W. Weidlich

Fokker-Planck equation, distribution and correlation functions for laser noise

As an application of a preceding paper we set up a Fokker-Planck equation with quantum mechanically defined dissipation and fluctuation coefficients for a distribution function of the atomic variables (dipole moments and level occupation numbers) as well as of the lasing light amplitude in a laser with a homogeneously broadened line. Since the nonlinear coefficients can be linearized in appropriate coordinates well below and well above threshold, the equation can be solved with the Wang-Uhlenbeck method. Then it is easy to obtain correlation functions, spectral densities and expressions for linewidth.

Quantum mechanical solutions of the laser masterequation: III. Exact equation for a distribution function of macroscopic variables

We treat a laser consisting of one mode described by a running wave and a set of atoms with two optically active levels which are homogeneously broadened. We start from the laser density matrix equations ofWeidlich andHaake and define a distribution functionf for lightfield and atomic variables, where we use for the lightfield the coherent state representation and for the atomic system a modified version of the distribution function used bySchmid andRisken in a previous paper. We derive a partial differential equation forf which is completely exact and is of the type of a generalized Fokker-Planck equation, i.e. it contains higher derivatives. Using a recently stated theorem ofHaken andWeidlich we show that this distribution function allows to calculate single-time as well as multitime quantum mechanical correlation functions. If the leading terms of the generalized Fokker-Planck equation are retained we find the semiclassical Fokker-Planck equation ofRisken,Schmid andWeidlich. Our treatment can be extended to several modes connected with standing waves and multilevel atoms.

Quantummechanical solutions of the laser masterequation. II

A Fokker-Planck equation for a distribution function over the macroscopic observables of the laser essentially equivalent to that recently obtained byRisken,Schmid andWeidlich is derived from the fundamental quantummechanical laser masterequation. The general method used is the expansion of the statistical operator in a complete set of projection operators of the atoms and the lightfield. The assumptions leading from the microscopic equation of motion to the macroscopic semiclassical Fokker-Planck equation are explicitly introduced and discussed.

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.

Quantum mechanical correlation functions for the electromagnetic field and quasi-probability distribution functions

We give new formulae permitting the calculation of time ordered but otherwise arbitrary correlation functionsK of electromagnetic field operators in terms of a class of quantum mechanical quasi-probability distribution functions. This class contains among others the socalledQ- andP-functions as well as the Wigner function.

Quantum fluctuations, master equation and Fokker-Planck equation

In order to describe quantum fluctuations a general method is developed, which also may be applied to nonstationary systems as well as to states far from thermodynamic equilibrium. After a concise derivation of the master equation quantum mechanically determined dissipation and fluctuation coefficients are introduced, for which several theorems and relations are given. By using these coefficients there is set up a general Fokker-Planck equation for the diffusion of the statistical operator due to quantum fluctuations.

NUCLEAR REACTIONS AS AN EXAMPLE OF THE QUANTUM THEORY OF IRREVERSIBLE PROCESSES.

A density matrix formalism is developed for nuclear reactions, where the nuclear system is described by an effective Hamiltonian in second quantisation, including potential scattering and different compound nucleus reactions. On the other hand, the nuclear system is coupled to reservoirs injecting and absorbing particles, such that the stationary solution of the irreversible density matrix equation describes a “fluxequilibrium”. Taking appropriate expectationvalues, the set of coupled equations is solved for potential scattering and especially for a compound nucleus reaction. The cross-sections are calculated, showing the Breit-Wigner resonance structure in the latter case, where the level width is expressed in terms of the matrix elements contained in the fundamental Hamiltonian.

A model for the measuring process in quantum theory

A simple model is presented (N two level atoms in interaction with a field mode) which shows explicitely some of the main features of a real quantum mechanical measurement, like the channel structure of the Hilbertspace of the measuring device with respect to macroscopic observables. Especially in its quasi exact and approximate treatment the model accounts for the irreversibility of the motion in the measuring system and for the destruction of the initial state of the measured object during the interaction with the measuring device.

Mode interaction in a spatially inhomogeneous laser

In this paper the influence of spatial matter inhomogeneities on laser action is investigated theoretically. The calculations are based on nonlinear laser theory for the amplitudes of the empty-cavity modes. The nonlinear mode interactions like mode suppression or phase locking are modified by spatial inhomogeneities. There also arises a linear coupling among the cavity modes in general leading to monochromatically oscillating states which are superpositions of the empty-cavity eigenfunctions.

Flux equilibria in quantum systems far away from thermal equilibrium

Abstract Quasi-probability functions make it possible to study flux equilibria and phase transitions in quantum systems far from thermal equilibrium using concepts of classical statistical mechanics.