Propagation of waves and chaos in transmission line with strongly anharmonic dangling resonator

Delayed differential equation of motion with multiple lags is derived for an anharmonic stub resonator coupled to a monomode transmission line. Transmission and reflection coefficients are found analytically in the harmonic approximation. Nonlinear response of the system is analysed by an electric circuit obeying the same equations of motion. Enhanced second harmonic generation is found at the frequencies, which in the harmonic approximation correspond to the zeros of transmission. An aperiodic (chaotic) response is found mainly in the frequency range close to the resonance of the dangling resonator. Zeros of transmission and total transmissions are shown to be lifted by the anharmonicity nearly in the same frequency region. Higher harmonics are preferentially transmitted at the zero transmission points in the presence of anharmonicity. Received 14 March 2002 / Received in final form 25 November 2002 Published online 14 March 2003     

Approaching the strongly anharmonic limit with ab initio calculations of materials’ vibrational properties – a colloquium*

Despite ab initio computational techniques have opened new possibilities to interpret experimental results and predict the properties of new materials, their applications are limited by the adopted approximative schemes. Consequently, the first-principles calculation of many physical properties and phenomena is hindered and ab initio methods need to be further developed to overcome such limits. For example, the standard harmonic approximation used to assess the vibrational properties of materials often completely breaks down, so that the vibrational properties need to be calculated including strong anharmonic effects. The harmonic approximation has also intrinsic failures as it cannot estimate the lattice thermal conductivity of materials nor the temperature dependence of the phonon frequencies, crucial to account for temperature driven second-order phase transitions. Several methods developed in the last years to account for anharmonicity in the non-perturbative regime and overcome such difficulties are briefly reviewed in this colloquium paper. In particular, the stochastic self-consistent harmonic approximation, a variational method that allows calculating vibrational properties in strongly anharmonic systems, is described in further detail. Applications of the latter method to superconducting palladium, platinum, and sulfur hydrides are discussed, where anharmonicity has a huge impact on their vibrational and superconducting properties.     

Über den Einfluß der Anharmonizität auf die thermische Röntgenstreuung an Kristallen

The effect of vibrational anharmonicity on the thermal scattering of X-rays from crystals is calculated for temperatures above the Debye temperature. A revised method of interpreting the experimental data is suggested. The (temperature dependent) dispersion curves (frequency vs. wave-vector) for small wave-vectors are determined by the isothermal rather than the adiabatic elastic constants. A procedure is outlined to extrapolate from the (temperature dependent) scattering data the (temperature independent) dispersion curves which correspond to the harmonic approximation.     

On the displacements of centers of vibration-rotation bands under isotope substitution in polyatomic molecules

The effect of isotope substitution in polyatomic molecules on the displacements of absorption vibration-rotation band centers is considered. Formulas are obtained that allow the band-center displacements to be calculated in harmonic approximation and with anharmonicity. The effect of resonance interactions is also considered. The results obtained are illustrated by numerical calculation for a number of molecules.     

Generation of Elastic Waves by a Magnetic Field Associated with Reversible Rotations in Triaxial Magnets

In the first-order anharmonicity approximation of Hooke's law, the amplitudes of elastic waves, absorption coefficients, wave numbers of the fundamental wave and of the first harmonic generated by an alternating magnetic field with a preset orientation relative to the basis axes of a crystal having arbitrary dimensions are calculated for multidomain magnets with rigidly fixed domain boundaries in terms of concentrations of magnet phases and magnetic structure parameters with allowance for the wave equation and angular momenta.     

Modified Perturbation Theory of an Anharmonic Oscillator

A modification of the perturbation theory of a symmetrical anharmonic oscillator is suggested. A more complex zero-order approximation of perturbation theory that considers to a certain degree anharmonicities is chosen rather than a harmonic oscillator model. This approximation is an analog of the self-consistent field model well known in the theory of many-particle systems. A comparison of modified and conventional perturbation theories demonstrates that the modified perturbation theory has much wider applicability range. It can be used for larger values of the parameters at which the conventional perturbation theory becomes inapplicable, namely, for strong anharmonicity and upper energy levels.     

Approximate reconstruction of torsional potential energy surface based on voronoi tessellation

Torsional modes within a complex molecule containing various functional groups are often strongly coupled so that the harmonic approximation and one-dimensional torsional treatment are inaccurate to evaluate their partition functions. A family of multi-structural approximation methods have been proposed and applied in recent years to deal with the torsional anharmonicity. However, these methods approximate the exact “almost periodic” potential energy as a summation of local periodic functions with symmetric barrier positions and heights. In the present theoretical study, we illustrated that the approximation is inaccurate when torsional modes present non-uniformly distributed local minima. Thereby, we proposed an improved method to reconstruct approximate potential to replace the periodic potential by using information of the local minima and their Voronoi tessellation. First, we established asymmetric barrier heights by introducing two periodicity parameters and assuming that the exact barrier positions are at the boundaries of Voronoi cells. Second, we used multiplicatively weighted Voronoi tessellation to refine the barrier heights and positions by defining a structure-related distance metric. The proposed method has been tested for a few higher-dimensional cases, all of which show promising improved accuracy.     

Nonlinear dynamics of a Bose-Einstein condensate in a magnetic waveguide

We have studied the internal and external dynamics of a Bose-Einstein condensate in an anharmonic magnetic waveguide. An oscillating condensate experiences a strong coupling between the center of mass motion and the internal collective modes. Because of the anharmonicity of the magnetic potential, not only the center of mass motion shows harmonic frequency generation, but also the internal dynamics exhibit nonlinear frequency mixing. Thereby, the condensate shows shape oscillations with an extremely large change in the aspect ratio of up to a factor of 10. We describe the data with a theoretical model to high accuracy. For strong excitations we test the experimental data for indications of a chaotic behavior.     

Harmonic balance approach to the periodic solutions of the (an)harmonic relativistic oscillator

The first-order harmonic balance method via the first Fourier coefficient is used to construct two approximate frequency-amplitude relations for the relativistic oscillator for which the nonlinearity (anharmonicity) is a relativistic effect due to the time line dilation along the world line. Making a change of variable, a new nonlinear differential equation is obtained and two procedures are used to approximately solve this differential equation. In the first the differential equation is rewritten in a form that does not contain a square-root expression, while in the second the differential equation is solved directly. The approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that, in the second procedure, application of the harmonic balance method produces an infinite set of harmonics, while in the first procedure only two harmonics are produced. Both approximate frequencies are valid for the complete range of oscillation amplitudes, and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed. The discrepancy between the first-order approximate frequency obtained by means of the second procedure and the exact frequency never exceeds 1.6%. We also obtained the approximate frequency by applying the second-order harmonic balance method and in this case the relative error is as low 0.31% for all the range of values of amplitude of oscillation A.     

Polaron on a one-dimensional lattice: II. A moving polaron

Continuum-limit equations for moving polarons on a one-dimensional lattice with a harmonic interaction potential between adjacent particles and a simple nonlinear potential with a cubic nonlinearity are derived for the first time; for some particular cases, their solutions are obtained. For a harmonic lattice in the continuum limit, a system of integrable nonlinear partial differential equations is derived. A one-soliton solution to this system describes a polaron moving with a constant velocity. The speed of this polaron is uniquely related to its amplitude, with its values ranging from zero to the speed of sound. For a nonlinear lattice, the resulting system of differential equations is integrable at a certain ratio of the problem parameters. The one-soliton solution to this system, as in the harmonic case, describes a polaron moving with a constant velocity. At arbitrary values of the lattice parameters, the nonlinear lattice was studied by numerical methods. It turned out that, in the entire range of parameters, the nonlinear lattice gives rise to moving polarons, with the speed of the polaron being determined by the competition between the electron-photon interaction parameter α and the nonlinearity parameter β. At α ? β, the behavior of the polaron is very close to the dynamics on the harmonic lattice. In the opposite case, the dynamic nonlinearity begins to dominate, giving rise to dynamics inherent to solitons, so that speed of the polaron can exceed the speed of sound. In a certain range of α and β, numerical calculations revealed a family of polaron-type stable solutions, the envelope of which can have several peaks. The numerical and exact analytical solutions are in very good agreement for a sufficiently large radius of the polaron, when the system of equations obtained in the continuum approximation has a solution.     

Equations of state of crystals with strong sixth-order anharmonicity

In a nonsymmetrized approximation of a self-consistent field, the equations of state of densely packed crystals with strong anharmonicity of up to sixth order are considered. These equations include two coefficients which are implicit functions of two dimensionless combinations of the temperature and force parameters of second, fourth, and sixth order. The properties of these functions are investigated; their values are tabulated for broad ranges of variation of the arguments. In particular, it follows from these properties that, as the temperature is reduced, sixth-order anharmonicity is first switched off, and the equations pass over into the already studied equations of state of crystals with strong fourth-order anharmonicity, which is switched off at still lower temperature. The method of obtaining corrections to the given approximation is also considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 77–82, August, 1979.     

Nonlinear lamb waves in a metal plate with defects

Experimental results on the propagation of finite-amplitude Lamb waves in a solid plate made of polycrystalline aluminum alloy with defects are presented. The Lamb waves are recorded and visualized using a scanning laser vibrometer. The dependences of the higher harmonic amplitudes, both averaged over the plate surface and measured at a point far from the defect and at the site of the defect, on the fundamental frequency amplitude are studied. A threshold character of the higher harmonic generation and a power-law behavior of their amplitudes are revealed, the latter feature being unconventional for the nonlinearity associated with the anharmonicity of the crystal lattice of the material. The possibility of locating the distribution of individual defects from the measured spatial distribution of structural nonlinearity in the material under study is experimentally demonstrated. The results of the experiments are explained in terms of the bilinear medium model.     

Thermal expansion of nanostructured PbS films and anharmonicity of atomic vibrations

The dependences of the coherent scattering region size and thermal expansion coefficient α of a PbS nanofilm on the annealing temperature in the range of 293–473 K and on the duration of annealing at a constant temperature of 423 K have been measured. It has been found that the thermal expansion coefficient α of the PbS nanofilm is almost twice as much as the coefficient α of coarse-grained lead sulfide. It has been shown that the large difference in the coefficients α is associated with the small size of particles in the film, which leads to an increase in the anharmonicity of atomic vibrations. The contribution from the small size of particles to the thermal expansion coefficient of the PbS nanofilm has been evaluated theoretically.     

Photonic spectral density of coupled optical cavities

We study a pair of anharmonic optical cavities that is connected by an optical fiber. The photonic spectral density characterizes the evolution of the coupled cavities after the system has been prepared in a Fock or N00N state. We evaluate the photonic spectral density within the recursive projection method and find that the anharmonicity leads to a collapse of the low-energy spectrum. The level spacing of the remaining spectrum agrees quite well with that of the harmonic cavities, whereas the spectral weights are strongly affected by the anharmonicity.     

Variational study of the ground state energy of theE−b 1,b 2 Jahn-Teller system

In this paper a variational method for the ground state energy approximation of theE−b ₁,b ₂ Jahn-Teller system is presented. This method is based on the choice of a suitable variational ground state wave function. This trial wave function — a correlated squeezed state — is used to account for the correlation and anharmonicity of the interaction between the two vibrational modes; the anharmonicity of both modes is taken into account by the squeeze effects of these modes. The ground state of mode 1 in this trial wave function is considered as a linear combination of the two displaced harmonic oscillators. The ground state energies for the linearE - e Jahn-Teller system calculated by this method are not only in good agreement with the exact diagonalization results, but they are also better than those from the previous analytical studies. Another conclusion which results from the presented model is the following one: the squeezing effect of mode 1 for the linearE - e Jahn-Teller system is substantially smaller, in contrast with the results which are presented in the previous analytical studies.     

Dynamics of non-linear optical processes

We study the statistical properties of parametric processes (parametric amplification, frequency conversion, splitting of frequency, etc.) described by the trilinear Hamiltonian and of the second harmonic generation including the lossy mechanism and rotating terms. In a short-time approximation some conservation laws are derived and the existence of the Glauber-Sudarshan weighting function is discussed. It is shown in the second harmonic generation case that good coherence of the incident radiation is lost proportionally to the intensity in interaction while the second harmonic radiation has tendency to be coherent. The effects of the spontaneous emission of the medium and of the physical vacuum are also discussed. In this approximation the statistics is generally described by the superposition of coherent and chaotic fields with correlated components.     

High-order optical harmonic generation from solid surfaces

During the interaction of an intense ultrashort laser pulse with solid targets, a thin layer of surface plasma is generated in which the density drops to the vacuum level in a distance much shorter than the wavelength. This sharp plasma-vacuum boundary performs an oscillatory motion in response to the electromagnetic forces of the intense laser light. It is shown that the generation of reflected harmonics can be interpreted as a phase modulation experienced by the light upon reflection from the oscillating boundary. The modulation side-bands of the reflected frequency spectrum correspond to odd and even harmonics of the laser frequency. Retardation effects lead to a strong anharmonicity for high velocities of the plasma-vacuum boundary. As a result, harmonic generation is strongly enhanced in the relativistic regime of laser intensities.Prof. F. P. Schäfer on the occasion of his 65th birthday.     

A generator co-ordinate approach to the breathing mode

In the harmonic approximation of the generator co-ordinate method (GCM) with the Skyrme interaction we show that the frequency of the breathing mode is essentially the same as the semiclassical result. The accuracy of the harmonic approximation is tested in ¹⁶O. Also, a comparison is made between the adiabatic time-dependent Hartree-Fock (ATDHF) theory and an anharmonic version of the GCM.     

用F-G-H方法研究玻色-爱因斯坦凝聚体基态性质

用F-G-H方法数值求解描述BEC凝聚体的非线性薛定谔方程-Gross-Pitaevskii方程.研究总粒子数、粒子间相互作用、谐振频率和一般幂指数外势对玻色凝聚体粒子数密度分布、基态能量的影响.结果表明,增大幂指数外势、谐振频率,降低粒子间的排斥作用会增加凝聚体中心的粒子数密度、缩小凝聚体半径;增大总粒子数、谐振频率、粒子间的排斥作用和幂指数外势的指数会增大体系的基态能量;随着总粒子数增大,数值结果与托马斯-费米近似结果渐趋一致,托马斯-费米近似在大粒子数条件下是一种较好的近似方法,在粒子数有限时,结果与真实情形偏差较大,应采用数值解法.