Quartic Oscillator Research Articles

Classical and quantum behavior of the harmonic and the quartic oscillators

In a previous paper a formalism to analyze the dynamical evolution of classical and quantum probability distributions in terms of their moments was presented. Here the application of this formalism to the system of a particle moving on a potential is considered in order to derive physical implications about the classical limit of a quantum system. The complete set of harmonic potentials is considered, which includes the particle under a uniform force, as well as the harmonic and the inverse harmonic oscillators. In addition, as an example of anharmonic system, the pure quartic oscillator is analyzed. Classical and quantum moments corresponding to stationary states of these systems are analytically obtained without solving any differential equation. Finally, dynamical states are also considered in order to study the differences between their classical and quantum evolution.

On the fly nodal searches in importance sampled fixed-node diffusion Monte Carlo using a parallel, fine-grained, genetic algorithm

A method for finding nodal surfaces on the fly in importance-sampled, fixed-node diffusion Monte Carlo calculations is described. The procedure relies on minimizing the difference between the nodal functions of the guiding wave function, ΨT, and HˆΨT, where Hˆ is the Hamiltonian. This is done by allowing the trial function to depend on a set of parameters whose values are then optimized using a parallel genetic algorithm (e.g., the Pikaia code developed in astrophysics). Application is made to the calculation of several excited states of a non-integrable two-dimensional quartic oscillator and to excited states of the He–C2H2 complex.

Bohmian trajectory from the "classical" Schrödinger equation.

The quantum-classical correspondence is studied for a periodically driven quartic oscillator exhibiting integrable and chaotic dynamics, by studying the Bohmian trajectory of the corresponding "classical" Schrödinger equation. Phase plots and the Kolmogorov-Sinai entropy are computed and compared with the classical trajectory as well as the Bohmian trajectory obtained from the time dependent Schrödinger equation. Bohmian mechanics at the classical limit appears to mimick the behavior of a dissipative dynamical system.

Photoionization of a hydrogen atom in a uniform electric field as visualization of the Stokes phenomenon

To describe the photoionization of a hydrogen atom in a static electric field, we considerably rely on the properties of a global asymptotic of the Stark wave functions in momentum representation. The accurate analytical photoionization cross section is found in terms of the Stokes multipliers for the quartic oscillator.

A simulated annealing based study of the effect of Gaussian perturbation in quartic oscillator and quadratic double well potentials

Perturbation theory based model can be used to locate the quasi-degeneracy in an arbitrary double well potential. In this work, unconstrained optimisation has been done using Simulated Annealing to calculate the energy spectrum of double well potential. Using this calculation the author has studied the effect of a Gaussian perturbation on single and double well potential. A comparative study of quartic double well potential and Gaussian double well potential has also been done on the basis of chemical and statistical point of view. The efficiency of this method is notable. Numerical calculation shows that the proposed method can give extremely accurate results for symmetric double well potentials. Repulsive Gaussian perturbation and attractive Quadratic Perturbation have been applied to a Quartic oscillator potential to compare the nature of double well. Both the perturbation leads to quasi degenerate potentials. Again, application of proper attractive Gaussian field removes normal quasi degeneracy but also generates accidental quasi degeneracy. Where lower state is an odd parity state and higher is an even parity state.

Nonlinearity-induced entanglement stability in a qubit-oscillator system

We propose a system composed of a qubit interacting with a quartic (undriven) nonlinear oscillator (NLO) through a conditional displacement Hamiltonian. We show that even a modest nonlinear perturbation in the NLO potential can enhance and stabilize the quantum entanglement dynamically generated between the qubit and the NLO. In absence of the nonlinearity the entanglement between the qubit and the oscillator is known to periodically oscillate between 0 and 1, whereas the nonlinearity suppresses the dynamical decay of entanglement once it is generated. While the generation of entanglement is due to the superposition principle combined with conditional displacements in the NLO, as noted in several works before, the improvements in this entanglement is because of the squeezing and other complex processes induced by two- and four-phonon interactions. Finally, we have solved the Markovian master equation, and even in this open case the system preserves the previous features and remain robust.

Confinement in 3D polynomial oscillators through a generalized pseudospectral method

Spherical confinement in three-dimensional (3D) harmonic, quartic and other higher oscillators of even order is studied. The generalized pseudospectral (GPS) method is employed for accurate solution of relevant Schrödinger equation in an optimum, non-uniform radial grid. Eigenvalues, eigenfunctions, position expectation values, radial densities in low- and high-lying states are presented in case of small, intermediate and large confinement radius. The degeneracy breaking in confined situation as well as correlation in its energy ordering with respect to the respective unconfined counterpart is discussed. For all instances, current results agree excellently with the best available literature results. Many new states are reported here for the first time. In essence, a simple, efficient method is provided for accurate solution of 3D polynomial potentials enclosed within the spherical impenetrable walls.

Hypoelliptic Heat Kernel Over 3-Step Nilpotent Lie Groups

In this paper, we provide explicitly the connection between the hypoelliptic heat kernel for some 3-step sub-Riemannian manifolds and the quartic oscillator. We study the left-invariant sub-Riemannian structure on two nilpotent Lie groups, namely, the (2,3,4) group (called the Engel group) and the (2,3,5) group (called the Cartan group or the generalized Dido problem). Our main technique is noncommutative Fourier analysis, which permits us to transform the hypoelliptic heat equation into a one-dimensional heat equation with a quartic potential.

Quartic oscillator potential in the γ-rigid regime of the collective geometrical model

A prolate γ-rigid version of the Bohr-Mottelson Hamiltonian with a quartic anharmonic oscillator potential in β collective shape variable is used to describe the spectra for a variety of vibrational-like nuclei. Speculating the exact separation between the two Euler angles and the β variable, one arrives at a differential Schrödinger equation with a quartic anharmonic oscillator potential and a centrifugal-like barrier. The corresponding eigenvalue is approximated by an analytical formula depending only on a single parameter up to an overall scaling factor. The applicability of the model is discussed in connection to the existence interval of the free parameter, which is limited by the accuracy of the approximation, and by comparison with the predictions of the related X(3) and X(3)-β 2 models. The model is applied to qualitatively describe the spectra for nine nuclei which exhibit near-vibrational features.

Variational truncated Wigner approximation.

In this paper we reconsider the notion of an optimal effective Hamiltonian for the semiclassical propagation of the Wigner distribution in phase space. An explicit expression for the optimal effective Hamiltonian is obtained in the short-time limit by minimizing the Hilbert-Schmidt distance between the semiclassical approximation and the real state of the system. The method is illustrated for the quartic oscillator.

Semiclassical initial-value representation of the transfer operator

The ability of semiclassical initial-value representation (IVR) methods to determine approximate energy levels for bound systems is limited due to problems associated with long classical trajectories. These difficulties become especially severe for large or classically chaotic systems. This work attempts to overcome such problems by developing an IVR expression that is classically equivalent to Bogomolny's formula for the transfer matrix [E. B. Bogomolny, Nonlinearity 5, 805 (1992); Chaos 2, 5 (1992)] and can be used to determine semiclassical energy levels. The method is adapted to levels associated with states of desired symmetries and applied to two two-dimensional quartic oscillator systems, one integrable and one mostly chaotic. For both cases, the technique is found to resolve all energy levels in the ranges investigated. The IVR method does not require a search for special trajectories obeying boundary conditions on the Poincaré surface of section and leads to more rapid convergence of Monte Carlo phase space integrations than a previously developed IVR technique. It is found that semiclassical energies can be extracted from the eigenvalues of transfer matrices of dimension close to the theoretical minimum determined by Bogomolny's theory. The results support the assertion that the present IVR theory provides a different semiclassical approximation to the transfer matrix than that of Bogomolny for ℏ≠0. For the chaotic system investigated the IVR energies are found to be generally more accurate than those predicted by Bogomolny's theory.

Stark problem in terms of the Stokes multipliers for the triconfluent Heun equation

The solution of the Stark problem is obtained in terms of the Stokes multipliers for the triconfluent Heun equation (the quartic oscillator equation). The Stokes multipliers are found in an analytical form at positive energies. For negative energies, the Stokes parameters are calculated in frames of a consistent asymptotic approach. The scattering phase, positions, and widths of the Stark resonances are determined as solutions of an implicit equation.

Classical limit of quantum mechanics induced by continuous measurements

We investigate the quantum–classical transition problem. The main issue addressed is how quantum mechanics can reproduce results provided by Newton’s laws of motion. We show that the measurement process is critical to resolve this issue. In the limit of continuous monitoring with minimal intervention the classical limit is reached. The Classical Limit of Quantum Mechanic, in Newtonian sense, is determined by two parameters: the semiclassical time (τsc) and the time interval between measurements (Δτu). If is Δτu small enough, comparing with the τsc, then the classical regime is achieved. The semiclassical time for Gaussian initial states coincides with the Ehrenfest time. We also show that the classical limit of an ensemble of Newtonian trajectories, the Liouville regime, is approximately obtained for the quartic oscillator model if the number of measurements in the time interval is large enough to destroy the revival and small enough to not reach the Newtonian regime. Namely, the Newtonian regime occurs when τsc≫Δτu and the Liouvillian regime is mimicked, for the position observable, if Δτu∈[τsc,TR], where TR is the revival time.

Quantum dynamics in the partial Wigner picture

Recently we have shown how the partial Wigner representation of quantum mechanics can be used to study hybrid quantum models where a system with a finite number of energy levels is coupled to linear or nonlinear oscillators (Beck and Sergi 2013 Phys. Lett. A 377 1047). The purpose of this work is to provide a detailed derivation of the partially Wigner-transformed quantum equations of motion for nonlinear oscillator subsystems under the action of general polynomial potentials. Such equations can be written in terms of a propagator, which can then be expanded in a power series. The linear terms of the series describe quantum-classical dynamics while the nonlinear terms provide the corrections needed to restore the fully quantum character of the evolution. In the case of polynomial potentials and position dependent couplings, the number of nonlinear terms is finite and the corrections can be calculated explicitly. In this work we show how to implement numerically the above scheme where, in principle, no assumption about the strength of the coupling must be taken. We illustrate the formalism by studying a two-level system interacting with an asymmetric quartic oscillator. We integrate the quantum dynamics of the total system and provide a comparison with the case of the quantum-classical dynamics of the quartic oscillator. The approach presented here is expected to be effective for studying hybrid quantum circuits in quantum information theory and for witnessing the quantum-to-classical transition in nano-oscillators coupled to pseudo-spins.

Improving the accuracy and efficiency of time-resolved electronic spectra calculations: Cellular dephasing representation with a prefactor

Time-resolved electronic spectra can be obtained as the Fourier transform of a special type of time correlation function known as fidelity amplitude, which, in turn, can be evaluated approximately and efficiently with the dephasing representation. Here we improve both the accuracy of this approximation-with an amplitude correction derived from the phase-space propagator-and its efficiency-with an improved cellular scheme employing inverse Weierstrass transform and optimal scaling of the cell size. We demonstrate the advantages of the new methodology by computing dispersed time-resolved stimulated emission spectra in the harmonic potential, pyrazine, and the NCO molecule. In contrast, we show that in strongly chaotic systems such as the quartic oscillator the original dephasing representation is more appropriate than either the cellular or prefactor-corrected methods.

Relation of exact Gaussian basis methods to the dephasing representation: Theory and application to time-resolved electronic spectra

We recently showed that the dephasing representation (DR) provides an efficient tool for computing ultrafast electronic spectra and that further acceleration is possible with cellularization [M. Šulc and J. Vaníček, Mol. Phys. 110, 945 (2012)]. Here, we focus on increasing the accuracy of this approximation by first implementing an exact Gaussian basis method, which benefits from the accuracy of quantum dynamics and efficiency of classical dynamics. Starting from this exact method, the DR is derived together with ten other methods for computing time-resolved spectra with intermediate accuracy and efficiency. These methods include the Gaussian DR, an exact generalization of the DR, in which trajectories are replaced by communicating frozen Gaussian basis functions evolving classically with an average Hamiltonian. The newly obtained methods are tested numerically on time correlation functions and time-resolved stimulated emission spectra in the harmonic potential, pyrazine S0∕S1 model, and quartic oscillator. Numerical results confirm that both the Gaussian basis method and the Gaussian DR increase the accuracy of the DR. Surprisingly, in chaotic systems the Gaussian DR can outperform the presumably more accurate Gaussian basis method, in which the two bases are evolved separately.

An asymptotic expansion for energy eigenvalues of anharmonic oscillators

In the present contribution, we derive an asymptotic expansion for the energy eigenvalues of anharmonic oscillators for potentials of the form V(x)=κx2q+ωx2,q=2,3,… as the energy level n approaches infinity. The asymptotic expansion is obtained using the WKB theory and series reversion. Furthermore, we construct an algorithm for computing the coefficients of the asymptotic expansion for quartic anharmonic oscillators, leading to an efficient and accurate computation of the energy values for n≥6.

Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with self-steepening and self-frequency shift effects

In this paper, we have obtained the optical solitary wave solutions for the nonlinear Schrödinger equation which describes the propagation of femtosecond light pulses in optical fibers in the presence of self-steepening and a self-frequency shift terms. The solitary wave ansatz method was used to carry out the derivations of the solitons. The parametric conditions for the formation of soliton pulses were determined. Using the 1-soliton solution, a number of conserved quantities have been calculated for Hirota and Sasa-Satsuma cases. We have also constructed some periodic wave solutions of the higher order nonlinear Schrödinger equation by reducing it to quartic anharmonic oscillator equation and by using projective Ricatti equations. Moreover by using He's semi-inverse method, variational formulation was established to obtain exact soliton solutions. The 1-soliton solutions of time dependent form of this equation was also obtained. To visualize the propagation characteristics of dark-bright soliton solutions, few numerical simulations are given.

Topology of classical molecular optimal control landscapes in phase space

Optimal control of molecular dynamics is commonly expressed from a quantum mechanical perspective. However, in most contexts the preponderance of molecular dynamics studies utilize classical mechanical models. This paper treats laser-driven optimal control of molecular dynamics in a classical framework. We consider the objective of steering a molecular system from an initial point in phase space to a target point, subject to the dynamic constraint of Hamilton's equations. The classical control landscape corresponding to this objective is a functional of the control field, and the topology of the landscape is analyzed through its gradient and Hessian with respect to the control. Under specific assumptions on the regularity of the control fields, the classical control landscape is found to be free of traps that could hinder reaching the objective. The Hessian associated with an optimal control field is shown to have finite rank, indicating the presence of an inherent degree of robustness to control noise. Extensive numerical simulations are performed to illustrate the theoretical principles on (a) a model diatomic molecule, (b) two coupled Morse oscillators, and (c) a chaotic system with a coupled quartic oscillator, confirming the absence of traps in the classical control landscape. We compare the classical formulation with the mathematically analogous quantum state-to-state transition probability control landscape.

Orthogonal polynomial projection quantization: a new Hill determinant method

Accurate energy eigenvalues are obtained by simply projecting the unknown bound state wave function on, essentially, arbitrary sets of orthogonal polynomials, and setting a subset of these to zero. The projection integrals are represented in terms of the power moments of the wave function, obtained recursively by transforming Schrödinger’s equation into a moment equation. Because unbounded wave functions do not have power moments, all solutions are guaranteed to be L2, resulting in a more robust, rapidly converging and stable method when compared with configuration space Hill determinant methods. More importantly, our approach permits the use of arbitrary, nonanalytic, positive reference functions, including those that manifest the true asymptotic behavior of the discrete states. These advantages are not usually possible with the standard Hill approach. The formulation presented here can be applied to any problem for which Schrödinger’s equation can be transformed into a moment equation. In this regard it is related to the L2 quantization prescription developed by Tymczak et al (1998 Phys. Rev. Lett. 80 3674; 1998 Phys. Rev. A 58 2708) corresponding to the Hill determinant method in momentum space, and to the eigenvalue moment method (Handy and Bessis 1985 Phys. Rev. Lett. 55 931; Handy et al 1988 Phys. Rev. Lett. 60 253), the first use of semidefinite programming related analysis in quantum physics. The latter method can be used to generate the ground state, whose orthogonal polynomials can serve to generate even more rapidly converging estimates for the excited discrete state energies. We demonstrate the power of this new approach on the sextic and quartic anharmonic oscillators, as well as on recently studied one- and two-dimensional pseudo-Hermitian Hamiltonians.