Hierarchic superradiant phases in anisotropic Dicke model

TL;DR

The paper addresses the dynamical phase structure of the anisotropic Dicke model (ADM) by identifying exceptional points (EPs) in the thermodynamic limit and showing that the conventional superradiant phase splits into three hierarchic subphases SP$_1$–SP$_3$. By performing a Holstein-Primakoff mapping and diagonalizing two effective quadratic Hamiltonians $H_1$ and $H_2$, the authors map each region to one of three canonical forms: the harmonic oscillator $H_{ho}$, the inverted harmonic oscillator $H_{iho}$, and the anti-harmonic oscillator $H_{aho}$, with region-specific spectra controlled by $\Omega_1$ and $\Omega_2$ (defined as $\Omega_1=\omega|\sqrt{(1+g_1/\omega)^{2}- (g_{2}/\omega)^{2}}|$ and $\Omega_2=\omega|\sqrt{(1-g_1/\omega)^{2}- (g_{2}/\omega)^{2}}|$). They diagnose the dynamical phases using the Loschmidt echo $L(t)$ after quenches, deriving analytic expressions that yield oscillations, decays, or combinations depending on the SP region, and they validate these predictions with finite-$N$ diagonalization. The results reveal a dynamical phase transition structure not captured by traditional ground-state QPTs and provide a framework for experimentally probing hierarchic SPs via dynamical observables.

Abstract

We revisit the phase diagram of an anisotropic Dicke model by revealing the non-analyticity induced by underlying exceptional points. We find that, from a dynamical perspective, the conventional superradiant phase can be further separated into three regions, in which the systems are characterized by different effective Hamiltonians, including the harmonic oscillator, the inverted harmonic oscillator, and their respective counterparts. We employ the Loschmidt echo to characterize different quantum phases by analyzing the quench dynamics of a trivial initial state. Numerical simulations for finite systems confirm our predictions about the existence of hierarchic superradiant phases.

Figures (3)

• Figure 1: Phase diagrams of the Hamiltonian in Eq. (\ref{['H_ADM']}) on the parameter $g_{1}g_{2}$ plane, indicating the main conclusion of this work. Different colors in the diagram distinguish different phases of the system. (a) The traditional phase diagram of the anisotropic Dicke model (ADM), obtained by the mean field method, shows that the region $g_{1}+g_{2}<\omega$ corresponds to the normal phase (NP), and the region $g_{1}+g_{2}>\omega$ corresponds to the superradiant phase (SP). (b) The phase diagram of the ADM, revealed by the underlying exceptional points (EPs) of the effective Hamiltonian in Eq. (\ref{['Heff']}) of the system, shows that the original superradiant phase (a) can be further divided into three distinct phases. We label these phases as $\text{SP}_{1}$, $\text{SP}_{2}$, and $\text{SP}_{3}$, respectively. The corresponding equivalent Hamiltonians of the effective Hamiltonian in each region are indicated in the panel. Here, we assume $\omega =\omega _{0}$.
• Figure 2: The plots of the decay rate $\lambda$ in (a), given by Eq. (\ref{['decay_rate']}) and frequency $f$ in (b), given by Eq. ( \ref{['frequency']}) of the effective Hamiltonian on the $g_{1}/\omega$-$g_{2}/\omega$ plane. It can be seen from the figures that there are clear distinctions between different phases in terms of $\lambda$ and $f$ . Four representitive points in each regions are selected, indicated by red dots at the same positions in both panels, with coordinates a$(0.4,0.4)$, b$(1.6,0.4)$, c$(1.2,0.8)$, and d$(0.4,1.6)$. The corresponding quench dynamical behaviors of the original ADM in finite systems at these points, obtained by numerical simulations, are presented in Fig. \ref{['fig3']}.
• Figure 3: The plots of $D(t)$, given by Eq. (\ref{['D_t']}), and their characteristics for the original ADM, given by Eq. (\ref{['H_ADM']}) and effective Hamiltonian $H_{\text{eff}}$ given by Eq. ( \ref{['Heff']}) in finite systems at the represented points indicated in Fig. \ref{['fig2']}. The plots in (a1)-(d1) are obtained by numerical simulations, the solid black line represents the numerical results obtained from the full ADM, whereas the red dashed line corresponds to the analytical result obtained from Eq. (\ref{['Lo']}). The corresponding decay rates $\lambda$ and frequencies $f$, plotted in (a2)-(d2), are extracted from the plots of $D(t)/\omega$. The number of atoms in the system is $N=100$, and the bosonic Hilbert space is truncated at $n_{\text{max}}=140$. Employing a larger bosonic cutoff $n_{\text{max}}$ or increasing the number of atoms $N$ does not alter the system's dynamical behavior over any finite time interval. These results are in accordance with the predictions from the analysis of the effective Hamiltonians. The time in the figure is in units of $\omega^{-1}$