Theory of laser noise in the phase locking region

We start from quantum mechanical laser equations which were derived in a previous paper for an inhomogeneously broadened laser and which contain in particular the noise sources due to cavity losses, vacuum fluctuations, interaction with phonons and nonlasing photons and the pump. For the example of frequency locking caused by the nonlinear polarization we derive a quantum mechanical Langevin equation for the relative phase angleψ=2ψ ₂ —ψ ₁-ψ ₃, whereψ ₁,ψ ₂,ψ ₃, are the total phases of three axial modes which would be equally spaced in the unloaded cavity. In the resulting equation

$$\dot \psi = \delta - \beta \sin \psi + f(t)$$     

At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point

We treat a set of coupled ordinary nonlinear differential equations and show that for each trajectory which belongs to an attractor (or to its basin) and which does not contain a fixed point, at least one Lyapunov exponent vanishes.     

Reconstruction of spatio-temporal signals of complex systems

We propose a novel method to analyze spatiotemporal signals emerging from synergetic systems. By this approach we are able to reconstruct the spatial modes, as well as their dynamic interaction close to instabilities. Our method is an extension of the principal component analysis to the case of nonlinear self-organizing systems. We demonstrate our method by an example of a codimension one instability, apply the algorithm to a simulated Bénard instability and present a generalization to bifurcations with several order parameters.     

Some exact results on discrete noisy maps

Using the Chapman-Kolmogorov type equation introduced by H. Haken and G. Mayer-Kress for discrete time processes we derive forward and backward equations for the corresponding transition probability and obtain an integral equation for the conditional first passage time. In the case of linear dynamics with Gaussian noise we present the exact solution of the Chapman-Kolmogorov equation.     

Generalized Onsager-Machlup function as solution of a multi-dimensional Fokker-Planck equation

The multi-dimensional Fokker-Planck equation is solved by Feynman path integrals. The solution may be interpreted as generalized Onsager-Machlup function.     

Generalized Ginzburg-Landau equations,slaving principle and center manifold theorem

The Generalized Ginzburg-Landau equations, introduced by one of us (H.H.), are considered in a simplified version to clarify their relation to the center manifold theorem.     

Quantum theory of the two-photon laser

We start from a master equation for the density operator of the atoms and the field mode, and apply the operator method of adiabatic elimination of the atomic variables, recently developed by Haake and Lewenstein for the usual single mode laser, to the case of a degenerate two-photon laser. A Fokker-Planck equation for the Wigner distribution function of the lightfield and its steady state solution are derived. With a Gaussian approximation to the solution, analytical and numerical results on the photon statistics are calculated.     

Slaving principle for stochastic differential equations with additive and multiplicative noise and for discrete noisy maps

We first treat multidimensional nonlinear noisy maps. We assume that the variables can be split into two classes of variablesu ands so that the linearized equations would give rise to growth or decay foru ands, respectively. We show how the slaved variabless can be explicitly expressed by the order parametersu by making use of the fully nonlinear equations. By taking the limit of vanishing time steps and using a Wiener process and the Îto calculus we derive the corresponding formulas for stochastic differential equations (including multiplicative noise). In this way a high-dimensional problem can be reduced to a problem of much lower dimensions described again by stochastic equations of theÎto type. A similar procedure holds for theStratonovich calculus.     

Quantum theory of noise in gas and solid state lasers with an inhomogeneously broadened line. I

We present a quantum mechanical nonlinear treatment of the phase and amplitude flucutations of gas lasers, i.e. lasers with moving atoms, and of solid state lasers with an inhomogeneously broadened line. The atoms may possess an arbitrary number of levels. As in our preceding papers the noise due to the pump, incoherent decay, lattice vibrations or atomic collisions, as well as due to the thermal and zero point fluctuations of the cavity is completely taken into account. The linewidth (due to phase diffusion), and the intensity fluctuations (due to amplitude noise) are essentially expressed by the threshold inversion, the unsaturated inversion and the saturated population numbers of the two atomic levels, which support the laser modes. Our results apply to the whole threshold region and above up to essentially the same photon number, to which the previous semiclassical theories of inhomogeneously broadened lasers were applicable. For the example of a two-level system we also demonstrate the application of a new technique which allows us to eliminate rigorously the atomic variables (operators), yielding a set of nonlinear coupled equations for the lightfield operators alone. If the elimination procedure is carried out only partially and additional approximations are made, we find essentially the rate equations ofMcCumber, in a form derived byLax. When we neglect noise, the nonlinear equation may be solved exactly in the case of single mode operation. By a suitable expansion of the exact multimode equations we find a convenient set of equations, which reduce in the noiseless case to those derived and used previously byHaken andSauermann as well asLamb.