Zur Theorie des Halbleiter-Lasers

The semiconductor laser is treated by a simple model which assumes parabolic bands. The optical transitions are supposed to obbey the usualk-selection rule. The position of the first running laser mode is considered both as a function of the degree of inversion and of temperature. The treatment includes also the effects of the response of the electrons to the lightfield (hole burning) as derived generally in a preceding paper.     

Higher order corrections to generalized Ginzburg-Landau equations of non-equilibrium systems

We develop a systematic iteration procedure which allows us to determine all higher corrections to our generalized Ginzburg-Landau equations derived in a preceding paper.     

The Fokker-Planck equation of a laser with many modes and multi-level atoms

Using the Glauber-SudarshanP-representation for the field modes and a quasi-distribution function recently presented for arbitrary quantum systems we derive an exact generalized Fokker-Planck equation for a multi-mode laser containing a set of multi-level atoms with homogeneous and inhomogeneous level broadening. By introduction of suitable collective atomic coordinates this generalized Fokker-Planck equation is reduced to an ordinary one which may serve as a basis for the adequate treatment of laser light statistics.     

An unavoidable set of configurations in planar triangulations

Let T be a normal triangulation (considered in the context of the four-color problem). Assume that no two vertices of degree 5 are adjacent. Then T contains at least 1 of the 47 configurations in Table I all of which are likely to be 4-color reducible.     

Exact stationary solution of the master equation for systems far from thermal equilibrium in detailed balance

If the master equation (Chapman-Kolmogorov equation) has a unique solution and fulfills the principle of detailed balance it may be solved explicitly by mere summations or, in the continuous case, by quadratures.     

Theory of ultra-short laser pulses

Using generalized Ginzburg-Landau equations derived recently we solve the laser equations analytically for the case of spontaneous ultra-short laser pulses. The results are compared with previous computer solutions by Risken and Nummedal and very good agreement is found.     

Exact generalized Fokker-Planck equation for arbitrary dissipative and nondissipative quantum-systems

We start from a density matrix equation in its most general form. It comprises the action of external fields on the system, internal interactions, as well as the action of dissipative mechanisms (heat-baths or reservoirs), which may be Markoffian or non-Markoffian. We then define a distribution function of a type introduced previously byHaken, Risken, Weidlich for atoms. This distribution function,f, which is now formulated quite generally with aid of projection operators,P _ik , establishes a connection between theP _ik 's and classical variablesv _ik . By means off it is possible to exactly calculate all quantum mechanical expectation values by purec-number procedures. If the basic density matrix equation is Markoffian, it is even possible to calculate all time-ordered multitime averages byc-number procedures usingf, as had been demonstrated byHaken, Risken andWeidlich. In the present paper we derive in an explicit way an exactc-number partial differential equation forf. It contains derivatives of arbitrarily high order. In important classes of problems, it can be reduced to an ordinary FokkerPlanck equation, however. Our new equation has many applications, e.g. in the quantum theory of lasers, nonlinear quantum optics, spinresonance, and spin-wave-theory, as will be demonstrated in forthcoming papers. We wish to thank Prof. W.Weidlich and Dipl. Phys. H.Vollmer for several valuable discussions. In addition, H.Vollmer has kindly checked our calculations.     

Functional quantum statistics of light propagation in a two-level system

The functional Fokker-Planck formalism developed in a preceding paper is applied to the problem of a radiation field propagating in a medium, which contains resonant two-level atoms. Besides the electromagnetic field also the medium is described by continuous space dependent fields. We give the masterequation and transform it into ac-number functional differential equation for a characteristic functional. This equation is reduced considerably by the projection onto one dimension and the introduction of the diffusion approximation. It forms a solid basis for the study of all types of light propagation in resonant media including classical and quantum noise. We give an approximate solution of this equation by considering the problem of an externally pumped optical transmission line, in the case that saturation effects are absent. The spectral function of the electric field strength is obtained which describes a statistical mixture of photons with the quasiparticles of the polarization field. It shows the onset of a condensation of the quasiparticles into a single state. Self excitation of the transmission line is obtained at a certain threshold of the atomic inversion. This threshold is characterized by a finite occupation number of one single quasiparticle state. The influence of a finite length of the transmission line is briefly considered.