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.     

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.     

Theory of intensity and phase fluctuations of a homogeneously broadened laser

We extend our previous quantum mechanical nonlinear treatment of laser noise to the following problem: We consider a set of atoms each with three levels, which support laser action of one or several modes. The laser action can take place either between the upper or the lower two levels. The atomic line is assumed to be homogeneously broadened. The broadening can be caused by the decay into the nonlasing modes, by the pumping process, lattice vibrations and other, non specified sources. The fluctuations of the atomic variables (or operators) are taken into account in a quantum mechanically consistent way using results of previous papers byHaken andWeidlich as well asSchmid andRisken. The laser modes are coupled to the thermal resonator noise usingSenitzky's method. In the first part of the present paper, we treat quite generally multimode laser action. It is shown, that each light mode chooses a specificcollective atomic “mode” to interact with. We introduce a set of suitable collective atomic “modes”, which leads to a simplification of the equations of motion for theHeisenberg operators of the light field and the atomic operators. From the new equations we can eliminate all atomic operators. We are then left with a set of coupled nonlinear, integro-differential equations for the light field operators alone. These equations, which are completely exact and valid both for running and standing waves, represent a considerable simplification of the original problem. In the second part of this paper, these equations are specialized to single mode operation, which is studied above laser threshold. In the vicinity of the threshold the laser equation can be simplified to an operator-equation, whose classical analogue is vander-Pol's equation with a noisy driving force. With increasing inversion, the full equation must be treated, however. Using the method of our previous paper, we decompose the light amplitude into a phase-factor and a real amplitude, which is expanded around its stable value. We determine the Fourier-transform of the intensity correlation function and the total intensity of the fluctuating part of the amplitude. Somewhat above threshold this intensity drops down with the inverse of the photon output power,P, while the inherent relaxation frequency increases withP. The noise intensity stems in this region from the off-diagonal elements of the noise operators and not from the diagonal elements, which are responsible for the shot noise. This result is insofar remarkable, as a rate equation treatment would include only the latter ones. Under certain conditions the intensity fluctuations can show resonances with increasing output power,P. At high inversion the vacuum fluctuations of the light field are dominant, while the other noise sources give rise to contributions which vanish with the inverse of the output power. As a by-product our treatment yields the following formula for the linewidth (half width at half power) which is caused by phase fluctuations:

$$\Delta \nu = \frac{{\gamma _{3 2}^2 \kappa ^2 }}{{(\kappa + \gamma _{3 2} )^2 }}\frac{{\hbar \omega }}{P}\left( {\frac{1}{2}\frac{{(N_3 + N_2 )}}{{N_3 - N_2 }} + n_{Th} + \frac{1}{2}} \right)$$     

Der mittlereg-Faktor der ersten angeregten Rotationszustände der geraden Wolframisotope in natürlichem Wolfram

The average gyromagnetic ratio of the first excited 2⁺ states of the Tungsten isotopes was measured using a target of natural Tungsten. An external magnetic field of 40950±200 Gauss was employed for observing the precession of the angular distribution of the 115 keV deexcitationγ-rays following Coulomb-excitation with an atomic hydrogen beam of 2·18 MeV. The measurement of the angular distribution was carried out with a proton beam at 6 different energies between 1·5 and 3·7 MeV, and was found to be slightly perturbed by internal fields. The measured attenuation coefficients areG ₂=0·966±0·049,G ₄=1·07±0·15. The gyromagnetic ratio was found be\(\bar g = 0 \cdot 264_{ - 0 \cdot 019}^{ + 0 \cdot 023}\), in agreement with the average value of the measurements ofGoldring andVager and the average theoretical value ofNilsson andPrior, but lower than the measurements ofBodenstedt et al.     

Difficulties with a unified treatment of nonleptonic weak interactions

The isospin selection rule in a unified treatment ofCP-conserving andCP-violating nonleptonic weak interactions proposed byNishijima andSwank is investigated and compared to that of a modified version.     

Relativistic Slater's integrals for theK andL shells

TheZ-dependent form of theSlater's integrals for electrons in theK andL shells is studied by expanding the relativistic hydrogenic radial wave functions in the manner explained byLayzer andBahcall. When screening is not taken into consideration, theseSlater's integrals can be put in the form AZ(1+a Z²) wherea is positive for all integrals considered except G¹(1s, 2p) and G²(2¯p, 2p). Values ofA anda are given.     

Polarization of atomic line radiation

In order to test the predictions of the general theory for the polarization of atomic radiation excited by electron impact, we measured the polarization of atomic lines in electron-alkali atom crossed-beam experiments. The results show a typical behaviour for the polarization of lithium and sodium resonance lines, which is contrary to the earlier measurements such as those ofSkinner andAppleyard: the polarization of the resonance lines increases monotonically from higher energies to the excitation threshold. The threshold polarization of the first resonance lines of Li⁶, Li⁷ and Na²³ is in good agreement with the calculations byFlower andSeaton. These results demonstrate that polarization is very sensitive to the natural level width, and to the fine and hyperfine structure separations of the excited states. Polarization of further lines has been investigated: an alkali line from anS to aP state is unpolarized as expected, but the polarization of a line from aD to aP state decreased in an unexpected manner near the threshold energy.     

Quantummechanical solutions of the laser masterequation I

The masterequation for the statistical operatorW of a Laser mode andA active two-level atoms 1 is solved by using the coherent state representation 2 of the lightfield. The ansatz forW represents the most general symmetrical coupling of the light mode to all atoms and therefore contains the full influence of quantum fluctuations of the atomic system on the light mode. The system of equations can be solved practically exactly in the stationary case and leads to a photon number distribution in the laser valid for arbitrary pumping. This distribution agrees with that found by a Fokker-Planck equation 3 for not too high pumping and approaches the Poisson distribution for very high pumping. The smooth transition of the inversion from σ₀ (below threshold) to σ (above threshold) can also be calculated.     

Mobility of electrons in nondegenerate semiconductors considering electron-electron scattering

Spitzer andHärm [1] have investigated the velocity distribution of electrons in the presence of a weak electric field in an ionized gas. Introducing the concept of time of relaxation τ′, due to electron-ion and electron-electron scattering, and using the results ofSpitzer andHärm [1], the authors have obtained an expression for τ′ which is applicable to semiconductors with little modification. In this paper the authors have used this expression for τ′ to obtain the mobility of electrons in nondegenerate semiconductors, taking into account the scattering by lattice vibrations, electron-ion interactions and electron-electron interactions.     

Separation of surface- and volume-vibrations in quadrupole-deformed nuclei and consequences on isotope shifts

Starting from the complete hamiltonian for a system ofN nucleons, the problem of separating volume and surface vibrations inpermanently deformed nuclei is investigated. The required “volume” variable which allows this separation turns out to be the mean square distance of a nucleon with reference to the center of mass. Our investigation confirms the concept ofpure surface vibrations, as is used in the rotationvibration model ofFässler andGreiner, and leads to a theoretical argument in favour ofFradkin's idea of “compressibility under deformation” forced upon him by experimental evidence in calculating isotope shifts in deformed nuclei.     

Untersuchungen zur Mehrfach-Emission von Sekundärelektronen

CuBe- and NaCl-targets are bombarded by single electrons (100–600 eV). The secondary electrons accelerated by 40 kV strike the crystal of a scintillation counter, backed by a multichannel analyser. The probabilityP _n of emission ofn=0, 1, 2, 3, ... secondaries can be found from the pulse height distribution. The probability distributionP _n =f(n) shows a characteristic deviation from aPoisson's distribution. There was no evidence that there is a preference for even numbers ofn as found byBarrington andAnderson.     

Messungen des van der Waals-Potentials zwischen Alkaliatomen

Total collision cross sections for the interaction between alkali atoms have been measured using a modulated atomic beam technique. Since the determination of the density in the scattering chamber is the main source of error in absolute scattering measurements, particular attention is given to this problem: The cross section for partnersA andB is measured first withA as beam particles anB as target particles, and then withB as beam particles andA as target particles. The data are used to deduce the long rangevan der Waals (inverse sixth power) potential constants, which can be compared with theoretically calculated values of different authors. The interaction constants resulting from the present measurements are in good agreement with those calculated byFontana, while other calculations give much larger values.     

Einfluß der Pion-Pion-Wechselwirkung auf Pion-Nukleon-Streuphasen

The isospin dependent part of theS-wave pion-nucleon amplitude is calculated according to the CGLN approach, but recoil and projection on the partial waves are treated exactly. Then the difference against the experimental data is approximately a constant up to 100 MeV and amounts to 1/3 of the total value. It is shown, that at low energies the main corrections of the CGLN equations are taken into account by a subtraction of theπ+π→?+\(\overline \Re \),J=I=1 amplitudesf _± ¹ (t) as proposed byMandelstam andFrazer. Calculations have been done forS, P andD pion-nucleon amplitudes. The main result is a prediction of theS-wave amplitude obtained without adjusting free parameters up to 100 MeV. The relations of this method to the equations derived byBowcock, Cottingham andLurié according to the Cini-Fubini approach are 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.     

Zur Theorie der infinitesimalen Holonomiegruppe in der Allgemeinen Relativitätstheorie

Starting from the infinitesimal holonomy groupH _i of aV ⁴, (+++?) the spinholonomy group\(\tilde H_i \equiv \bar \sigma ^1 (H_i )\) defined by the covering isomorphism\(\sigma :G \to L_ + ^ \uparrow \) is introduced. In Einstein-spaces we may replace its real Lie-algebra by a complex one. With the complex calculus we may reproduce the results ofSchell, Goldberg andKerr with very much simplified proofs. A theorem on non-empty Einstein-spaces is given. In part 4 we prove a theorem on the connection between theH _i -behaviour of a vector (spinor) and its covariant derivative in aV ⁴. With its help we get in a simple manner the metiics of aV ⁴ with givenH _i and Dim (H _i ) <6; our results agree with those given byGoldberg andKerr, Cahen andDebever. Finally we make some new statements on imperfect holonomy groups.     

Paramagnetische Resonanz von Zentren mit den Symmetrienn,\bar n undn/mundn/m

The consequences of applying the spin-Hamiltonians ofAbragam andPryce to paramagnetic centers with the low symmetriesn,\(\bar n\) andn/m(C _n ,C _nh ,S _n ) are examined. The asymmetry of theg- and the hfs-tensors, described in an earlier paper, is considered in this context. The point is that for the symmetries under consideration the coordinate systems are not determined by the symmetry elements. For this reason it is possible to introduce separate coordinate systems for the magnetic field, the electron spin and the spins of the nuclei and this allows the symmetrisation of the tensors under certain conditions. This procedure also leeds to an understanding of the independent parameters found in the Hamiltonians. Ambiguities between the spin-Hamiltonians and the esr-spectra indicate the limits of theAbragam andPryce formalism. The application of the theory to paramagnetic centers under the influence of external, electrical fields is discussed and reveals interesting aspects.     

Quasielastic kaon knockout from a proton and momentum distribution of virtual kaons in the proton

The momentum distribution of kaons in the proton and the coupling-constant ratio fKΛN/fKΣN are determined on the basis of the ³P₀ quark model of meson-baryon coupling. The longitudinal cross section for kaon electroproduction is calculated. The results of the present study are compatible with available experimental data, but more detailed data on the longitudinal cross section would make it possible to refine the absolute values of the coupling constants fKΛN and fKΣN.     

Der Einfluß der Krümmung eines Schirmrandes auf die Beugung elektromagnetischer cm-Wellen

The diffraction of electromagnetic cm-waves passing by the straight edge of a metal screen was experimentally investigated. The diffraction which depends on the polarisation of the waves and the thickness of the screen was measured within a range of about 20 wavelengths around the edge of the screen. The results agree with the theory ofSommerfeld and the investigations ofFranz andDeppermann. The intensity function of the waves near the edge of the screen shadow depends — according to our measurements — on the curvature of the screen edge. This agrees with the theoretical postulation ofArtmann: if the radius of the screen edge isa>λ the diffraction fringes appearing around the shadow of the screen edge are shifted towards the light area whenσ-polarized radiation is used. When usingπ-polarized radiation a shift towards the shadow area ist observed.     

Vergleich der Photoerzeugung von π0 am Proton mit der Voraussage nach der Dispersionstheorie

The real parts of the photoamplitudesE 1S _1/2,M 1P _1/2,M 1P _3/2 have been calculated from the angular distribution of the reactionγ+p→p+π⁰ recently measured byGoldansky et al. at 160 to 240 MeV. One of the solutions fits pretty well to the theoretical prediction for theM 1P _3/2-amplitude according to the dispersion method ofChew, Goldberger, Low andNambu. There is a discrepancy forM 1P _1/2 ifα ₁₁ is taken from the effective range formula, but the positive values ofα ₁₁, necessary to give agreement, are not excluded by the results of the phase shift analysis, especially sincePontecorvo et al. have recently found positive values at higher energies. The prediction for the real part of theE 1 S_1/2-amplitude agrees with the experimental data, if pretty large recoil corrections are added which had been neglected byChew et al.     

Polarization of recoil nucleons in π0-photoproduction on protons below 500 MeV. I

Using the model of pion photoproduction recently discussed byBall andSchmidt a prediction for the polarization of the recoil protons is given for unpolarized photons. The strong dependence on the assumed values ofE ₀₊ andM _1? is shown. Taking into account this result and assuming thatM ₁₊ and the higher multipoles are sufficiently known, the possibility of determining the small multipoles from measurements is discussed.