Atom-only descriptions of the driven-dissipative Dicke model

We investigate how to describe the dissipative spin dynamics of the driven-dissipative Dicke model, describing $N$ two-level atoms coupled to a cavity mode, after adiabatic elimination of the cavity mode. To this end, we derive a Redfield master equation which goes beyond the standard secular approximation and large detuning limits. We show that the secular (or rotating wave) approximation and the large detuning approximation both lead to inadequate master equations, that fail to predict the Dicke transition or the damping rates of the atomic dynamics. In contrast, the full Redfield theory correctly predicts the phase transition and the effective atomic damping rates. Our work provides a reliable framework to study the full quantum dynamics of atoms in a multimode cavity, where a quantum description of the full model becomes intractable.

Figure 1

Cartoon of the Dicke model. Left: many two-level systems placed in a lossy cavity. Right: Raman driving scheme; transitions between two states $|\downarrow \rangle$ and $|\uparrow \rangle$ involve a virtual transition due to scattering between a pump and a cavity photon.

Figure 2

Evolution of the steady-state value of $\langle S^z\rangle$ with matter-light coupling $g\sqrt{N}$ for ${\omega }_0=0.1$, and $\kappa =1$ (in units chosen so that $\omega =1$). Curves are rescaled by $N/2$ so as to produce a finite limit as $N\to \infty$. The blue curves show the exact numerical simulations [based on the resolution of the master equation (22)] for various values of $N$ as indicated in the legend. The dashed black curve shows the semiclassical solution (valid as $N\to \infty$). The position of $g_c\sqrt{N}$, which marks the transition between the normal state and the superradiant state, is indicated by a red vertical dashed line.