Characterization of random features of chaotic eigenfunctions in unperturbed basis

In this paper we study random features manifested in components of energy eigenfunctions of quantum chaotic systems, given in the basis of unperturbed, integrable systems. Based on semiclassical analysis, particularly on Berry's conjecture, it is shown that the components in classically allowed regions can be regarded as Gaussian random numbers in a certain sense, when appropriately rescaled with respect to the average shape of the eigenfunctions. This suggests that when a perturbed system changes from integrable to chaotic, deviation of the distribution of rescaled components in classically allowed regions from the Gaussian distribution may be employed as a measure for the “distance” to quantum chaos. Numerical simulations performed in the Lipkin-Meshkov-Glick model and the Dicke model show that this deviation coincides with the deviation of the nearest-level-spacing distribution from the prediction of random-matrix theory. Similar numerical results are also obtained in two models without classical counterpart.

Figure 1

(a) The average shape of EFs, $\langle |C_{\alpha \mathit{n}}{|}^2\rangle$, in the chaotic regime of the LMG model with $\mathrm{\Omega }=500$ and $\lambda =1$. The average is taken over 500 EFs of $|E_{\alpha }\rangle$ with $\epsilon =0.4175$. (b) ${\mathrm{\Pi }}_N\left(E_{\alpha },{\mathit{I}}_{\mathit{n}}\right)$, which is the normalized $\mathrm{\Pi }(E_{\alpha },{\mathit{I}}_{\mathit{n}})$, as a classical analog of $\langle |C_{\alpha \mathit{n}}{|}^2\rangle$ [see Eq. (24)].

Figure 2

Similar to Fig. 1, but for the Dicke model with $N=500, \lambda =1$, and $\epsilon =0.007$.

Figure 3

The distribution $f\left(R\right)$ (open circles) and $g\left(R^{\prime }\right)$ (solid blocks with dashed lines) for $\lambda =1$ in the LMG model with (a) $\mathrm{\Omega }=80$ and (b) $\mathrm{\Omega }=1000$. The solid curves indicate the Gaussian distribution $f_G\left(R\right)$.

Figure 4

Similar to Fig. 3, but for the Dicke model with $\lambda =1$, (a) $N=80$, (b) $N=1000$.

Figure 5

“Distance” to chaos in the LMG model and the Dicke model. The measures ${\mathrm{\Delta }}_{EF}$ (solid squares) in Eq. (30) and ${\mathrm{\Delta }}_{EF}^{\prime }$ (open circles) in Eq. (59) are computed from statistical properties of EFs and the measure ${\mathrm{\Delta }}_E$ (open triangles) in Eq. (32) is computed from the statistics of spectra. (a) The LMG model with $\mathrm{\Omega }=80$, (b) the LMG model with $\mathrm{\Omega }=1000$, (c) the Dicke model with $N=80$, and (d) the Dicke model with $N=1000$. The effective Planck constants are given by $1/\mathrm{\Omega }$ and $1/N$, respectively, in the two modes. The two measures ${\mathrm{\Delta }}_{EF}$ and ${\mathrm{\Delta }}_E$ give almost the same results for the “distance” to chaos, when the systems are not far from chaos.

Figure 6

“Distances” to classical and quantum chaos in the LMG model and the Dicke model. The measure ${\mathrm{\Delta }}_{EF}$ (solid squares) is the same as that in Fig. 5. The measure ${\mathrm{\Delta }}_{cl}$ (open diamonds) is defined in Eq. (60) and was computed from properties of the corresponding classical phase spaces. (a) The LMG model with $\mathrm{\Omega }=80$, (b) the LMG model with $\mathrm{\Omega }=1000$, (c) the Dicke model with $N=80$, and (d) the Dicke model with $N=1000$.

Figure 7

Poincaré surfaces of section in the LMG model for $E=12+\sqrt{2}$ and $q_1=0$. (a) $\lambda =0.1$; (b) $\lambda =0.3$; (c) $\lambda =0.4$; (d) $\lambda =1.0$.

Figure 8

Poincaré surfaces of section in the Dicke model for $E=\frac{\sqrt{3}-1}{2}$ and $q_1=0$. (a) $\lambda =0.2$; (b) $\lambda =0.4$; (c) $\lambda =0.6$; (d) $\lambda =1.0$.

Figure 9

The distribution $f\left(R\right)$ (open circles) and $g\left(R^{\prime }\right)$ (solid blocks with dashed lines) for $\lambda =0.5$ in the defect Ising model and the defect XXZ model. The solid curve indicates the Gaussian distribution $f_G\left(R\right)$. (a) The defect Ising model with $N=10$; (b) the defect Ising model with $N=15$; (c) the defect XXZ model with $N=12$ and $S_z=-1$; (d) the defect XXZ model with $N=19$ and $S_z=-3.5$.

Figure 10

Similar to Fig. 5, but for the defect Ising model with (a) $N=10$ and (b) $N=15$.

Figure 11

Similar to Fig. 5, but for the defect XXZ model with (a) $N=12, S_z=-1$, and (b) $N=19, S_z=-3.5$.

Figure 12

(a) The distributions of $f\left(R\right)$ (open circles) and $g\left(R^{\prime }\right)$ (solid blocks with dashed lines) in the defect Ising model with $N=15$ and $\lambda =0.06$. (b) The corresponding cumulative distribution of the nearest-level-spacing distribution (open circles). The solid curve indicates the cumulative distribution given by the Wigner surmise.

Figure 13

Similar to Fig. 12, but for the defect XXZ model with $N=19, S_z=-7$, and $\lambda =0.12$.