Majorana stellar representation for mixed-spin (s,12) systems
Abstract
By describing the evolution of a quantum state with the trajectories of the Majorana stars on a Bloch sphere, Majorana's stellar representation provides an intuitive geometric perspective to comprehend a quantum system with high-dimensional Hilbert space. However, the problem of the representation of a two-spin coupling system on a Bloch sphere has not been solved satisfactorily yet. Here, we present a practical method to resolve the problem for the mixed-spin (s, 1/2) system. The system can be decomposed into two spins: spin-(s+1/2) and spin-(s-1/2) at the coupling bases, which can be regarded as independent spins. Besides, we may write any pure state as a superposition of two orthonormal states with one spin-(s+1/2) state and the other spin-(s-1/2) state. Thus, the whole state can be regarded as a state of a pseudo spin-1/2. In this way, the mixed spin decomposes into three spins. Therefore, we can represent the state by (2s+1)+(2s-1)+1=4s+1 sets of stars on a Bloch sphere. Finally, to demonstrate our theory, we give some examples that indeed show laconic and symmetric patterns on the Bloch sphere, and unveil the properties of the high-spin system by analyzing the trajectories of the Majorana stars on a Bloch sphere.