Quantum Mechanical Hamiltonian Research Articles

Exceptional Entanglement Phenomena: Non-Hermiticity Meeting Nonclassicality.

Non-Hermitian (NH) extension of quantum-mechanical Hamiltonians represents one of the most significant advancements in physics. During the past two decades, numerous captivating NH phenomena have been revealed and demonstrated, but all of which can appear in both quantum and classical systems. This leads to the fundamental question: what NH signature presents a radical departure from classical physics? The solution of this problem is indispensable for exploring genuine NH quantum mechanics, but remains experimentally untouched so far. Here, we resolve this basic issue by unveiling distinct exceptional entanglement phenomena, exemplified by an entanglement transition, occurring at the exceptional point of NH interacting quantum systems. We illustrate and demonstrate such purely quantum-mechanical NH effects with a naturally dissipative light-matter system, engineered in a circuit quantum electrodynamics architecture. Our results lay the foundation for studies of genuinely quantum-mechanical NH physics, signified by exceptional-point-enabled entanglement behaviors.

High-order SUSY-QM, the quantum XP model and zeroes of the Riemann Zeta function

Abstract Making use of the first- and second-order algorithms of supersymmetric quantum mechanics (SUSY-QM), we construct quantum mechanical Hamiltonians whose spectra are related to the zeroes of the Riemann Zeta function ζ(s). Inspired by the model of Das and Kalauni (DK) A Das and P Kalauni (2019 Physics Letters B 791, 265-269) which corresponds to this function in the strip 0 < Re [ s ] < 1 , and taking the factorization energy equal to zero, we use the wave function ∣x∣−S , S ∈ C , as a seed solution for our algorithms, obtaining XP-like operators. Thus, we construct SUSY-QM partner Hamiltonians whose zero energy mode locates exactly the nontrivial zeroes of ζ(s) along the critical line Re [ s ] = 1 / 2 in the complex plane. We further find that unlike the DK case, where the SUSY-QM partner potentials correspond to free particles, our partner potentials belong to the family of inverse squared distance potentials with complex couplings.

Quantum simulation of quantum mechanical system with spatial noncommutativity

Quantum simulation has become a promising avenue of research that allows one to simulate and gain insight into the models of high energy physics whose experimental realizations are either complicated or inaccessible with current technology. We demonstrate the quantum simulation of such a model, a quantum mechanical system with spatial noncommutativity, which is inspired by the works in noncommutative geometry and noncommutative field theory for a universal quantum computer. We use the novel group theoretical formalism to map the Hamiltonian of such a noncommutative quantum system into the ordinary quantum mechanical Hamiltonian and then carry out the quantum simulation using the Trotter–Suzuki product formula. Furthermore, we distinguish the impact of the noncommutativity parameter on the quantum simulation, especially on the Trotter error, and point out how its sizable value affects the simulation.

Intensification Dimension of Prices Cryptocurrency with Hamiltonian Quantum Mechanics

Aims: Cryptocurrency (CC) is a digital currency innovation that has impacted the financial sector's distribution mechanism since December 2013. This research aims to discover a non-linear mathematical Quantum relationship between the escalation of the development of cryptocurrency price fluctuations and prices in CC predictors.
 Study Design: The research uses simulation techniques to operate numerical models that correspond to the process of dynamic observation behavior.
 Place and Duration of Study: Types of crypto chosen based on this research data are Bitcoin (BTC), Ethereum (ETH), USD Coin (USDC), and Binance (BNB). The data collection period coverage is required with a number per week from 2019 to 2021 or 52 weeks.
 Methodology: From the collection of crypto prices that have been selected, there are 157 data samples taken using a systematic sampling strategy with elements that are randomly selected and then followed by the next element from the column on the table after the first choice. The conceptual form of the research background is modeled by a multiple regression term equation which predicts a continuous variable unit as a non-linear mathematical function.
 Results: The results of the research study found that the most prominent altcoins traded in the market are not affected by the price of Bitcoin securities. Percentage-wise, there were 40.33% of factors that influenced the movement of crypto coin price progress, with 59.67% being the remaining limiting factors in the study. Partial results show Bitcoin and altcoin assets have arbitrage potential with significant positives on put option volatility with asset discounting producing negative results and martingale strategies arising from a Hamiltonian perspective.
 Conclusion: By time limitations, BTC and USDC have the most potential in martingale conditions, while the cryptocurrency ETH might be a solution in picking assets with growing values at medium risk. Meanwhile, crypto BNB is an asset that offers new data on many market indices.

Electrical circuit simulation of non-Hermitian lattice models

<sec>Quantum simulation serves as a significant tool for studying and understanding novel phenomena in the quantum world. In recent years, it has be realized that apart from quantum platforms, classical systems like photonic crystals, phononic crystals, and mechanical oscillators can also be used to simulate quantum models by analogizing the Schrödinger equation. Among these systems, electrical circuits have emerged as a promising simulation platform owing to their low cost, technological maturity, and ease of scalability, successfully simulating numerous important quantum phenomena. Meanwhile, non-Hermitian physics breaks the Hermiticity of systems’ Hamiltonians in traditional quantum mechanics, providing a fresh perspective for understanding the physics of quantum systems, particularly open quantum systems. Non-Hermitian systems, owing to their manifestation of unique phenomena absent in Hermitian systems, have become emerging research subjects in various fields of physics. However, many non-Hermitian phenomena require specialized configurations that pose relatively high technical thresholds on quantum platforms. For instance, the non-Hermitian skin effect typically requires systems to possess non-reciprocal hopping between lattice sites. Therefore, utilizing flexible electrical circuits to simulate non-Hermitian physics becomes a natural choice.</sec><sec>This paper provides a short review of the current experimental progress in simulating non-Hermitian lattice models by using electrical circuits. It offers a brief introduction to the relevant knowledge of non-Hermitian physics, including mathematical concepts and novel phenomena, as well as the simulation theory of electrical circuits, including the mapping theory of the lattice models, the introduction of non-Hermiticity, and the measurement of physical quantities. The aim is to provide readers with a reference for better understanding or engagement in related researches, thus promoting further development in this field.</sec>

On generalizing trace minimization principles

Various trace minimization principles have interplayed with numerical computations for the standard eigenvalue and generalized eigenvalue problems in general, as well as important applied eigenvalue problems including the linear response eigenvalue problem from electronic structure calculation and the symplectic eigenvalue problem of positive definite matrices that play important roles in classical Hamiltonian dynamics, quantum mechanics, and quantum information, among others. In this paper, Ky Fan's trace minimization principle is extended along the line of the Brockett cost function tr(DXHAX) in X on the Stiefel manifold, where D of an apt size is positive definite. Specifically, we investigate infX⁡tr(DXHAX) subject to XHBX=Ik (the k×k identity matrix) or XHBX=Jk, where Jk=diag(±1). We establish conditions under which the infimum is finite and when it is finite, analytic solutions are obtained in terms of the eigenvalues and eigenvectors of the matrix pencil A−λB, where B is possibly indefinite and possibly singular, and D is also possibly indefinite.

Hermitian and non-Hermitian normal-mode splitting in an optically-levitated nanoparticle

Normal-mode splitting is a hallmark of strong coupling between two coupled harmonic oscillators. Here, we report the realization of strong coupling in the optically-levitated nanoparticle system via feedback. A silica nanoparticle is trapped by a tightly focused laser travelling in free space, which is regarded as a harmonic oscillators. An external electric oscillator is then phase-locked to the nanoparticle’s motion as another harmonic oscillator, which is modulated on the trapping laser to feedback and interact with the nanoparticle. Therefore, a highly manipulatable coupled-harmonic oscillator system is built in our platform and the normal-mode splitting is realized with strong coupling in both Hermitian and non-Hermitian cases. Moreover, since the coupling between the two harmonic oscillators induced by the feedback is flexibly manipulated, the normal-mode splitting following the cooling or heating effect is simultaneously observed. This method could be useful for further studying quantum mechanical Hamiltonian and non-Hermitian phenomena of an optically-levitated nanoparticle.

Construction of a Few Quantum Mechanical Hamiltonians via Lévy-Leblond Type Linearization: Clifford Momentum, Spinor States and Supersymmetry

Construction of a Few Quantum Mechanical Hamiltonians via Lévy-Leblond Type Linearization: Clifford Momentum, Spinor States and Supersymmetry

Geometric integration of classical spin dynamics via a mean-field Schrödinger equation

The Landau-Lifshitz equation describes the time evolution of magnetic dipoles and can be derived by taking the classical limit of a quantum mechanical spin Hamiltonian. To take this limit, one constrains the many-body quantum state to a tensor product of coherent states, thereby neglecting entanglement between sites. Expectation values of the quantum spin operators produce the usual classical spin dipoles. One may also consider expectation values of polynomials of the spin operators, leading to quadrupole and higher-order spin moments, which satisfy a dynamical equation of motion that generalizes the Landau-Lifshitz dynamics [Zhang and Batista, Phys. Rev. B 104, 104409 (2021)]. Here we reformulate the dynamics of these ${N}^{2}\ensuremath{-}1$ generalized spin components as a mean-field Schr\"odinger equation on the $N$-dimensional coherent state. This viewpoint suggests efficient integration methods that respect the local symplectic structure of the classical spin dynamics.

A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics

We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection yielding the (generalized) geometric phase. This bundle provides in addition a natural generalization of the quantum geometric tensor and derived tensors, and we show how it can incorporate the non-geometric dynamical phase as well. We finish by demonstrating how the bundle can be recast as a principal bundle, so that both the geometric phases and the permutations of eigenstates can be expressed simultaneously by means of standard holonomy theory.

Squeezed states generation in an array of Linear and Nonlinear Waveguides

We demonstrate the generation of squeezed states of light due to the second harmonic generation and Kerr effect in an array of nonlinear waveguides mediated through a linear one. We characterized the electromagnetic field by a quantum mechanical Hamiltonian and the density operator time evolution is obtained from the Von-Neumann equation of motion. Using the quasiprobability positive P of phase space representation, the classical Fokker-Planck equation is obtained from the master equation and translated to its classical matching set of nonlinear differential equations. We showed that because of the new possibilities of correlation between the linear and nonlinear channel waveguides, highly nonclassical light may be produced.

Matrix model: Emergence of a quantum number in the strong coupling regime

We continue here to study simple matrix models of quantum mechanical Hamiltonians. The eigenvalues and eigenfunctions were associated energy levels and wave functions. Whereas previously, we considered the weak coupling limits of our models, we here address the more difficult strong coupling limits. We find that the wave functions fall into two classes and we can assign a quantum number to distinguish them. Implications for transition rates are also discussed.

Theory of a superluminous vacuum quanta as the fabric of Space

Our observations of magneton with Ferrolens shows evidence pointing to such magneton entity, and more evidently in recent results with the synthetic vacuum unipole experiments. Physics formalism ansatz novel model analyses demonstrate how vacuum quanta may have sufficient energy for vacuum genesis, by constructing eigen spinors of zero_point microblackhole Hamiltonian quantum mechanics with Helmholtz decomposition matrix of gradient and rotational tensors, that are characteristic of translational vortex fields. With these mathematical physics processes, we obtain resulting energy fields spatial property partial differential equations characterizing eigen state energetics of zero point vacuum quagmire, as well as eigen state vortex fields of micro black hole, both together making up plasmodial zones within quagmire. Specific eigen spinors Hamiltonian partial differential equations quantifying energy and fields eigen functions. Vacuum that is dipole vacuum may have superposition of complex input of quagmire vortex fields acting to create non–hermitian quantum relativistic physics.

The Non-Relativistic Many-Body Quantum-Mechanical Hamiltonian with Diamagnetic Current-Current Interaction

We extend the standard solid-state quantum mechanical Hamiltonian containing only Coulomb interactions between the charged particles by inclusion of the (transverse) current-current diamagnetic interaction starting from the non-relativistic QED restricted to the states without photons and neglecting the retardation in the photon propagator. This derivation is supplemented with a derivation of an analogous result along the non-rigorous old classical Darwin-Landau-Lifshitz argumentation within the physical Coulomb gauge.

Comments on the quantum field theory of the Coulomb gas formalism

The holomorphic Coulomb gas formalism, as developed by Feigin-Fuchs, Dotsenko-Fateev and Felder, is a set of rules for computing minimal model observables using free field techniques. We attempt to derive and clarify these rules using standard techniques of quantum field theory. We begin with a careful examination of the timelike linear dilaton. Although the background charge of the model breaks the scalar field’s continuous shift symmetry, the exponential of the action remains invariant under a discrete shift because the background charge is imaginary. Gauging this symmetry makes the dilaton compact and introduces winding modes into the spectrum. One of these winding operators corresponds to the anti-holomorphic completion of the BRST current first introduced by Felder, and the full left/right cohomology of this BRST charge isolates the irreducible representations of the Virasoro algebra within the degenerate Fock space of the linear dilaton. The “supertrace” in the BRST complex reproduces the minimal model partition function and exhibits delicate cancellations between states with both momentum and winding. The model at the radius R=sqrt{pp^{prime }} has two marginal operators corresponding to the Dotsenko-Fateev “screening charges”. Deforming by them, we obtain a model that might be called a “BRST quotiented compact timelike Liouville theory”. The Hamiltonian of the zero-mode quantum mechanics of this model is not Hermitian, but it is PT-symmetric and exactly solvable. Its eigenfunctions have support on an infinite number of plane waves, suggesting an infinite reduction in the number of independent states in the full quantum field theory. Applying conformal perturbation theory to the exponential interactions reproduces the Coulomb gas calculations of minimal model correlation functions. In contrast to spacelike Liouville, these “resonance correlators” are finite because the zero mode is compact. We comment on subtleties regarding the reflection operator identification, as well as naive violations of truncation in correlators with multiple reflection operators inserted. This work is part of an attempt to understand the relationship between the JT model of two dimen- sional gravity and the worldsheet description of the (2, p) minimal string as suggested by Seiberg and Stanford.

Super Hamiltonian in superspace for incommensurate superlattices and quasicrystals

Infinite quasiperiodic arrangements in space, such as quasicrystals, are typically described as projections of higher-dimensional periodic lattices onto the physical dimension. The concept of a reference higher-dimensional space, called a superspace, has proved useful in relation to quasiperiodic systems. Although some quantum-mechanical systems in quasiperiodic media have been shown to admit quasiperiodic states, any sort of general Hamiltonian formalism in superspace is lacking to this date. Here, we show how to extend generic quantum-mechanical Hamiltonians to higher dimensions in such a way that eigenstates of the original Hamiltonian are obtained as projections of the Hamiltonian in superspace, which we call the super Hamiltonian. We apply the super Hamiltonian formalism to a simple, yet realistic one-dimensional quantum particle in a quasiperiodic potential without the tight-binding approximation, and obtain continuously labelled eigenstates of the system corresponding to a continuous spectrum. All states corresponding to the continuum are quasiperiodic. We also obtain the Green’s functions for continuum states in closed form and, from them, the density of states and local density of states, and scattering states off defects and impurities. The closed form of this one-dimensional Green’s function is equally valid for any continuum state in any one-dimensional single-particle quantum system admitting continuous spectrum. With the basis set we use, which is periodic in superspace, and therefore quasiperiodic in physical space, we find that Anderson-localised states are also quasiperiodic if distributional solutions are admitted, but circumvent this difficulty by generalising the superspace method to open boundary conditions. We also obtain an accurate estimate of the critical point where the ground state of the system changes from delocalised to Anderson localised, and of the critical exponent for the effective mass. Finally, we calculate, within the superspace formalism, topological edge states for the semi-infinite system, and observe that these exist, in the delocalised phase, within all spectral gaps we have been able to resolve. Our formalism opens up a plethora of possibilities for studying the physics of electrons, atoms or light in quasicrystalline and other aperiodic media.

Discreteness of space from GUP in strong gravitational fields

A large class of quantum theories of gravity show that the Heisenberg's uncertainty principle is modified to the “Generalised Uncertainty Principle” (GUP) near the Planckian scale. It has also been shown that the GUP induces perturbative corrections to all quantum mechanical Hamiltonians, even at low energies, and thereby introduces Planck scale corrections to the Schrödinger equation and to the relativistic quantum mechanical equations. Some of these corrections give rise to potentially measurable effects in the low-energy laboratory. Another prediction of these corrections is that a measured length must be quantized, as seen by studying the solutions of the GUP modified Schrödinger, Klein-Gordon, and Dirac equations in a one, two, and three dimensional box. This result was subsequently extended to spacetimes with weak gravitational fields. In this work, we further extend this length quantization to spacetimes with strong gravitational fields and show that this result continues to hold, thereby showing that it is robust.

-symmetric potentials having continuous spectra

One-dimensional -symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form H = p2 + V(x), where −∞ < x < ∞. The potential V(x) is odd in x, pure imaginary, and vanishes as |x| → ∞. Five -symmetric potentials are studied: the Scarf-II potential V1(x) = iA1 sech(x)tanh(x), which decays exponentially for large |x|; the rational potentials V2(x) = iA2x/(1 + x4) and V3(x) = iA3x/(1 + |x|3), which decay algebraically for large |x|; the step-function potential V4(x) = iA4 sgn(x)θ(2.5 − |x|), which has compact support; the regulated Coulomb potential V5(x) = iA5x/(1 + x2), which decays slowly as |x| → ∞ and thus may be viewed as a long-range potential. The real parameters An measure the strengths of these potentials. Numerical techniques for solving the time-independent Schrödinger eigenvalue problems associated with these potentials reveal that, in general, the spectra of these Hamiltonians are partly real and partly complex. The real eigenvalues form the continuous part of the spectrum and the complex eigenvalues form the discrete part of the spectrum. The real eigenvalues range continuously in value from 0 to +∞. The complex eigenvalues occur in discrete complex-conjugate pairs and for Vn(x) (1 ⩽ n ⩽ 4) the number of these pairs is finite and increases as the value of the strength parameter An increases. However, for V5(x) there is an infinite sequence of discrete eigenvalues with a limit point at the origin. While this sequence is complex, it resembles the Balmer series for the hydrogen atom because it has inverse-square convergence.

The fundamental connections between classical Hamiltonian mechanics, quantum mechanics and information entropy

We show that the main difference between classical and quantum systems can be understood in terms of information entropy. Classical systems can be considered the ones where the internal dynamics can be known with arbitrary precision while quantum systems can be considered the ones where the internal dynamics cannot be accessed at all. As information entropy can be used to characterize how much the state of the whole system identifies the state of its parts, classical systems can have arbitrarily small information entropy while quantum systems cannot. This provides insights that allow us to understand the analogies and differences between the two theories.

Self-Adjoint Extension Approach for Singular Hamiltonians in (2 + 1) Dimensions

In this work, we review two methods used to approach singular Hamiltonians in (2+1) dimensions. Both methods are based on the self-adjoint extension approach. It is very common to find singular Hamiltonians in quantum mechanics, especially in quantum systems in the presence of topological defects, which are usually modelled by point interactions. In general, it is possible to apply some kind of regularization procedure, as the vanishing of the wave function at the location of the singularity, ensuring that the wave function is square-integrable and then can be associated with a physical state. However, a study based on the self-adjoint extension approach can lead to more general boundary conditions that still gives acceptable physical states. We exemplify the methods by exploring the bound and scattering scenarios of a spin 1/2 charged particle with an anomalous magnetic moment in the Aharonov-Bohm potential in the conical space.