Anharmonic Potential Well Research Articles

Mixing in anharmonic potential well

We prove phase-space mixing for solutions to Liouville’s equation for integrable systems. Under a natural non-harmonicity condition, we obtain weak convergence of the distribution function with rate ⟨time⟩−1. In one dimension, we also study the case where this condition fails at a certain energy, showing that mixing still holds but with a slower rate. When the condition holds and functions have higher regularity, the rate can be faster.

Long time oscillation of solutions of nonlinear Schrödinger equations near minimal mass ground state

In this paper, we consider the long time dynamics of radially symmetric solutions of nonlinear Schrödinger equations (NLS) having a minimal mass ground state. In particular, we show that there exist solutions with initial data near the minimal mass ground state that oscillate for long time. More precisely, we introduce a coordinate defined near the minimal mass ground state which consists of finite and infinite dimensional part associated to the discrete and continuous part of the linearized operator. Then, we show that the finite dimensional (2–D) part approximately obeys Newton's equation of motion for a particle in an anharmonic potential well. Showing that the infinite dimensional part is well separated from the finite dimensional part, we will have long time oscillation.

Rattling-Induced Ultralow Thermal Conductivity Leading to Exceptional Thermoelectric Performance in AgIn5S8.

Rattling has emerged as one of the most significant phenomenon for notably reducing the thermal conductivity in complex crystal systems. In this work, using first-principles density functional theory, we found that rattlers can be hosted in simpler crystal systems such as AgIn5S8 and CuIn5S8. Rattlers Ag and Cu exhibit weak and anisotropic bonding with the neighboring In and S and reside in a very shallow anharmonic potential well. The phonon spectra of these compounds have multiple avoided crossing of optical and acoustic modes, which are a signature of rattling motion. This leads to ultralow thermal conductivity, which is inversely proportional to mass and frequency span of rattling modes. Even though Ag atoms contribute to the valence band states, the rattler modes of Ag do not scatter carriers significantly, leaving the electronic transport virtually unaffected. Moreover, AgIn5S8 possesses a combination of heavy and light valence bands resulting in a very high power factor. A combination of favorable thermal and electronic transport results in a very high figure of merit of 2.2 in p-doped AgIn5S8 at 1000 K. The proposed idea of having rattlers in simpler systems can be extended to a wider class of materials, which would accelerate the development of thermoelectric modules for waste energy harvesting.

Structure Change and Rattling Dynamics in Cu12Sb4S13 Tetrahedrite: an NMR Study.

We present a 63Cu and 65Cu NMR study of Cu12Sb4S13, the basis for tetrahedrite thermoelectric materials. In addition to electronic changes observed at the Tc = 88 K metal-insulator transition, we find that locally there are significant structural changes occurring as the temperature extends above Tc, which we associate with Cu atom displacements away from symmetry positions. Spin-lattice relaxation rates (1/ T1) are dominated by a quadrupolar process indicating anharmonic vibrational dynamics both above and below Tc. We used a quasiharmonic approximation for localized anharmonic oscillators to analyze the impact of Cu rattling. The results demonstrate that Cu-atom rattling dynamics extends unimpeded in the distorted structural configuration below Tc and provide a direct measure of the anharmonic potential well.

Effective quantum dynamics in a weakly anharmonic interaction in the vicinity of a focusing point

We describe here the dynamic behaviors of a quantum ensemble of particles in a one-dimensional anharmonic potential well and of the corresponding classical ensemble in the vicinity of a focusing point. The anharmonicity of the interaction is chosen to be very small and small. Accordingly, the perturbation theory is used, and the accurate analytical solutions of the problem in both the quantum and classical cases are obtained. The dynamics of the quantum ensemble is considered through a detailed morphological analysis of the families of amplitudes squared and phases of the particle wave functions. We explore the dependence of the quantum solution of the problem on the anharmonicity and its relation to the classical solution. In addition, a solution of the problem based on catastrophe theory is given and compared with the quantum solution. It has been proven that an effective quantum dynamics of the ensemble can be clearly observed. In addition, we have concluded that, when the anharmonicity increases, the difference between the effective quantum behavior and classical behavior becomes larger.

Microscopic optical buffering in a harmonic potential.

In the early days of quantum mechanics, Schrödinger noticed that oscillations of a wave packet in a one-dimensional harmonic potential well are periodic and, in contrast to those in anharmonic potential wells, do not experience distortion over time. This original idea did not find applications up to now since an exact one-dimensional harmonic resonator does not exist in nature and has not been created artificially. However, an optical pulse propagating in a bottle microresonator (a dielectric cylinder with a nanoscale-high bump of the effective radius) can exactly imitate a quantum wave packet in the harmonic potential. Here, we propose a tuneable microresonator that can trap an optical pulse completely, hold it as long as the material losses permit, and release it without distortion. This result suggests the solution of the long standing problem of creating a microscopic optical buffer, the key element of the future optical signal processing devices.

Controlling trapping potentials and stray electric fields in a microfabricated ion trap through design and compensation

Recent advances in quantum information processing with trapped ions have demonstrated the need for new ion trap architectures capable of holding and manipulating chains of many (>10) ions. Here we present the design and detailed characterization of a new linear trap, microfabricated with scalable complementary metal-oxide-semiconductor (CMOS) techniques, that is well-suited to this challenge. Forty-four individually controlled dc electrodes provide the many degrees of freedom required to construct anharmonic potential wells, shuttle ions, merge and split ion chains, precisely tune secular mode frequencies, and adjust the orientation of trap axes. Microfabricated capacitors on dc electrodes suppress radio-frequency pickup and excess micromotion, while a top-level ground layer simplifies modeling of electric fields and protects trap structures underneath. A localized aperture in the substrate provides access to the trapping region from an oven below, permitting deterministic loading of particular isotopic/elemental sequences via species-selective photoionization. The shapes of the aperture and radio-frequency electrodes are optimized to minimize perturbation of the trapping pseudopotential. Laboratory experiments verify simulated potentials and characterize trapping lifetimes, stray electric fields, and ion heating rates, while measurement and cancellation of spatially-varying stray electric fields permits the formation of nearly-equally spaced ion chains.

Trajectory compensation in an autoresonant trap mass spectrometer

The auto-resonant trap mass spectrometer, ART-MS, utilizes electrostatic ion trapping within an anharmonic potential well. Ions are detected after mass selective trap ejection with auto-resonant driving employing only low-power rf electronics. We identify the major limiting factor in the mass resolution of these instruments. Whilst keeping in the spirit of maintaining a rapid scan rate, low cost, lightweight instrument, with minimal required machining tolerances, we introduce a method for much improving the mass resolutions of an ART-MS. The addition of two electrodes has enabled an improvement in the mass resolution by a factor of ∼4. The scheme significantly reduces the effects of a finite sized trap and compensates for the influence of radial variation in natural oscillating frequencies within the trap. Compensation can be implemented with a wide range of designs and is not limited by the size of the trap.

Nonlinear dynamics and quasibound state formation in a driven Gaussian potential

We examine the classical and quantum dynamics of an electron in an anharmonic atomic potential well in the presence of monochromatic laser radiation. The atomic potential is modeled by the inverted Gaussian potential. The radiation induces nonlinear resonances and chaos in the electron phase space. As the laser amplitude increases, regions of chaos grow and a pitchfork bifurcation of a key electron periodic orbit occurs leading to the appearance of three dominant periodic orbits which move apart with increasing laser amplitude. Electron quasibound states in the quantum system are found using complex coordinate scaling and Floquet theory. We show that the bifurcation of periodic orbits in the classical phase space is a precursor to the creation of a new quasibound state in the quantum system. As we increase the laser amplitude, the residues of the quasibound states become contracted into regions near the fixed points of the bifurcation and cause stabilization of the electron.

Investigation of resonant and transient phenomena in Josephson junction flux qubits

We present an analytical and computational study of resonances and transient responses in a classical Josephson junction system. A theoretical basis for resonances in a superconducting loop with three junctions is presented, outlining both the direct relationship between the dynamics of single- and multijunction systems and the direct relationships between observations of the classical counterparts to Rabi oscillations, Ramsey fringes, and spin-echo oscillations in this class of systems. We show simulation data along with analytical analyses of the classical model, and the results are related to previously reported experiments conducted on three junction loops. We further investigate the effect of off-resonant microwave perturbations to, e.g., the Rabi-type response of the Josephson system, and we relate this response back to the nonlinear and multivalued resonance behavior previously reported for a single Josephson junction. The close relationships between single and multijunction behaviors demonstrate the underlying dynamical mechanism for a whole class of classical counterparts to expected quantum-mechanical observations in a variety of systems, namely, the resonant and transient behavior of a particle in an anharmonic potential well with subsequent escape.

Simulating Ramsey-Type Fringes in a Pulsed Microwave-Driven Classical Josephson Junction

We present evidence for a close analogy between the nonlinear behaviour of a pulsed microwave-driven Josephson junction at low temperature and the experimentally observed behaviour of Josephson systems operated below the quantum transition temperature under similar conditions. We specifically address observations of Ramsey-type fringe oscillations, which can be understood in classical nonlinear dynamics as results of slow transient oscillations in a pulsed microwave environment. Simulations are conducted to mimic experimental measurements by recording the statistics of microwave-induced escape events from the anharmonic potential well of a zero-voltage state. Observations consistent with experimentally obtained Ramsey-type oscillations are found in the classical model.

Formula omitted]: Superconducting rattler

Electronic structure calculations and experimental data point to soft local vibrations of alkali ions in AOs 2 O 6 (A=K, Rb, Cs). Due to its size the potassium ion experiences an extremely anharmonic and broad potential well. We use ab initio energies and forces to derive the Hamiltonian for the dynamics of K ions and solve the single site problem.

Raman scattering from the peroxide ion in Cs2O2

AbstractThe Raman spectrum of Cs2O2 excited at 632.8 nm shows a single O2−2 fundamental band at wavenumber of 742.5 cm−1 and second and third overtones at 1468.5 and 2176.8 cm−1, respectively. The harmonic wavenumber ωe = 759.5 ± 0.7 cm−1 and anharmonicity constant ωexe = 8.4 ± 0.3 cm−1 were calculated directly from the measurements. These results are compared with ωe = 1151.0 ± 1.5 cm−1 and ωexe = 9.6 ± 0.5 cm−1 recently found for O2− ion in CsO2 by others. The lower ωexe of O2−2 relative to that of O2− indicates a broader (smaller curvature) anharmonic oscillator potential well. Copyright © 2002 John Wiley & Sons, Ltd.

Relaxation of classical particles in two-dimensional anharmonic single-well potentials.

The canonical ensemble relaxation function of a particle in a symmetric anharmonic potential well in D=1 is known to exhibit slow algebraic behavior [S. Sen, R. S. Sinkovits and S. Chakravarti, Phys. Rev. Lett. 77, 4855 (1996); R. S. Sinkovits, S. Sen, J. C. Phillips, and S. Chakravarti, Phys. Rev. E 59, 6497 (1999)]. In the present work, we report a study of relaxation of a particle in symmetric and asymmetric quartic anharmonic potential wells of the form V(x,y)=1 / 2 (x(2)+Cy2)+1 / 4 (x(2)+Cy2)(2) in D=2. The relaxation in the above system is identical to that in D=1 wells when C=0 (since it is then a D=1 system) and C=1. However, for 0<C<1 and for C>>1, the frequencies associated with well dynamics are strongly affected and hence the power spectra are altered as a function of C. Our calculations suggest that the exponents of the long-time tails associated with the relaxation processes are insensitive to D. In closing, we comment on the consequences of our analysis for the study of slow dynamics in interacting many-particle systems that are connected by harmonic springs with the individual particles in anharmonic potential wells.

Computer Simulation Study of tert-Butyl Alcohol. 1. Structure in the Pure Liquid

MD simulations of neat liquid tert-butyl alcohol (TBA) are carried out. Two very different potential models have been constructed for TBA: a simple rigid three-site model and a fully flexible all-atom 15-site model, with the O−H stretching described with an anharmonic potential well. Pure liquid TBA is investigated to evaluate the two potential models and to study the liquid structure in more detail. An analysis based upon both radial distribution functions and spatial distribution functions is employed. Generally it is found that the 15-site model is superior, providing reasonable agreement with experimentally derived data, while the three-site model is only able to reproduce qualitatively the competitive hydrophobic and hydrophilic aspects of the intermolecular interactions. Overall the structures observed reflect a frustrated hydrogen-bonded system.

Feynman path centroid dynamics for Fermi–Dirac statistics

In this communication the centroid molecular dynamics (CMD) method is tested for systems of fermions using a modified version of the expression for the fermion Feynman path centroid density derived in a recent publication [J. Chem. Phys. 110, 3647 (1999)]. The method is applied to a model system consisting of two noninteracting fermions in an anharmonic potential well and the position autocorrelation function is computed. The exact fermion position autocorrelation function exhibits a longer period of oscillation when compared to the bosonic case. This shift is due to the anharmonic nature of the given potential function where the level spacing decreases with the increase of energy. The CMD results agree very well with the exact quantum mechanical results for this example. This indicates, along with the previous results on a boson system, that quantum statistics can in principle be incorporated into the CMD method without any degradation of the accuracy of the approximation.

Electron-phonon coupling and properties of dopedBaBiO3

We report density-functional calculations based on the local-density approximation (LDA) of the properties of doped barium bismuthates. Using the linear-response approach developed in the framework of the linear muffin-tin-orbital method the phonon spectrum of the ${\mathrm{Ba}}_{0.6}{\mathrm{K}}_{0.4}{\mathrm{BiO}}_{3}$ system is calculated and is compared with the results of the neutron-diffraction measurements. The effect of doping in the calculation is modeled by the virtual crystal and mass approximations. The electron-phonon coupling constant $\ensuremath{\lambda}$ is then evaluated for a grid of phonon wave vectors using the change in the potential due to phonon distortion found self-consistently. A large coupling of the electrons to the bond-stretching oxygen vibrations and especially to the breathinglike vibrations is established. Also, a strongly anharmonic potential well is found for the tiltinglike motions of the oxygen octahedra. This mode is not coupled to the electrons to linear order in the displacements; therefore an anharmonic contribution to $\ensuremath{\lambda}$ is estimated using the frozen-phonon method. Our total (harmonic plus anharmonic) $\ensuremath{\lambda}$ is found to be 0.34. This is too small to explain high-temperature superconductivity in ${\mathrm{Ba}}_{0.6}{\mathrm{K}}_{0.4}{\mathrm{BiO}}_{3}$ within the standard mechanism. Finally, based on standard LDA and LDA+$U$ like calculations, a number of properties of pure ${\mathrm{BaBiO}}_{3}$ such as tilting of the octahedra, breathing distortion, charge disproportionation, and semiconducting energy gap value is evaluated and discussed in connection with the negative-$U$ extended Hubbard model frequently applied to this compound.

Interstitial oxygen in germanium and silicon

The microscopic structure of interstitial oxygen in germanium and its associated dynamics are studied both experimentally and theoretically. The infrared absorption spectrum is calculated with a dynamical matrix model based on first-principles total-energy calculations describing the potential energy for the nuclear motions. Spectral features and isotope shifts are calculated and compared with available experimental results. From new spectroscopic data on natural and on quasimonoisotopic germanium samples, new isotope shifts have been obtained and compared with the theoretical predictions. The low-energy spectrum is analyzed in terms of a hindered rotor model. A fair understanding of the center is achieved, which is then compared with interstitial oxygen in silicon. The oxygen atom is nontrivially quantum delocalized both in silicon and in germanium, but the physics is shown to be very different: while the Si-O-Si quasimolecule is essentially linear, the Ge-O-Ge structure is puckered. The delocalization in a highly anharmonic potential well of oxygen in silicon is addressed using path-integral Monte Carlo simulations, for comparison with the oxygen rotation in germanium. The understanding achieved with this new information allows us to explain the striking differences between both systems, in both the infrared and the far-infrared spectral regions, and the prediction of the existencemore » of hidden vibrational modes, never directly observed experimentally, but soundly supported by the isotope-shift analysis. {copyright} {ital 1997} {ital The American Physical Society}« less

Optical response of conjugated polymers.

We present a simple model for the description of the linear and nonlinear optical response of conjugated polymers, where the \ensuremath{\pi} electrons are confined in a periodic system composed of anharmonic potential wells. We calculate the optical band gap, the polarizability, and the second hyperpolarizability as a function of the molecular length for the oligomers of thiophene and benzene, adjusting the model parameters to two band-gap measurements. We find them to be in fair agreement with experimental data. Furthermore, the length dependencies are discussed and compared with experimental and theoretical reported values. The model is very suitable for studying the sensitivity of the optical response to conformational disorder.

Energy diffusion of a weakly damped and periodically driven particle in an anharmonic potential well.

The dynamics of a particle in an anharmonic potential well is studied for the case where the oscillations of the particle are disturbed by the dissipative and random forces caused by a thermal bath and by an external driving force. The former interaction is characterized by a damping coefficient with arbitrary frequency dependence. The damping is assumed to be weak so that the average energy loss of the particle per oscillation period is much less than the energy. Also the energy change per oscillation period due to the external force is assumed to be small compared to the energy, at least for large-amplitude oscillations. Under these weak-coupling conditions the non-Markov process of the position and momentum variables of the particle can be reduced to a Markov process for the energy and phase variables. The Markov process is described by a Fokker-Planck equation, the drift and diffusion coefficients of which are calculated systematically.