Fitting of the experimental data

**Figure 3.14:** Anticrossing of the cavity (C) and exciton (X) photoluminescence lines as reported by Reithmaier et al. (2004) in Nature, demonstrating SC in their system. Energies are given in meV. The red lines are our superimposed fits with the best global fit parameters in the top left corner. Such a good agreement cannot be obtained neglecting the pump-induced decoherence.
$\includegraphics[width=.66\linewidth]{chap3/Fig-1.eps}$

The most striking feature of strong coupling is the splitting of the spectral shape when the system is at resonance: the line of the cavity and that of the emitter, both at the same frequency, do not superimpose but anticross with a splitting related to the coupling strength. In Fig. 3.14 we can see the central result of Reithmaier et al. (2004) work. Here, it is claimed that the doublet found at 21K, when the two modes are expected to be resonant, demonstrates SC. However, this claim and the coupling strenght extracted, are based on the SE picture and a Lorentzian fitting of the spectra.

**Figure 3.15:** Theoretical SE spectra: anticrossing of the cavity ( $\omega_a=0$ ) and exciton ( $\omega_b=-\Delta$ ) modes varying detuning from to . The first row corresponds to the cavity emission $S_a^\mathrm{SE}(\omega)$ for the SE of (a) one exciton and (b) one photon as initial conditions. The second row corresponds to the lateral excitonic direct emission $S_b^\mathrm{SE}(\omega)$ for the SE of (c) one exciton and (d) one photon. The parameters are those given by Reithmaier et al. (2004): $g=80\mu$ eV, $\gamma_a=180\mu$ eV, $\gamma_b=50\mu$ eV. In (e) we put together the four cases at resonance [in solid-blue (a) and (d), dashed-purple (b) and dotted-brown (c)], and their common Lorentzian contribution (blue filling). The different dispersive contributions are plotted in (f) with the same color code.
$\includegraphics[width=.45\linewidth]{chap3/newFigs/ReithExall.eps}$ $\includegraphics[width=.45\linewidth]{chap3/newFigs/ReithPhall.eps}$ $\includegraphics[width=.45\linewidth]{chap3/newFigs/ReithExallLat.eps}$ $\includegraphics[width=.45\linewidth]{chap3/newFigs/ReithPhallLat.eps}$ $\includegraphics[width=.45\linewidth]{chap3/newFigs/ReithRes.eps}$ $\includegraphics[width=.45\linewidth]{chap3/newFigs/ReithResDisp.eps}$

First, let us try to analyze this experimental data with the general SE expressions we obtained, Eqs. (3.37) and (3.47). In Fig. 3.15, we can see four examples of anticrossing with detuning. The first row correspond to the cavity emission and the second to the exciton direct emission. Cases (a), (c) correspond to the decay of an exciton and (b), (d) of a photon. An exciton decay detected in the SE of the exciton, plot (c), is the situation most commonly considered in the literature Andreani et al. (1999); Carmichael et al. (1989)e.g., although, it does not correspond to the experimental reality. Instead, in what concerns Reithmaier et al. (2004)'s experiments, we prefer to consider the cavity emission only. Changing the channel of detection provides results qualitatively different, contrary to what one could naively think. This is because, although the system is in strong coupling and photons and excitons should convert into each other rather quickly making all the cases equivalent, $\gamma_a$ , $\gamma_b$ are still quite different. This becomes even more obvious out of resonance. Given that $\gamma_a>\gamma_b$ , if, for instance, the initial state is an exciton [see Fig. 3.15(a) and (c)], the polaritons realized in the system and the emission have always a stronger excitonic character (that decays slower). On the other hand, the bare cavity mode only survives resonance when photons are detected through the cavity emission [Fig. 3.15(b)]. The two cases of detection of one mode through the opposite channel [(a) and (d)], result in the same spectra by symmetry. The anticrossing (b) is the closest to the experimental one in Fig. 3.14, showing a more intense cavity mode out of resonance. Still, a fitting of the data with these formulas would not be very good. The resemblance between Fig. 3.14 and Fig. 3.15(b) is only a hint about the state realized in the system, that seems to be more photonic in character. Finally, it is illustrative to see in Figs. 3.15(e) how misleading is to reduce the spectra to the Lorentzian contribution (plotted in blue filling). At resonance, this part of the spectral shapes is common to the four cases (because $n_{ab}^0=0$ ), which anyway are very different from each other due to the dispersive part, plotted in (f). The quantum interference between the modes can result in closing the Rabi doublet, cases (a) and (d) in solid-blue, or in separating the peaks, case (b) in dashed-purple. It can also contribute very little as in case (c), in dotted-brown. In most cases, it is therefore an indispensable element.

**Figure 3.16:** Theoretical fit (in semi-transparent red) of the data digitized from Reithmaier et al. (2004) (blue). The data has been fitted on rescaled axes for numerical stability by a Levenberg-Marquardt method with $\mathcal{N}S(\omega)-c$ , with $\mathcal{N}$ and to account for the normalization and the background. Beside these two necessary parameters regardless of the model, each panel only has $P_{a/b}$ and $\omega_{a/b}$ (c) as fitting parameters. and $\gamma_{a/b}$ have been optimized globally, with best fits for $g=61\mu$ eV, $\gamma_a=220\mu$ eV and $\gamma_b=140\mu$ eV. (a) shows again the anticrossing from theoretical curves 1-15 put together. (b) keeps all fitting parameters the same but with and vanishing . The dot emission now dominates and no anticrossing is observed, although the system is still in strong-coupling.
$\includegraphics[width=\linewidth]{chap3/laussy_icps_rio.eps}$

Now, we will turn to our SS model under incoherent continuous pump, that is closer to the semiconductor experimental reality, and see that it can successfully reproduce Reithmaier et al. (2004) data. In this experiment, both large QDs and low excitations were used, so a Fermionic model might be less appropriate than the bosonic picture of this chapter. In Fig. 3.15 and in the more detailed Fig. 3.16, we show in red the results of optimizing the global nonlinear fit of the experimental data (in black) with Eqs. (3.37) and (3.50). That is, the detuning ( $\omega_a$ and $\omega_b$ ) and pumping rates ( and ) are the fitting parameter from one curve to the other, while and $\gamma_{a,b}$ have been optimized but kept constant for all curves. We find an excellent overall agreement, that instructs on many hidden details of the experiment.

First, the model provides more reliable estimations of the fitting parameters than a direct reading of the line-splitting at resonance or of the linewidths far from resonance: The best-fitting coupling constant is $g=61\mu$ eV. The value for $\gamma_a=220\mu$ eV is consistent with the experiment^3.4 and the value for $\gamma_b$ , that is the most difficult to estimate experimentally, is reasonable in the assumption of large QDs, as is the case of those that have been used to benefit from their large coupling strength. However, the point here is not to conduct an accurate statistical analysis of this particular work but to show the excellent agreement that is afforded by our model with one of the paradigmatic experiment in the field. Such a good global fit cannot be obtained without taking into account the effect of pumping, even when it is small. More interestingly, it is necessary to include both the exciton pumping (expected from the experimental configuration) but also the cavity pumping . The latter requirement confirms the idea that the photonic contribution is important in the system that we could extract from the comparison with the SE. Experimentally, we already discussed in Chapter 2 that there are numerous QDs weakly-coupled to the cavity in addition of the one that undergoes SC. Beyond this QD of interest, a whole population of ``spectator'' dots contributes an effective cavity pumping, which looms up in the model as a nonzero . The fitting pumping rates (Fig. 3.16) vary slightly with detuning, as can be explained by the change in the effective coupling of both the strongly-coupled dot with the cavity (pumping tends to increase out of resonance) and the spectator QDs that drift in energy with detuning. We find as best fit parameters at resonance $P_a\approx0.44g=0.12\gamma_a$ and $P_b\approx0.42g=0.18\gamma_b$ (the mean over all curves is $\bar P_a\approx0.15\gamma_a$ and $\bar P_b\approx0.28\gamma_b$ with rms deviations of $\approx10\%$ ). The existence of in an experiment with electronic pumping is supported by Reithmaier et al. (2004) who observed a strong cavity emission with no QD at resonance. We shall see in the following the considerable importance of this fact to explain the success of their experiment.

**Figure 3.17:** The same phase space of SC/WC as in Fig. 3.9 with $\gamma_b\approx2.3g$ and $P_a\approx0.44g$ fitting the experiment of Reithmaier et al. Reithmaier et al. (2004), marked by a plain blue point ( $\gamma_a\approx3.6g$ , $P_b\approx0.42g$ ). In inset, the same phase space but for , in which case the line-splitting of Reithmaier et al. (2004) would not be resolved. (b) and (c): Spectra of emission with increasing exciton pumping marked by the hollow points in panel . For $\gamma_a=2g$ in (b), SC is retained throughout and made more visible. For the best fit parameter, $\gamma_a\approx3.6g$ in (c), line-splitting is lost increasing pumping, first because it is not resolved and then because the system goes into WC.
$\includegraphics[width=.9\linewidth]{chap3/Fig-2.eps}$

The position where we estimate Reithmaier et al. (2004) result in the SC-WC diagram for this system (Fig. 3.17), validates that SC has indeed been observed in this experiment. In inset, however, one sees that in the case where the cavity pumping is set to zero keeping all other parameters the same, the point falls in the dark region where, although still in SC, the line-splitting cannot be resolved. Even if it is possible in principle to demonstrate SC through a finer analysis of the crossing of the lines [see Fig. 3.16(b)], it is obviously less appealing than a demonstration of their anticrossing. This is despite the fact that the case of is equally, if not more, relevant as far as SC is concerned, as it corresponds to the case where only the QD is excited, whereas in the case of Fig. 3.14, it also relies on cavity photons. With the populations involved in the case of the best fit parameters that we propose-- $n_a\approx0.15$ --one can still read in Reithmaier et al.'s experiment a good vacuum Rabi splitting, so the appearance of the line-splitting with is not due to the photon-field intensity. Rather, the system is maintained in a quantum state that is more photon-like in character, which is more prone to display line splitting in the cavity emission, as we already discussed. In this sense, there is indeed an element of chance involved in the SC observation, as one sample can fall in or out of region 2 depending on whether or not the pumping scheme is forcing photon-like states. Understanding the excitation scheme drastically reduces this element of hazard. The shortcoming of downplaying the importance of the quantum state that is realized in the system owing to pumping, has as its worst consequence a misunderstanding of the results, the most likely being the qualification of WC for a system in SC that cannot be spectrally resolved because of decoherence-induced broadening of the lines.

A natural experiment to build upon our results is to tune pumping. In our interpretation, it is straightforward experimentally to change , but it is not clear how would then be affected, as it is due to the influence of the crowd of spectator QDs, not directly involved in SC. In Fig. 3.17, we hold to its best fit parameters and vary in the best fit case $\gamma_a=3.6g$ on panel , then for a better cavity with $\gamma_a=2g$ on panel , where the system is in SC for all possible values of . Spectra are displayed for the values of marked by the points in . Two very different behaviors are observed for two systems varying slightly only in $\gamma_a$ . In one case (), strong renormalization of the linewidths and splitting results from Bose effects in a system that retains SC throughout. In the other (), line-splitting is lost and transition towards WC then follows. At very high pumping, the model breaks down. A careful study of pump-dependent PL can tell much about the underlying statistics of the excitons and the precise location of one experiment in the SC diagram.

The confrontation of the theory with the experimental data is an illustration only, as the raw experimental data was not available by the time of our investigation. We have therefore digitized the data, what forbids an in-depth statistical analysis, since the experimental points are required rather than the interpolated curves published, if only to know the number of degrees of freedom. Note that one expects better still results as our procedure added noise. We found the best agreements near resonance, which might be due to the exciton that, when it is less-strongly coupled at larger detunings, may go below the resolution of the detector, resulting in an apparently broader line. All these limitations can be circumvented with a careful statistical analysis (and treatment of the data to reconstruct linewidths below the experimental resolution). This is a standard procedure to validate a theoretical model over another by statistical analysis of the experiment. It would also provide a meaningful and quantitative comparison between the various implementations (micropillars, microdisks and photonic crystals). Lacking the full experimental data, we have not been able to provide a confidence interval to our most-likelihoods estimators. Doing so, progress will be meaningfully quantified, and claims--rather than ranging between likely and convincing--will become unambiguously proven (within their interval of confidence). Going in this direction, Laucht, Hauke, Villas-Bôas, Hofbauer, Böhm, Kaniber & (2009) successfully fitted their SC experimental data (with a photonic crystal) by extending the model we present here to add pure dephasing in the linear regime. They obtained reliable estimations of the SC parameters, such as the coupling strength , and showed the clear disagreement with results from a Lorentzian fitting of the lineshapes. They investigated the great influence of pure dephasing when increasing the excitation power or the temperature. Münch et al. (2009) also studied the transition into WC with the excitation power, obtaining extraordinarily good fittings from our model which allowed them to construct the SC phase space of their sample.