{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Majorana Stars and Structured Gaussian Beams\n", "\n", "In this notebook, we'll be exploring yet another fascinating application of our favorite construction: the \"Majorana stars.\" \n", "\n", "We know well how we can view constellations on the 2-sphere as SU(2) representations, in other words, as spin-j states (up to phase). These states represent the intrinsic angular momentum of a massive particle. On the other hand, massless particles also have a form of intrinsic angular momentum: polarization. Indeed, we've seen how we can use the sphere to represent polarization states: a trick that was known even to Poincare. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To refresh your memory:\n", "\n", "Given a qubit quantized along the $Z$ axis, we can obtain the corresponding polarization ellipse simply by rotating the overall phase of the state, and taking the real part of its two components. If we treat these two real numbers as $(x, y)$ coordinates in the plane, they trace out the correct ellipse. \n", "\n", "In this basis, $Z+$/$Z-$ correspond to horizontal/vertical polarization, $X+$/$X-$ correspond to diagonal/antidiagonal polarization, and $Y+$/$Y-$ correspond to clockwise/counterclockwise circular polarization. Any arbitrary point on the sphere, thus, corresponds to some ellipse in the plane. \n", "\n", "In this case, we'd take the $Y$ axis to coincide with the linear momentum axis of the massless particle, and consider its polarization state to consist of a possible superposition of clockwise/counterclockwise states, which are sufficient to describe *any ellipse*. We could imagine that this polarization state is telling us how the photon is \"corkscrewing\"/oscillating in the plane orthogonal to its motion. \n", "\n", "A photon is of course a spin-1 particle, but since it's massless, it's $m=0$ state, which would correspond to longitudinal polarization (in the direction of its motion), must be 0, and so it reduces down to a two state system. Don't ask me for the [details](https://physics.stackexchange.com/questions/46643/why-is-the-s-z-0-state-forbidden-for-photons)!\n", "\n", "Instead, let's visualize this correspondence between the sphere and the ellipse:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "from spheres import *\n", "import vpython as vp\n", "\n", "qubit = basis(2, 0, 'z')\n", "xyz = spinor_xyz(qubit)\n", "\n", "scene = vp.canvas(background=vp.color.white)\n", "vsphere = vp.sphere(pos=vp.vector(2,0,0), color=vp.color.blue, opacity=0.3)\n", "varrow = vp.arrow(pos=vsphere.pos, axis=vp.vector(*xyz))\n", "\n", "x, y = components(qubit).real\n", "vpt = vp.sphere(pos=vp.vector(x,y,0), radius=0.1, color=vp.color.yellow, make_trail=True)\n", "\n", "for t in np.linspace(0, 2*np.pi, 8000):\n", " x, y = components(np.exp(1j*t)*qubit).real\n", " vpt.pos = vp.vector(x,y,0)\n", " vp.rate(1000)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "More geometrically:\n", "\n", "\n", "\n", "On the other hand, it's less well known, but one can interpret a full spin-$1$ state of two stars geometrically in terms of polarization. According to [Bliokh, Alonso, and Dennis (and Hannay)](https://arxiv.org/abs/1903.01304): the unit vector which bisects the two stars picks out a direction normal to a plane. Projecting the two points onto this plane, one obtains the two foci of the polarization ellipse. In other words, a spin-1 state specifies an ellipse *oriented in 3D*. The direction normal to the plane is taken to be the direction of propagation of the photon. In the degenerate case of the two stars in the same location, we get circular polarization; and beautifully, the $m=0$ case, when two stars are opposite, never comes up, since the polarization plane rotates along with them, and we get linearly polarized light instead.\n", "\n", "\n", "\n", "(As the paper points out, however, this construction gets the size of ellipse wrong, and they suggest a slight variant as a remedy.) \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "