Figure 1

Sketch of the considered system of $N$ three-level atoms identically coupled to a single cavity mode. Each three-level system is represented by the transition frequencies ${\omega }_{10}={\omega }_1-{\omega }_0>0$, ${\omega }_{21}={\omega }_2-{\omega }_1>0$, and ${\omega }_{20}={\omega }_{21}+{\omega }_{10}>0$ and by the oscillator strengths $f_{01}\ge 0$, $f_{12}\ge 0$, and $f_{02}\ge 0$ [see definition (3)].Reuse & Permissions

Figure 2

Results for the ladder configuration ($f_{02}=0$). Photonic order parameter $x=\langle a\rangle /\sqrt{N}$ in the ground state versus $f_{01}$ and $f_{12}$ for each three-level atom. The area within the dashed (red) lines indicates the region described by the TRK inequalities. Parameters: $D=3{\omega }_{\mathrm{cav}}$, ${\omega }_{10}=0.1{\omega }_{\mathrm{cav}}$, ${\omega }_{21}={\omega }_{\mathrm{cav}}$. The collective vacuum Rabi frequencies ${\Omega }_{01}$ and ${\Omega }_{12}$ are obtained via Eq. (2).Reuse & Permissions

Figure 3

$\mathsf{V}$-type configuration ($f_{12}=0$). Photonic order parameter versus $f_{01}$ and $f_{02}$. Parameters: $D={\omega }_{\mathrm{cav}}$, ${\omega }_{10}={\omega }_{\mathrm{cav}}$, ${\omega }_{21}=0.1{\omega }_{\mathrm{cav}}$. In this configuration, the superradiant phase is always incompatible with the TRK inequality boundaries.Reuse & Permissions

Figure 4

Results for the $\Lambda$ configuration ($f_{01}=0$). Photonic order parameter versus $f_{12}$ and $f_{02}$. Parameters: $D=3{\omega }_{\mathrm{cav}}$, ${\omega }_{10}=0.1{\omega }_{\mathrm{cav}}$, ${\omega }_{21}=0.9{\omega }_{\mathrm{cav}}$.Reuse & Permissions

Figure 5

Top: Filled red area in the $(f_{01},f_{02},f_{12})$ space represents the superradiant part of the diagram compatible with the TRK inequalities $0\le f_{01}+f_{02}\le 1$ and $0\le f_{10}+f_{12}\le 1$. Parameters: $D=5{\omega }_{\mathrm{cav}}$, ${\omega }_{10}=0.17{\omega }_{\mathrm{cav}}$, ${\omega }_{21}={\omega }_{\mathrm{cav}}$. Middle: Photonic order parameter $x$ versus $f_{01}$ and $f_{02}$ for $f_{12}=0.735$. The area below the dashed (red) line corresponds to the region satisfying the TRK inequalities. Bottom: The black line represents the spatial profile of an example 1D potential with squared wells in units of the energy $E_c=\frac{{\hslash }^2}{2m{L}^2}$, where the spatial coordinate is expressed in units of the length $L$. The horizontal lines depict the energies of the three bound states with their corresponding wave functions (other states are in the continuum). For this potential shape, $f_{01}=0.3995,f_{02}=0.4069,f_{12}=0.735$, and anharmonicity ratio ${\omega }_{10}/{\omega }_{21}=0.1709$, corresponding to the point depicted by the (blue) X in the middle panel.Reuse & Permissions