Anharmonic Research Articles

Coexistence and fluctuations phenomena with Davidson-like potentials in quadrupole–octupole deformed nuclei

The energy spectra of three types of two-dimensional potentials (we will call them ‘Davidson-like potentials’ (DLPs)), characterized by four minima separated by barriers, are investigated. The predictions for spectra and wave functions are obtained by using the nine-point finite-difference method. For these potentials, with the existence of a single configuration, a transition of spectra, as a function of barrier height, is covered from tunneling splitting modes to fluctuations phenomena, with equal peaks wave functions, crossing to the spectra of purely anharmonic oscillator potentials (AOPs). A different type of phase transition occurs when two (or more) configurations coexist. With the change of the parameters, a transition of spectra is covered from coexistence phenomena of two distinct quantum tunneling modes to fluctuations phenomena, with unequal peaks wave functions, crossing to the spectra of purely AOPs. Using DLPs, a particular application of the coherent quadrupole–octupole model to describe energy bands with alternating parity of the nuclei 100Mo, 146,148Nd, 148,150Sm, 220Ra, 220,222Rn, and 220,222Th is presented. The global parametrization of the model for the selected nuclei is achieved.

Analytic ground state energy and wave function of a quantum system in an exponential-form anharmonic interaction potential

An anharmonic oscillator with a perturbed quadratic potential which is coupled with an exponential term is being investigated in this paper by isolating an anharmonic oscillator interaction potential from the actual interaction experienced by the quantum system studied, and using standard perturbative techniques. The anharmonic potential considered is of interest because of its usefulness in the study of non-centrosymmetric materials which have applications in piezoelectricity. The ground state energy eigenvalue and its associated eigenstates were calculated for the quantum system using an analytical approach. Results obtained are compared to those of quantised harmonic oscillator to show the effect of the perturbation.

Time-dependent propagator for an-harmonic oscillator with quartic term in potential

In this work, we present the analytical approach for the evaluation of the conditional measure Wiener path integral. We consider the time-dependent model parameters. We find the differential equation for the variable, determining the behavior of the harmonic as well as the an-harmonic parts of the oscillator. We present the an-harmonic part of the result in the form of the operator function.

Superatomic Signature and Reactivity of Silver Clusters with Oxygen: Double Magic Ag 17 – with Geometric and Electronic Shell Closure

Understanding the stability and reactivity of silver clusters toward oxygen provides insights to design new materials of coinage metals with atomic precision. Herein, we report a systematic study o...

Role of phonon anharmonicity and electronic free energy on thermodynamic properties of fcc ytterbium

Although many thermophysical properties of lanthanide family members are similar, ytterbium is an exception. The coefficient of thermal expansion for Yb is almost three times as large as the coefficients for other rare-earth metals, a clear manifestation of anharmonicity. In the present study, therefore, the influence of the phonon anharmonicity and the electronic free energy on thermal expansion and other thermodynamic properties of ytterbium has been investigated from absolute zero to the melting temperature (T). We used first principles density functional perturbation theory combined with thermal perturbative treatment for including intrinsic anharmonicity. Modeling Yb as an anharmonic oscillator as suggested by Oganov and Dorogokupets [J. Phys. Condens. Matter 16, 1351 (2004)], an anharmonic vibrational contribution is included in a parametric way. It has allowed us to unveil the effect of anharmonicity connecting the low-temperature quantum correction up to the high-temperature classical value. Furthermore, due to the complex behavior of 4f-shell electrons, the electronic excitation was computed through the fixed electronic density-of-state approximation. It remains two orders of magnitude higher than the anharmonic lattice term. Combining these contributions, we can evaluate several T-dependent but zero-pressure thermodynamic properties of Yb in its fcc phase. Since, at lower pressure, anharmonicity increases with temperature, one of the objectives of this study is to examine the importance of anharmonicity in determining these properties. For instance, an excellent agreement is found for the linear thermal expansion for the entire temperature range, whereas other properties such as entropy, bulk moduli, thermal Grüneisen parameter, and the phonon frequency shifts are also in agreement with the reported findings; notably, the discrepancy observed in the enthalpy and specific heats at high-T is discussed. A detailed analysis has suggested that an additional contribution from point defects is needed, like vacancy formation, for an accurate calculation of specific heat, while higher-order terms in temperature-dependent perturbative series are mandatory for enthalpy. Electronic contribution remains positive for caloric properties. The theory of anharmonic phonon–phonon interaction and computed thermal expansion of the crystal was finally used to analyze renormalized phonon frequency. The significant objective of the study is to elucidate the role of electronic agitations and intrinsic phonon thermal stress as a physical mechanism over and above the dominating volume expansion effect, which is responsible for restricting an overwhelming quasi-harmonic thermal expansion and a rapid decrease in bulk moduli close to melting. We propose that a quantitative agreement for entropy and enthalpy requires a delicate balance between T2-dependent and higher-order terms in an anharmonic perturbation series.

A Model Effective Mass Quantum Anharmonic Oscillator and Its Thermodynamic Characterization

Using a model anharmonic oscillator with asymptotically decreasing effective mass to study the effect of compositional grading on the quantum mechanical properties of a semiconductor heterostructure, we determine the exact bound states and spectral values of the system. Furthermore, we show that ordering ambiguity only brings about a spectral shift on the quantum anharmonic oscillator with spatially varying effective mass. A study of thermodynamic properties of the system reveals a resonance condition dependent on the magnitude of the anharmonicity parameter. This resonance condition is seen to set a critical value on the said parameter beyond which a complex valued entropy which is discussed, emerges.

Quantum mechanical out-of-time-ordered-correlators for the anharmonic (quartic) oscillator

Out-of-time-ordered correlators (OTOCs) have been suggested as a means to study quantum chaotic behavior in various systems. In this work, I calculate OTOCs for the quantum mechanical anharmonic oscillator with quartic potential, which is classically integrable and has a Poisson-like energy-level distribution. For low temperature, OTOCs are periodic in time, similar to results for the harmonic oscillator and the particle in a box. For high temperature, OTOCs exhibit a rapid (but power-like) rise at early times, followed by saturation consistent with 2〈x2〉T〈p2〉T at late times. At high temperature, the spectral form factor decreases at early times, bounces back and then reaches a plateau with strong fluctuations.

Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity

We obtain explicit characterization of orbital and spectral stability of solitary wave solutions to the \begin{document}$ {\bf{U}}(1) $\end{document} -invariant 1D Klein–Gordon equation coupled to an anharmonic oscillator. We also give the complete analysis of the spectrum of the linearization at a solitary wave.

Energy Eigenvalues of Quantum Anharmonic Oscillators: Exact Expression for the Pure λx2m Oscillators and a Simple Expression for x2 + λx2m Oscillators

We propose an exact expression for the energy eigenvalues of the pure λx2m quantum anharmonic oscillators. The only information needed for the purpose is K0(m,n), which is the first term of strong-coupling expansion for oscillator with potential λx2m for the nth excited state. We then propose simple expression for the eigenvalues of the x2 + λx2m oscillators which reproduce both the ground state and excited states energies of these oscillators with a fairly good accuracy. The present formula is found to better reproduce the energy eigenvalues than those calculated by earlier authors.

Quantum classical correspondence for the statistical moments of eigenfunctions in closed systems with chaotic dynamics

Quantum classical correspondence, QCC, is a wide subject of interest from the historical approaches to current studies in simple and complex systems, including time dependent Hamiltonians. Quantum eigenfunctions, or wave-functions, are part of the quantities analysed in recent years, including the implementation due to M V Berry and A Voros in 1977 of a suitable quantity to be considered as the classical eigenfunction counterpart. Here we present the calculation of the statistical moments of eigenfunctions in energy representation for anharmonic oscillators. The classical counter part of the system is fully chaotic. The comparison of the quantum version in order to test for quantum classical correspondence is analysed using a slightly modified formula of the proposed by Berry and Voros. The agreement is good with the corresponding classical counterpart being at the bottom of the quantum version for all the energy range considered.

Generalized Evolution Equation of Wigner Function for an Arbitrary Linear Quantization

The arbitrary linear quantization is considered for definition of Wigner function. The evolution equation of this function is derived with the use of inversion of linear quantization kernel. In general case the Moyal equation depends on the characteristic function of quantization. It is shown that only for Weyl quantization this equation does not consist the source of quasi-probability. The stationary solutions of Moyal equation are constructed for an arbitrary linear quantization of a harmonic oscillator. The case of anharmonic oscillator is presented as a practical example of quantum system model in the so-called post-newtonian approximation. This model exhibits the intersection effect between quantization dependences of Hamiltonian and Wigner function.

Prediction of thermodynamic properties of formation of HCl at temperatures 200–1000 K

In this work, the thermodynamic properties of the formation reaction of hydrogen chloride from the reactants chlorine and hydrogen are studied by employing the developed Manning-Rosen potential. A closed-form expression for the vibration part of the molecular partition function is used to calculate the vibrational contribution of the thermodynamic properties. The results obtained are used to investigate the change in the thermodynamic functions for the formation reaction of HCl at a high temperature range. The predicted results are compared with the ones obtained from the other theoretical approaches based on the harmonic and an-harmonic oscillator models.

Nonadiabatic coupling of the dynamical structure to the superconductivity in YSr2Cu2.75Mo0.25O7.54 and Sr2CuO3.3

A crucial issue in cuprates is the extent and mechanism of the coupling of the lattice to the electrons and the superconductivity. Here we report Cu K edge extended X-ray absorption fine structure measurements elucidating the internal quantum tunneling polaron (iqtp) component of the dynamical structure in two heavily overdoped superconducting cuprate compounds, tetragonal YSr2Cu2.75Mo0.25O7.54 with superconducting critical temperature, Tc = 84 K and hole density p = 0.3 to 0.5 per planar Cu, and the tetragonal phase of Sr2CuO3.3 with Tc = 95 K and p = 0.6. In YSr2Cu2.75Mo0.25O7.54 changes in the Cu-apical O two-site distribution reflect a sequential renormalization of the double-well potential of this site beginning at Tc, with the energy difference between the two minima increasing by ∼6 meV between Tc and 52 K. Sr2CuO3.3 undergoes a radically larger transformation at Tc, >1-Å displacements of the apical O atoms. The principal feature of the dynamical structure underlying these transformations is the strongly anharmonic oscillation of the apical O atoms in a double-well potential that results in the observation of two distinct O sites whose Cu-O distances indicate different bonding modes and valence-charge distributions. The coupling of the superconductivity to the iqtp that originates in this nonadiabatic coupling between the electrons and lattice demonstrates an important role for the dynamical structure whereby pairing occurs even in a system where displacements of the atoms that are part of the transition are sufficiently large to alter the Fermi surface. The synchronization and dynamic coherence of the iqtps resulting from the strong interactions within a crystal would be expected to influence this process.

Generalized measure of quantum synchronization

We present a generalized information-theoretic measure of synchronization in quantum systems. This measure is applicable to dynamics of anharmonic oscillators, few-level atoms, and coupled oscillator networks. Furthermore, the new measure allows us to discuss synchronization of disparate physical systems such as coupled hybrid quantum systems and coupled systems undergoing mutual synchronization that are also driven locally. In many cases of interest, we find a closed-form expression for the proposed measure.

Evolution of the Bessel–Gaussian beam modeled by the fractional Schrödinger equation

We investigate the evolution of Bessel–Gaussian (BG) optical beams in using the fractional Schrödinger equation (FSE) without potential or with nonlocal nonlinear media, theoretically and numerically. We find that, as the propagation distance increases, the linear propagation dynamics of the 1D BG beams undergo an initial compression phase, after which each of the beams splits into two sub-beams; these sub-beams then separate from each other, forming a saddle shape as the propagation distance continues to increase; in addition, their interval also increases linearly with propagation distance. However, when the nonlocal nonlinear term is included in the FSE, 1D BG beams follow a zigzag trajectory in real space, which corresponds to a modulated anharmonic oscillation in momentum space. In the 2D case, the input chirped BG beam first evolves into a filament in real space and then into a ring structure; if the input is a superposed BG beam carrying orbital angular momentum, the rule fulfilled in evolution is similar to that for a single one, and it forms a funnel-like structure, with periodic inversion and variable rotation.

First-principles analysis of cross-resonance gate operation

We present a comprehensive theoretical study of the cross-resonance gate operation covering estimates for gate parameters and gate error as well as analyzing spectator qubits and multi-qubit frequency collisions. We start by revisiting the derivation of effective Hamiltonian models following Magesan et al. (arXiv:1804.04073). Transmon qubits are commonly modeled as a weakly anharmonic Kerr oscillator. Kerr theory only accounts for qubit frequency renormalization, while adopting number states as the eigenstates of the bare qubit Hamiltonian. Starting from the Josephson nonlinearity and by accounting for the eigenstates renormalization, due to counter-rotating terms, we derive a new starting model for the cross-resonance gate with modified qubit-qubit interaction and drive matrix elements. Employing time-dependent Schrieffer-Wolff perturbation theory, we derive an effective Hamiltonian for the cross-resonance gate with estimates for the gate parameters calculated up to the fourth order in drive amplitude. The new model with renormalized eigenstates lead to 10-15 percent relative correction of the effective gate parameters compared to Kerr theory. We find that gate operation is strongly dependent on the ratio of qubit-qubit detuning and anharmonicity. In particular, we characterize five distinct regions of operation, and propose candidate parameter choices for achieving high gate speed and low coherent gate error when the cross-resonance tone is equipped with an echo pulse sequence. Furthermore, we generalize our method to include a third spectator qubit and characterize possible detrimental multi-qubit frequency collisions.

Vibrational–translational relaxation in nitrogen discharge plasmas: Master equation modeling and Landau–Teller model revisited

Vibrational–translational (V–T) relaxation is quite common in molecular nitrogen discharge non-equilibrium plasmas. In this paper, the energy relaxation in V–T transition is investigated by master equation modeling on all vibrational levels below the dissociation limit. The state-to-state transition rates are calculated by a forced harmonic oscillator (HO)-free rotating model. Meanwhile, the classic Landau–Teller model based on the HO of vibrational levels is revisited. First, the V–T relaxation in a heat bath is compared between the HO model, Morse’s anharmonic oscillator (AHO) model, and realistic vibrational levels by a direct-potential-fit analysis of spectroscopic data. The relaxation of average vibrational energy using the AHO model is faster than that using the HO model. Then, the influence of more frequent vibrational–vibrational (V–V) collision on the V–T transition in the heat bath is investigated by using different numbers of vibrational levels. The anharmonic effect is significant with more vibrational levels. Finally, the V–T energy transfer is modeled by a coupled solution to master equations and gas heating. The stronger the non-equilibrium between vibrational and translational temperature in the beginning, the larger the difference that can be obtained between the HO model (Landau–Teller theory) and realistic vibrational levels.

Numerical Solutions of Quantum Mechanical Eigenvalue Problems

A large class of problems in quantum physics involve solution of the time independent Schrodinger equation in one or more space dimensions. These are boundary value problems, which in many cases only have solutions for specific (quantized) values of the total energy. In this article we describe a Python package that ``automagically'' transforms an analytically formulated Quantum Mechanical eigenvalue problem to a numerical form which can be handled by existing (or novel) numerical solvers. We illustrate some uses of this package. The problem is specified in terms of a small set of parameters and selectors (all provided with default values) that are easy to modify, and should be straightforward to interpret. From this the numerical details required by the solver is generated by the package, and the selected numerical solver is executed. In all cases the spatial continuum is replaced by a finite rectangular lattice. We compare common stensil discretizations of the Laplace operator with formulations involving Fast Fourier (and related trigonometric) Transforms. The numerical solutions are based on the NumPy and SciPy packages for Python 3, in particular routines from the \btt{scipy.linalg}, \btt{scipy.sparse.linalg}, and \btt{scipy.fftpack} libraries. These, like most Python resources, are freely available for Linux, MacOS, and MSWindows. We demonstrate that some interesting problems, like the lowest eigenvalues of anharmonic oscillators, can be solved quite accurately in up to 3 space dimensions on a modern laptop --- with some patience in the 3-dimensional case. We demonstrate that a reduction in the lattice distance, for a fixed the spatial volume, does not necessarily lead to more accurate results: A smaller lattice length increases the spectral width of the lattice Laplace operator, which in turn leads to an enhanced amplification of the numerical noise generated by round-off errors.

Link between volume, thermal expansion, and bulk metallic glass formability

The specific volumes and thermal expansion coefficients of 41 transition-metal based alloy liquids, which include both bulk and marginal glass-formers, are presented. Those parameters are compared with their values either in the corresponding crystal phases or in their constituent elemental liquids. The volume differences in both cases at the liquidus temperature, ${T}_{l}$, correlate well with the critical thicknesses, ${d}_{\mathrm{crit}}$ of the corresponding glass, taken either from literature or from estimates. The estimates are based on a recent study [Dai et al., J. Non-Cryst. Solids 525, 119673 (2019)], which requires knowledge of the liquid expansion coefficient and viscosity. The volume differences between the better glass-forming liquids and the crystal phases are smaller; they are larger when compared with the elemental liquids. The thermal expansion coefficients of the liquids correlate well with the cohesive energies and fragilities. Their differences with the crystal phases and estimates from elemental liquids correlate well with ${d}_{\mathrm{crit}}$. The differences are smaller for the former and larger for the latter for alloy liquids having larger values of ${d}_{\mathrm{crit}}$. This parameter then appears to be a prime indicator of glass formability. The results are explained in terms of changes in the anharmonicity and structural contributions to the expansion coefficients on alloying.

Critical potentials and fluctuations phenomena with quartic, sextic, and octic anharmonic oscillator potentials

This is an investigation of the energy levels of three classes of two-dimensional anharmonic oscillator potentials characterized by free-, one-, and two- parameters with quartic, sextic, and octic anharmonicity. The wave functions and energy eigenvalues are found through the discretization using the nine-point finite-difference method. It is proved that the free-parameter potentials provide a “bridge” between the two-dimensional harmonic oscillator potential and two-dimensional infinite square well potential. However, the one- and two- parameter potentials show a transition from tunneling splitting mode to the spectra of free-parameter potentials crossing to the spectrum of infinite square well potential and in the same time present new features. One can distinguish cases of fluctuations phenomena in the ground state and excited states. We obtained, for the first time, a ground state with a four-peak structure. The quantum tunneling between the four minima of one- and two- parameter potentials depends on the characteristics of the separating barrier. These phase transitions are not of the ordinary thermodynamic category, but truly they are phase transitions in the structure of the low-lying energy spectra. The concept of critical potentials is associated with the potentials at the phase-transition points. The coherent quadrupole-octupole motion model, using these potentials, is applied to describe yrast and non-yrast energy sequences with alternating parity in several even-even nuclei from different regions, namely, in 100Mo, 146Ba, 226Ra, and 226Th.