Background.

Gaining a good understanding of the relaxation processes of many particle (or many wave) systems is a difficult task. A successful approach requires many ingredients. First, one must make approximations concerning the relative strength and uniformity (in momentum space) of the nonlinear coupling. Second, one needs to understand how the infinite hierarchy of moment equations can effectively decouple and give rise to a closed kinetic Boltzmann equation for the redistribution of particles (waves) and energy in momentum space. One of the goals of this paper is to show that one can indeed, as one can do for classical systems, derive the quantum kinetic Boltzmann equation in a self consistent manner without resorting to a priori statistical hypotheses or cumulant discard assumptions. Third, systems are rarely in isolation and frequently involve sources (some instability or forcing that injects particles and energy, often at specific locations in momentum space) and sinks (regions of absorption of particles and energy). Moreover, and in analogy with hydrodynamic turbulence and classical wave systems, the presence of sources and sinks dramatically changes the nature of the equilibria reached by the quantum kinetic equation. A second and major goal of this paper is to describe these equilibria and show how they can affect the output of semiconductor lasers.

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Fermionic quantum systems have been studied intensively for more than five decades. Many theoretical approaches (see, for example, [1]-[14]) have been developed but the common feature of all is the derivation of the quantum mechanical Boltzmann kinetic equation (henceforth referred to as the quantum kinetic equation or QKE) which describes the evolution and relaxation of the particle number, $<a_k^\dagger a_k>$, due to collisions. It is the analogue of the classical kinetic equation (KE) of wave turbulence (see, e.g., [15]) and its natural quantum extension to boson gases. The QKE accounts for phase space blocking effects (Pauli exclusion principle) and is equivalent to what one would obtain by treating the scattering cross-section using the second Born approximation. Examples of variations of this equation and its derivation include the Lenard-Balescu equation which accounts for dynamical screening effects (see, e.g., [7]-[8]). Other extensions include generalized scattering cross sections, such as the exchange effects (crossed diagrams), as well as T-matrix effects (see, e.g., [2],[9]). T-matrix approaches are especially important in systems that allow for bound states. Further generalizations of the quantum Boltzmann equation are the various forms of the Kadanoff-Baym equations. The most general Kadanoff-Baym equations are two-time equations which describe charge-carrier correlations consistently with relaxation dynamics. The Markov approximation, which is the lowest-order gradient expansion with respect to macroscopic times, yields the familiar form of the Lenard-Balescu equation, and the additional static screening and small momentum transfer approximation yields the Landau kinetic equation. In contrast to the slow relaxation dynamics for which the Kadanoff-Baym gradient expansion is applicable, recent investigations have addressed the issue of ultrafast relaxation and the related problems of memory effects as contained in the Kadanoff-Baym equations (see, e.g., [11] and all references therein) and in the generalized Kadanoff-Baym equations of Lipavski, Spicka, and Velicky (see, e.g., [12]-[14]).

In all of the derivations, the principal obstacle to be overcome is the closure problem. Due to the nonlinear character of the quantum mechanical Coulomb interaction Hamiltonian, the time evolution of the expectation values of two operator products such as $<a_k^\dagger a_k>$ are determined by four operator expectation values $<{a^\dagger_{k_1}a^\dagger_{k_2}a_{k_3}a_{k_4}}>$. The problem compounds. The time evolution of the N operator product expectation value is determined by (N+2) operator expectation values and one is left with an infinite set of moment operator equations known as the BBGKY hierarchy. The closure problem is to find a self consistent approximation of this infinite hierarchy which reduces the infinite set of coupled equations to an infinite set of equations which are essentially decoupled. To do this, one needs to make approximations and these almost always involve the introduction of small parameters.

In what follows we introduce one such small parameter, the relative strength $\epsilon , 0<\epsilon \ll 1$, of the coupling coefficient $T_{k k_1,k_2 k_3}$ in the system Hamiltonian ($V=$ d-dimensional system volume)

$$H = \frac{V}{(2\pi)^d} \int d {\bf k} \hbar \omega _k a_k^\dagger a_k$$

$$+ \frac{V^3}{(2 \pi)^{3d}} \frac{1}{2} \int d {\bf k_1} d{\bf k_2... ...dagger_{k_1} a_{k_2} a_{k_3} \delta ({\bf k}+{\bf k_1} -{\bf k_2} -{\bf k_3}) .$$

By relative, we mean the ratio of $T_{k k_1,k_2 k_3}$ to $\omega _k$. In other words, the linear response of the Hamiltonian is dominant over short times. This is not a trivial approximation because in order to develop a consistent asymptotic closure we need $T_{k k_1,k_2 k_3}$ to be small compared to $\omega _k$ uniformly in $k$. There are many important examples where this is not the case. For instance, for the classical nonlinear Schroedinger system (see, e.g., [16]), which describes the weak turbulence of optical waves of diffraction in a nonlinear medium, $\omega _k\propto k^2$ and $T_{k k_1, k_2 k_3}=a/(4 \pi ^2)$ where $a$ is a positive (negative) constant for the focusing (defocusing) case and it is clear that the ratio of $T_{k k_1,k_2 k_3}$ to $\omega _k$ increases dramatically as $k\to 0$. This has real physical consequences. The asymptotic closure for the classical case, analogous to one we are about to derive for fermions, whose central feature is the classical kinetic equation describing resonant four-wave energy exchange, is only valid when the particles and energy are at finite $k$ values. But the dynamics of the classical kinetic equation are such that most of the energy travels to a sink at large $k$ and most of the particles and some energy travels to low $k$. Unless there is a strong damping near ${\bf k}=0$, the flow of particles and energy towards ${\bf k}=0$ will trigger collapsing filaments in the focusing case or build condensates in the defocusing case. In either case, unless there is strong damping near ${\bf k}=0$, the weak turbulence theory of four wave interactions breaks down. Likewise in plasmas and and highly excited semiconductors, while the frequency $\omega _k$ is $\alpha k^2$, the bare coupling coefficient $T_{k k_1,k_2 k_3}$ behaves as $1/k^2$ reflecting Coulomb interactions. Again, even though the carriers may be initially excited at ${\bf k}$ values much greater than zero, the natural dynamics of either the classical or quantum kinetic equations will lead to particle deposit at small ${\bf k}$. However, in both of these cases, there is another effect which modifies the $T_{k k_1,k_2 k_3}$ and brings the theory closer to the nonlinear Schroedinger case. The physical origin of this effect is screening and it is manifested as a weakened potential at long distances or small ${\bf k}$ values. The modification is a renormalization of the coupling coefficient $T_{k k_1,k_2 k_3}$ near ${\bf k}=0$ which effectively cancels the singularity, or equivalently replaces $k^{-2}$ by $(k^2+\kappa^2)^{-1}$, where $\kappa$ is the inverse screening length. This is the small $k$ limit of the dynamically screened effective interaction in the Lenard-Balescu approach (see, e.g., [7]). In this paper, we do not take issue ¹ with the self consistency of this approximation because we will be introducing a sink at small $k$ values (greater than $\kappa$) which remove the potential danger in the nonuniformity of the ratio of $T_{k k_1,k_2 k_3}$ to $\omega _k$. This is not at all artificial. The sink is, in the semiconductor laser context, nothing other than the lasing output. In what follows, then, we assume that in the $k$ regions of interest, the ratio of $T_{k k_1,k_2 k_3}$ to $\omega _k$ is uniformly small.

There are essentially two reasons for the successful closure of the hierarchy over long times (asymptotic closure). First, to leading order in $\epsilon$, the cumulants (the non Gaussian part) corresponding to the expectation values of products of $N$ operators ($N>2$) play no role in the long time behavior of cumulants of order $r<N$. This is a consequence of phase mixing and due to the nondegenerate nature of the dispersion relation $\omega _k=\omega (\vert{\bf k}\vert)$. In fact, we will find that these (zeroth order in $\epsilon$) cumulants slowly decay. The net result is that the initial state (subject to certain smoothness conditions) plays no role in the long time dynamics. Second, and more important, the higher order (in $\epsilon$) contributions to these $N$-operator cumulants, which are generated directly by the nonlinearities in the system, are dominated by products of lower order cumulants. Some of these dominant terms are supported only on certain well defined low dimensional (resonant) manifolds in momentum space. These are responsible for the redistribution of particle number among the different momentum states and for the slow decay in time of the leading order approximation to the cumulants of order $N>2$. The other dominant terms are nonlocal and lead to a nonlinear frequency renormalization of $\omega _k$.

As a result, we are led to a very simple and natural asymptotic closure of the BBGKY hierarchy.

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