Dissipative Phase Transition in a Parametrically Amplified Quantum Rabi Model with Two-photon decay

Abstract

We investigate dissipative phase transitions (DPTs) in a parametrically amplified open quantum Rabi model (QRM) with both single- and two-photon decay. In the classical oscillator limit, four composite phases emerge, arising from the possible normal or superradiant regimes across the upper and lower spin branches. A mean-field analysis reveals an ``inverted" regime where superradiance emerges only at sufficiently low spin-boson coupling. This regime features first- and second-order DPTs separated by a tricritical point, while two-photon dissipation preserves the stability of the superradiant phase. Utilizing an adiabatic approach and the semi-classical Langevin formalism, we further study the steady-state structure beyond the mean-field level. We show that the tricriticality stems from the intrinsic nonlinearity of QRM, unveiled by the interplay of coherent and dissipative two-photon processes. The universality classes of the DPTs are identified, with the corresponding critical and finite-size scaling exponents derived and a scaling ansatz proposed to describe the critical behavior.

Figures (11)

• Figure 1: (a) Phase diagram in the $(\mu,g)$ parameter space with $\gamma_1=\gamma_2=1$. In each regime the phases of the $(\pm)$ spin branches are listed in the order {phase ($-$), phase (+)}. The dashed (solid) black curves marks the critical condition for DPTs of second- (first-)order. The TCP at $(\mu=2,g=\sqrt{2})$ is highlighted by a yellow star. The purple dot-dashed line at $\mu_c$ marks the boundary of the inverted regime. The dotted line marks the $\mu=1$ case where the two branches stay in NP for all $g$. The four black triangles on the $\mu$ axis marks the selected $\mu$ for generating the plots in Fig. \ref{['fig_mf_n_sz']}. In (b) and (c) we plot $F_-(\bar{x})$ without ($\gamma_2=0$) and with ($\gamma_2=1$) nonlinear decay, respectively, in the inverted regime with $\mu=2$ and different values of $g$. We have set $\gamma_1=1$.
• Figure 2: Mean-field predictions for the steady-state normalized boson number $\bar{n}_{ss}$ (solid lines, left marker) and spin polarization $s_z$ (dashed lines, right marker) as a function of $g$. The plots illustrate the specific branch undergoing DPT for selected values of $\mu$ (marked by black triangles on the $\mu$ axis of Fig. \ref{['fig_phase_diag']}). Legends indicate the specific value of $\mu$ and the plotted branch. All other parameters are identical to those used in Fig. \ref{['fig_phase_diag']}.
• Figure 3: Wigner functions of the steady state at the TCP. (a). Steady-state wigner function of the complete model (Eq. \ref{['eq_lind']}) calculated numerically. (b). Steady-state Wigner function of the $(-)$ branch, evaluated using Eq. \ref{['eq_W_ss']}. Parameters: $\omega_0=1,\eta=2500,g\sim1.73,$$\gamma_1=0.1,\gamma_2\sim44.26$. The exact values of $g,\gamma_2$ are determined by the TCP conditions in Eqs. \ref{['eq_gc']},\ref{['eq_tcp']}.
• Figure 4: Verification of the finite-frequency scaling at critical point (Eq. \ref{['eq_finite_scale']}) and the generalized scaling ansatz (Eq. \ref{['eq_ansatz_scale']}).(a). $\Delta x^2$ as a function of $\eta$ at critical point $g=g_c$. The two values of $\gamma_1$ correspond to the second-order (orange) and tricritical (blue) DPTs. The dots represent numerical results and the dashed lines are linear fits in log-log scale, with the extracted slope ($\zeta_i$) displayed in the labels. Parameters: $\omega_0=1,\mu=2,$$\gamma_2\sim44.26$. (b). Normalized scaling ansate $\tilde{F}$ as a function of $\Theta$. Data points of different shapes and colors show numerical results calculated for several fixed values of $\Lambda$. The red dashed line represents a linear fit in log-log space performed in the regime $\Theta>0.8$, indicating the corresponding asymptotic power-law behavior described by Eq. \ref{['eq_exp_nu']}. The numerical simulations use fixed parameters $\omega_0=1,\mu=2,\gamma_1$ and samples from $\eta\in[10^{4},10^{5}]$, $(g-g_c)\in[10^{-5},10^{-2}]$. The value of $\gamma_2$ is determined accordingly such that $\Lambda$ equals to the target fixed value.
• Figure 5: Normalized order parameter $\bar{n}_{ss}$ as a function of $g$ for different $\gamma_2$. Dots represent the numerical solutions of the complete model (steady state of Eq. \ref{['eq_lind']} of the main text) while dashed lines show mean-field predictions incorporating the classical mixture ansatz (see Appendix \ref{['ap_spin_pop']}). Colors denote the nature of the DPT: first-order (blue), tricritical (green), and second-order (red). The black dashed line marks the critical coupling for the second-order DPT. The deviation from MF predictions near the critical point demonstrates finite-size effectsnote_finite_eta. The parameters used for simulations are: $\omega_0=1,\eta=2500,\mu=2,$$\gamma_1=0.1.$ and the corresponding $\gamma_{2,c}\sim44.26$.