2D Harmonic Oscillator Research Articles

Active Brownian information engine: Self-propulsion induced colossal performance.

The information engine is a feedback mechanism that extorts work from a single heat bath using the mutual information earned during the measurement. We consider an overdamped active Ornstein-Uhlenbeck particle trapped in a 1D harmonic oscillator. The particle experiences fluctuations from an inherent thermal bath with a diffusion coefficient (D) and an active reservoir, with characteristic correlation time (τa) and strength (Da). We design a feedback-driven active Brownian information engine (ABIE) and analyze its best performance criteria. The optimal functioning criteria, the information gained during measurement, and the excess output work are reliant on the dispersion of the steady-state distribution of the particle's position. The extent of enhanced performance of such ABIE depends on the relative values of two underlying time scales of the process, namely, thermal relaxation time (τr) and the characteristic correlation time (τa). In the limit of τa/τr → 0, one can achieve the upper bound on colossal work extraction as ∼0.202γ(D+Da) (γ is the friction coefficient). The excess amount of extracted work reduces and converges to its passive counterpart (∼0.202γD) in the limit of τa/τr → high. Interestingly, when τa/τr = 1, half the upper bound of excess work is achieved irrespective of the strength of either reservoirs, thermal or active. Finally, we look into the average displacement of active Brownian particles in each feedback cycle, which surpasses its thermal analog due to the broader marginal probability distribution.

Late time dynamics in SUSY saddle-dominated scrambling through higher-point OTOC

In this article, we propose higher-point out-of-time-order correlators (OTOCs) as a tool to differentiate chaotic from saddle-dominated dynamics in late times. As a model, we study the scrambling dynamics in supersymmetric quantum mechanical systems. Using the eigenstate representation, we define the 2N-point OTOC using two formalisms, namely the ’Tensor Product formalism’ and the ’Partner Hamiltonian formalism’. We analytically find that the 2N-point OTOC for the supersymmetric 1D harmonic oscillator is in exact agreement with that of the 1D bosonic harmonic oscillator system. We show that the higher-point OTOC is a more sensitive measure of scrambling than the usual 4-point OTOC. To demonstrate this, we analyze a supersymmetric sextic 1D oscillator, for which the bosonic partner system has an unstable saddle in the phase space, while the saddle is absent in the fermionic counterpart. For such a system, we show that the saddle-dominated scrambling, higher anharmonic potential effects, and the supersymmetric OTOC exhibit similar dynamics due to supersymmetry constraints. Finally, we illustrate that the late-time dynamics of the higher-point OTOC become oscillatory after the peak for saddle-dominated scrambling and anharmonic oscillator systems. We propose the higher-point OTOC as a probe of late-time dynamics in non-chaotic systems that exhibit fast early-time scrambling.

Landau Levels versus Hydrogen Atom

The Landau problem and harmonic oscillator in the plane share a Hilbert space that carries the structure of Dirac’s remarkable so(2,3) representation. We show that the orthosymplectic algebra osp(1|4) is the spectrum generating algebra for the Landau problem and, hence, for the 2D isotropic harmonic oscillator. The 2D harmonic oscillator is in duality with the 2D quantum Coulomb–Kepler systems, with the osp(1|4) symmetry broken down to the conformal symmetry so(2,3). The even so(2,3) submodule (coined Rac) generated from the ground state of zero angular momentum is identified with the Hilbert space of a 2D hydrogen atom. An odd element of the superalgebra osp(1|4) creates a pseudo-vacuum with intrinsic angular momentum 1/2 from the vacuum. The odd so(2,3)-submodule (coined Di) built upon the pseudo-vacuum is the Hilbert space of a magnetized 2D hydrogen atom: a quantum system of a dyon and an electron. Thus, the Hilbert space of the Landau problem is a direct sum of two massless unitary so(2,3) representations, namely, the Di and Rac singletons introduced by Flato and Fronsdal.

Exploring the parameter space of an endohedral atom in a cylindrical cavity.

Endohedral fullerenes, or endofullerenes, are chemical systems of fullerene cages encapsulating single atoms or small molecules. These species provide an interesting challenge of Potential Energy Surface determination as examples of non-covalently bonded, bound systems. While the majority of studies focus on C60 as the encapsulating cage, introducing some anisotropy by using a different fullerene, e.g., C70 can unveil a double well potential along the unique axis. By approximating the potential as a pairwise Lennard-Jones (LJ) summation over the fixed C cage atoms, the parameter space of the Hamiltonian includes three tunable variables: (M, ɛ, σ) representing the mass of the trapped species, the LJ energy, and length scales respectively. Fixing the mass and allowing the others to vary can imitate the potentials of endohedral species trapped in more elongated fullerenes. We choose to explore the LJ parameter space of an endohedral atom in C70 with ɛ ∈ [20, 150 cm-1], and σ ∈ [2.85, 3.05 Å]. As the barrier height and positions of these wells vary between [1, 264 cm-1] and [0.35, 0.85 Å] respectively, using a 3D direct product basis of 1D harmonic oscillator (HO) wavefunctions centred at the origin where there is a local maximum is unphysical. Instead we propose the use of a non-orthogonal basis set, using 1D HO wavefunctions centred in each minimum and compare this to other choices. The ground state energy of the X@C70 is tracked across the LJ parameter space, along with its corresponding nuclear translational wavefunctions. A classification of the wavefunction characteristics, namely the prolateness and "peanut-likeness" based on its statistical moments is also proposed. Excited states of longer fullerenes are assigned quantum numbers, and the fundamental transitions of Ne@C70 are tracked across the parameter space.

Twisted-bilayer FeSe and the Fe-based superlattices

We derive BM-like continuum models for the bands of superlattice heterostructures formed out of Fe-chalcogenide monolayers: (I) a single monolayer experiencing an external periodic potential, and (II) twisted bilayers with long-range moire tunneling. A symmetry derivation for the inter-layer moire tunnelling is provided for both the \GammaΓ and MM high-symmetry points. In this paper, we focus on moire bands formed from hole-band maxima centered on \GammaΓ, and show the possibility of moire bands with C=0C=0 or ±1±1 topological quantum numbers without breaking time-reversal symmetry. In the C=0C=0 region for \theta→0θ→0 (and similarly in the limit of large superlattice period for I), the system becomes a square lattice of 2D harmonic oscillators. We fit our model to FeSe and argue that it is a viable platform for the simulation of the square Hubbard model with tunable interaction strength.

Machine-learning Kohn–Sham potential from dynamics in time-dependent Kohn–Sham systems

The construction of a better exchange-correlation potential in time-dependent density functional theory (TDDFT) can improve the accuracy of TDDFT calculations and provide more accurate predictions of the properties of many-electron systems. Here, we propose a machine learning method to develop the energy functional and the Kohn–Sham potential of a time-dependent Kohn–Sham (TDKS) system is proposed. The method is based on the dynamics of the Kohn–Sham system and does not require any data on the exact Kohn–Sham potential for training the model. We demonstrate the results of our method with a 1D harmonic oscillator example and a 1D two-electron example. We show that the machine-learned Kohn–Sham potential matches the exact Kohn–Sham potential in the absence of memory effect. Our method can still capture the dynamics of the Kohn–Sham system in the presence of memory effects. The machine learning method developed in this article provides insight into making better approximations of the energy functional and the Kohn–Sham potential in the TDKS system.

Stability of Finite Difference Solution of Time-Dependent Schrodinger Equations

In this paper, the stability of finite difference methods for time-dependent Schrodinger equation with Dirichlet boundary conditions on a staggered mesh was considered with explicit and implicit discretization. Using the matrix representation for the numerical algorithm, it is shown that for the explicit finite difference method, the solution is conditionally stable while it becomes unconditionally stable for implicit finite difference methods. A 1D Harmonic Oscillator problem shall be used to illustrate this behaviour.

Families of Orbits Produced by Three-Dimensional Central and Polynomial Potentials: An Application to the 3D Harmonic Oscillator

We study three-dimensional potentials of the form V=U(xp+yp+zp), where U is an arbitrary function of C2-class, and p∈Z, which produces a preassigned two-parametric family of spatial regular orbits given in the solved form f(x,y,z) = c1, g(x,y,z) = c2 (c1, c2 = const). These potentials have to satisfy two linear PDEs, which are the basic equations of the 3D inverse problem of Newtonian dynamics. The functions f and g can be represented uniquely by the ”slope functions” α(x,y,z) and β(x,y,z). The orbital functions α(x,y,z) and β(x,y,z) have to satisfy three differential conditions according to the theory of the inverse problem. If these conditions are satisfied, then we can find such a potential analytically. We offer pertinent examples of potentials that are mainly used in physical problems. The values obtained for p lead to cases of well-known potentials, such as the Newtonian, cored, logarithmic, polynomial and quadratic ones. New families of orbits produced by the 3D harmonic oscillator are found. Pertinent examples are given and cover all cases. Two-dimensional potentials belong to a special category of potentials and are studied separately. The families of straight lines in 3D space are also examined.

Exact solutions of the generalized Dunkl oscillator in the Cartesian system

In this paper, we use the generalized Dunkl derivatives instead of the standard partial derivatives in the Schrödinger equation to obtain an explicit expression of the generalized Dunkl–Schrödinger equation in 3D. It was found that this generalized Dunkl–Schrödinger equation for the 3D harmonic oscillator is exactly solvable in the Cartesian coordinates. From the relevant commutation relations, it is evident that the symmetry possessed by the original Dunkl Harmonic oscillator is broken by the generalized Dunkl derivative. Finally, we show that energy levels can be affected by considering a deformation parameter ɛ.

Vibrational spectrum of a 1D oscillator: The quantum, the Wigner, and the classical ways

AbstractInfrared spectra have been used in many chemical applications, and theoretical calculations have been useful for analyzing these experimental results. While quantum mechanics is used for calculating the spectra for small molecules, classical mechanics is used for larger systems. However, a systematic understanding of the similarities and differences between the two approaches is not clear. Previous studies focused on peak position and relative intensities of the spectra obtained by various quantum and classical methods, but here, we included “absolute” intensities in the evaluation. The infrared spectrum of a one‐dimensional (1D) harmonic oscillator (HO) and Morse oscillator were examined using four treatments: quantum, Wigner, truncated Wigner, and classical microcanonical treatments. For a 1D HO with a linear dipole moment function (DMF), the quantum and Wigner treatments give nearly the same spectra. On the other hand, the truncated Wigner underestimates the fundamental transition's intensity by half. In the case of cubic DMF, the truncated Wigner and classical methods fail to reproduce the relative intensity between the fundamental and second overtone transitions. Unfortunately, all the Wigner and classical methods fail to agree with the quantum results for a Morse oscillator with just 1% anharmonicity.

Rydberg exciton-polaritons in a magnetic field

We theoretically investigate exciton-polaritons in a two-dimensional (2D) semiconductor heterostructure, where a static magnetic field is applied perpendicular to the plane. To explore the interplay between magnetic field and a strong light-matter coupling, we employ a fully microscopic theory that explicitly incorporates electrons, holes and photons in a semiconductor microcavity. Furthermore, we exploit a mapping between the 2D harmonic oscillator and the 2D hydrogen atom that allows us to efficiently solve the problem numerically for the entire Rydberg series as well as for the ground-state exciton. In contrast to previous approaches, we can readily obtain the real-space exciton wave functions and we show how they shrink in size with increasing magnetic field, which mirrors their increasing interaction energy and oscillator strength. We compare our theory with recent experiments on exciton-polaritons in GaAs heterostructures in an external magnetic field and we find excellent agreement with the measured polariton energies. Crucially, we are able to capture the observed light-induced changes to the exciton in the regime of very strong light-matter coupling where a perturbative coupled oscillator description breaks down. Our work can guide future experimental efforts to engineer and control Rydberg excitons and exciton-polaritons in a range of 2D materials.

Conformal bridge transformation, mathcal{PT} - and supersymmetry

Supersymmetric extensions of the 1D and 2D Swanson models are investigated by applying the conformal bridge transformation (CBT) to the first order Berry-Keating Hamiltonian multiplied by i and its conformally neutral enlargements. The CBT plays the role of the Dyson map that transforms the models into supersymmetric generalizations of the 1D and 2D harmonic oscillator systems, allowing us to define pseudo-Hermitian conjugation and a suitable inner product. In the 1D case, we construct a mathcal{PT} -invariant supersymmetric model with N subsystems by using the conformal generators of supersymmetric free particle, and identify its complete set of the true bosonic and fermionic integrals of motion. We also investigate an exotic N = 2 supersymmetric generalization, in which the higher order supercharges generate nonlinear superalgebras. We generalize the construction for the 2D case to obtain the mathcal{PT} -invariant supersymmetric systems that transform into the spin-1/2 Landau problem with and without an additional Aharonov-Bohm flux, where in the latter case, the well-defined integrals of motion appear only when the flux is quantized. We also build a 2D supersymmetric Hamiltonian related to the “exotic rotational invariant harmonic oscillator” system governed by a dynamical parameter γ. The bosonic and fermionic hidden symmetries for this model are shown to exist for rational values of γ.

Fractional quantum oscillator and disorder in the vibrational spectra

We study the role of disorder in the vibration spectra of molecules and atoms in solids. This disorder may be described phenomenologically by a fractional generalization of ordinary quantum-mechanical oscillator problem. To be specific, this is accomplished by the introduction of a so-called fractional Laplacian (Riesz fractional derivative) to the Scrödinger equation with three-dimensional (3D) quadratic potential. To solve the obtained 3D spectral problem, we pass to the momentum space, where the problem simplifies greatly as fractional Laplacian becomes simply k^mu , k is a modulus of the momentum vector and mu is Lévy index, characterizing the degree of disorder. In this case, mu rightarrow 0 corresponds to the strongest disorder, while mu rightarrow 2 to the weakest so that the case mu =2 corresponds to “ordinary” (i.e. that without fractional derivatives) 3D quantum harmonic oscillator. Combining analytical (variational) and numerical methods, we have shown that in the fractional (disordered) 3D oscillator problem, the famous orbital momentum degeneracy is lifted so that its energy starts to depend on orbital quantum number l. These features can have a strong impact on the physical properties of many solids, ranging from multiferroics to oxide heterostructures, which, in turn, are usable in modern microelectronic devices.

On the perturbative formalism and a possible quantum Discrete spectrum for the Regge–Wheeler equation of a De Sitter spacetime

In this paper, we study the perturbative regime in the static patch of de Sitter metric in the Regge–Wheeler formalism. After realizing that perturbative regime in a de Sitter spacetime depicted in terms of usual spherical coordinates cannot be extended up to the cosmological horizon, we study perturbative equations, in particular the axial ones, in terms of the tortoise coordinate [Formula: see text]. We show that perturbative regime can be extended up to the cosmological horizon, provided that suitable boundary conditions are chosen. As an application, we explore the Regge–Wheeler equation at short distances by performing a Taylor expansion. In order to study some possible quantum effects at short distances, we impose to the equation so-obtained the same boundary conditions suitable for a quantum 3D harmonic oscillator. As a result, a discrete spectrum can be obtained. The aforementioned spectrum is analyzed and a relation with possible effects denoting quantum behavior of gravitons is suggested.

Simple Approximation for the Ideal Reference State of Gases Adsorbed on Solid-State Surfaces

Reference states are useful as models for facilitating calculations of equilibrium constants, and they may also serve as standard states that are convenient for organizing and tabulating thermodynamic data; however, standard state conventions and appropriate reference states for adsorbed species have received less attention than those for pure substances and solutes. Here, we compare seven choices of reference states for calculations of equilibrium constants and transition state theory rate constants for flat surfaces, in particular (1) an ideal 2D harmonic oscillator, (2) an ideal rigid-molecule harmonic oscillator, (3) an ideal 2D harmonic oscillator with separable surface modes, (4) a 2D ideal gas, (5) an ideal 2D hindered translator, (6) an ideal 2D hindered translator with lowest-order barriers, and (7) a simple ideal 2D hindered translator proposed in this work. The advantage of models 5-7 is that they can treat both mobile and localized adsorbates in a consistent way, whereas models 1-3 are only appropriate for localized adsorbates, and model 4 is only appropriate for a freely translating adsorbate. Furthermore, models 6 and 7 reduce the computational cost without the user having to calculate barrier heights for diffusion. An advantage of the simple ideal 2D hindered translator is that it has a physical high-temperature limit. We also propose a reference state for nonflat surfaces. The user is encouraged to choose a reference state based on the appropriateness of the model and the practicality of the calculations.

Configuration localised states from orthogonal polynomials for effective potentials in 3D systems vs. algebraic DVR approaches

Configuration localised states (CLS) based on the zeroes of orthogonal polynomials were introduced some time ago for 1D systems [Phys. Rev. A 61, 042504 (2000)]. Recently two algebraic approaches based on a discrete variable representation were proposed to describe 3D systems: the 3D harmonic oscillator discrete variable representation (HO-DVR) approach and the U(4)-UGA [Mol. Phys. 118, e1662959 (2020)]. Both of them together with the CLS are based on the harmonic oscillator basis. A comparison between the three ways of generating a DVR is timely. For that purpose, here, first the CLS method based on zeroes of orthogonal polynomials is generalised so as to be applied to 3D systems and, then, its results are compared with the algebraic DVR methods. It is shown that the CLS states obtained from the zeroes of orthogonal polynomials are equivalent to the localised states of the HO-DVR approach, while the U(4)-DVR method does not provide localised states. As applications, the ro-vibrational spectra of O molecules and berylium dimers, Be, are analysed. Excellent results are obtained with relatively small CLS and HO-DVR basis size.

Asymptotic representation of projector kernel for 2D harmonic oscillator in a strip

Asymptotic representation of projector kernel for 2D harmonic oscillator in a strip

Connes distance of 2D harmonic oscillators in quantum phase space* *Project supported by the Key Research and Development Project of Guangdong Province, China (Grant No. 2020B0303300001), the National Natural Science Foundation of China (Grant No. 11911530750), the Guangdong Basic and Applied Basic Research Foundation, China (Grant No. 2019A1515011703), the Fundamental Research Funds for the Central Universities, China (Grant No. 2019MS109),

We study the Connes distance of quantum states of two-dimensional (2D) harmonic oscillators in phase space. Using the Hilbert–Schmidt operatorial formulation, we construct a boson Fock space and a quantum Hilbert space, and obtain the Dirac operator and a spectral triple corresponding to a four-dimensional (4D) quantum phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements. We construct the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of 2D quantum harmonic oscillators. We prove that these two-dimensional distances satisfy the Pythagoras theorem. These results are significant for the study of geometric structures of noncommutative spaces, and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.

Ladder Operators for the Spherical 3D Harmonic Oscillator

The Schrödinger equation for an isotropic three-dimensional harmonic oscillator is solved using ladder operators. The starting point is the shape invariance condition, obtained from supersymmetric quantum mechanics. Generalized ladder operators can be constructed for the three spherical spatial coordinates. Special emphasis is given to the adaptation made to each of these coordinates. The approach used is general and is indicated as an alternative method to solve the Schrödinger equation.

Whispering gallery and surface mode of electrons in lateral and corrugated quantum dots

Quantum dots (QDs) are fundamental elements in the applications related to light–matter interaction, such as solar cells, lasers and sensors. Moreover, some modes of electrons including whispering galley modes (WGMs) and surface modes can be incorporated in many electronic systems, in high Q-resonators, and quantized reflection/transmission. Therefore, controlling and manipulating their energy spectra is vital. Here, we investigate energy spectra as well as WGM and surface mode of electrons in lateral and corrugated QDs. Although, lateral QDs are usually modelled by a 2D harmonic oscillator (with zero thickness), we show that even very small thickness of dots can change their energy spectra, also they can contain surface mode of electrons. Moreover, we investigate WGM in deformed QDs, and the dots which contain corners in their outermost region, and show that some degenerated points would be created. Meanwhile, in the corrugated QDs, the wavefunction would be distributed in the specific teeth based on its energy level.