Nonergodicity in the Anisotropic Dicke Model

We study the ergodic-nonergodic transition in a generalized Dicke model with independent corotating and counterrotating light-matter coupling terms. By studying level statistics, the average ratio of consecutive level spacings, and the quantum butterfly effect (out-of-time correlation) as a dynamical probe, we show that the ergodic-nonergodic transition in the Dicke model is a consequence of the proximity to the integrable limit of the model when one of the couplings is set to zero. This can be interpreted as a hint for the existence of a quantum analogue of the classical Kolmogorov-Arnold-Moser theorem. In addition, we show that there is no intrinsic relation between the ergodic-nonergodic transition and the precursors of the normal-superradiant quantum phase transition.

Figure 1

The normalized distribution of level spacings $s_n=E_{n+1}-E_n$ for the ADM with $\omega ={\omega }_0=1$ and $j=10$ at $g_1=0.1$ (left) and $g=g_1=g_2$ (right). The histograms are drawn from the sorted energy levels $E_n$ with $n$ ranging from 200 to 1000, where the lowest levels are left out to account for the nonuniform density of states. As a reference, the Poissonian and Wigner-Dyson distributions are shown in gray.

Figure 2

(a) The average $⟨r⟩$ of $r_n$ taken over the lowest 1000 energy levels of the ADM with $\omega ={\omega }_0=1$ and $j=10$ as a function of $g_{1,2}$. The Dicke model and the quantum phase transition (QPT) between the normal (NP) and superradiant (SP) phases are indicated by a dashed and a dash-dotted line, respectively. The lower left-hand corner, where the data artificially suggest nonergodic behavior (see main text), has been masked. (b) The dependence of $⟨r⟩$ on the upper energy window cutoff $\mathrm{\Lambda }$ for various values $g_{1,2}$. The cutoffs for which the energy windows contain the lowest 1000 energy levels are indicated by black lines. (c) The dependence of $⟨r⟩$ on $j$ for various values $g_{1,2}$.

Figure 3

(a) Density plot of $1-F(t)/F(0)$ at $t=100$ and $\beta =1/10$ for the ADM with $\omega ={\omega }_0=1$ and $j=5$. Annotations are similar to Fig. 2. Compared to Fig. 2, $j$ is chosen smaller to account for the computational cost of evolving $W(t)$ over time. (b) The temperature dependence of $1-F(100)/F(0)$ for various values $g_{1,2}$. (c) The dependence of $1-F(t)/F(0)$ on time for various values $g_{1,2}$.

Figure 4

The quantity $m$ as defined in the main text for the ADM along the line $g_1=g_2=g$ ($g_1=1$) with $g=0$ ($g_2=0$) denoting the integrable Hamiltonian in the upper (lower) part. The results are shown for system size $j=10$ (left) and $j=15$ (right).