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Linear model with excitonic interactions

In the previous Section, I have shown how, even for bosonic excitons, a multiplet structure and some fermionic signatures arise in the emission spectrum when intra-dot exciton-exciton interactions are included. In this Section, we will study how these interactions change the spectral properties of the linear model that we studied in Chapter 3 in a quantum regime. The results, that we obtain in the steady state only, are not analytical anymore as in the previous Section. We describe light-matter interaction in a large QD including exciton-exciton interactions with the Hamiltonian

$\displaystyle H= \omega_a\ud{a}a+\omega_b\ud{b}b+g(\ud{a}b+a\ud{b})+\frac{U}{2}\ud{b}\ud{b}bb\, .$ (5.10)

The model couples an HO (photons) with an AO (excitons). In this Section, $ g$ is the unit and $ \omega_b=0$ the reference energy. Let us start by diagonalizing Hamiltonian (5.10) by manifolds, in the spirit of Eq. (2.65). The structure of eigenenergies at resonance is sketched in Fig. 5.5 up to a maximum of two excitations (i.e., up to the second manifold) for three different cases. First, the bare levels corresponding to noninteracting and uncoupled (or weakly-coupled) modes ($ g=0$, $ U=0$). Second, the eigenenergies arising from the coupling ($ g\ne0$, $ U=0$), as we studied them in Chapter 2 and 3, and finally, the blueshifted lines that result from including the interactions ($ g\ne0$, $ U\ne0$). All the levels but those in the manifold $ n=1$, involving only one particle, change with the interactions because of the excitonic part of their corresponding eigenvectors. In order to keep track of the excitonic character of each level, in Fig. 5.6, we plot the excitonic component of the eigenvectors of manifold $ n=2$ and their corresponding eigenenergies as a function of the interaction $ U$. We can see that, starting from a situation completely symmetric between the photonic and excitonic fractions, that the highest level gets more and more excitonic-like with $ U$ and blueshifts strongly. The other two energy levels are only slightly affected, as follows from their more photonic character. This characterization of the levels, which also depends on the detuning, plays an important role when identifying the spectral lines, as we show in the following sections.

Figure 5.5: Energy levels of the eigenstates of the light-matter coupling Hamiltonian with interactions, Eq. (5.10) up to the second manifold (two excitations) at resonance ( $ \Delta=0$), left panel for weak (or no-) coupling ($ g=0$), central and right panel in strong-coupling, with right panel also including interactions $ U$ varying on the $ x$ axis. The transitions between levels account for the spectral features. Red lines correspond to the vacuum Rabi doublet, that turns into a singlet in WC regime (transitions in green). Blue lines superimpose to the Rabi doublet when higher manifolds are probed. Without interactions, $ U=0$, these transitions are not distinguishable in the spectra. Transitions $ 2\rightarrow 1$ in presence of interactions are plotted in Fig. 5.7 as a function of the detuning and $ U$. New qualitative features appear thanks to the interactions. Black dashed lines are new transitions previously forbidden, although they remain weak.
\includegraphics[width=0.6\linewidth]{chap5/brasilia/figure1.ps}


Subsections
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Elena del Valle ©2009-2010-2011-2012.