A detailed study on various phases in dissipative anisotropic Dicke model

TL;DR

This work investigates how dissipation and anisotropy affect phases in the dissipative anisotropic Dicke model, including the quantum phase transition and ergodic-to-nonergodic transition. It combines Liouvillian-spectrum analysis, finite-size scaling of the Liouvillian gap, and biorthogonal participation ratios to map the phase structure, showing that ENET survives dissipation and exhibits distinct gap scalings similar to the closed model. The driven extension with a Thue-Morse quasi-periodic drive reveals that bosonic dissipation stabilizes a long-lived prethermal plateau and can suppress Floquet heating, highlighting a mechanism for reservoir-assisted control of driven open quantum systems. Overall, the paper provides a phase diagram of dissipative ADM with anisotropy, connects eigenvalue and eigenvector diagnostics, and demonstrates practical implications for stabilization of nontrivial dynamical phases under drive.

Abstract

We present a comprehensive study of different phases in the Dicke model incorporating both anisotropy and dissipation. We begin with a concise review of the quantum phase transition in this setting, highlighting how these two parameters shift the critical point. We then perform a detailed investigation of the transition from ergodic to nonergodic phases by analyzing the eigenvalue and eigenvector properties of the Liouvillian with the aid of scaling of the Liouvillian gap and the average participation ratio. Our results show that the eigenvector properties of the Liouvillian are consistent with its spectral characteristics, leading to a phase diagram that has similarities with the closed counterpart. Furthermore, we demonstrate that the Liouvillian gap exhibits distinct scaling behaviors in these two phases. Finally, we extend our study to the driven case by applying a Thue-Morse quasiperiodic drive. In this case, we find that bosonic dissipation plays a crucial role in stabilizing the prethermal plateau, offering an effective mechanism to halt the heating effect arising from the quasi-periodic drive.

Figures (9)

• Figure 1: (a) $\ln\Delta$ vs $\ln N$ plot for $g_1=1.0$, and $\kappa=0.1$. (b) The slope $z$ (extracted from the scaling of Liouvillian gap $\Delta$ as a function of atom number) as a function of $g_2$ for fixed $g_1$ value. We consider the dissipation strength, $\kappa=0.1, 0.01$. Here, $n_{\text{max}}=26$, and the atom number $N$ is changing from $4-10$. We find that the scaling is different for the ergodic and non-ergodic phases. We also show that for changing $\kappa$, the qualitative features of ENET remain the same.
• Figure 2: Average of participation ratio ($P_{\text{avg}}$), using biorthonormal basis considering both left and right eigenvectors of the Liouvillian for dissipative ADM as a function of the rotating and counter-rotating coupling parameters $g_1$ and $g_2$. The atom number: $N=10$, the bosonic cut-off: $n_{\text{max}} =26$. For thisfigure $\omega_0 = \omega=1,\ \kappa=1$.
• Figure 3: (a, b, e, f, i, j, m, n) Average boson number, $N_\text{av}(t)$, and (c, d, g, h, k, l, o, p) mutual information between spins and bosons, $I_{\text{spin-boson}}$, as a function of time $t_n=2^n T$ for the dissipative ADM under the Thue-Morse quasiperiodic drive. The initial states have low energies, so that $\langle E_{\text{in}}\rangle = 0.25$. Panels (a, c, e, g, i, k, m, o) represent the data for a fixed driving frequency $\omega_{ \text{d}} = 50$, $g_1 = 1.25$, and various values of $g_2$. Panels (b, d, f, h, j, l, n, p): $g_1 = 1.25$, $g_2 = 0.7$ and various values $\omega_{\text{d}}$. In all panels, the driving amplitude is $\Omega=1.0$, $N=6,\ n_{\text{max}}=20,\ \omega_0 = \omega=1$. For (a-d) $\kappa=0.0$, (e-h) $\kappa=0.05$, (i-l) $\kappa=0.6$, (m-p) $\kappa=1.0$. The time t is in units of $\omega_0^{-1}$. In this figure, the dashed line represents the page value
• Figure 4: Scatter plot of the complex spectrum of the Liouvillian $\mathcal{L}$ of the dissipative anisotropic Dicke model for the atom number $N=10$ and $\omega=\omega_0=\kappa=1$, for (a) a point in the non-ergodic phase: $g_1=1.25,\ g_2=0.1$, and (b) a point in the ergodic phase: $g_1=1.25,\ g_2=1.0$. The bosonic cut-off: $n_{\text{max}} = 26$.
• Figure 5: Level-spacing distribution of the complex spectrum of the Liouvillian $\mathcal{L}$ in (a) the non-ergodic phase with $g_1=1.25,\ g_2=0.1$ and (b) the ergodic phase with $g_1=1.25,\ g_2=1.0$. We find remarkable agreement with the 2D Poisson distribution $P_{\text{2D-P}}(s)$ in Eq. and that of the GinUE RMT prediction $P_{\text{GinUE}}(s)$. The atom number $N=10$, bosonic cut-off: $n_{\text{max}} = 26$. The other parameters are: $\omega=\omega_0=1$, $\kappa=1$.