One-photon lasing

In this Section we analyze the lasing properties of the system. A practical motivation is the significant improvements--as far as low threshold is concerned--obtained by Strauf et al. (2006) in a system having just a few (from 2 to 4) QDs embedded in a single-mode microcavity, with respect to previous attempts using quantum wells or high density QDs, namely those of Slusher et al. (1993), Rohener et al. (1997), Painter et al. (1999), Ryu et al. (2000), Zhang & (2003), Park et al. (2004), etc... An important finding in this Section is that the presence of a second dot in strong coupling with the cavity, even far from resonance with the cavity, changes substantially the emission of a single one.

As opposed to the previous Section, now we focus on the quantum state of the cavity photons, rather than on that of the QDs. Nevertheless, the light state depends on the coherences established between the levels of the system and therefore depends on the QD parameters. Unlike atoms, QDs can be differently detuned with respect to the cavity mode. In what follows, we therefore study the dependence on the detuning configuration of the mean photon number and of the second-order coherence function in the case of zero delay $ g^{(2)}$. Only in the case where the two dots are equally detuned, one can compare the two limiting pumping schemes described previously; as mentioned above, such symmetry is a necessary condition for the common pumping bath.

Given the structure of Eq. (6.9), all off-diagonal elements of the density matrix between levels with equal QD states but different number of photons have been washed away in the steady state. As in the case of one QD in the cavity under incoherent pump (Chapter 5), the photon reduced density matrix $ \rho_\mathrm{ph}=\mathrm{Tr}_\mathrm{QD}\rho$ is diagonal in the number of photons: it is thus impossible for the system to achieve a coherent state, with density matrix $ \rho_{\alpha}$ of the form (2.23), at the steady state. However, the system can reach a mixture of number states, $ \rho_{\vert\alpha\vert}$ from Eq. (2.25), with the same Poissonian distribution, as it happens for a laser much above threshold. In both cases, $ \langle n\rangle =\vert\alpha\vert^2$ and $ g^{(2)}=1$.

Figure 6.12: Mean number of photons stored in the cavity as a function of pumping $ P$ (in units of $ g_1$), for $ \gamma_a=0.2g_1$, $ \gamma=5\times10^{-3}g_1$, $ g_1=g_2$. Both cases, with independent ( $ P_\mathrm{ind}=P$ and $ P_\mathrm{com}=0$, red) and common pumping ( $ P_\mathrm{ind}=0$ and $ P_\mathrm{com}=P$, green), are presented for the resonant case ( $ \Delta_1=\Delta_2=0$). These are compared with the emission of a single QD in resonance with coupling $ g=g_1$ (blue line), and the sum of emissions of two independent dots in resonance with renormalized coupling constants $ g=\sqrt{2}g_1$ (black line).
\includegraphics[width=0.7\linewidth]{chap6/fig_7.ps}

Fixing the leaky modes to $ \gamma=5\times10^{-3}g_1$ and the coupling constants $ g_2 = g_1$, we first compare the number of photons  $ \langle n\rangle $ in the cases where one or two QDs are in resonance with the cavity. This is shown on Fig. 6.12, where the growth of the occupation number with pumping is seen to be limited by the self-quenching effect. This results in a maximum cavity population, corresponding to an optimum value of the pumping intensity. Further increase of the pumping results in a decrease of the mean number of photons and saturation of the dots. We already discussed in the previous Chapter how this effect is due to the incoherent nature of the pump, which destroys the coherences established between QDs and cavity driving the system into a thermal state ($ g^{(2)}=2$). However, we are interested in the behavior at much lower pumps, where the number of photons does not yet saturate as is the case of Strauf et al. (2006).

In the pumping range plotted in Fig. 6.12, the self-quenching region is reached only for the case of a common pumping bath (green line). Therefore, this case is the least suitable for lasing properties. There are several reasons for the enhancement of the self-quenching effect in this case. The first reason is that, neglecting the leaky modes, the QD system is reduced from four to three levels, diminishing the range of pump available before reaching the saturation of the ensemble. Taking into account leaky modes, the second reason is that, although $ g_1=g_2$, and thus the singlet is not coherently coupled to the triplet, the decay of state $ \ket{\mathrm{T}_1,0}$ into $ \ket{\mathrm{S},0}$ via those leaky modes populates the singlet, thus hindering the storage of photons. A third drawback is the presence of the coherent coupling between states $ \ket{E1}$ and $ \ket{E2}$, which prevents the distribution of photons from being Poissonian. The resulting distribution is a sum of the contribution of the singlet subspace (with high probabilities around zero photons) and the triplet (Poissonian like distributions as found in the other cases plotted here).

On the other hand, in the case of independent pumpings, the emission of two QDs approximately corresponds to the sum of the individual emissions with coupling constants renormalized by a factor $ \sqrt2$. This approximation improves when both dots are close to resonance, as shown by the red curve in Fig. 6.12. The second-order coherence function is also similar at low pumping.

As the common-bath of excitations is detrimental to lasing, we now consider the case of independently and equally pumped dots only ( $ P_\mathrm{com}=0$, $ P_\mathrm{ind}=P$), where QDs are coupled uniquely through the cavity mode. Results are given in Fig. 6.13, which shows the behavior of $ \langle n\rangle $ and $ g^{(2)}$ as a function of the pump, for different detuning configurations: equal ( $ \Delta_1=\Delta_2$), opposite ( $ \Delta_1=-\Delta_2$), and mixed ( $ \Delta_1=0 \neq \Delta_2$) detunings.

Figure 6.13: (a) Mean number of photons $ \langle n\rangle $ stored in the cavity and (b) second-order coherence function $ g^{(2)}$ of the cavity field, as a function of $ P_\mathrm{ind}=P$ (in units of $ g_1$), for $ P_\mathrm{com}=0$, $ \gamma_a=0.2g_1$, $ \gamma=5\times10^{-3}g_1$, $ g_2 = g_1$. Several detuning cases are presented, with equal ( $ \Delta_1=\Delta_2=5g_1$), opposite ( $ \Delta_1=-\Delta_2=5g_1$) and mixed ( $ \Delta_1=0,\Delta_2=5g_1$) configurations. There is a qualitative change with two dots as, even when none is in resonance with the cavity mode, the threshold for lasing (with linear increase of  $ \langle n\rangle $ with $ P$) is low also in the case where the single dot alone would not lase.
\includegraphics[width=0.49\linewidth]{chap6/fig_8a.ps} \includegraphics[width=0.49\linewidth]{chap6/fig_8b.ps}

In the ideal case, if the QDs are in resonance (Fig. 6.12), the production of photons is very efficient, and $ g^{(2)}$ is always one until the self-quenching begins. With detuning, a threshold for linear production of photons (as a function of pumping) appears, as can be seen in the figure. It occurs approximately when the pumping compensates the losses: the number of photons becomes larger than one and the stimulated exceeds the spontaneous emission. This transition into lasing is accompanied by the decrease of the second-order coherence function to a value of one [Fig. 6.13(b)], and a Poissonian distribution of the photon number. As we have seen in the previous Chapter and we can see now in this figure, lasing is also present in the case with one dot, but the threshold of the transition is considerably lowered by the presence of a second strongly coupled dot.

The optimal configurations, i.e., the ones with lowest threshold, are those where at least one of the dots is in resonance. In these case, the presence of the second dot makes a great difference even if its detuning is large. So, neglecting the role of strongly coupled QDs when they are out of resonance is not a good approximation. We can see this by comparing $ \langle n\rangle $ of one QD in resonance [thin blue line in Fig. 6.13(a)] with two dots, one in resonance and the second highly detuned ( $ \Delta_1=0$ and $ \Delta_2=5$ [magenta line in Fig.6.13(a)]). Whether the detunings are identical ( $ \Delta_1=\Delta_2$) or opposite ( $ \Delta_1=-\Delta_2)$, makes no qualitative difference, although the two cases are not strictly equal.

In these results we find a possible explanation for the experimental findings of Strauf et al. (2006) on lasing with unexpected low laser thresholds and high photon production efficiency from a cavity containing a few dots out of resonance. The experimental parameters in that case are comparable to ours (with a pump threshold of $ P=0.08$meV), as well as the detunings of the dots, $ \Delta_1\approx-\Delta_2\approx5$meV. In our scheme, several dots result in qualitative changes of the emission, enhancing it significantly even when dots are off-resonance. In this sense, our model predicts still better cavity emission with extremely low threshold if one dot could be matched in resonance with the cavity.

Elena del Valle ©2009-2010-2011-2012.