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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Wideband Impedance Passivation of MMCs for Suppressing Harmonic Oscillations

Harmonic instability and associated oscillation events have become one of the main concerns in MMC-based HVdc systems. These oscillations can appear in the range from a few hundred hertz to several kilohertz, and the root cause is identified as delay-induced negative damping of MMC impedance appearing at the system resonance frequency. Here, this paper introduces a wideband impedance reshaping method through MMC control to eliminate such negative damping for both grid-following (GFL) and grid-forming (GFM) MMCs, which makes the MMC impedance completely passive from the second harmonic frequency upward, thus preventing all harmonic oscillations. First, simplified impedance models of MMCs are derived for the harmonic stability analysis and the impedance reshaping control design. Next, a passivity-based impedance reshaping method is presented, as well as practical considerations for its implementation. In addition, to maintain the MMC's disturbance ride-through capability, an adaptive activation scheme is developed, which enables the wideband impedance reshaping control only in the presence of harmonic oscillation events. The effectiveness of the proposed method is validated by frequency domain analysis and by electromagnetic transient (EMT) simulations of two typical MMC-based power systems.

17 WIND ENERGY↗
DOE OSTI2021

The multi-faceted inverted harmonic oscillator: Chaos and complexity

The harmonic oscillator is the paragon of physical models; conceptually and computationally simple, yet rich enough to teach us about physics on scales that span classical mechanics to quantum field theory. This multifaceted nature extends also to its inverted counterpart, in which the oscillator frequency is analytically continued to pure imaginary values. In this article we probe the inverted harmonic oscillator (IHO) with recently developed quantum chaos diagnostics such as the out-of-time-order correlator (OTOC) and the circuit complexity. In particular, we study the OTOC for the displacement operator of the IHO with and without a non-Gaussian cubic perturbation to explore genuine and quasi scrambling respectively. In addition, we compute the full quantum Lyapunov spectrum for the inverted oscillator, finding a paired structure among the Lyapunov exponents. We also use the Heisenberg group to compute the complexity for the time evolved displacement operator, which displays chaotic behaviour. Finally, we extended our analysis to N-inverted harmonic oscillators to study the behaviour of complexity at the different timescales encoded in dissipation, scrambling and asymptotic regimes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Algebraic discrete quantum harmonic oscillator with dynamic resolution scaling

We develop an algebraic formulation for the discrete quantum harmonic oscillator (DQHO) from the Hamiltonian for two, coupled QHOs and provide a physical picture for the Kravchuk function eigenstates of the oscillator. The familiar $\mathfrak{su}(2)$ structure of the coupled QHO Hamiltonian divides its spectrum into sets corresponding to the DQHO at different resolutions. In addition to energy ladder operators, the formulation allows for the introduction of resolution ladder operators connecting all DQHOs with different resolutions, thus enabling the dynamic scaling of the resolution of finite degree-of-freedom quantum simulations. The coherent state of the DQHO is constructed, and its expected position is proven to oscillate as a classical harmonic oscillator. The DQHO coherent state recovers that of the quantum harmonic oscillator at large resolution.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Angular momentum eigenstates of the isotropic 3-D harmonic oscillator: Phase-space distributions and coalescence probabilities

The isotropic 3-dimensional harmonic oscillator potential can serve as an approximate description of many systems in atomic, solid state, nuclear, and particle physics. In particular, the question of 2 particles binding (or coalescing) into angular momentum eigenstates in such a potential has interesting applications. Here we compute the probabilities for coalescence of two distinguishable, non-relativistic particles into such a bound state, where the initial particles are represented by generic wave packets of given average positions and momenta. We use a phase-space formulation and hence need the Wigner distribution functions of angular momentum eigenstates in isotropic 3-dimensional harmonic oscillators. These distribution functions have been discussed in the literature before but we utilize an alternative approach to obtain these functions. Along the way, we derive a general formula that expands angular momentum eigenstates in terms of products of 1-dimensional harmonic oscillator eigenstates.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Equivalence of a harmonic oscillator to a free particle and Eisenhart lift

Highlights: • Equivalence of a harmonic oscillator to a free particle is considered. • The Eisenhart lift is applied to most general harmonic and linear potentials. • In geometric language, equivalence is the result of the conformal flatness of the Eisenhart metric. It is widely known in quantum mechanics that solutions of the Schrödinger equation (SE) for a linear potential are in one-to-one correspondence with the solutions of the free SE. The physical reason for this correspondence is Einstein’s principle of equivalence. What is usually not so widely known is that solutions of the Schrödinger equation with harmonic potential can also be mapped to the solutions of the free Schrödinger equation. The physical understanding of this equivalence is not known as precisely as in the case of the equivalence principle. We present a geometric picture that will link both of the above equivalences with one constraint on the Eisenhart metric.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Physics of the Inverted Harmonic Oscillator: From the lowest Landau level to event horizons

In this work, we present the inverted harmonic oscillator (IHO) Hamiltonian as a paradigm to understand the quantum mechanics of scattering and time-decay in a diverse set of physical systems. As one of the generators of area preserving transformations, the IHO Hamiltonian can be studied as a dilatation generator, squeeze generator, a Lorentz boost generator, or a scattering potential. In establishing these different forms, we demonstrate the physics of the IHO that underlies phenomena as disparate as the Hawking–Unruh effect and scattering in the lowest Landau level (LLL) in quantum Hall systems. We derive the emergence of the IHO Hamiltonian in the LLL in a gauge invariant way and show its exact parallels with the Rindler Hamiltonian that describes quantum mechanics near event horizons. This approach of studying distinct physical systems with symmetries described by isomorphic Lie algebras through the emergent IHO Hamiltonian enables us to reinterpret geometric response in the lowest Landau level in terms of relativistic effects such as Wigner rotation. Further, the analytic scattering matrix of the IHO points to the existence of quasinormal modes (QNMs) in the spectrum, which have quantized time-decay rates. We present a way to access these QNMs through wave packet scattering, thus proposing a novel effect in quantum Hall point contact geometries that parallels those found in black holes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Evidence of nonlinear coupling in the edge harmonic oscillation sustaining quiescent high confinement in a tokamak plasma

In a tokamak plasma, we measure nonlinear coupling between harmonics comprising a saturated edge oscillation, as predicted by nonlinear magnetohydrodynamic simulations. The coherent structure formed by this coupling suppresses bursty edge-localized modes that cause energy and particle loss. We also measure nonlinear coupling of the edge harmonic oscillation to core-resonant magnetic tearing modes. When tearing modes are present, the edge oscillation harmonics decouple, and edge-localized modes return. In this article, we conclude that nonlinear interaction with tearing modes interrupts the formation of the edge harmonic oscillation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Vibrational heat-bath configuration interaction with semistochastic perturbation theory using harmonic oscillator or VSCF modals

Vibrational heat-bath configuration interaction (VHCI)—a selected configuration interaction technique for vibrational structure theory—has recently been developed in two independent works [J. H. Fetherolf and T. C. Berkelbach, J. Chem. Phys. 154, 074104 (2021); A. U. Bhatty and K. R. Brorsen, Mol. Phys. 119, e1936250 (2021)], where it was shown to provide accuracy on par with the most accurate vibrational structure methods with a low computational cost. Here, we eliminate the memory bottleneck of the second-order perturbation theory correction using the same (semi)stochastic approach developed previously for electronic structure theory. This allows us to treat, in an unbiased manner, much larger perturbative spaces, which are necessary for high accuracy in large systems. Stochastic errors are easily controlled to be less than 1 cm−1. We also report two other developments: (i) we propose a new heat-bath criterion and an associated exact implicit sorting algorithm for potential energy surfaces expressible as a sum of products of one-dimensional potentials; (ii) we formulate VHCI to use a vibrational self-consistent field (VSCF) reference, as opposed to the harmonic oscillator reference configuration used in previous reports. Our tests are done with quartic and sextic force fields, for which we find that with VSCF, the minor improvements to accuracy are outweighed by the higher computational cost associated the matrix element evaluations. We expect VSCF-based VHCI to be important for more general potential representations, for which the harmonic oscillator basis function integrals are no longer analytic.

Chemistry↗
DOE OSTI2022

Impact of edge harmonic oscillations on the divertor heat flux in NSTX

Previously, ELM-free and inter-ELM divertor peak heat flux reduction induced by an edge harmonic oscillation (EHO) was observed in NSTX (K.F. Gan et al., 2017 Nucl. Fusion 57 126053). This paper introduces new analysis of the EHO impact on the divertor heat flux. It was found that enhanced edge turbulence significantly altered the divertor heat flux footprint in NSTX H-modes, relative to ELM-free discharges. When the background edge fluctuation level was low, the EHO significantly increased the heat flux width and decreased the divertor peak heat flux. When the background edge fluctuation level was high, the EHO actually increased the divertor peak heat flux. As a result, it was also found that the heat flux width increased with increasing frequency of the EHO.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Axially-deformed solution of the Skyrme-Hartree-Fock-Bogoliubov equations using the transformed harmonic oscillator basis (IV) HFBTHO (v4.0): A new version of the program

We describe the new version 4.0 of the code HFBTHO that solves the nuclear Hartree-Fock-Bogoliubov problem by using the deformed harmonic oscillator basis in cylindrical coordinates. In the new version, we have implemented the restoration of rotational, particle number, and reflection symmetry for even-even nuclei. The restoration of rotational symmetry does not require using bases closed under rotation. Furthermore, we added the SeaLL1 functional and improved the calculation of the Coulomb potential. Finally, we refactored the code to facilitate maintenance and future developments.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

Spatiotemporal structure of edge harmonic oscillation and its role in ELM-free QH-mode at KSTAR

In Quiescent H-mode (QH-mode), edge-localized modes (ELMs) are naturally replaced by a low-n edge harmonic oscillation (EHO), yet the self-regulating transport mechanism driven by the EHO remains insufficiently understood. Using high-spatiotemporal-resolution imaging diagnostics on KSTAR—electron cyclotron emission imaging and broadband electron cyclotron emission—we resolve the eigenmode structure of the EHO and elucidate its regulatory role in edge transport. The EHO is localized within the pedestal near the maximum pressure gradient, propagates in the ion-diamagnetic direction, and its radial envelope expands with increasing shear, suggesting that rotational shear is closely associated with the structural evolution of the EHO. Information-theoretic Transfer Entropy analysis identifies a distinct ‘dual-stabilization’ regulatory interaction pattern: the EHO is associated with enhanced outward energy transport to limit the pressure gradient, while the background shear flow is associated with preferential regulatory influence on the EHO saturation amplitude and energy flux. These results demonstrate that the QH-mode pedestal is sustained by a shear-associated regulation mechanism, where the coupling between rotational shear and mode structure appears closely linked to the sustained ELM-free state.

ECEI↗
DOE OSTI2026

MARS-Q modeling of low- n resistive kink-peeling modes and edge harmonic oscillations in DIII-D

Linear and quasilinear magnetohydrodynamic (MHD) modeling is carried out for two DIII-D discharges that both featured a transition from quiescent H-mode (QH) to wide-pedestal QH (WPQH). The MHD perturbations, associated with the edge harmonic oscillations (EHOs) observed during the QH-phase in both discharges, are identified as low-n (n is the toroidal mode number) resistive kink-peeling instabilities, with the ideal MHD counterpart remaining stable. The quasilinear model successfully simulates EHO-like perturbations during the QH phase in both discharges, confirming the experimental observations. A less intuitive finding is the EHO-like behavior involving the n = 2 perturbation, simulated for one of the discharges during the WPQH-phase. This result, while consistent with experimental observations, is obtained despite the fact that the initial perturbation is linearly stable. The eventual growth of the perturbation is solely due to nonlinear interaction between the MHD perturbation and the plasma toroidal flow. The occurrence of the EHO-like perturbation during the WPQH-phase is found to be sensitive to the initial profile of the plasma edge rotation. For DIII-D plasmas considered, the neoclassical toroidal viscosity, generated by three-dimensional low-n perturbations, is found to play a dominant role in modifying the edge flow during EHOs.

EHO↗
DOE OSTI2025

Global optimization of harmonic oscillator basis in covariant density functional theory

The present investigation focuses on the improvement of the accuracy of the description of binding energies within moderately sized fermionic basis. Using the solutions corresponding to infinite fermionic basis it was shown that in the case of meson exchange (ME) covariant energy density functionals (CEDFs) the global accuracy of the description of binding energies in the finite $N_F$ = 16 - 20 bases can be drastically (by a factor ranging from ~3 up to ~9 dependent on the functional and $N_F$) improved by a global optimization of oscillator frequency of the basis. This is a consequence of the unique feature of the ME functionals in which with increasing fermionic basis size fermionic and mesonic energies approach the exact (infinite basis) solution from above and below, respectively. As a consequence, an optimal oscillator frequency $\hbar\omega_0$ of the basis can be defined which provides an accurate reproduction of exact total binding energies by the ones calculated in truncated basis. This leads to a very high accuracy of the calculations in moderately sized $N_F=20$ basis when mass dependent oscillator frequency is used: global rms differences $\delta B_{rms}$ between the binding energies calculated in infinite and truncated bases are only 0.025 MeV and 0.031 MeV for the NL5(Z) and DD-MEZ functionals, respectively. Optimized values of the oscillator frequency $\hbar\omega_0$ are provided for three major classes of CEDFs, i.e. for density dependent meson exchange functionals, nonlinear meson exchange ones and point coupling functionals.

Binding energy & masses↗
DOE OSTI2021

Shadow Lagrangian dynamics for superfluidity

Motivated by a similar approach for Born-Oppenheimer molecular dynamics, this paper proposes an extended "shadow" Lagrangian density for quantum states of superfluids. The extended Lagrangian contains an additional field variable that is forced to follow the wave function of the quantum state through a rapidly oscillating extended harmonic oscillator. By considering the adiabatic limit for large frequencies of the harmonic oscillator, we can derive the two equations of motions, a Schrödinger-type equation for the quantum state and a wave equation for the extended field variable. The equations are coupled in a nonlinear way, but each equation individually is linear with respect to the variable that it defines. The computational advantage of this new system is that it can be easily discretized using linear time stepping methods, where we propose to use a Crank-Nicolson-type approach for the Schrödinger equation and an extended leapfrog scheme for the wave equation. Furthermore, the difference between the quantum state and the extended field variable defines a consistency error that should go to zero if the frequency tends to infinity. By coupling the time-step size in our discretization to the frequency of the harmonic oscillator we can extract an easily computable consistency error indicator that can be used to estimate the numerical error without additional costs. The findings are illustrated in numerical experiments.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

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. Here, the user is encouraged to choose a reference state based on the appropriateness of the model and the practicality of the calculations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗