Non-Gaussian Phase Transition and Cascade of Instabilities in the Dissipative Quantum Rabi Model

Abstract

The open quantum Rabi model describes a two-level system coupled to a harmonic oscillator. A Gaussian phase transition for the nonequilibrium steady states has been predicted when the bosonic mode is soft and subject to damping. We show that oscillator dephasing is a relevant perturbation, which leads to a non-Gaussian phase transition and an intriguing cascade of instabilities for $k$-th order bosonic operators, as well as a jump in the steady-state qubit polarization. For the soft-mode limit, the equations of motion form a closed hierarchy and spectral properties can be efficiently studied. To this purpose, we establish a fruitful connection to non-Hermitian Hamiltonians. The results for the phase diagram, stability boundaries, and relevant observables are based on mean-field analysis, exact diagonalization, perturbation theory, and Keldysh field theory.

Figures (8)

• Figure 1: Stability boundaries of the $k$-th order bosonic operators in the thermodynamic limit $\eta \to \infty$. The color map indicates the smallest $k$ for which, at the given point in the phase diagram, the non-Hermitian Hamiltonian $\hat{H}^{(k)}$ from Eq. \ref{['eq:eff_Hk']} is unstable (not a Hurwitz matrix). The black line indicates $g^\infty_c$ from Eq. \ref{['eq:gcinfty']}, below (above) which the whole system is stable (unstable). The star-shaped markers indicate the parameters used in Fig. \ref{['fig:hierarchy']}. Here we set $\kappa/\omega_0 = 1$. Inset: Infinitesimal dephasing $\gamma>0$ causes nonanalytic jumps in the qubit polarization and the normal-phase boundary, leading to the non-Gaussian DPT at $g_c^\infty$.
• Figure 2: Steady-state expectation values for the 2nd, 4th, and 6th order operators $\hat{n}$, $\hat{n}^2$, and $\hat{n}^3$ at $\kappa/\omega_0=1$, $g=1.05$ for (a)$\gamma=0.1$ such that $g<g_c^{(6)}$ and (b)$\gamma=0.3$ such that $g_c^{(4)}<g<g_c^{(2)}$. In (a), the expectation values are well converged for $N_{\max}\gtrsim 300$. For (b), we used $N_{\max}=300$, 400, and 500.
• Figure 3: (a) Steady-state expectation values of $\hat{\sigma}^z$ for various values of $g$ and $\gamma$. Numerical results for $\eta=10^3$ are compared with the $\eta\to\infty$ limit \ref{['eq:sigmazs']}. (b)(c) Numerically calculated steady-state expectation values of (b) $\hat{\sigma}^z$ and (c) $\hat{n}$ for various values of $\eta$ and $\gamma$, where $g=g_c^{(2)}$ for each $\gamma$ value. Expectation values converged well with $N_{\max}=500$. In (c), the dashed line represents $\propto \eta^{1/2}$ and solid lines show fits of the data to the crossover function $\langle \hat{n}\rangle_s = {B\eta^{1/2}}/{(1+A\eta^{1/2})}$. For all plots, $\kappa/\omega_0=1$.
• Figure 4: Stability boundaries of the $k$-th order bosonic operators in the of thermodynamic limit $\eta \to \infty$ for $\mathcal{L}_{\downarrow}$ in panels (a) and (b) as well as $\mathcal{L}_{\uparrow}$ in panels (c) and (d). As in Fig. \ref{['fig:boundary']}, which is reproduced in panel (a), the color map always indicates the smallest even or odd $k$ for which, at the given point in the phase diagram, the corresponding non-Hermitian Hamiltonian $\hat{H}^{(k)}$, governing the Green's function evolution, is unstable. For this data, we have set $\kappa/\omega_0 = 1$.
• Figure 5: Trotterized evolution of the Wigner function with respect to $\mathcal{L}_{\downarrow}$ in panels (a) to (e) and with respect to $\mathcal{L}_{\uparrow}$ in panels (f) to (j), starting from the vacuum state. Here we consider the case of pure dephasing ($\gamma=1$). The white dashed lines represent the direction along which the Wigner function is stretched by the squeezing Hamiltonian. The solid, dashed, and dotted contour lines represent the values $10^{-1}$, $10^{-2}$, and $10^{-3}$, respectively. Parameter values are set as $g=1.05$ and $\kappa/\omega_0 = 0.1$, and the evolution time between consecutive columns is $\Delta t =1/\omega_0$.