Strong coupling of two quantum emitters to a single light mode: The dissipative Tavis-Cummings ladder

A criterion for strong coupling between two quantum emitters and a single resonant light mode in a cavity is presented. The criterion takes into account the escape of cavity photons and the spontaneous emission of the emitters, which are modeled as two-level systems. By using such a criterion, the dissipative Tavis-Cummings ladder of states is constructed, and it is shown that the inclusion of one more emitter with respect to the Jaynes-Cummings (single emitter) case increases the effective parameter region in which $n\text{th}$-order Rabi splitting is observed.

Figure 1

Ladder of bare states at resonance ($\Delta =0$) for two 2-level systems coupled to a cavity mode for the first three excitation manifolds and for two excitation manifolds containing $n$ and $n$+1 excitations. The double-headed solid green horizontal arrows depict the light-matter coupling $g$, dashed red diagonal arrows the spontaneous emission process ${\gamma }_{\sigma }$, and blue dotted vertical arrows the coherent emission process ${\gamma }_a$. It is interesting to note that, because the spontaneous emission process acts independently on each two-level system, it is the only process that couples the singlet state $|S\rangle$ to the dynamics. The spacing between each excitation manifold ${\Lambda }_n$ is given by ${\omega }_0$. In the nonresonant case the atomic states with one excitation $|n,0\rangle ,|n,S\rangle$ will be shifted downwards by an amount $\Delta$ and the state with zero atomic excitations $|n,-1\rangle$ will be shifted by an amount $2\Delta$.

Figure 2

Positions (thick lines) and widths (shadowing) of the complex eigenenergies at resonance for the first two rungs of the system as a function of the inverse lifetime of the photon ${\gamma }_a$. The spontaneous decay rate of the system has been set to zero (${\gamma }_{\sigma }=0$). Note that, in experiments, typically ${\omega }_0\gg g$ and, thus, the widths between different excitation manifolds do not overlap. Also note that for more than one excitation there are always two eigenenergies that are degenerate but that differ in their associated width which is illustrated here for the second rung of the ladder.

Figure 3

The black thick line is the contour of $\left|\mathrm{Im}\right({\mathcal{Q}}_n\left)\right|=1$ at resonance $\Delta =0$, and $n$ is taken as a real number whose scale is in the right of the figure. The purple thin lines are the different Rabi splittings $\left|\mathrm{Re}\right({\epsilon }_n-{\omega }_{0}n\left)\right|$ at resonance $\Delta =0$ for the first four excitation manifolds (the scale of the different splittings is in the left of the figure). Note that whenever the contour of $\left|\mathrm{Im}\right({\mathcal{Q}}_n\left)\right|=1$ crosses an integer value of $n$ as ${\gamma }_{-}/g$ is varied, one of the Rabi splittings becomes zero. Finally, as expected from the purely Hamiltonian picture, the $n\text{th}$-order Rabi splitting reduces to $\sqrt{4n-2}$ as the dissipation effects become negligible, ${\gamma }_{-}/g\to 0$.