Point 1: very good systems and the incoherent Mollow triplet

Point 1, is best suited to explore quantum effects. Its spectral shape is unambiguously evidencing transitions in the Jaynes-Cummings ladder, as shown in Fig. 5.17 with a clear ``Jaynes-Cummings fork'' (a quadruplet). The outer peaks at $\pm1$ are the conventional vacuum Rabi doublet, whereas the two inner peaks correspond to higher transitions in the ladder. Observation of a transition from outer to inner peaks with pumping such as shown in Fig. 5.17 would be a compelling evidence of a quantum exciton in SC with the cavity. Fig. 5.18 shows another multiplet structure of this kind for Point 1. The intensity of emission is presented in log-scale and for a broader range of frequencies, so that small features can be revealed. Transitions from up to the third manifold can be explicitly identified. The decay from the second manifold, that manifests distinctly with peaks labelled 2 (although it also contributes to peaks labelled 0), is already weak but still might be identifiable in an experimental PL measurement. Higher transitions have decreasing strenght. This can be checked with the probability to have photons in the cavity, $\mathrm{p}(n)$ , computed from Eqs. (5.13). Whenever the mean number is low (as is the case here), this probability is maximum for the vacuum ( $\mathrm{p}(n)>\mathrm{p}(n+1)$ for all ), independently of the nature of the photon distribution (sub, super or Poissonian). Only when , in the best of cases (for a Poissonian distribution), does this trend start to invert and $\mathrm{p}(1)=\mathrm{p}(0)$ .

**Figure 5.17:** Jaynes-Cummings forks as they appear in the luminescence spectrum of a QD in a microcavity with system parameters given by Point 1 of Fig. 5.13 and for pumping rates $(P_a,P_\sigma)/g$ given by (a), ; (b), and (c), . The two outer peaks at $\pm1$ correspond to the vacuum Rabi doublet. Inner peaks correspond to transitions with states of more than one excitation. Although the underlying structure is the same, many variations of the actual lineshapes can be obtained.
$\includegraphics[width=.45\linewidth]{chap5/JC/fig7-jc-fork.ps}$

**Figure 5.18:** Expanded view in logarithmic scale of a spectrum similar to those of Fig. 5.17, this time with $(P_a,P_\sigma)/g=(0.002,0.076)$ . Transitions up to the third manifold (shown in insets) are resolvable. Others are lost in the broadening. The transition energies of the Jaynes-Cummings ladder are shown by vertical lines (up to the third manifold). The Rabi peaks that corresponds to transitions from the first manifold to vacuum (line 1) is in this case dominated by higher transitions that accumulate close to the center (line 0).
$\includegraphics[width=.66\linewidth]{chap5/JC/fig8-jc-multiplets.ps}$

This makes it impossible, even in the very good system of Point 1, to probe clearly and independently transitions between manifolds higher than

, as their weak two outer peaks (approximately at $\pm(\sqrt{n}+\sqrt{n-1})$ ) are completely hidden by the broadening. A stronger manifestation of nonlinear emission is to be found in the pool of pairs of inner peaks from all high-manifold transitions (labelled 0 in Fig. 5.19), at approximately $\pm (\sqrt{n}-\sqrt{n-1})$ . Not only the inner peaks coming from different manifolds are close enough to sum up, but also they are more intense than their outer counterparts. This can be easily understood by looking at the probability, $\mathrm{I}_{c}$ , of transition between eigenstates $\ket{\pm,n}$ through the emission of a photon,

, or an exciton, $c=\sigma$ . This probability, $\mathrm{I}_{c}^{(i\rightarrow f)}\propto\vert\bra{f}c\ket{i}\vert^2$ , estimates the relative intensity of the peaks depending on the initial, $\ket{i}$ , and final, $\ket{f}$ , states of the transition and on the channel of emission, $c=a,\sigma$ . A discussion in terms of the eigenstates of the Hamiltonian is still valid in the regime of Point 1 (very good system) at very low pump. At resonance, neglecting pumps and decays, the eigenstates for manifold

, are $\ket{n,\pm}=(\ket{n,0}\pm\ket{n-1,1})/\sqrt{2}$ . The outer peaks arise from transitions between eigenstates of different kind, $\ket{n,\pm}\rightarrow\ket{n-1,\mp}$ , while the inner peaks arise from transitions between eigenstates of the same kind, $\ket{n,\pm}\rightarrow\ket{n-1,\pm}$ . Their probability amplitudes in the cavity emission,

$\begin{subequations}\begin{align}&\mathrm{I}_{a}^{(\pm\rightarrow\mp)}\propto\ve... ...n,\pm}\vert^2=\vert\sqrt{n}+\sqrt{n-1}\vert^2/4\,, \end{align}\end{subequations}$

evidence the predominance of the inner peaks versus the outer ones, given that one expect the same weighting of both transitions from the dynamics of the system. The doublet formed by the inner peaks is therefore strong and clearly identifiable in an experiment. On the other hand, in the direct exciton emission, the counterparts of Eqs. (5.43) are manifold-independent and equal for both the inner and outer peaks:

$\begin{subequations}\begin{align}&\mathrm{I}_{\sigma}^{(\pm\rightarrow\mp)}\prop... ...o\vert\bra{n-1,\pm}\sigma\ket{n,\pm}\vert^2=1/4\,. \end{align}\end{subequations}$

In this case, therefore, one can expect similar strength of transitions for both the inner and outer peaks with a richer multiplet structure for the direct exciton emission.

In Fig. 5.19, we give an overview of the PL spectra as $P_\sigma$ is varied from very small to very large values. For point 1, as we already noted, the cavity pumping plays a relatively minor quantitative role. Therefore we only show two cases, of no-cavity pumping (first row) and high-cavity pumping (second row). As can be seen, there is no strong difference from one spectra with no cavity pumping to its counterpart with large cavity pumping. Third row shows the direct exciton emission that, with no cavity pumping, corresponds to the first row. Indeed, one can observe the richer multiplet structure up to $P_\sigma\approx 0.5g$ in the direct exciton emission, whereas only inner peaks are neatly manifest in the cavity emission. This region corresponds to a quantum regime with a few quanta of excitations (and subpoissonian particle number distribution, $g^{(2)}<1$ ) giving rise to clearly resolvable peaks, attributable to the Hamiltonian manifolds. Therefore, a good system (high and ) and a good QD (two-level) emitter suffice to easily and clearly observe quantum effects. There is no need of pumping harder than it has been done in present systems so far.

$\begin{sidewaysfigure} % latex2html id marker 11289\centering \includegraphi... ..._\sigma<30g$) and is finally quenched ($P_\sigma>30g$).} \end{sidewaysfigure}$

**Figure 5.20:** Point 1 of Fig. 5.13. Details of the loss of the multiplet structure with increasing exciton pumping and zero cavity pumping. The two upper rows (blue) correspond to the cavity emission $S_a(\omega)$ and the two lower (violet) to the exciton direct emission $S_\sigma(\omega)$ . The spectral structure is richer in the exciton spectra that develops a Mollow triplet-like emission.
$\includegraphics[width=\linewidth]{chap5/JC/fig10-SpecsPt1details.ps}$

In the region $g<P_\sigma<30g$ , the photon fluctuations are those of a coherent, classical state, $g^{(2)}=1$ . Increasing pumping with the intention to penetrate further into the nonlinearity, merely collapses the multiplet structure into a single line, as far as cavity emission is concerned. However, this does not mean that the system is in weak coupling. In the direct exciton emission, the rich SC fine structure has turned into a Mollow triplet, like the one found by Mollow (1969), that we discuss in depth below. In this region, the first manifolds have crossed to WC but higher manifolds retain SC, bringing the system into lasing. At this point, a change of realm should be performed favoring a classical description, as we already pointed out with the AO. A last transition into thermal light and WC, due to saturation and self-quenching, takes place at $P_\sigma\approx30g$ that leads to a single central peak in the spectra.

**Figure 5.21:** Point 1 of Fig. 5.13. Incoherent Mollow triplets are observed in the exciton direct emission with broad satellite peaks at approximately $\pm2\sqrt{n_a}$ and a strong narrow central peak taking over a narrow resonance. Three upper rows show the spectra over the interval $\vert\omega\vert\le15g$ allowing to see the satellites. Three lower rows are the same in the window $\vert\omega\vert\le3g$ , allowing to see the narrow resonance and peak that sit at the origin. Values of the electronic pumping are given in the frame of the first three rows. Cavity pumping is zero but influences very little the Mollow triplets. Rightmost figure superposes various spectra at increasing electronic pumping, showing the drift and broadening of the satellites, and putting to scale the very strong coherent feature at the origin. The incoherent Mollow triplet appears thus very differently from its counterpart under coherent excitation.
$\includegraphics[width=\linewidth]{chap5/JC/fig11-mollow-triplet.ps}$

In Fig. 5.20, we take a closer look into Fig. 5.19 in the region of the loss of the doublet of inner peaks with increasing electronic pumping, where the system starts to cross from the quantum to the classical regime. In the cavity emission, the doublet of inner peaks collapses into a single line that is going to narrow as the system lases. At the same time, a strikingly richer structure and regime transition is observed in the direct exciton emission. As the peaks are more clearly resolved as explained before [cf. Eqs. (5.43) and (5.44)], the ``melting'' of the Jaynes-Cummings ladder into a classical structure is better tracked down. Indeed, as pumping is increased, broadening of these lines starts to unite them together into an emerging structure of a much less reduced complexity, namely a triplet. This is completely equivalent to the transition from quantum to classical AO that we studied in Sec. 5.2. In Fig. 5.21, we provide another zoom of the overall picture given by Fig. 5.19, this time for the direct exciton emission exclusively. First three rows show the evolution with electronic pumping $P_\sigma$ (values in inset) over a wide range of frequencies, up to $\pm15g$ , while the three last rows show the very same spectra, with a one-to-one mapping with previous rows, only in the range of frequencies $\pm3g$ . The transition manifests to different scales, with a rich fine multiplet structure in the quantum regime, as seen in the zoomed-in region, to a monolithic triplet at higher pumpings, as seen in the enlarged region. On the right, spectra are superimposed to follow their evolution with pumping. The two broad satellites peaks, at approximately $\pm2\sqrt{n_a}$ (in the AO the broad peak was placed at ), drift apart from the main central one with increasing excitation, and in this sense behave as expected from a Mollow triplet. Various deviations are however observed, of a more or less striking character. The most astonishing feature is the emergence of a very sharp and narrow peak in the center, that has been plotted with its total intensity on the right panel to give a sense of its magnitude. It is clearly seen in the zoomed-region how this peak arises on top of the broad mountain of inner peaks, surviving the collapse of the fine structure in the classical regime. This thin central resonance appears when a large truncation is needed. It is a sum of many contributing peaks centered at zero, most of them with very small intensities. This region therefore shows all the signs of a transition from a quantum to a classical system. At low pumping, the inner peaks of all quadruplets coming from low order manifolds are placed approximately at $\pm(\sqrt{n}-\sqrt{n-1})\neq0$ . Even when they are summed up to produce the total spectrum, the nonlinear doublet is still resolved. At around $P_\sigma \approx 1.5g$ , manifolds high enough are excited so that for them $\pm(\sqrt{n}-\sqrt{n-1})\approx 0$ . This is a feature of a classical field resulting in a Mollow triplet. Note that nothing of this sort is observed in the cavity emission. The Mollow triplet, whether in atomic physics with coherent excitation or in semiconductor physics with incoherent pumping, is a feature of the quantum emitter itself, when it is directly probed. There is therefore a strong motivation here to detect leak emission of semiconductor structures. The overall features of this ``incoherent Mollow triplet'' differ from its counterpart namesake in the strong asymmetry of the satellites and their increased broadenings with pumping.

In order to appreciate more precisely the structure of the Mollow triplet and how it emerges from the quantum regime, we consider a even better system than Point 1: $\gamma_a$ of the order of $10^{-3}$ , $10^{-2}$ and $\gamma_\sigma=0$ . With very small pumping rates, in Fig 5.22 we can see the very sharp (owing to the small decay rates) individual lines from each transition. In the plot of the cavity emission, Fig 5.22, we have marked each peak with its corresponding transition between two quantized, dressed states of the JC Hamiltonian. These peaks correspond one-to-one with those of the exciton emission, that are, as we just explained, weighted differently. In both cases, the Rabi doublet dominates strongly over the other peaks. In Fig. 5.22, for instance, the peaks at $\pm1$ extend for about 9 times higher than is shown, and already the outer transitions are barely noticeable. This is because the pumping is small and so also the probability of having more than one photon in the cavity (it is in this configuration of about 10% to have photons, see Fig. 5.25). One could spectrally resolve the window over a long integration time and obtain the multiplet structure of nonlinear inner peaks, with spacings $\{\sqrt{n+1}-\sqrt{n},\, n>1\}$ (in units of ), observing direct manifestation of single photons renormalizing the quantum field. Or one could increase pumping (as we do later) or use a cavity with smaller lifetime. In this case, less peaks of the JC transitions are observable because of broadenings mixing them together, dephasing and, again, reduced probabilities for the excited states, but the balance between them is better. In Fig. 5.23, where $\gamma_a$ is now , the vacuum Rabi doublet (marked ) is dominated by the nonlinear inner peaks in the cavity emission, and a large sequence of peaks is resolved in the exciton emission.

Going back to the case of Fig. 5.22, but increasing pumping, we observe the effect of climbing higher the Jaynes-Cummings ladder. Results are shown in Fig. 5.24 in logarithmic scale, so that small features are magnified. First row is Fig. 5.22 again but in log-scale, so that the effect of this mathematical magnifying glass can be appreciated. Also, we plot over the wider range . Note how the fourth outer peak, that was not visible on the linear scale, is now comfortably revealed with another three peaks at still higher energies. As pumping is increased, we observe that the strong Rabi doublet is receding behind nonlinear features, with more manifolds indeed being probed, with their corresponding transitions clearly observed (one can track up to the 19th manifold in the last row). This demonstrates obvious quantization in a system with a large number of photons. The distribution of photons in these three cases is given in Fig. 5.25, going from a thermal-like, mostly dominated by vacuum, distribution, to coherent-like, peaked distribution stabilizing a large number of particles in the system. At the same time, note the cumulative effect of all the side peaks from the higher manifolds excitations, absorbing all quantum transitions into a background that is building up shoulders, with the overall structure of a triplet. This is the mechanism through which the system bridges from a quantum to a classical system with the Mollow triplet. These are obtained this time, both in the cavity and the exciton emission, but much more so in the latter.

**Figure 5.22:** Fine structure of the ``light-matter molecule'': emission spectra in the cavity (a) and direct exciton emission (b) of the strongly-coupled system with $(\gamma_a,\gamma_\sigma)/g=(10^{-3},0)$ at $P_\sigma/g=10^{-3}$ .
$\includegraphics[width=\linewidth]{chap5/ICTOPON/spec1.ps}$

**Figure 5.23:** Same as Fig. 5.22 but now with $\gamma_a/g=10^{-2}$ . Less peaks are resolved because of broadening but nonlinear peaks are neatly observable. In fact, now inner nonlinear peaks dominate in the cavity emission (the vacuum Rabi peaks are denoted ). In the exciton direct emission, the Rabi doublet remains the strongest.
$\includegraphics[width=\linewidth]{chap5/ICTOPON/spec2.ps}$

**Figure 5.24:** Spectra of emission in log-scales as a function of pumping $P_\sigma/g$ , for $10^{-3}$ (upper row), $5\times10^{-3}$ (middle row) and $10^{-2}$ (lower row).
$\includegraphics[width=\linewidth]{chap5/ICTOPON/specs3.ps}$

**Figure 5.25:** Probability of having photon(s) in the cavity, for the three cases shown in Fig. 5.24. Quite independently of the distribution of photon numbers in the cavity, field-quantization is obvious.
$\includegraphics[width=.5\linewidth]{chap5/ICTOPON/distribution.ps}$