Two-photon lasing

The analysis of the system in terms of the coherent Hamiltonian dynamics is essential in order to characterize the configurations where there is resonance with the cavity mode and the strength of the couplings giving rise to 1P and 2P oscillations. However, the efficiency and actual possibility of 2P lasing versus 1P lasing or simply against the decoherence, must be investigated taking into account pump and decay with the master equation:

$\displaystyle \frac{d \rho}{dt}=i\lbrack \rho, H \rbrack +\frac{\gamma_a}{2} \m... ...o+\frac{P}{2}( \mathcal{L}^{\ud{\sigma_1}}+ \mathcal{L}^{\ud{\sigma_2}})\rho\,.$

(6.31)

As we know, one of the effects of pump and decay on the Hamiltonian dynamics, is to average out the Rabi oscillations from the matrix elements of the density matrix, and in particular from the populations of all the states. The steady state can only be represented by a mixture of all the possible final outcomes. Different manifolds of excitation are involved in the dynamics and the mixture of states with some probability. It is no longer possible to observe the 1P/2P oscillations in the average populations. However, the 1PR and 2PR are still an intrinsic feature of the system and can be studied in the spirit of Fig. 6.19. When tuning the cavity mode over the 1PR-2PR transitions, the intensity of the emission $\langle n\rangle$ and the 2P sensitive quantities $g^{(2)}$ and $C^{(2)}$ , change dramatically as we can see in Fig. 6.20-(a) (in solid-blue, dashed-purple and dashed-brown lines respectively). Here, we kept the previous configuration for the QD levels ( $\chi=20g$ ) and some reasonable parameters for pump and decay ( $\gamma_a=0.2g$ , $\gamma=0.05g$ and ). The resonant energies (where the intensity increases) are approximately the same as those given by the Hamiltonian analysis in the case where the initial state is closer to the biexciton (see Fig. 6.19-(b) and 6.19-(d)). Let us discuss these results in more detail.

**Figure 6.20:** Steady state properties of the system as a function of the cavity detuning $\Delta$ for a biexciton energy of $\chi=20g$ (see Fig. 6.17). (a) Mean value of photons $\langle n\rangle$ in the cavity (solid-blue line), $g^{(2)}(0)$ (dashed-purple) and $C^{(2)}(0)$ (dashed-brown). (b) Populations of the states: $\ket{B}$ (solid-blue), $\ket{E}$ (dashed-purple) and $\ket{G}$ (dotted-brown). The peak corresponding to 1PR can be seen at $\Delta=2\chi$ (between $\ket{E}$ and $\ket{B}$ ) and that of 2PR at $\Delta=\chi$ . In a logarithmic scale (c), also the 1PR at $\Delta=0$ (between $\ket{G}$ and $\ket{E}$ )
$\includegraphics[width=0.48\linewidth]{chap6/2P/Mean/Mean.eps}$ $\includegraphics[width=0.48\linewidth]{chap6/2P/Mean/Pop.eps}$ $\includegraphics[width=0.48\linewidth]{chap6/2P/Mean/Pop-detail.eps}$

**Figure 6.21:** Same as Fig. 6.20 for a better cavity with $\gamma_a=0.1g$ (upper) and for a worse cavity with $\gamma_a=0.3g$ (lower).
$\includegraphics[width=0.48\linewidth]{chap6/2P/Mean/Mean-gamma_0.1.eps}$ $\includegraphics[width=0.48\linewidth]{chap6/2P/Mean/Pop-gamma_0.1.eps}$ $\includegraphics[width=0.48\linewidth]{chap6/2P/Mean/Mean-gamma_0.3.eps}$ $\includegraphics[width=0.48\linewidth]{chap6/2P/Mean/Pop-gamma_0.3.eps}$

**Figure 6.22:** Same as Fig. 6.20 for less pumping .
$\includegraphics[width=0.48\linewidth]{chap6/2P/Mean/Mean-P_1.eps}$ $\includegraphics[width=0.48\linewidth]{chap6/2P/Mean/Pop-P_1.eps}$

**Figure 6.23:** Same as Fig. 6.20 for a system with more biexciton binding energy $\chi=40g$ but also better cavity $\gamma_a=0.1g$ (upper) and simply with less biexciton energy $\chi=10g$ (lower).
$\includegraphics[width=0.48\linewidth]{chap6/2P/Mean/Mean-gamma_0.1-chi_40.eps}$ $\includegraphics[width=0.48\linewidth]{chap6/2P/Mean/Pop-gamma_0.1-chi_40.eps}$ $\includegraphics[width=0.48\linewidth]{chap6/2P/Mean/Mean-chi_10.eps}$ $\includegraphics[width=0.48\linewidth]{chap6/2P/Mean/Pop-chi_10.eps}$

When the cavity mode is close to a resonance with some excitonic transition, the system can enter a lasing regime, where $\langle n\rangle >1$ and $g^{(2)}\approx 1$ as we have already seen. This depends on the strength of the couplings, either for 1P lasing or the effective coupling $g_\mathrm{eff}$ for 2P lasing, relatively to the decoherence in the system. If the cavity is good enough, $\gamma_a\ll g$ (like in the case under study), the excitations created inside can be sustained for a long time thanks to the light-matter exchange allowing for an efficient storage of photons. Also the QD must be far from saturation (in the biexciton in this case) or self-quenching.

In Fig. 6.20-(a) it is clear from the increase in $\langle n\rangle$ and decrease in $g^{(2)}$ towards one, that both resonances bring the system into a lasing regime. The 2PR is less efficient as the coupling associated is weaker ( $g_\mathrm{eff}=0.2g$ ) than in the case of the 1PR ( with two excitonic states involved). This manifests in three ways: the maximum intensity achieved at 2PR resonance is lower, the state is less Poissonian-like^6.2 and the width of the lasing regime in terms of the detuning $\Delta$ is much narrower. When the detuning brings the cavity mode too far from any QD level, $\langle n\rangle$ decays, no lasing is produced and $g^{(2)}$ becomes . We can see a clear transition between the two lasing regimes although the two peaks overlap and none is purely 1P or 2P lasing. Moreover, $C^{(2)}$ decreases towards zero close to $\Delta\approx 2\chi$ evidencing 1P exchange while it is enhanced around $\Delta \approx \chi$ evidencing 2P exchange.

In order to understand up to which extent the 2P lasing regime is dominated by 2P dynamics, in Fig. 6.20-(b) we plot the total populations of states $\ket{B}$ (solid-blue line), $\ket{G}$ (dotted-brown) and $\ket{E}$ (dashed-purple) as a function of detuning. Out of any excitonic resonance, the biexciton state is saturated by the pump. The transition into 2PR and 1P lasing is evidenced by the deviation from this saturation ( $\rho_{B}=1$ ). This is why resonances excited in Fig. 6.20 are basically those of Fig. 6.19-(b) and 6.19-(d) where the coherent dynamics started from biexcitonic states. The other QD states populated through the interplay with the biexciton, are the levels involved in each lasing regime, as it was the case with the Rabi oscillations. At $\Delta=2\chi=40g$ , only $\ket{E}$ and $\ket{B}$ are involved. This implies that the dynamics are driven by 1P processes only.

On the other hand, at $\Delta=\chi=20g$ , not only $\ket{G}$ and $\ket{B}$ participate, which would imply a 2P dynamics, but also $\ket{E}$ . The dynamics inside $\mathcal{H}_\mathrm{0P}$ alone does not populate the cavity (as we could see in the derivation of the effective Hamiltonian), but this subspace can exchange 1P with states $\ket{G}$ and $\ket{B}$ separately even being so far from the 1PR (the same far for both transitions $\ket{G}$ - $\ket{E}$ and $\ket{E}$ - $\ket{B}$ ). At the 2PR, this 1P exchange is inefficient, as we know, by it results in some contribution, not negligible in this case. Therefore, the 2PR does not lead to 2P dynamics exclusively in general, as there will always be some weak nonresonant 1P process still present. In this case, the transition $\ket{B}$ and $\ket{E}$ is the one providing single photons as the ground state is less probably occupied. We can conclude that the population of the ground state $\rho_G$ is another good magnitude to identify the 2P versus 1P lasing at the 2PR.

Finally, at $\Delta=0$ , some signature of the 1PR with the transition $\ket{G}$ and $\ket{E}$ is expected. It appears in the logarithmic plot of the population $\rho_G$ , Fig. 6.20-(c), as a small perturbation. For a case with stronger coupling (smaller $\gamma_a$ ) and less pumping (less saturation) this resonance would be more evident.

This discussion leads us the problem of maximizing the 2P processes so that there is a truly 2P laser operating in some clearly defined regime. There are the following points to take into account:

The system should be in strong 2P-coupling which means a large coupling together with a good cavity so that $\gamma_a<4g_\mathrm{eff}$ for detunings $\Delta$ as large as needed. However the cavity should not be so good that even at large detuning for the 1PR, the system is still sensitive to it and therefore the 2P lasing gets polluted with single photons. We can see this effect in Fig. 6.21 where the cavity quality is improved (upper figures) or worse (lower ones). In the first case, the lasing is more efficient but both for 1P and 2P processes making it so that the separation is not clear enough. In the second case, the separation between the two regimes is larger but the 2P resonance does not lead to lasing ( $g^{(2)}>2$ ).
The pump should be not so strong that the system saturates for all detunings, quenching the production of photons at resonance. But it should be strong enough so that in the vicinity of the 2PR, and in particular in the transition from 2P to 1P lasing, the biexciton is populated and the probability to be in excitonic states (that involve 0P or 1P processes) is low. This makes the transition clearer experimentally and makes the lasing regimes purely 1P or 2P.
The binding energy $\chi$ should be large enough so that the 1P and 2P lasing regions (resonances) are far from each other and can be resolved and considered independently. The 2P lasing efficiency is reduced in this case, but the character of the emission seems to be more defined (see how $C^{(2)}$ in upper Fig. 6.23-(a) is large and distinct around $\Delta \approx \chi$ ). $\chi$ cannot be so large, however, that the effective coupling becomes negligible at the 2PR. We can see in lower Fig. 6.21 the effect of reducing the biexciton binding energy to $\chi=10g$ . The effective 2P coupling is stronger and therefore at the 2PR ( $\Delta=10g$ ) the lasing is more intense with $g^{(2)}=1$ , but the two lasing regions are superimposed making it impossible to assert that the system is dominated by 2P processes.

All these effects together could be summarized in the conditions for the best system: A large biexcitonic extra energy $\chi$ balanced with a reasonable good cavity. The optimum pumping depends on the goal. More efficient lasing requires high pumping while a more quantum 2P emission happens at low pump.