非线性两模玻色子系统的Majorana表象

利用Majorana表象,从平均场模型和二次量子化模型两方面研究了非线性双模玻色子系统的动力学问题.得到了Majorana点在球面上的运动方程,分析了平均场模型和二次量子化模型之间的区别及其在Majorana点运动方程中的体现.研究了二次量子化模型中量子态在少体和多体情况下的动力学演化及其与平均场量子态的区别和联系.以平均场模型和二次量子化模型量子态之间的保真度和Majorana点之间的关联为手段,讨论了在不同玻色子间相互作用强度、不同玻色子数下量子态的演化及相应的自囚禁效应.

Majorana representation for the nonlinear two-mode boson system

By presenting the quantum evolution with the trajectories of points on the Bloch sphere, the Majorana representation provides an intuitive way to study a high dimensional quantum evolution. In this work, we study the dynamical evolution of the nonlinear two-mode boson system both in the mean-field model by one point on the Bloch sphere and the second-quantized model by the Majorana points, respectively. It is shown that the evolution of the state in the mean-field model and the self-trapping effect can be perfectly characterized by the motion of the point, while the quantum evolution in the second-quantized model can be expressed by an elegant formula of the Majorana points. We find that the motions of states in the two models are the same in linear case. In the nonlinear case, the contribution of the boson interactions to the formula of Majorana points in the second quantized model can be decomposed into two parts:one is the single point part which equals to the nonlinear part of the equation in mean-field model under lager boson number limit; the other one is related to the correlations between the Majorana points which cannot be found in the equation of the point in mean-field model. This means that, the quantum fluctuation which is neglected in the mean-field model can be represented by these correlations. To illustrate our results and shed more light on these two different models, we discussed the quantum state evolution and corresponding self-trapping phenomenon with different boson numbers and boson interacting strength by using the fidelity between the states of the two models and the correlation between the Majoranapoints and the single points in the mean-field model. The result show that the dynamics evolution of the two models are quite different with small boson numbers, since the correlation between the Majorana stars cannot be neglected. However, the second-quantized evolution and the mean-field evolution still vary in both the fidelity population difference between the two boson modes and the fidelity of the states in the two models. The difference between the continuous changes of the second quantized evolution with the boson interacting strength and the critical behavior of the mean-field evolution which related to the self-trapping effect is also discussed. These results can help us to investigate how to include the quantum fluctuation into the mean-field model and find a method beyond the mean field approach.