Steady State under continuous incoherent pumping

In the case where the system is excited by a continuous, incoherent pumping, a steady state is reached and the boundary conditions are given by the stationary limit, as time tends to infinity, of the dynamical equation (whose solution is unique). The parameter, Eq. (3.40), is defined in this case as:

$\displaystyle D^\mathrm{SS}=\frac{n_{ab}^\mathrm{SS}}{n_a^\mathrm{SS}}= \frac{\... ...elta}2)} {g^2\Gamma_+(P_a+P_b)+P_a\Gamma_b(\Gamma_+^2+(\frac{\Delta}{2})^2)}\,.$

(3.47)

There is a clear analogy between Eq. (3.50)--that corresponds to the SS--and Eq. (3.48)--that corresponds to SE when $n_{ab}^0=0$ . In this case, the spectrum can also be written in terms of the ratio, counterpart of Eq. (3.49),

$\displaystyle \alpha=\frac{P_a}{P_b}\,,$

(3.48)

in which case Eqs. (3.48) and (3.50) assume the same expression, keeping in mind the definition of Eqs. (3.4). Table 3.1 displays this common expression of

in terms of $\alpha$ . The limiting cases when $\alpha\rightarrow0$ or $\infty$ are also given. They correspond to only photons or excitons as the initial state for the SE, or to the presence of only one kind of incoherent pumping for the SS case.

Table 3.1: Expression of

, Eq. (3.40), as a function of $\alpha$ , Eqs. (3.49) and (3.51), in the SE (with $\Gamma _{\pm ,a,b}\rightarrow \gamma _{\pm ,a,b}$ and $n_{ab}^0=0$ ) and SS cases.

embodies in the luminescence spectrum the influence of the quantum state of the system. The latter is specified by the initial condition in SE, or the pumping/decay interplay in the SS.

$\displaystyle\alpha=\frac{n_a^0}{n_b^0}=\frac{P_a}{P_b}$
0	$\displaystyle\frac{-\frac{g}2(i\Gamma_+-\frac{\Delta}2)\gamma_b}{g^2\Gamma_++\Gamma_b(\Gamma_+^2+(\frac{\Delta}{2})^2))}$
$0<\alpha<\infty$	$\displaystyle\frac{\frac{g}2(i\Gamma_+-\frac{\Delta}2)(\gamma_a-\gamma_b\alpha)}{g^2\Gamma_+(1+\alpha)+\alpha\Gamma_b(\Gamma_+^2+(\frac{\Delta}{2})^2))}$
$\infty$	$\displaystyle\frac{(i\Gamma_{+}-\frac{\Delta}{2})\gamma_a}{2g\Gamma_{+}}$

The analogies and differences between $D^\mathrm{SE}$ and $D^\mathrm{SS}$ reflect in the spectra $S^\mathrm{SE}$ and $S^\mathrm{SS}$ . For the same $\alpha$ , they become identical when the pumping rates are negligible as compared to the decays, $P_{a,b}\ll\gamma_{a,b}$ . In this case, where $\Gamma_{\pm,a,b}\approx\gamma_{\pm,a,b}$ , the SS system indeed behaves like that of the SE of particles that decay independently and that are, at each emission, either a photon or an exciton, with probabilities in the ratio $\alpha$ .

However, in the most general case, $D^\mathrm{SS}$ depends on more parameters than $D^\mathrm{SE}$ . Moreover, the pumping rates $P_{a,b}$ affect $S^\mathrm{SS}$ not only through $\alpha$ and $D^\mathrm{SS}$ , but also in the position and broadening of the peaks (given by $\Gamma_{\pm}$ and ). Therefore, the SS is a more general case, from which the SE with $n_{ab}^0=0$ can be obtained, but not the other way around. On the other hand, as seen in Table 3.1, the SS case cannot recover the SE case when $n_{ab}^0\neq0$ . Further similarities could be found if cross Lindblad pumping terms like those we will discuss in Sec. (3.7) were introduced in Eq. (3.3) with parameters $P_{ab}$ in analogy to the cross initial mean value $n_{ab}^0$ , but this describes a different system where polaritons can also be directly excited. In the present system, none of the SE and SS cases comprises all the possibilities of the other.

Anyhow, an important fact for the semiconductor community is that a SS with nonvanishing pumping rates is out of reach of the SE of any initial state, which has been the case studied by Carmichael et al. (1989) and Andreani et al. (1999), and that even in this limiting case, the effective quantum state obtained in the SS should still be resolved self-consistently, rather than assuming for $\alpha$ the particular case 0 or $\infty$ .