Superradiant phase transition in a model of three-level-$\Lambda$ systems interacting with two bosonic modes

We consider an ensemble of three-level particles in Lambda configuration interacting with two bosonic modes. The Hamiltonian has the form of a generalized Dicke model. We show that in the thermodynamic limit this model supports a superradiant quantum phase transition. Remarkably, this can be both a first- and a second-order phase transition. A connection of the phase diagram to the symmetries of the Hamiltonian is also given. In addition, we show that this model can describe atoms interacting with an electromagnetic field in which the microscopic Hamiltonian includes a diamagnetic contribution. Even though the parameters of the atomic system respect the Thomas-Reiche-Kuhn sum rule, the system still shows a superradiant phase transition.

Figure 1

$\Lambda$ configuration of energy levels of a single particle: The two bosonic modes with energies ${\omega }_1$ and ${\omega }_2$ couple the excited state with energy $E_3$ to either of the two ground states with energies $E_1$ and $E_2$ of the particles, respectively. The corresponding coupling strengths are given by $g_1$ and $g_2$. The geometry of the setup where the particles are atoms and the bosonic modes are modes of an electromagnetic field (cf. Appendix 6) is shown at the top. In our numerical analysis we assume that the polarization vectors ${\varepsilon }_n$ are parallel to the dipole matrix elements ${\mathbf{d}}_{3n}$, respectively. In addition, the polarization vectors must not be orthogonal.

Figure 2

Top: Mean fields ${\Psi }_2$ and ${\Psi }_3$ for parameters $\delta =0.1$, $\Delta =1$, ${\omega }_1=0.5$, ${\omega }_2=0.6$, $\alpha =\sqrt{0.5}$, ${\kappa }_1={\kappa }_2=1$, ${\kappa }_3=\sqrt{\alpha }$, and ${\alpha }_{11}={\alpha }_{22}=1$. The green line illustrates the phase boundary and is separated into two segments: The solid (dashed) green line corresponds to a first-order (second-order) superradiant phase transition. The point where these two segments meet is marked with the red ring. The numerical value of $g_1$ for the boundary of the second-order phase transition is given by $1.03g_{1,\text{TRK}}$, which coincides with the value obtained by an analytical analysis. The blue (dark) region below the green line in both diagrams represents the normal phase (${\psi }_1=1$, ${\Psi }_2={\Psi }_3={\phi }_1={\phi }_2=0$). The complementary region corresponds to the superradiant phase where all mean fields are finite. The accessible parameter region for atoms is indicated by the solid yellow box. The small inset clarifies that the superradiant phase is within this region. Bottom: Mean field ${\Psi }_2$ as a function of $g_1$ for different values of $g_2$, indicating the change of the order of the superradiant phase transition. The value of $g_2$ for each single line increases in the direction of the arrow from 0 to $1.5$ with an increment of $0.15$. In the range $0\le g_2\le 0.45$, ${\Psi }_2$ is continuous as a function of $g_1$ (solid blue lines), whereas for $g_2\ge 0.6$, ${\Psi }_2$ is discontinuous (dashed green lines). The dashed red line marks the critical value for $g_1$ of Eq. (8) above which the normal phase is unstable and corresponds to the red ring in the top panel. In both panels, the couplings $g_n$ are scaled with $g_{n,\text{TRK}}$. These are the largest possible couplings for the analogous atomic model given by the TRK sum rules, Eq. (5). The diagrams for ${\phi }_1$ and ${\phi }_2$ look qualitatively the same as the diagram for ${\Psi }_3$. In addition, ${\psi }_1$ is obtained by using the relation ${\psi }_1=\sqrt{1-{\Psi }_2^2-{\Psi }_3^2}$.

Figure 3

Critical coupling $g_c$ at different points on the phase boundary as a function of $\chi \equiv {\chi }_1={\chi }_2$ and for different values of $\kappa \equiv {\kappa }_1={\kappa }_2={\kappa }_3$. The solid lines correspond to points on the phase boundary with $(g_1=g_c,g_2=0)$, dashed lines to points with $(g_1=0,g_2=g_c)$, and dash-dotted lines to points with $(g_1=g_c,g_2=g_c)$. The red lines (bottom three) correspond to $\kappa =0$, the green lines (three lines in the center) to $\kappa =0.6$, and the blue lines (top three) to $\kappa =1.2$. The remaining parameters are given by $\Delta ={\omega }_1=1$, $\delta =0.1$, ${\omega }_2=0.9$.