Quantum Phases in a Quantum Rabi Triangle

The interplay of interactions, symmetries, and gauge fields usually leads to intriguing quantum many-body phases. To explore the nature of emerging phases, we study a quantum Rabi triangle system as an elementary building block for synthesizing an artificial magnetic field. We develop an analytical approach to study the rich phase diagram and the associated quantum criticality. Of particular interest is the emergence of a chiral-coherent phase, which breaks both the ${\mathbb{Z}}_2$ and the chiral symmetry. In this chiral phase, photons flow unidirectionally and the chirality can be tuned by the artificial gauge field, exhibiting a signature of broken time-reversal symmetry. The finite-frequency scaling analysis further confirms the associated phase transition to be in the universality class of the Dicke model. This model can simulate a broad range of physical phenomena of light-matter coupling systems, and may have an application in future developments of various quantum information technologies.

Figure 1

(a) The schematic diagram of quantum Rabi triangle system with artificial gauge field. (b) The analytic phase diagram in the $g_1\text{−}\theta$ parameter space. The second order critical lines (red dash) from the iCP to nCP and cCP join with the first order line (black sold) between nCP and cCP at the triple points (TPs) (black dot). The black dashed line separates the cCP according to its chirality. In all our calculations, we set $\omega =1$ as the units for frequency, and $\mathrm{\Delta }=50$, $J=0.05$.

Figure 2

(a) Energy spectrum in the iCP phase with $g_1=0.1$. Curves represent the analytic result. Symbols correspond to the lowest 4 eigenenergies numerically obtained from ED. (b) Energy spectrum in the nCP and cCP phases with $g_1=0.7$. The black curve corresponds to analytic ground-state energy. Symbols correspond to the lowest 6 eigenenergies numerically obtained from ED. Other parameters are the same as in Fig. 1. The two peaks are located at $\pm {\theta }_c=\pm 0.516\pi$.

Figure 3

(a) The order parameter $⟨a_n⟩$ as a function of the scaled coupling strength $g_1$ for the iCP-nCP transition with $\theta =2\pi /3>{\theta }_c$. (b) $|⟨a_n⟩|$ as a function of $g_1$ for the iCP-cCP transition with $\theta =\pi /3<{\theta }_c$. (c) Photon current $I_{\mathrm{ph}}$ and (d) the expected value of the chirality operator $C_{\mathrm{ph}}$ in the ground state as a function of the hopping phase $\theta$ for the nCP-cCP transition with $g_1=0.7>g_{1c}$. Other parameters are the same as in Fig. 1.

Figure 4

Scaling of the ground-state energy (a) and photon number (b) as a function of $\eta$ at the critical point for the different gauge field phases $\theta ={\theta }_c$, $1.2{\theta }_c$, and $0.96{\theta }_c$ for continuous QPTs of the iCP-nCp and iCP-cCP transitions. The insets show the corresponding slope versus $1/\eta$.