Wigner-Husimi phase-space structure of quasi-exactly solvable sextic potential

TL;DR

This work systematically contrasts the Wigner function $W(x,p)$, its modulus $|W(x,p)|$, and the Husimi distribution $H(x,p)$ for the quasi-exactly solvable sextic potential, tracking the transition from a single-well to a double-well and its impact on phase-space structure. By employing variationally optimized low-lying states and analyzing full 2D distributions, marginals, Shannon entropies, mutual information, and Cumulative Residual Jeffreys divergences, the authors reveal a robust hierarchy $W>|W|>H$ in capturing coherence, localization, and interference during tunneling. The modulus $|W|$ acts as a practical intermediate descriptor that preserves geometric features while suppressing interference, whereas $H$ yields the broadest, most classical-like portraits due to Gaussian smoothing. The results provide a quantitative framework for selecting phase-space representations in systems exhibiting bimodality or tunneling and point to broad applicability in quantum-control and decoherence studies, with potential extensions to time-dependent dynamics and other anharmonic or QES models.

Abstract

In this study, we compare the Wigner function $W$, its modulus, and the Husimi distribution $H$ 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 $W$, $|W|$, and $H$ 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.

Figures (15)

• Figure 1: QES confining sextic potential $V^{\rm QES}(x;\lambda)$ in (\ref{['vqes']}) for different values of the parameter $\lambda$. For $\lambda>-\frac{1}{2}$ it develops two symmetric degenerate minima, and the $\lambda-$independent maximum located at $x = 0$ corresponds to a potential energy $V = 0$.
• Figure 2: Wigner quasiprobability distribution $W(x,p)$ (top), modulus of the Wigner function $|W(x,p)|$ (middle), and Husimi distribution $H(x,p)$ (bottom) for the ground state $n=0$ of the QES sextic oscillator at $\lambda=-\tfrac{3}{4}$ and $\lambda=4$.
• Figure 3: Wigner quasiprobability distribution $W(x,p)$ (top), modulus of the Wigner function $|W(x,p)|$ (middle), and Husimi distribution $H(x,p)$ (bottom) for the first excited state $n=1$ of the QES sextic oscillator at $\lambda=-\tfrac{3}{4}$ and $\lambda=4$.
• Figure 4: Ground state $n=0$. Position-space marginals $Q_x$ of the three distributions: (a)–(d) Wigner function $W$, (e)–(h) its modulus $|W|$, and (i)–(l) the Husimi distribution $H$, shown for different values of the parameter $\lambda$.
• Figure 5: Position-space marginals $Q_x$ of $W$, $|W|$, and $H$ for the ground state ($n=0$), shown on a common scale for (a) $\lambda = -\tfrac{3}{4}$ and (b) $\lambda = 4$.