Thermodynamics and superradiant phase transitions in a three-level Dicke model

We analyze the thermodynamic properties of a generalized Dicke model, i.e., a collection of three-level systems interacting with two bosonic modes. We show that at finite temperatures the system undergoes first-order phase transitions only, which is in contrast to the zero-temperature case where a second-order phase transition exists as well. We discuss the free energy and prominent expectation values. The limit of vanishing temperature is discussed as well.

Figure 1

Single-particle energy levels in $\mathrm{\Lambda }$ configuration: The excited state $|3\rangle$ is coupled via the two bosonic modes with frequencies ${\omega }_1$ and ${\omega }_2$ to either of the two ground states, $|1\rangle$ and $|2\rangle$, respectively. The corresponding coupling strengths are given by $g_1$ and $g_2$.

Figure 2

Graphical analysis of Eq. (21) for $g_n<g_{n,c}$ (left) and $g_n>g_{n,c}$ (right) for a certain $n$ and fixed temperature. The red straight line corresponds to the first term, ${\left(\frac{g_{n,c}}{g_c}\right)}^2\mathrm{\Omega }$, the other blue curved line corresponds to $q\left(\mathrm{\Omega }\right)$ of Eq. (21) involving the hyperbolic functions. Of physical relevance is the region $\mathrm{\Omega }>1$ only. Thus, for $g<g_{n,c}$ no physical solution is possible, whereas for $g_n>g_{n,c}$ a physical solution might exist.

Figure 3

The function $f$ of Eq. (9) with $z_n=0=y_2$, and $y_1=y$ for fixed temperature. The coupling strength $g$ increases from the upper to the lower curves. The units are arbitrary and $f$ has been rescaled for comparison, such that $f\left(0\right)=0$ for all coupling strengths $g$. On increasing $g$, a local minimum forms distant from the origin. For large $g$ this minimum eventually becomes a global minimum.

Figure 4

Numerical computation of the scaled occupations $\langle {\widehat{a}}_n^{†}{\widehat{a}}_n\rangle /\mathcal{N}$ of the two bosonic modes for $k_{\mathrm{B}}T=0.001\mathrm{\Delta }$ (left) and $k_{\mathrm{B}}T=0.25\mathrm{\Delta }$ (right). The parameters are set to $\mathrm{\Delta }=1, \delta =0.1, {\omega }_1=1.1$, and ${\omega }_2=0.8$. The computation has been done in the thermodynamic limit using Laplace's method.

Figure 5

Numerical computation of the scaled occupations $\langle {\widehat{A}}_n^n\rangle /\mathcal{N}$ of the single-particle energy levels of the three-level systems for $k_{\mathrm{B}}T=0.001\mathrm{\Delta }$ (left) and $k_{\mathrm{B}}T=0.25\mathrm{\Delta }$ (right). Parameters as in Fig. 4.

Figure 6

Numerical computation of the scaled polarizations $\langle {\widehat{A}}_1^3\rangle /\mathcal{N}, \langle {\widehat{A}}_2^3\rangle /\mathcal{N}$ of the three-level systems for $k_{\mathrm{B}}T=0.001\mathrm{\Delta }$ (left) and $k_{\mathrm{B}}T=0.25\mathrm{\Delta }$ (right). Parameters as in Fig. 4.

Figure 7

Numerical computation of the scaled polarization $\langle {\widehat{A}}_1^3\rangle /\mathcal{N}$ of the transition $|1\rangle \leftrightarrow |3\rangle$ of the three-level system. Parameters as in Fig. 4 and $g_2=0.2g_{2,c}$.

Figure 8

Occupations $\langle {\widehat{a}}_1^{†}{\widehat{a}}_1\rangle$ of the first bosonic mode as a function of the coupling strength $g_1$ for fixed coupling strength $g_2=0.2g_{2,c}$ and rising temperature (left to right, blue to red, respectively). The other parameters are set to $\mathrm{\Delta }=1, \delta =0.1, {\omega }_1=1.1$, and ${\omega }_2=0.8$.