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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

Lai Zhen-jian

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

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.

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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.
Abstract We present a comparative phase-space analysis of the Wigner function W , the Husimi distribution H , and a normalized auxiliary diagnostic derived from the Wigner function, | W | , for a one-dimensional quantum system undergoing a transition from a single-well to a symmetric double-well potential. Using the quasi-exactly solvable sextic oscillator as a benchmark, we analyze the two-dimensional phase-space structure of these quantities, together with their marginals and associated information-theoretic measures. We show that W uniquely captures phase-space interference, negativity, and the strongest information-theoretic signatures associated with the emergence of the double-well structure, whereas | W | and H primarily highlight geometric splitting and coarse-grained delocalization. In particular, | W | provides a sign-suppressed structural diagnostic that lies between the fully oscillatory Wigner description and the Gaussian-smoothed Husimi picture. These results clarify the distinct information retained by standard and derived phase-space characterizations of the state under continuous deformation of the confining potential.
Photon statistics and phase properties of two-mode squeezed number states are studied. It is shown that photon number distribution and Pegg-Barnett phase distribution for such states have similar (N + 1)-peak structure for nonzero value of the difference in the number of photons between modes. Exact analytical formulas for phase distributions based on different phase approaches are derived. The Pegg-Barnett phase distribution and the phase quasiprobability distribution associated with the Wigner function are close to each other, while the phase quasiprobability distribution associated with the Q function carries less phase information.
We compare quantum phase space distributions of individual quantum states of the diamagnetic hydrogen atom obtained by means of Wigner functions with those given by Husimi functions. The comparison is carried out at effective h(cross) approximately 0.035 and at a fixed scaled energy ( epsilon =-0.316) which corresponds to an especially interesting mixed classical phase-space structure. The object of the comparison is to establish which of the two distributions best correlates with the classical Poincare surface of section for a representative set of single states. As expected, the states investigated display the strongest positive intensity on a single local invariant phase-space structure (such as a scar or torus) and at this level there is little difference between the Husimi and the Wigner function. However the weak structures (fringes of the Wigner function or zeros of the Husimi function) are shown here to be radically different for the Wigner relative to the Husimi representation. In both cases the weak structures permeate all of phase space. In addition they show different character for integrable and chaotic dynamics and so reflect the global structure of phase-space to a much greater extent than the localized strong structure. The Wigner functions of selected states are shown to have additional dynamically significant structures which are not apparent in the Husimi functions. These include negative intensity features seen in the Wigner-but not the Husimi-functions which are very well correlated with structures of the classical Poincare surfaces of section. Despite its complex oscillatory nature the Wigner function delineates better classical features, for example showing the outline of islands of stability avoided by states supported by chaotic regions with greater sharpness than the Husimi function.
Based on the two-mode squeezing operator which is the quantum version of the symplectic transformation, we find that the corresponding squeezed two-mode number state is just a two-variable Hermite polynomial excited state, which possesses well-behaved features, e.g. its Wigner function is a direct product of two Laguerre polynomials; its Husimi function, a Gaussian broaden version of the Wigner function, is related to two-variable Hermite polynomials. Moreover, its quantum statistical features, such as squeezing properties and the inter-mode photon bunching, are discussed.
In this study, we compare the Wigner function WWW, its modulus, and the Husimi distribution HHH in a one-dimensional quantum system exhibiting a transition from a single-well to a double-well configuration, using the quasi-exactly solvable sextic oscillator as a representative example. High-accuracy variational wavefunctions for the lowest states are used to compute two-dimensional phase-space structures, one-dimensional marginals, and the corresponding Shannon entropies, mutual information, and Cumulative Residual Jeffreys divergences. The analysis shows that the Wigner representation is uniquely responsive to interference effects and displays clear, nonmonotonic entropic behavior as the wells separate, whereas the modulus-Wigner and Husimi distributions account only for geometric splitting or coarse-grained delocalization. These findings establish a quantitative hierarchy in the ability of WWW, ∣W∣|W|∣W∣, and HHH to resolve structural changes in a quantum state and provide a general framework for assessing the descriptive power of different phase-space representations in systems with emerging bimodality or tunneling.
The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.