Abstract
Winding number is a topologically significant quantity that has found valuable applications in various areas of mathematical physics. Here, topological qubits are shown capable of formation from winding number superpositions and so of being used in the communication of quantum information in linear optical systems, the most common realm for quantum communication. In particular, it is shown that winding number qubits appear in several aspects of such systems, including quantum electromagnetic states of spin, momentum, orbital angular momentum, polarization of beams of particles propagating in free-space, optical fiber, beam splitters, and optical multiports.
Highlights
Topological quantities, such as winding numbers, Chern numbers, and Euler characteristics [1,2], depend on global properties of a system and are unchanged by smooth, local deformations of the space on which they are defined
In the study of quantum electromagnetic states of nonzero orbital angular momentum, it is most common to consider those arising in paraxial laser light beams, that is, beams propagating nearly parallel to a given axis
Recall that quantum optical beams of photons can be divided by beam splitters (BS)—producing superposition states of paths for photons in ordinary space—and by their polarized versions (PBS), which selectively direct them in beams according to polarization characteristics
Summary
Topological quantities, such as winding numbers, Chern numbers, and Euler characteristics [1,2], depend on global properties of a system and are unchanged by smooth, local deformations of the space on which they are defined They have become prominent in areas of physics ranging from particle physics and condensed matter physics to quantum communications and optics. The winding number might count the number of rotations of a polarization vector, the amount of orbital angular momentum present, or the number of times a phase rotates as the photon momentum circles the Brillioun zone of a spatially periodic system This characterization can be generalized to include the winding number relative to any point p in the plane, the winding number about the origin is considered here without loss of generality; see Figure 1. We show how winding number qubits, the natural description of which is in the so-called Bloch sphere S2 (cf., e.g., [21], Section 1.2), appear in a sequence of three progressively more useful examples
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