Majorana representation for the nonlinear two-mode boson system

Abstract

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.

Discrete modulational instability and bright localized spin wave modes in easy-axis weak ferromagnetic spin chains involving the next-nearest-neighbor coupling*

“…Thus, the bright nonlinear localized mode in ferromagnets may have potential applications in quantum-information storage or transmission. [57] With the development of the experiment and theory, [58][59][60][61][62][63][64][65][66][67][68][69][70] we expect that our theoretical results will be confirmed in the related experiments.…”

Majorana stellar representation for mixed-spin (s, 1/2) systems*

“…Majorana's perspective is that the evolution of a spin-J state can be intuitively described by trajectories of 2J points on the two-dimensional (2D) Bloch sphere, with these 2J points generally coined as Majorana stars (MSs), rather than one point on an intricate high-dimensional geometric structure. Therefore, this representation spontaneously provides an intuitive way to study high spin systems from geometrical perspectives, which has made the MSR a useful tool in many different fields, e.g., classification of entanglement in symmetric quantum states, [5][6][7][8][9][10][11][12] analyzing the spectrum of the Lipkin-Meshkov-Glick model, [13,14] studying Bose condensate with high spins, [15][16][17][18][19][20][21][22] and calculating geometrical phases of large-spin systems. [23][24][25] In addition, the MSR can be employed in quantum metrology [26,27] and in describing polarization states of N photons.…”