Point 2: Good systems, the nonlinear doublet and role of cavity pumping

In Fig. 5.26, we show for Point 2 a similar overall picture as Fig. 5.19 does for Point 1. Point 2 has larger dissipation and to current estimates, corresponds more closely to the best systems available at the time of writing. As opposed to Point 1, a small cavity pumping has a strong influence on the result, so we display more cases, namely those that range from no cavity pumping (first row) to large cavity pumping (, 4th row) with two intermediate cases showing the transfer of the emission from the vacuum Rabi doublet to the inner peaks arising from transitions between higher manifolds.

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**Figure 5.27:** Point 2 of Fig. 5.13. Details of the loss of the multiplet structure with increasing exciton pumping. Two upper rows (blue) correspond to the cavity emission $S_a(\omega)$ and two lower rows (violet) to the corresponding exciton direct emission $S_\sigma(\omega)$ for (), $\approx0.076$ () and $\approx0.15$ () (higher pumping corresponds to innermost peaks). Cavity pumping is essential in such a system to reveal the Fermionic nature of the QD emitter.
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**Figure 5.28:** Detail of $S_a(\omega)$ for Point 2 of Fig. 5.13 at vanishing $P_\sigma$ for the values of indicated (higher pumpings correspond to innermost peaks). In a reasonably good QD-cavity system, strong deviations from the linear regime are observed in the emission spectrum, revealing the Jaynes-Cummings fork. The quantum features are made more obvious by increasing the cavity pumping, with a neat renormalization of the dominant doublet even if the quadruplet cannot be resolved experimentally.
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The fifth row shows the corresponding direct exciton emission, for the extreme cases of no (1st row) and highest (4th) cavity pumping. The cavity pumping has the important role of revealing the quantum nonlinearity of the system, that was obvious for reference point 1 in any case but is now invisible in the first row, where at increasing electronic pumping, the vacuum Rabi doublet undergoes a rather dull collapse. The same spectra could be expected from a linear (bosonic) model, in the appropriate range of parameters. This is particularly evident in Fig. 5.27 where three cases of cavity pumping (none, intermediate, and large) are shown for various electronic pumping, both for the cavity and direct emission. Outer lines correspond to zero and inner lines to larger cavity pumping. Note how the intermediate cavity pumping cases display obvious deviation from a bosonic model, that has essentially the shape of a doublet of Lorentzian peaks (with a dispersive correction that has little bearing on the qualitative aspect of the final result). Cavity pumping literally unravels the nonlinearity. The case of intermediate pumping is the most determining in this aspect as far as cavity emission is concerned, while higher cavity pumpings are more favorable for uncovering quantum features from the direct exciton emission. This is mainly for two reasons. One has to do with the influence of what effective quantum state is realized in the system, that we will discuss in more details in connection with the third reference point. The other being the excitation of higher manifolds from the Jaynes-Cummings ladder, that are now less accessible because of the larger dissipation rates. Note how the disappearance of the vacuum Rabi doublet with increasing $P_\sigma$ (with no cavity pumping), is of a different character than for Point 1, where higher $P_\sigma$ resulted in an excitation of the upper manifolds and a transfer of the dynamics higher in the Jaynes-Cummings ladder, whereas in this case it essentially results in a competition between only the first and second manifold transitions. Cavity pumping can help climbing the ladder with no prejudice to broadening. Finally, even if blurry resolution or statistical noise of an actual experiment would cast doubt on the presence of a quadruplet in such a structure, the transfer with increasing cavity pumping of the emission from outer (vacuum Rabi) to inner peaks (from the second manifold transitions in this case) makes it clear that the underlying statistics is of a Fermi rather than of a Bose character. In Fig. 5.28, we show the case $P_\sigma=10^{-3}g$ for such an increasing cavity pumping for a detailed appreciation of the previous statement. A very close look might still suggest that the case (outer peaks) still has a small deviation from the linear model that would betray, in a very finely resolved experiment, its nonbosonic or nonlinear character. Counter to intuition, this is better seen for vanishing electronic pumping, as otherwise the lines are broadened according to Eq. (5.16) and this dampens the inner nonlinear peaks. Note, on the other hand, how cavity pumping unambiguously settles the issue.