Comparative Analysis of Spin, Paths, Temporal, and Spatial Modes

Abstract

Research on optical modes, such as orbital, temporal, or parity, bring much attention, for these new degrees of freedom allow larger quantum communication alphabets. Each lab usually adapts the Bloch or Poincare sphere to their experiment or light mode. This takes an extra effort and time and produces a plethora of spheres and notations. Yet, we miss a common framework or convention valid among diverse physical-modes. We aim to unite in one representation the best points from many different spheres. Such common-sphere could also help to compare distant experiments, for an intuitive understanding of quantum optical states. We built a common representation by mathematically aligning the Hilbert space and a three dimensional color space. We define a unique color for each one of the three Poincare axes and positive Pauli vectors. Beyond three primary colors and states, our equations associate each Hilbert state to a specific tonality, among the infinite combinations in Color space. These maths achieve a new ability to unequivocally represent any quantum state by its precise combination of colors. Thus, with these equations, quantum states ‘yellow’ or ‘magenta’ are not mere names, rather each one denotes an exact superposition in Hilbert space. To handle disparities between SO3 vs. SU2 space operations, we propose a darkness bit and a Hermite-inspired shape. A simulation of HG modes let us align distinct shapes to quantum optical states. Three examples of applications show our color sphere in practice. First, we apply the Hilbert-Color mapping in Polarization. Then, the same color-space is shown in Orbital Angular Momentum. We also represent location paths in this color-mapping. The simulations and practical comparisons let us refine the proposed color sphere convention. For higher-order and path-to-industry, any sphere section serves as color constellation diagram. One color-space sphere served as common ground to represent coexisting concepts among diverse physical areas. The introduced change diagrams are visual tools to communicate setups and operators. The examples showed a unique notation matches many physical processes. The resulting diagram of superposition of spatially separated optical paths is coherent with a Plate on Polarization or cylinder lens on OAM Hermite Gaussian modes. A unique change diagram describes the three examples. The meaning persist despite the physical implementation. Found also how this color space let us grasp visually some meaning. Thus, the amount of blue in a state representation indicated the degree of its phase shift. Overall, we presented math and visual tools to display and compare experiments. We showed examples in different physical modes, all linked to a unique color sphere.