Overcoming the No-Go Theorem Yields a Rich Dissipative Phase Diagram in the Open Quantum Rabi Model

TL;DR

The paper addresses the emergence of dissipative phase transitions in the open quantum Rabi model by explicitly including the $A^{2}$ term mandated by the TRK sum rule. Through a mean-field semiclassical analysis and an effective master equation framework valid in the large qubit–cavity frequency ratio, it demonstrates that anisotropy parametrized by $\tau$ stabilizes superradiant behavior and unlocks a rich, asymmetric steady-state phase diagram featuring normal, superradiant, and bistable phases with tricritical points. The authors derive critical lines $\tilde{g}_{c}^{\pm}$ and $\tilde{g}_{c}^{b}$, analyze stability, compute quantum fluctuations and universal exponents $\beta$ and $\nu$, and validate these findings with numerical spin-dynamics and Wigner-function simulations, revealing nonclassical states such as squeezed-vacuum and displaced-squeezed cat-like fields. The work provides a realistic route to observing and controlling nonequilibrium critical phenomena in open quantum systems, with potential implementations in circuit QED and trapped-ion platforms, while clarifying how the $A^{2}$ term shapes tricritical behavior and scaling properties.

Abstract

The open quantum Rabi model is studied in this work, with the explicit $\mathbf{A}^{2}$ term incorporated as required by the Thomas-Reich-Kuhn sum rule. It is shown that anisotropy provides a generic and robust mechanism for overcoming the no-go theorem in dissipative quantum systems, thereby establishing a genuine platform for observing dissipative phase transitions. The inclusion of the $\mathbf{A}^{2}$ term yields a significantly richer and asymmetric steady-state phase diagram, consisting of normal, superradiant, and bistable phases that intersect at tricritical points, while isolated bistable phases also emerge and the number of tricritical points is reduced. Notably, it is near the intersection of the two critical-line branches enclosing the superradiant phases, rather than at the tricritical points, that the $\mathbf{A}^{2}$ term fundamentally alters the scaling of photon-number fluctuations. Given the inherent role of the $\mathbf{A}^{2}$ term in light-matter interactions, our findings open a realistic route toward the experimental investigation and dynamical control of nonequilibrium critical phenomena in practical open quantum platforms.

Figures (5)

• Figure 1: Upper panels (a)--(c) present the steady-state phase diagrams of the dissipative anisotropic QRM in the $\tilde{g}-\tau$ plane for $\kappa = 0$, $1$, and $3$, respectively. Regions corresponding to the NP (white), SRP (blue), and NP+SRP (purple) are marked accordingly. The critical couplings $\tilde{g}_c^{-}$ and $\tilde{g}_c^{+}$ are depicted as solid black and red curves, respectively. A tricritical point is indicated by a red circle, and the intersection of $\tilde{g}_c^{+}$ and $\tilde{g}_c^{-}$ is marked by a green pentagram. The boundary between the NP+SRP and NP phases is shown as a black dashed curve, with the dissipation rate fixed at $\gamma = 0.5\omega$. Lower panels (d)--(f) display the ground-state phase diagrams of the dissipationless anisotropic QRM for the same $\kappa$ values as in the upper row. Here, dark-purple and dark-blue regions correspond to the $x$-type and $p$-type SRPs, respectively. The critical coupling strengths $\tilde{g}_c^{x}$ and $\tilde{g}_c^{p}$ are shown as solid and dashed blue lines. A first-order phase boundary between the two SRPs is plotted as a red dashed line, with red circles marking the tricritical points.
• Figure 2: The left panels display the order parameter $|\alpha|$ along three representative parameter paths: (a) a path with fixed CRW coupling strength $g_{cr}=\tau \tilde{g} = 3$, corresponding to the black dotted line in Fig. \ref{['fig:phase_diagram1']}(c); (c) a path that crosses the intersection point $\tilde{g}_{c0}$ of $\tilde{g}_c^{\pm}$ lines at fixed $\tau = \tau_{c}^s \approx 2.603$; and (e) a path passing through the tricritical point at fixed $\tau = \tau_{\mathrm{tri}} \approx 2.414$. The right panels show the photon-number fluctuations along the same parameter paths, with (b), (d), and (f) corresponding to panels (a), (c), and (e), respectively. Solid blue and red curves represent the NP and the SRP, respectively. Other parameters are fixed at $\kappa = 3$ and $\gamma = 0.5\omega$.
• Figure 3: Bloch-sphere trajectories of the spin dynamics for different coupling strengths $\tilde{g}$ and anisotropy parameters $\tau$. Upper panels correspond to a distinct parameter set: (a) the NP at $(\tau = 0.5, \tilde{g} = 1)$ with the initial state $\alpha_0 = 0.2 + 0.2i$, (b) the SRP at $(\tau = 2, \tilde{g} = 1)$ with the initial state $\alpha_0 = 0.2 + 0.2i$, and (c–d) the bistable phase at $(\tau = 6, \tilde{g} = 0.5)$ with initial states $\alpha_0 = 0.3 + 0.05i$ (c) and $\alpha_0 = 0.7 + 0.1i$ (d). The trajectories evolve from the initial coherent states (green dots) toward their corresponding steady states (red dots), with the dynamical evolution represented by the blue curves. Lower panels (e–h), all trajectories start from initial conditions with $s_z > 0$ and converge to the spin-inverted steady state $s_z = 1$, with eachpanel corresponding to the same parameter settings as in the upper panels (a–d), respectively. Other system parameters are fixed at $\kappa = 3$ and $\gamma = 0.5\omega$.
• Figure 4: Steady-state Wigner functions of the cavity field for the anisotropic QRM including the $\mathbf{A^2}$ term. Each panel corresponds to a distinct phase: (a)–(b) NP, (c) SRP, and (d) bistable phase. Upper panels (a)–(b) depict the symmetric case ($\tau = 1$) with $\tilde{g} = 0.5$ and $1$, respectively, where the superradiant phase transition is forbidden. Panels (c) and (d) correspond to the asymmetric case with $\tau = 2.4, \tilde{g} = 0.9$ and $\tau = 3.3, \tilde{g} = 0.6$, respectively. Other parameters are fixed at $\kappa = 3$ and $\gamma = 0.5\omega$.
• Figure 5: The phase diagram from Fig. \ref{['fig:phase_diagram']} is replotted in the $(g_{cr}=\tau\tilde{g}>0,\ g_{r}=\tilde{g}>0)$-plane, keeping the same notation as in the original figure. In panel (c), the black dotted line traces a path at fixed CRW coupling strength $g_{cr}=3$, which is used in the main text to analyze fluctuations and universal critical behavior.