Representation of Berry Phase by the Trajectories of Majorana Stars

The Majorana’s stellar representation, which represents the evolution of a quantum state with the trajectories of the Majorana stars on a Bloch sphere, provides an intuitive way to study a physical system with a high dimensional projective Hilbert space. In this Letter, we study the Berry phase by these stars and their loops on the Bloch sphere. It is shown that the Berry phase of a general spin state can be expressed by an elegant formula with the solid angles of Majorana star loops. Furthermore, these results can be used to a general state with arbitrary dimensions. To demonstrate our theory, we study a two mode interacting boson system. Finally, the relation between stars’ correlations and quantum entanglement is discussed.

Figure 1

A schematic illustration of (a) the solid angles subtended by the parallel transports of ${\mathit{u}}_i({\theta }_i,{\varphi }_i)$ and ${\mathit{u}}_j({\theta }_j,{\varphi }_j)$ on the Bloch sphere: ${\mathrm{\Omega }}_{{\mathit{u}}_i}$ (areas subtended by the blue solid loop) and ${\mathrm{\Omega }}_{{\mathit{u}}_j}$ (areas subtended by the red solid loop); (b) the solid angle in moving frame [$X^{\prime }((\pi /2)+{\theta }_j,{\varphi }_j),Y^{\prime }((\pi /2),-(\pi /2)+{\varphi }_j),Z^{\prime }({\theta }_j,{\varphi }_j)$] subtended by the parallel transports of the relative evolution path between the two stars in (a): ${\mathrm{\Omega }}_{{\mathit{u}}_i^{\prime }}$ (areas subtended by the blue solid loop)

Figure 2

(a) The trajectories and arrangements of stars for the eigenstates of ten bosons with $\lambda /R=0.3$. Dependence of the Berry phase on $\lambda /R$ and trajectories of Majorana stars (red loops on spheres) for (b) the ground state of the interaction boson Hamiltonian $H$ with two bosons, (c) the first excited state of three bosons, and (d) the second excited state of four bosons. The inset in (b) shows the evolution of $\theta$ and $\phi$ in parameter space.