Spectrum with detuning

**Figure 5.31:** Positions $\omega_p/g$ of the lines around $\omega_a=0$ in the luminescence spectrum with detuning and in the absence of pump. Columns correspond to various detunings, first column being the case of resonance (in Fig. 5.12). First three rows show in isolation the first, second and third manifold, respectively. First manifold corresponds to the bosonic or linear case. Fourth row shows all manifolds together. Left-bottom panel is detailed for positive $\omega_p$ in Fig. 5.12.
$\includegraphics[width=\linewidth]{chap5/JC/fig18-eig-det.ps}$

**Figure 5.32:** Anticrossing of the luminescence lines as detuning $\Delta=\omega_a-\omega_\sigma$ is varied. Here, $\omega_a=0$ is fixed and the QD bare energy is tuned from below the cavity (positive detuning) to above (negative detuning). Panels (a)-(d) correspond to Point 1 and panel (e) to Point 2. (a)-(d) are at zero cavity pumping, . (a) and (d) are for $P_\sigma=0.03g$ [(d) is a zoom of (a)] and (b)-(c) for $P_\sigma=0.3g$ . (a), (b), (d) are the cavity emission $S_a(\omega)$ and (c) the direct exciton emission $S_\sigma(\omega)$ . (e) is for $P_\sigma=10^{-3}g$ and (cf., 7th row, 1st column of Fig. 5.26). The nonlinear central peaks give rise to very characteristic anticrossing profiles.
$\includegraphics[width=\linewidth]{chap5/JC/fig19-anticrossing.ps}$

In semiconductors, the detuning between bare modes is a parameter that can easily be varied and which provides useful information of the SC physics. Strong coupling is better studied at resonance, and detuning is mainly used to help locate it, by finding the point where anticrossing is maximum and level repulsion stationary. In a fitting analysis of an experiment, it brings a lot of additional data at the cost of only one additional fitting parameter. In the Fermion case, it also has the benefit of uncovering new qualitative behavior of the PL lineshapes, that are strongly restricted by symmetry at resonance.

Fig. 5.31 shows the vanishing pumping case of $\omega_p$ in Eq. (5.24) with detuning, i.e., the imaginary part of Eq. (5.26) for the first row that corresponds to the first manifold (also, the boson case) and of Eq. (5.27) for the second and third rows, that corresponds to the second and third manifold, respectively. Fourth row is a superposition of all manifolds up to the 15th one. Detuning is varied in columns, from no detuning (first column) to twice the coupling strength (fifth column). Negative detunings are symmetric with respect to the axis.

The line opening is common to all manifolds, but note the different behavior of the first manifold (linear or boson case) and higher manifolds: in the first case, one line collapses towards the center (on the cavity mode) while the other recedes away, towards the exciton mode. In the nonlinear case, there is up to four lines, and outer lines are both repelled away while inner lines get both attracted towards the cavity mode, at the center. As we discussed, the total doublet of inner peaks is intense and will dominate. For cases with high dissipation, there is little or no particular insights to be gained from detuning, as, again, most features are lost in broadening. We restrict out attention to Points 1 and 2 in what follows. In Fig. 5.32, PL with detuning are shown for Point 1 in panels (a)-(d) and for Point 2 in panel (e). Panel (d) is a magnified view of panel (a). It is seen clearly how the doublet of inner peaks essentially remains fixed at its resonance position independently of the exciton position. Only at very high detunings does the doublet collapse onto the center. The vacuum Rabi doublet however appears as an anticrossing of the exciton bare mode with the doublet of inner peaks (that eventually becomes the cavity bare mode). Panel (a) is at small electronic pumping and (b), (c) at ten time larger electronic pumping (both no cavity pumping), for the cavity and direct exciton emission, respectively. Again, lower electronic pumping is more prone to reveal rich quantum features. In panel (b) only the inner nonlinear doublet is visible, with a transfer of the emission intensity from one peak (essentially fixed) to the other. The resonance case is plotted in the third panel on first row (third row for the exciton emission) of Fig. 5.20. The linear Rabi doublet, which trace is seen faintly undergoing anticrossing with the pinned central peaks, provides small shoulders. In general, PL with detuning in the Fermi case shows a very characteristic behavior, that cannot be mistaken with a conventional (bosonic) anticrossing experiment.

In panel (e), the case of a more realistic system is shown with detuning. The pinning of the inner peaks is less obvious in this case, although if one draws a vertical line at the resonance, through the minimum of the doublet, one observes that this minimum is fixed. As a result, triplets are obtained in the cavity emission spectra, that are of a very distinct nature than the Mollow triplet observed in the side (exciton) emission of Point 1. The triplets involving the nonlinear doublet are a manifestation of the quantum regime overcoming broadening while the Mollow triplet is a manifestation of the lasing regime.