Dicke Transition in Open Many-Body Systems Determined by Fluctuation Effects

We develop an approach to describe the Dicke transition of interacting many-particle systems strongly coupled to the light of a lossy cavity. A mean-field approach is combined with a perturbative treatment of fluctuations beyond mean field, which becomes exact in the thermodynamic limit. These fluctuations completely change the nature of the steady state, determine the thermal character of the transition, and lead to universal properties of the emerging self-organized states. We validate our results by comparing them with time-dependent matrix-product-state calculations.

Figure 1

Mean-field order parameter $\lambda$ as a function of temperature computed for $\hslash g=3J$, $\hslash \mathrm{\Gamma }=J$, $U=2J$, $L=10$. The temperature of the stationary state is obtained by searching for the point (red dot) with $(\partial /\partial t)⟨H_b⟩=0$. Here fluctuations beyond mean field determine $(\partial /\partial t)⟨H_b⟩$. The inset shows a schematic picture of the model: a Bose-Hubbard chain coupled to a cavity with loss rate $\mathrm{\Gamma }$.

Figure 2

Phase diagram as function of $\mathrm{\Gamma }$ and $g$ for $\hslash \delta =2J$, $U=2J$, $n=1/2$. The color and contour lines encode $⟨a^{†}a⟩/L$ which vanishes in the disordered phase for $L\to \infty$ and is finite in the ordered phase. Remarkably, $⟨a^{†}a⟩/L$ reaches the universal value $3/8=0.375$ deep in the ordered phase due to heating effects [see Eq. (11)].

Figure 3

$g$ and $\mathrm{\Gamma }$ dependence of the photon number [panels (a) and (b)] and the effective temperature [panels (c) and (d)] for $U=2J$ and $\hslash \delta =2J$. The mean-field results [Eq. (5)] (red and orange for $L=12$ and $L=10$, respectively) are obtained using fluctuation effects to determine $T$ [Eq. (9)]. The resulting photon numbers deviate strongly from the $T=0$ mean-field result (blue squares) due to the heating effects shown in the two lower panels. We show numerically exact TMPS results for two system sizes and several final times. The TMPS results approach the fluctuation-corrected mean-field results for larger system sizes and times and, therefore, are fully consistent with the mean-field results. As expected, the sharp phase transition obtained within mean field (vertical dashed lines) is smeared in the finite-$L$ TMPS calculations.