Brief discussion of principal results

Here we list and briefly discuss the main and new results of this article.

• 1. We systematically derive evolution equations over long times $\epsilon ^{-2}$, of the order of the inverse square of the coupling strength, for the leading order approximations of all $N^{th}$ order cumulants $N\ge 2$. The first of these is the QKE, a closed equation for the particle number, and is consistent with what one obtains using the Hartree-Fock approximation ² when one takes proper account of the frequency renormalization to order $\epsilon ^2$. The remaining equations for the leading order behavior of the higher order cumulants decouple and can be solved by a simple renormalization of the frequency which depends only on particle number density. The first correction is the well known Hartree-Fock self energy correction whereas the second correction is new and moreover complex. The sign of the imaginary part of the latter is positive definite as the particle number approaches an equilibrium solution of the QKE. As a result, the leading order contribution to the cumulants of order $N>2$ (which contain information on the initial state) slowly decays. Not so the higher order corrections. We obtain specific expressions for these and show that they contain terms with cumulants of higher order, products of cumulants of order $N$ with products of particle number and products simply containing particle number alone. Only the latter two types of terms exert a long time influence and it is this fact that leads us to a natural asymptotic closure of the hierarchy.
• 2. The connections with classical systems and boson systems are discussed. By analogy with results obtained in the case of the classical kinetic equation, we introduce a new four parameter family of steady (equilibrium) solutions for the QKE. The four parameters are temperature $T$, chemical potential $\mu$, particle number flow rate $Q$ and energy density flow rate $P$. When $P=Q=0$, the equilibrium solution is the well known Fermi-Dirac distribution. For either $P$ or $Q$ nonzero, the distributions are the analogues of the Kolmogorov distributions of classical wave turbulence, in which particles (energy) flow from sources at intermediate momentum scales to sinks at low (high) momentum values.

  These solutions have not been considered in the fermionic context before. It turns out in fact, in contrast to the pure Kolmogorov solutions (for which $Q=0, P>0$, or $P=0, Q>0$) that the relevant equilibrium solutions of the QKE which describe (a) the finite flux of particles and a little energy between intermediate momenta at which the system is pumped and a sink at low momenta and (b) the finite flux of energy and a few particles between the intermediate momenta and the energy dissipation sink at high momenta, involve special relations between $Q$ and $P$. These solutions are also new for the cases of classical wave turbulence and boson systems.

• 3. Again by analogy with the classical case, we introduce a simple, local approximation to the collision integral in the QKE which gives excellent qualitative results. This is known as the differential approximation (see, e.g., [16],[17],[18]) and is strictly valid only when the principal transfer of energy and particles is between close neighbors in momentum space. Its most important feature, however, is that it allows us explore both analytically and numerically the behavior of the collision integral and thereby gain a clear qualitative picture about the nature of its steady (equilibrium) solutions.
• 4. We briefly examine the relevance of these results in the context of semiconductor lasers using a simplified model which assumes that any relaxation due to carrier-phonon interactions is negligible with respect to carrier-carrier interactions. We suggest that, because of Pauli blocking which results in a reduced pumping efficiency into those momentum states that are already filled, it may be advantageous to pump these lasers in a fairly narrow momentum window whose corresponding frequency is greater than the lasing frequency and allow the finite flux steady (equilibrium) solutions to carry particles and energy back to the lasing frequency via collision processes. Indeed we show that, at least for the model we use, this strategy is more efficient than pumping the system across a wide range of momenta as it is currently done. Indeed, in several numerical experiments, for equal amounts of energy pumping, we find that the laser turns on when the finite flux equilibrium is excited but fails to turn on under broadband pumping. When the pumping is increased so that the laser turns on under the latter conditions, the output power is significantly smaller than that emitted when the finite flux solution is operative.