Harmonic Potential Research Articles

Shannon entropy measurements for quantum oscillator system in the presence of a spiral dislocation

Shannon entropy measurements for quantum oscillator system in the presence of a spiral dislocation

Absorption and fluorescence spectroscopy of cold proflavine ions isolated in the gas phase.

Proflavine, a fluorescent cationic dye with strong absorption in the visible, has been proposed as a potential contributor to diffuse interstellar bands (DIBs). To investigate this hypothesis, it is essential to examine the spectra of cold and isolated ions for comparison. Here, we report absorption spectra of proflavine ions, trapped in a liquid-nitrogen-cooled ion trap filled with helium-buffer gas, as well as fluorescence spectra to provide further information on the intrinsic photophysics. We find absorption- and fluorescence-band maxima at 434.2 ± 0.1 and 434.7 ± 0.3nm, corresponding to a Stokes shift of maximum 48cm-1, which indicates minor differences between ground-state and excited-state geometries. Based on time-dependent density functional theory, we assign the emitting state to S2 as its geometry closely resembles that of S0, whereas the S1 geometry differs from that of S0. As a result, simulated spectra involving S1 exhibit long Franck-Condon progressions, absent in the experimental spectra. The latter displays well-resolved vibrational features, assigned to transitions involving in-plane vibrational modes where the vibrational quantum number changes by one. Dominant transitions are associated with vibrations localized on the NH2 moieties. Experiments repeated at room temperature yield broader spectra with maxima at 435.5 ± 1nm (absorption) and 438.0 ± 1nm (fluorescence). We again conclude that prevalent fluorescence arises from S2, i.e., anti-Kasha behavior, in agreement with previous work. Excited-state lifetimes are 5 ± 1ns, independent of temperature. Importantly, we exclude the possibility that a narrow DIB at 436.4nm originates from cold proflavine cations as the band is redshifted compared to our absorption spectra.

Weak-valued correlation functions: Insights and precise readout strategies

The correlation function in quantum systems plays a vital role in decoding their properties and gaining insights into physical phenomena. Its interpretation corresponds to the propagation of particle excitations between space-time, similar in spirit to the idea of quantum weak measurement in terms of recording the system information by interaction. By defining weak-valued correlation function, we propose the basic insights and the universal methods for recording them on the apparatus through weak measurement. To demonstrate the feasibility of our approach, we perform numerical experiments of perturbed quantum harmonic oscillators, addressing the intricate interplay between the coupling strength and the number of ensemble copies. Additionally, we extend our protocol to the domain of quantum field theory, where joint weak values encode crucial information about the correlation function. Hopefully, this comprehensive investigation can advance our understanding of the fundamental nature of the correlation function and weak measurement in quantum theories. Published by the American Physical Society 2024

ℏ 4 quantum corrections to semiclassical transmission probabilities

The combination of vibrational perturbation theory with the replacement of the harmonic oscillator quantization condition along the reaction coordinate with an imaginary action to be used in the uniform semiclassical approximation for the transmission probability has been shown in recent years to be a practical method for obtaining thermal reaction rates. To date, this theory has been developed systematically only up to second order in perturbation theory. Although it gives the correct leading order term in an ℏ2 expansion, its accuracy at lower temperatures, where tunneling becomes important, is not clear. In this paper, we develop the theory to fourth order in the action. This demands developing the quantum perturbation theory up to sixth order. Remarkably, we find that the fourth order theory gives the correct ℏ4 term in the expansion of the exact thermal rate. The relative magnitude of the fourth order correction as compared to the second order term objectively indicates the accuracy of the second order theory. We also extend the previous modified second order theory to the fourth order case, creating an ℏ2 modified potential for this purpose. The resulting theory is tested on the standard examples—symmetric and asymmetric Eckart potentials and a Gaussian potential. The modified fourth order theory is remarkably accurate for the asymmetric Eckart potential.

An approach to calculate Franck-Condon factors involving anharmonic potentials using harmonic oscillator bases

Abstract In this contribution an approach to calculate Franck-Condon factors (FCFs) is presented. The method is based on the knowledge of the quantum harmonic oscillator system, a familiar system included in any course of quantum mechanics. The advantage of this approach is that it is possible to calculate Franck-Condon factors associated with general potentials based on the knowledge of the harmonic oscillator theory. First the coordinate and momentum representations are introduced. Based on this concept, the discrete variable representation (DVR) is proposed, which allows the Franck-Condon factors to be calculated in a simple form. This method takes advantage of the momentum as generator of translations to include the displacement of the potentials. The calculation of the FCFs is exemplified for transitions involving the ground state X 1Σ+ g and the excited state a 1Πg of the diatomic molecule 14N2.

The spectra masses for heavy pentaquark using generalized fractional of the extended Nikiforov-Uvaro method

The Nikiforov-Uvarov method is an efficient technique for solving of heavy diquark systems. It has been used to derive analytic-exact energy eigenvalues and eigenfunctions in fractional forms, which are useful in describing such systems. The potentials employed include the Cornell potential, harmonic potential, and spin-spin interaction; have been updated with respect to previous studies. Mass spectra of heavy pentaquarks were also calculated using this method. Compared to previous studies, the present results exhibit good experimental data agreement and are improved. We deduce that the fractional models contribute greatly to the heavy pentaquark masses.

Superposition and interference states in q-deformed quantum oscillator

Superposition and interference states in q-deformed quantum oscillator

Intrinsic Information-Theoretic Models

With this follow-up paper, we continue developing a mathematical framework based on information geometry for representing physical objects. The long-term goal is to lay down informational foundations for physics, especially quantum physics. We assume that we can now model information sources as univariate normal probability distributions N (μ, σ0), as before, but with a constant σ0 not necessarily equal to 1. Then, we also relaxed the independence condition when modeling m sources of information. Now, we model m sources with a multivariate normal probability distribution Nm(μ,Σ0) with a constant variance–covariance matrix Σ0 not necessarily diagonal, i.e., with covariance values different to 0, which leads to the concept of modes rather than sources. Invoking Schrödinger’s equation, we can still break the information into m quantum harmonic oscillators, one for each mode, and with energy levels independent of the values of σ0, altogether leading to the concept of “intrinsic”. Similarly, as in our previous work with the estimator’s variance, we found that the expectation of the quadratic Mahalanobis distance to the sample mean equals the energy levels of the quantum harmonic oscillator, being the minimum quadratic Mahalanobis distance at the minimum energy level of the oscillator and reaching the “intrinsic” Cramér–Rao lower bound at the lowest energy level. Also, we demonstrate that the global probability density function of the collective mode of a set of m quantum harmonic oscillators at the lowest energy level still equals the posterior probability distribution calculated using Bayes’ theorem from the sources of information for all data values, taking as a prior the Riemannian volume of the informative metric. While these new assumptions certainly add complexity to the mathematical framework, the results proven are invariant under transformations, leading to the concept of “intrinsic” information-theoretic models, which are essential for developing physics.

Dynamics of Cosmological Scalar Fields Revisited

This paper reviews the dynamics of a single isotropic and homogeneous scalar field φ(t) in the context of cosmological models. A non-standard approach to the solution of the Einstein–Klein–Gordon equations is described which uses the scalar field as the evolution parameter for cosmic dynamics. General conclusions about the qualitative behaviour of the solutions can be drawn, and examples of how to obtain explicit solutions for some cosmological models of interest are given. For arbitrary potentials, analytical results can be obtained from the slow-roll approximation by using a series expansion for the Hubble parameter H[φ], from which a quantitative estimate for the number of e-folds of expansion is obtained. This approach is illustrated with the examples of quadratic potentials and hilltop models, with special consideration of Higgs-type potentials. The GUT-scale is shown to come out of such a model quite naturally. Finally, it is discussed how to find scalar potentials giving rise to a predetermined scalar-field behaviour and the associated evolution of the scale factor.

Exponential Stability and Hypoelliptic Regularization for the Kinetic Fokker–Planck Equation with Confining Potential

This paper is concerned with a modified entropy method to establish the large-time convergence towards the (unique) steady state, for kinetic Fokker–Planck equations with non-quadratic confinement potentials in whole space. We extend previous approaches by analyzing Lyapunov functionals with non-constant weight matrices in the dissipation functional (a generalized Fisher information). We establish exponential convergence in a weighted H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}-norm with rates that become sharp in the case of quadratic potentials. In the defective case for quadratic potentials, i.e. when the drift matrix has non-trivial Jordan blocks, the weighted L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-distance between a Fokker–Planck-solution and the steady state has always a sharp decay estimate of the order O((1+t)e-tν/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal O\\big ( (1+t)e^{-t\ u /2}\\big )$$\\end{document}, with ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} the friction parameter. The presented method also gives new hypoelliptic regularization results for kinetic Fokker–Planck equations (from a weighted L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-space to a weighted H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}-space).

On Solving the Quantum Oscillator Equation by Using the Differential Transformation

On Solving the Quantum Oscillator Equation by Using the Differential Transformation

Computational Issues of Quantum Heat Engines with Non-Harmonic Working Medium

In this work, we lay the foundations for computing the behavior of a quantum heat engine whose working medium consists of an ensemble of non-harmonic quantum oscillators. In order to enable this analysis, we develop a method based on the Schrödinger picture. We investigate different possible choices on the basis of expanding the density operator, as it is crucial to select a basis that will expedite the numerical integration of the time-evolution equation without compromising the accuracy of the computed results. For this purpose, we developed an estimation technique that allows us to quantify the error that is unavoidably introduced when time-evolving the density matrix expansion over a finite-dimensional basis. Using this and other ways of evaluating a specific choice of basis, we arrive at the conclusion that the basis of eigenstates of a harmonic Hamiltonian leads to the best computational performance. Additionally, we present a method to quantify and reduce the error that is introduced when extracting relevant physical information about the ensemble of oscillators. The techniques presented here are specific to quantum heat cycles; the coexistence within a cycle of time-dependent Hamiltonian and coupling with a thermal reservoir are particularly complex to handle for the non-harmonic case. The present investigation is paving the way for numerical analysis of non-harmonic quantum heat machines.

Dissipation-induced collective advantage of a quantum thermal machine

Do quantum correlations lead to better performance with respect to several different systems working independently? For quantum thermal machines, the question is whether a working medium (WM) made of N constituents exhibits better performance than N independent engines working in parallel. Here, by inspecting a microscopic model with the WM composed by two non-interacting quantum harmonic oscillators, we show that the presence of a common environment can mediate non-trivial correlations in the WM leading to better quantum heat engine performance—maximum power and efficiency—with respect to an independent configuration. Furthermore, this advantage is striking for strong dissipation, a regime in which two independent engines cannot deliver any useful power. Our results show that dissipation can be exploited as a useful resource for quantum thermal engines and are then corroborated by optimization techniques here extended to non-Markovian quantum heat engines.

Fortuitous Correlations in Molecular Dynamics Simulations: Their Harmful Influence on the Probability Distributions of the Main Principal Components.

Nonsense correlations frequently develop between independent random variables that evolve with time. Therefore, it is not surprising that they appear between the components of vectors carrying out multidimensional random walks, such as those describing the trajectories of biomolecules in molecular dynamics simulations. The existence of these correlations does not imply in itself a problem. Still, it can present a problem when the trajectories are analyzed with an algorithm such as the Principal Component Analysis (PCA) because it seeks to maximize correlations without discriminating whether they have physical origin or not. In this Article, we employ random walks occurring on multidimensional harmonic potentials to evaluate the influence of fortuitous correlations in PCA. We demonstrate that, because of them, this algorithm affords misleading results when applied to a single trajectory. The errors do not only affect the directions of the first eigenvectors and their eigenvalues, but the very definition of the molecule's "essential space" may be wrong. Additionally, the main principal component's probability distributions present artificial structures which do not correspond with the shape of the potential energy surface. Finally, we show that the PCA of two realistic protein models, human serum albumin and lysozyme, behave similarly to the simple harmonic models. In all cases, the problems can be mitigated and eventually eliminated by doing PCA on concatenated trajectories formed from a large enough number of individual simulations.

Vibrational ADAPT-VQE: Critical points lead to problematic convergence.

Quantum chemistry is one of the most promising applications for which quantum computing is expected to have a significant impact. Despite considerable research in the field of electronic structure, calculating the vibrational properties of molecules on quantum computers remains a relatively unexplored field. In this work, we develop a vibrational Adaptive Derivative-Assembled Pseudo-Trotter Variational Quantum Eigensolver (vADAPT-VQE) formalism based on an infinite product representation (IPR) of anti-Hermitian excitation operators of the Full Vibrational Configuration Interaction (FVCI) wavefunction, which allows for preparing eigenstates of vibrational Hamiltonians on quantum computers. In order to establish the vADAPT-VQE algorithm using the IPR, we study the exactness of disentangled Unitary Vibrational Coupled Cluster (dUVCC) theory and show that dUVCC can formally represent the FVCI wavefunction in an infinite expansion. To investigate the performance of the vADAPT-VQE algorithm, we numerically study whether the vADAPT-VQE algorithm generates a sequence of operators that may represent the FVCI wavefunction. Our numerical results indicate frequent appearance of critical points in the wavefunction preparation using vADAPT-VQE. These results imply that one may encounter diminishing usefulness when preparing vibrational wavefunctions on quantum computers using vADAPT-VQE and that additional studies are required to find methods that can circumvent this behavior.

Perturbation theory for the Gross–Pitaevskii equation modeling stationary Bose–Einstein condensates

In this article a perturbative solution of the Gross–Pitaevskii(GP) equation in the D-dimensional space RD with a general continuous non-negative external potential is studied. The solution describes the condensate wave-function of a gas containing N Bose particles. A criteria for the validity of the perturbative solution is developed. Furthermore expressions for the particle density, the chemical potential, the internal energy and the mean-square radius of the condensate are derived corrected to first order in the coupling constant. The scheme is then applied to obtain the solution of the GP equation in D=1,2,3 for external harmonic potentials, and the results are compared with the numerical solutions. It is shown, in each case, that if N exceeds a certain value the perturbative solution breaks down.

Tunneling splittings using modified WKB method in Cartesian coordinates: The test case of vinyl radical.

Modified WKB theory for calculating tunneling splittings in symmetric multi-well systems in full dimensionality is re-derived using Cartesian coordinates. It is explicitly shown that the theory rests on the wavefunction that is exact for harmonic potentials. The theory was applied to calculate tunneling splittings in vinyl radical and some of its deuterated isotopologues in their vibrational ground states and the low-lying vibrationally excited states and compared to exact variational results. The exact results are reproduced within a factor of 2 in most states. Remarkably, all large enhancements of tunneling splittings relative to the ground state, up to three orders in magnitude in some excited mode combinations, are well reproduced. It is also shown that in the asymmetrically deuterated vinyl radical, the theory correctly predicts the states that are localized in a single well and the delocalized tunneling states. Modified WKB theory on the minimum action path is computationally inexpensive and can also be applied without modification to much larger systems in full dimensionality; the results of this test case serve to give insight into the expected accuracy of the method.

Vibration to Vibration: Product Energy Distribution of F + HD Crossed Molecular Beam Experiments.

We investigated the F + HD(v = 1, j = 0) → HF + D reaction using the crossed molecular beam technique combined with the D atom Rydberg tagging time-of-flight spectroscopy. By detecting the products at various scattering angles for different collision energies in the range of 0.8-1.2 kcal/mol, we observed the forward-scattering products of HF(v' = 4) and determined the threshold energy for the opening of this reaction channel. Similar experiments were conducted for the F + HD(v = 0, j = 0) → HF + D reaction within the range of 1.1-1.6 kcal/mol, where forward-scattering products of HF(v' = 3) were observed, and the threshold energy for this reaction channel was determined as well. Furthermore, we measured the differential cross-sections for the F + HD → HF + D reaction in both the vibrational ground state and the excited state of HD and analyzed the vibrational quantum-state distribution of the HF products. It was found that the population of vibrational quantum states of the HF products increases synchronously with the excitation of the reactant HD vibrationally.

Exact instanton transseries for quantum mechanics

We calculate the instanton corrections to energy spectra of one-dimensional quantum mechanical oscillators to all orders and unify them in a closed form transseries description. Using alien calculus, we clarify the resurgent structure of these transseries and demonstrate two approaches in which the Stokes constants can be derived. As a result, we formulate a minimal one-parameter transseries for the natural nonperturbative extension to the perturbative energy, which captures the Stokes phenomenon in a single stroke. We derive these results in three models: quantum oscillators with cubic, symmetric double well and cosine potentials. In the latter two examples, we find that the resulting full transseries for the energy has a more convoluted structure that we can factorise in terms of a minimal and a median transseries. For the cosine potential we briefly discuss this more complicated transseries structure in conjunction with topology and the concept of the resurgence triangle.

Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential

In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, \[ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R}^n, \] where $n \geq 1$ , $0< s<1$ , $\omega >-\lambda _{1,s}$ , $2< p< {2n}/{(n-2s)^+}$ , $\lambda _{1,s}>0$ is the lowest eigenvalue of $(-\Delta )^s + |x|^2$ . The fractional Laplacian $(-\Delta )^s$ is characterized as $\mathcal {F}((-\Delta )^{s}u)(\xi )=|\xi |^{2s} \mathcal {F}(u)(\xi )$ for $\xi \in \mathbb {R}^n$ , where $\mathcal {F}$ denotes the Fourier transform. This solves an open question in [M. Stanislavova and A. G. Stefanov. J. Evol. Equ. 21 (2021), 671–697.] concerning the uniqueness of ground states.