Majorana Representation and Mean Field Approach for Interacting-Boson System

The Majorana representation, which represents a quantum state by stars on the Bloch sphere, provides us an intuitive tool to study the quantum evolution in high dimensional Hilbert space. In this work, we investigate the second quantized model and the mean-field model for the interacting-boson system in the Majorana representation. It is shown that the motions of states in the two models are same in the linear case. Furthermore, the contribution of the nonlinear interaction to the star motions in the second quantized model can be expressed by a single star part which is equal to the nonlinear part of the equation for the star in mean-field model under large boson number limit and an extra part caused by the correlation between stars. These differences and relations can not only be reflected by the population differences between the two boson modes in the two models, but also lie with the differences between the continuous changes of the second quantized evolution with the nonlinear interacting strength and the critical behavior of the mean-field evolution which related to the self-trapping effect. The reason of the difference between the two models is also discussed by an effective Hamiltonian.