Conclusions

In this Chapter, I have presented the study of nonlinearities that the excitons can bring, with different models. The main results of this Chapter are in the description of the spectra of emission for each case, that follows from the general expression (2.105), in WC or SC regimes.

In Sec. 5.2 we studied isolated interacting excitons, still considered as bosons, with the AO model. This fundamental model can be solved analytically for the SE case [Eq. (5.8)] but requires numerical computations in the SS case. The most interesting consequence of the nonlinearity then, is that the transition from a quantum (multimode lasing) to a classical (mean field lasing) regime can be easily tracked by the melting of the individual peaks (at ) that compose the spectra into a broad single peak (at ) (see Fig. 5.3). The transition can be induced by increasing the pump (``inverting'' the population) or decreasing the interactions (towards the ``classical'' HO).

In Sec. 5.3, we coupled the AO--representing the interacting excitons--with an HO--representing the cavity photons--to study the effect of nonlinearities in the LM. In this case, I have focused on the quantum regime (very low pump) of the SS case, where nonlinearities are weakly probed. In this limit, we could successfully analyze all the spectral features from the very well resolved lines, in terms of manifold transitions only. We paid special attention to the specific lines that the different kinds of pumping, excitonic or photonic, enhances in the optical emission, and how this is affected by detuning (Fig. 5.9). The spectral shapes observed are accounted mainly by the Coulomb energy blueshift on top of the vacuum Rabi doublet, with crossings and anticrossings of the lines with detuning, depending on their opposed (photon and exciton) or identical (photon or exciton) character. We can conclude in this model that the optimum experimental configuration to observe nonlinear effects in the PL spectra is at an intermediate, nonzero detuning, for instance at $\Delta\approx\pm g$ . There is an asymmetry with the sign of the detuning due to the interactions that further helps in characterizing the nature of the nonlinearity (e.g., to which extent it comes from the exciton-exciton Coulomb interaction). The presence of satellites with detuning demonstrates emission from quantized manifolds, and as such is a signature of the quantum regime. The spectral drift of these lines with detuning is a useful tool to explicit the exact form of the Hamiltonian that accounts for the exciton nonlinearities.

In Sec. 5.4, we turned to the most important Hamiltonian of quantum optics: the Jaynes-Cummings model. It takes into account the saturation of the QD due to Pauli blocking. This describes the case of a small QD where electron and hole wavefunctions are quantized separately. As in the previous Sections, manifestations of nonlinearities in the SC physics of a genuine quantum nature are no better sought at high pumpings, looking forward to large number of excitations. The quantum regime involves a few quanta only. It is achieved and better manifests with low pumpings in high quality samples (see JC forks in Fig. 5.18, with well identified transitions in the Jaynes-Cummings ladder). The incoherent nature of the pumping results in largely fluctuating distributions of the particle numbers, and small averages populations still succeed in probing appreciably high manifolds. As we did with the AO, we have tracked theoretically, by increasing the pump, the crossover from the quantum to the classical regime, where the cavity field can be considered a continuous field. The counterpart of a Mollow triplet is observed in this regime for the best samples, more clearly in the direct exciton emission (Fig. 5.21). It features a narrow resonance in the center of the spectrum that turns into a sharp emission line. When the Mollow triplet is fully formed, the cavity mode is in the lasing regime. The Mollow triplet is lost as the system is quenched with no return to quantum behaviors. This provides the general sequence of regimes with increasing electronic pumping for a good (strongly coupled) system: quantum regime, lasing (classical) regime and quenched (also classical but thermal) WC regime.

Again, the cavity pumping appears as an important factor to take into account. First, because of its relevance in an actual experiment, where it can arise due to secondary effects such as other dots (not in SC) emitting in the cavity, temperature, or a variety of other factors. It could conceivably also be input directly by the experimentalist. Cavity pumping has many virtues for the physics of SC in a semiconductor. Because the typical type of excitation is electronic and the typical channel of detection is photonic, SC is hampered as compared to the microwave cavity case where detection and excitation are on the same footing (both directly on the atom). A cavity pumping can help balance this situation and provide an effective photon character to the states realized in the semiconductor, enhancing or even revealing spectral structures. This phenomena manifests also in the LM and has been investigated and explained in its full detailed in Chapter 3. Also in the fermion case, cavity pumping is beneficial for the same reasons, and it can help go beyond the linear regime (with a vacuum Rabi doublet) to the nonlinear quantum regime, typically by making emerge additional quadruplets of the JCM, with a doublet of inner peaks to be sought as the strongest signature (as in Fig. 5.28).