Figure Captions.

• Figure 1
  Carriers and energy are added at $\omega _0$ at rates $Q_0$ and $\omega _0 Q_0$. Energy and some carriers are dissipated at $\omega _R>\omega _0$ (an idealization) and carriers and some energy are absorbed by the laser at $\omega _L$. (The carriers number will build until the laser switches on.) A little calculation shows $Q_L=Q_0 (\omega _R-\omega _0)/(\omega _R-\omega _L)$, $Q_R=Q_0(\omega _L-\omega _0)/(\omega _R-\omega _L)$, $P_R=Q_0 \omega _R (\omega _0-\omega _L)/(\omega _R-\omega _L)$, $P_L= \omega _L Q_0 (\omega _0-\omega _R)/(\omega _R-\omega _L)$. Finite flux stationary solutions are realized in the windows $(\omega _L,\omega _0)$ and $(\omega _0,\omega _R)$ although in practice there will be some losses through both these regions.
• Figure 2.
  To test accuracy we take some initial distribution function (thin line) and plot its time evolution as described by (6) with boundary conditions $P=Q=0$ at both ends. The distribution function relaxes to Fermi-Dirac state (thick line). Several intermediate states is shown by long-dashed and short-dashed lines (Figure 2a). We then modify boundary conditions to $P=0,Q=Q_0>0$ at both ends. Then initial distribution function (thick line) relaxes to finite-$Q$-equilibria as shown by long-dashed line. We then change boundary conditions to $P=0, Q=Q_1>Q_0$ at both ends, so that distribution function is shown by short-dashed line. Increasing $Q$ at boundaries even further, so that $P=0, Q=Q_2>Q_1$ at both ends, the distribution function is given by dotted line (Figure 2b).
• Figure 3.
  We now solve (2-4) with the collision term given by (6). We pump broadly, so that the effective carrier distribution has zero flux. The initial distribution function (thin line) builds up because of a global pumping (dashed lines), until the laser switches on. The final (steady) distribution function is shown by thick solid line (Figure 3a). The output power (in arbitrary units) as a function of time (measured in relaxation times $\simeq 100$fs) is also shown (Figure 3b).
• Figure 4.
  We pump in the narrow region around $\omega _0\simeq 200 {\rm meV}$ and we model this by specifying carrier and energy flux rates $Q_L$ and $P_L=-\omega _L Q_L$. The initial distribution function (thin line) builds up because of influx of particles and energy from right boundary (dashed lines), until the laser switches on. The final (steady) distribution function is shown by thick solid line and corresponds to a flux of particles and energy from right boundary (where we add particles and energy) to the left boundary, where the system lases (Figure 4a). The output power as a function of time is also shown (Figure 4b).

Figure 1