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Acta Sinica Quantum Optica

The Husimi Function of Eigenstates and its Statistical Properties For Bound Bosons in a Double Well Potential

赖振讲 LAI Zhen-jiang, 赖小伟 LAI Xiao-wei, 薛艳 XUE Yan, 常峻巍 CHANG Jun-wei

Published Jan 1, 2013• 0 citations
Publisher
DOI
10.3788/asqo20131904.0301

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With the aid of the two-mode entangled coherent state,and by precise mathematical derivation, we found a canonical number-phase difference Wigner function for the Bosons in a double well according to the feature and concept of Wigner function.Then through the method of Gaussian smoothing on the Wigner function we obtained its Husimi function.Under the quasilinear Rabi regime,the pictures of the probability density distributions in phase space were plotted with 8bosons 9eigenstates,calculated their expectations and variances of number difference of random variable.The statistical features of the system were discussed under the different way.It was found that there were the structures of the one-peak,two-peak,three-peak and even multi-peak of number-difference distribution,and as the number of particles increased,the number of peaks increased,too.The minimum of variance appeared in the ground state,and the next minimum was in the highest excited state and the most middle state between the ground and highest excited state had the maximum of variance.The phase-difference distribution was a periodic function and similar to the distribution of the interference fringes in the optics.It always presented the two-peak structure in one cycle,and the most probable probability of the phase had only two options:either 0,πor π/2,-π/2.
For the representation of eigenstates on a Poincare section at the boundary of a billiard different variants have been proposed. We compare these Poincare Husimi functions, discuss their properties, and based on this select one particularly suited definition. For the mean behavior of these Poincare Husimi functions an asymptotic expression is derived, including a uniform approximation. We establish the relation between the Poincare Husimi functions and the Husimi function in phase space from which a direct physical interpretation follows. Using this, a quantum ergodicity theorem for the Poincare Husimi functions in the case of ergodic systems is shown.