One quantum dot in a cavity with a biexcitonic state

In this Section, we consider the situation where up to two excitons can be created in a single QD, forming a biexciton. This is an interesting system for two-photon generation and several works exist on its coherent control through pulsed light, like that of Stufler et al. (2004) and Machnikowski (2008) or that of Flissikowski et al. (2004) and (2005). However, here we center our attention on the competition between two and one photon processes under incoherent continuous pump. With this excitation mechanism is more adequate to speak about 2P versus 1P lasing or gain, in a similar way than in the studies of Ning (2004) and Lambropoulos (1999).

**Figure 6.15:** QD levels as compared to the cavity mode $\omega_a$ when the biexcitonic binding energy $\chi=\omega_1+\omega_2-\omega_B>0$ is taken into account and the 2PR achieved. The two excitonic levels are considered equal in energy (to $\omega_E$ ) and detuned from the cavity mode by the negative quantity $\Delta_{1,2}=\omega_a-\omega_E=-\Delta/2$ . The 2PR condition $\Delta=\chi+\delta_\mathrm{2PR}$ is fulfilled here.
$\includegraphics[width=0.8\linewidth]{chap6/2P/Levels/levels-2PR.eps}$

This system can be modelled with the same Hamiltonian and master equation than the two QDs in a cavity but now we must take into account that the two excitons in the biexciton configuration form a molecule with a binding energy. As a consequence, the QD level structure is altered. This can be described with an energy correction to the Hamiltonian (6.1) of the form

$\displaystyle H_\mathrm{B}=-\chi\ket{B}\bra{B}\,.$

(6.22)

The biexciton binding energy defined as $\chi=\omega_1+\omega_2-\omega_B$ , is a positive number and can be as big as one order of magnitude larger than the couplings

. The resulting level structure is plotted in Fig. 6.15 for the case with similar energies for the excitonic states, $\omega_1=\omega_2=\omega_E$ , that we will analyze in what follows. In Section 6.3.1, we derive an effective Hamiltonian for large detuning, as was done by Fernandez-Vidal et al. (2007), in order to find analytically the conditions for 1P and 2P resonances and the effective couplings associated. In Section 6.3.2, we add a continuous incoherent pump of the QD and decay. Finally, in Section 6.3.2, we compute exactly the luminescence spectrum in a truncated scheme.

Subsections