. We also comment on the possibility of laboratory construction of an anyon beam.

Figure 1

(a) $J^0-\theta$ for different values of $\rho$. (b) Polar plot for $J^0$ with $\theta$ for fixed $\rho$ where the radial distance is the magnitude of $J^0$. (c) Density plot of $J^0$. (d) 3D plot of $J^0$. We set $\lambda =0.6$ and the integration ranges from $-\pi /2$ to $+\pi /2$.

Figure 2

$J^x$ ($\lambda =0.6$) with integration range $\pm \pi /2$, where (a)–(d) correspond to the similar meaning as described in Fig. 1.

Figure 3

$J^y$ ($\lambda =0.6$) with integration range $\pm \pi /2$, where (a)–(d) represent a similar meaning as described in Fig. 1.

Figure 4

Experimental setup: A slice (projection) of a 3D convex lens in the $x\text{−}y$ plane (left diagram) is taken out and shown separately as a thick lens in the right diagram. The red lines show the superposing wave vectors in the $x\text{−}y$ plane.