Results

We show the results in figures 2, 3, and 4. First, to test accuracy, we show, in Figure 2, the relaxation of (6) to a pure Fermi-Dirac spectrum in the window $\omega _L=1<\omega <\omega _0=2$. The boundary conditions correspond to $P=Q=0$ at both ends. We then modify boundary conditions to read $Q=Q_0>0$ and $P=0$ at both ends. Next, in Figure 3 and 4, we show the results of two experiments in which we compare the efficiencies of two experiments in which we arrange to (i) pump broadly so that the effective carrier distribution equilibrium has zero flux and (ii) pump carriers and energy into a narrow band of frequencies about $\omega _0$ and simulate this by specifying carrier and energy flux rates $Q_L$ and $P_L=-\omega _L Q_L$ ($P_L$ chosen so that the energy absorbed by the laser is consistent with the number of carriers absorbed there) at the boundary $\omega =\omega _0$. $\omega =\omega _L$ is the frequency at which the system lases.

In both cases, the rate of addition of carriers and energy is (approximately) the same. The results support the idea that it is worth exploring the exploitation of the finite flux equilibrium. The carrier density of the equilibrium solutions at $\omega _0$ is small thus making pumping more efficient there. The output of the laser is greater by a factor of 10. While we do not claim that, when all effects are taken account of, this advantage will necessary remain, we do suggest that the strategy of using finite flux equilibrium solutions of the Quantum Boltzmann equation is worth further exploration.