Phase transitions and dark-state physics in two-color superradiance

We theoretically study an extension of the Dicke model, where the single-particle Hamiltonian has three energy levels in Lambda configuration (i.e., the excited state is coupled to two nondegenerate ground states via two independent quantized light fields). The corresponding many-body Hamiltonian can be diagonalized in the thermodynamic limit with the help of a generalized Holstein-Primakoff transformation. Analyzing the ground-state energy and the excitation energies, we identify one normal and two superradiant phases, separated by phase transitions of both first and second oder. A phase with both superradiant states coexisting is not stable. In addition, in the limit of two degenerate ground states a dark state emerges, which seems to be analogous to the dark state appearing in the well-known stimulated Raman adiabatic passage scheme.

Figure 1

Level structure of the $\Lambda$-configuration. One particle has two ground states $|1\rangle$, $|2\rangle ,$ and one excited state $|3\rangle$, where the excited state is coupled to the two ground states via two independent scalar bosonic modes with in general different frequencies ${\omega }_1$, ${\omega }_2$ and coupling strengths $g_1$, $g_2$.

Figure 2

Physical interpretation of the bosons introduced via the generalized Holstein-Primakoff transformation [Eq. (5)]: The two bosonic operators ${\widehat{b}}_r^{†}$, with $r\ne m$, can be understood as collectively exciting the particles from the reference state $|m\rangle$ (left: $m=1$, right: $m=2$) to the state $|r\rangle$. The analog holds for the annihilation operators ${\widehat{b}}_r$.

Figure 3

Phase diagram for $\Delta >\delta >0$ showing the three different phases: the normal and the blue and the red superradiant phase. The (symmetric) normal phase is defined by ${\Psi }_2={\Psi }_3={\varphi }_1={\varphi }_2=0$. In the (symmetry-broken) blue superradiant phase ${\Psi }_2=0,{\Psi }_3\ne 0,{\varphi }_1\ne 0$, and ${\varphi }_2=0$. Finally, in the (symmetry-broken) red superradiant phase ${\Psi }_2\ne 0,{\Psi }_3\ne 0,{\varphi }_1=0$, and ${\varphi }_2\ne 0$ holds. The phase transition from the normal to the blue superradiant phase is of second order (dashed line), whereas the phase transition from the normal to the red superradiant phase and between the two superradiant phases is of first order (solid line). The normal state is metastable in the region of the red superradiant phase as long as $g_1<g_{1,c}$.

Figure 4

Ground-state energy ${\widehat{h}}^{\left(0\right)}$, ground-state occupation ${\varphi }_n^2$ of the first ($n=1$) and the second ($n=2$) bosonic mode, and the ground-state occupation ${\Psi }_n^2$ of the single-particle energy levels ($n=1,2,3$). Numerical values: $\Delta ={\omega }_1=1$, $\delta =0.75$, ${\omega }_2=0.25$ (on resonance).

Figure 5

Excitation energies ${\varepsilon }_{n,\sigma }$ from Eqs. (30) and (31). Numerical values: $\Delta ={\omega }_1=1$, $\delta =0.75$, ${\omega }_2=0.25$ (on resonance).

Figure 6

Phase diagram (a) as in Fig. 3 and energy surfaces ${\widehat{h}}^{\left(0\right)}$ (b)–(d) in the degenerate ($\delta \to 0$) limit. In (a), the solid and the broken red lines denote first- and second-order phase transitions, respectively. The red lines in (b)–(d) visualize the inequality [Eq. (37)] (i.e., the line where the dark state with ${\Psi }_3=0$ is stable). However, as the red line is flat,fluctuations along the ${\Psi }_2$ direction transform stable states on the red line into unstable states outside the red line. Eventually, these states decay to superradiant states with ${\Psi }_3\ne 0$.