Weights and Renormalization

To give a complete picture of the spectral structure, that we have obtained analytically in Section 5.4.2, we need to consider how this limiting case of vanishing pumpings evolves with finite pumping. Here again, we have to turn to numerical results.

**Figure 5.15:** Spectral structure in the cavity emission of the Jaynes-Cummings model as a function of $\gamma_a/g$ , with some electronic pumping ( $\Delta=0$ , $\gamma_\sigma=0$ , , $\omega_a=0$ ). Panel (a) is for $P_\sigma=g/50$ and (b)-(c) for $P_\sigma=g/10$ . Panel (c) is a zoom on the central peaks of the entire picture (b). In blue (resp., red) are the peaks with (resp., ).
$\includegraphics[width=.75\linewidth]{chap5/JC/fig5-bc-with-pump.ps}$

**Figure 5.16:** Spectral structure in the cavity emission at resonance as a function of $P_\sigma/g$ for Point 2 ( $\Delta=0$ , , $\omega_a=0$ ). In insets, we zoom over the central peaks and the region $0\le P_\sigma\le g$ (upper) and $0\le P_\sigma \le 10g$ (lower), showing the complex structures that arise. In blue (resp., red) are the peaks with (resp., ). At sufficiently high pumping, all eigenvalues have collapse to zero, defining an extreme case of weak coupling.
$\includegraphics[width=\linewidth]{chap5/JC/final-fig6.eps}$

Two cases of finite pumpings are shown in Fig. 5.15 for the finite pumping counterpart of Fig. 5.12(a), namely $P_\sigma=g/50$ , (a), and $P_\sigma=g/10$ , (b) and (c). We take $\omega_a=0$ as the reference energy. Panel (a) shows how the limiting case ( $P_\sigma \ll g$ ) is weighted and deviates rather lightly from the analytical result. The computation has been made to truncation order $n_\mathrm{max}=50$ and we checked that it had converged with other truncation orders giving exactly the same result. In the figure, only $\omega_p$ whose weighting in the cavity emission (Lorentzian part) is nonzero are shown, although most of them are very small. If we plot only those with $\vert L_p^a\vert\ge 0.01$ , only the usual vacuum Rabi doublet (in green in Fig. 5.12) would remain. In addition of the weight, also the degeneracy (number of peaks) at a given resonance should be taken into account to quantify the intensity of emission at a particular energy. This information is not apparent in the figures, where we only show in blue or red the cases of positive or negative, respectively, weighting. In some cases, many peaks superimpose with opposite signs, possibly cancelling each other. We plot negative values last so that a blue line corresponds to a region of only positive values, while a red line may come on top of a blue line. This figure gives nevertheless an insightful image of the underlying energy structure and how they contribute to the final spectrum as an addition of many emitting (or interfering) events. In (b) we show a case of higher pumping, with the same principal information to be found in the mapping of the eigenvalues. The characteristic branch-coupling of the JCM, still easily identifiable in Fig. 5.15(a), has vanished, and lines of external peaks directly collapse toward the center. A zoom of the central part, panel (c), shows the considerable complexity of the inner peaks, forming ``bubbles'' around the central line, due to intensity-aided SC fighting against increasing dissipation that ultimately overtakes. This is the counterpart of the second order and mixed coupling regimes in the SC phase space of 2LSs, where bubbles could form as a result of new asymmetric eigenstates ( $\ket{I}$ and $\ket{O}$ ) appearing in the system.

The origin of the lines can be better understood if we plot them as a function of pumping, as we commented in Section 5.4.2. In Fig. 5.16, the same weighted peak positions $\omega_p$ are shown (with the same color code) for Point 2 as electronic pumping is varied from $10^{-3}g$ to (). This last picture supports the idea that quantum effects (such as subpoissonian statistics, Fig. 5.14) are observed at small pumpings, with optimal range being roughly $P_\sigma<0.5g$ , where only the lowest manifolds are probed. This is the range of pumping where the Jaynes-Cummings manifold structure is still close to that without pump. Further pumping pushes the lines to collapse, starting by the vacuum Rabi splitting which closes, evidencing the loss of the first order SC at $P_\sigma\approx4g$ . Here again we observe this phenomenon of bubbling, with a sequence of lines opening and collapsing, that makes it impossible to specify the exact pump at which the transition takes place. From this point, SC is lost manifold by manifold similarly to the case where $\gamma_a$ was increased. When $P_\sigma\approx 40g$ , all lines have collapsed onto the center and will remain so at higher pumpings. The dot saturates and the cavity empties with thermal photons in a WC regime.

In these conditions, either from Fig. 5.15(b)-(c) or Fig. 5.16, a general definition of Strong-Coupling in presence of pumping is obviously very complex and remains to be established.