Nonequilibrium Quantum Phase Transitions in the Dicke Model

We establish a set of nonequilibrium quantum phase transitions in the Dicke model by considering a monochromatic nonadiabatic modulation of the atom-field coupling. For weak driving the system exhibits a set of sidebands which allow the circumvention of the no-go theorem which otherwise forbids the occurrence of superradiant phase transitions. At strong driving we show that the system exhibits a rich multistable structure and exhibits both first- and second-order nonequilibrium quantum phase transitions.

Figure 1

Stability diagram of the nonequilibrium normal phase on resonance $\omega ={\omega }_0$ for (a) weak driving, $\Delta g/\Omega =0.15$, and (b) strong driving, $\Delta g/\Omega =0.4$. The colored zones correspond to stability, white zones, instability. In the undriven case, the separatrix between stable and unstable zones is given by $g_c=\omega /2$ on resonance (dotted red line). For weak driving instability zones open up around the resonance conditions ${\epsilon }_{\pm }=k(\Omega /2)$ for $k=1,2,3$ (dotted black lines), which grow and begin to dominate for large driving.

Figure 2

Phase diagram as a function of static ($g$) and driving amplitudes ($\Delta g$) of effective Hamiltonian $h_0^{(0)}$, which describes the driven DM near the $k=0$ resonance. The labels indicate the number of local minima of the ground-state energy landscape $E_G$ in the corresponding zone. The normal phase is the region where there is just a single minimum ($X=Y=0$). The superradiant phase is a region with two global minima (nonzero order parameter). The boundary between these two regions marks a second-order QPT (dotted red curve). Outside these regions, the energy surface has an odd number $\ge 3$ of total minima, sometimes with a single global minima, sometimes with two. The boundary between these possibilities is a first-order QPT (solid blue line). The parameters are $\omega /\Omega ={\omega }_0/\Omega =0.05$.

Figure 3

Sections of the ground-state energy landscape $E_G(X,Y)$ for a fixed value of the static coupling ($2g/\Omega =0.195$) and increasing driving amplitudes ($\Delta g$) represented by dashed straight lines in Fig. 2. This sections correspond to the function $E_G(X[Y],Y)$, where $X[Y]$ is a line in the order-parameter space crossing all the critical points of the ground-state energy. The four panels show (a) the superradiant phase with two global minima, (b) the emergence of a local minimum at the origin, (c) multiple minima, but still two global minima, and (d) a single global minima at the origin plus many further local minima. Panels (c) and (d) are separated by the first-order phase transition. The parameters are $\omega /\Omega ={\omega }_0/\Omega =0.05$.