Quantum chaos in the nuclear collective model: Classical-quantum correspondence

Spectra of the geometric collective model of atomic nuclei are analyzed to identify chaotic correlations among nonrotational states. The model has been previously shown to exhibit a high degree of variability of regular and chaotic classical features with energy and control parameters. Corresponding signatures are now verified also on the quantum level for different schemes of quantization and with a variable classicality constant.

Figure 1

Energy levels of the GCM Hamiltonian for $A=-1$, $B=1.09$, $C=1$, and ${\hslash }^2/K=25\times {10}^{-4}$ drawn inside the $y=0$ section of the potential well. Energy is given in relative units. Panels (a) and (b), respectively, show even and odd states in the 2D quantization, panel (c) corresponds to the 5D quantization. Levels associated with the wave functions in Fig. 2 are marked with their respective ordinal numbers. In both 2D cases, levels of the harmonic-well approximation are drawn on the left.

Figure 2

Probability densities derived from the wave functions for selected levels from Fig. 1. Columns (a), (b), and (c) show the 2D even, 2D odd, and 5D cases, respectively. All distributions are constrained by the threefold symmetry of the potential. The dashed lines demarcate the kinematically accessible area at given energy; cf. the dashed lines in Fig. 1. The states in two upper rows belong to the region where the quadratic-well approximation of the potential is valid. The third and fourth rows, respectively, depict examples of regular and chaotic states with higher energies.

Figure 3

An example of the linear fit illustrating the determination of the Brody parameter from an unfolded spectrum of levels within a single bin [it belongs to the dotted line in Fig. 5b at $E\sim 1.33$]. The left-hand side of Eq. (11) is denoted as $T$. The NNS distribution and its fitted Brody approximation (with $\omega =0.62$) are shown (by the full line) in the inset. Dashed and dotted lines correspond to Poisson and Wigner distributions, respectively.

Figure 4

Regular fraction $f_{\text{reg}}$ of the classical phase space as a function of the control parameter $B$ and energy $E$ in a vicinity of $E\approx 0$. The degree of chaos is coded in shades of gray, with white (black) corresponding to complete order (chaos). One may notice an island with increased regularity near $B\approx 0.6$. The number of bins in the $B$ direction is much larger than that in the $E$ direction.

Figure 5

A comparison between the classical regular fraction $f_{\text{reg}}$ (panel a) and an adjunct $\left(1-\omega \right)$ of the Brody parameter (panels b and c) for $B=0.62$ (the main maximum of regularity at $E=0$). Results for different values of $\kappa$ in the 2D even case are shown in panel (b), while results for different types of quantization for $\kappa =25\times {10}^{-6}$ are presented in panel (c). A reasonable agreement of classical and quantum measures is observed. The jittering of individual curves is caused by statistical errors.

Figure 6

The same as in Fig. 5, but for $B=1.09$ (a minimum of regularity for $E=0$). Only the 2D even quantization is analyzed for different values of $\kappa$.

Figure 7

Classical (a) and quantum (b) chaotic measures for $B=0.24$ (a minimum of regularity for $E=0$). Only results with $\kappa =25\times {10}^{-6}$ are shown in panel (b) for various quantizations (fluctuations are caused by statistical errors).