Hamiltonian Framework Research Articles

Multiple Fano interferences due to waveguide-mediated phase coupling between atoms

We examine quantum interference effects due to absorption and emission from multiple atoms coupled to a waveguide and highlight the modifications they entail in regards to single-photon transport properties. A prominent upshot of these interference phenomena is the resonant suppression of the reflection amplitude, which leads to the observation of multiple Fano minima in the reflection spectrum. Such minima determine the points at which transparency is induced in the system. By taking recourse to the real-space Hamiltonian framework, we calculate analytically the reflectivity and transmissivity for a one-dimensional waveguide that evanescently couples to a chain of equally spaced quantum emitters. The inter-emitter spacing relative to the wavelength of the propagating photon, leading to a waveguide-mediated "phase-coupling" between the atoms, is found to fundamentally affect the existence of Fano minima. For a chain of $N$ atoms, the number of minima can be at most $N-1$. However, suitable choices of the phase can suppress the discernibility of the full range of roots in the reflection spectrum. A principal observation for the case of multiple emitters is the emergence of super-Gaussian characteristics close to zero-detuning and consequently, a plateau-shaped broadband spectrum in the region of high reflectivity. For a large chain size, the plateau gets transformed into a flat-topped quasi-rectangular profile in the frequency domain.

Precession of the non-rigid Earth: Effect of the mass redistribution

This research is focused on determining the contribution to the precession of the Earth’s equator due to the mass redistribution stemming from the gravitational action of the Moon and the Sun on a rotating solid Earth. In the IAU2006 precession theory, this effect is taken into account through a contribution of −0.960 mas cy−1for the precession in longitude (with the unspecific name of non-linear effect). In this work, the revised value of that second-order contribution reaches −37.847 mas cy−1when using the Love numbers values given in IERS Conventions, and −43.945 mas cy−1if those values are supplemented with the contributions of the oceanic tides. Such variations impose a change of the first-order precession value that induces relative changes of the Earth’s dynamical ellipticity of about 7.3 and 8.5 ppm, respectively. The corresponding values for the obliquity rate are 0.0751 and 0.9341 mas cy−1, respectively, in contrast to 0.340 mas cy−1considered in IAU2006. The fundamentals of the modeling have been revisited by giving a clear construction of the redistribution potential of the Earth through the corresponding changes in the Earth tensor of inertia. The dynamical problem is tackled within the Hamiltonian framework of a two-layer Earth model, introduced and developed by Getino and Ferrándiz. This approach allows for the achievement of closed-analytical formulae for the precession in longitude and obliquity. It makes it possible to obtain numerical values for different Earth models once a set of associated Love numbers is selected. The research is completed with a discussion on the permanent tide and the related estimation of the variation of the second degree zonal Stokes parameter,J2, and also the indirect effects on nutations arising from the relative change of the Earth’s dynamical ellipticity.

Simple Flux-Side Formulation of State-Resolved Thermal Reaction Rates for Ring-Polymer Surface Hopping.

Employing the recently developed isomorphic Hamiltonian framework for including nuclear quantum effects in mixed quantum-classical nonadiabatic dynamics, we present a flux-side formulation of state-resolved thermal reaction rates for ring-polymer surface hopping (iso-RPSH). An appealing aspect of the new approach is that calculation of multiple state-resolved nonadiabatic thermal reaction rates is enabled with only a single free-energy surface calculation, whereas previous nonadiabatic flux-side formulations for surface hopping involve multiple free-energy surface calculations. The method is shown to be robust and straightforwardly implemented, and numerical results reveal that RPSH in the isomorphic Hamiltonian framework leads to better dividing surface independence than alternative RPSH methods due to improved preservation of the path-integral statistics.

Recurrence and Resonance in the Cubic Klein-Gordon Equation

In a number of models for coupled oscillators and nonlinear wave equations primary resonances dominate the phase-space phenomena. A new feature is that in a Hamiltonian framework, the interaction of primary and higher order resonances is shown to be important and can be signaled by using recurrence properties. The interaction may involve embedded double resonance. We will demonstrate these phenomena for the cubic Klein-Gordon equation on a square with Dirichlet boundary conditions using normal form techniques. The results are qualitatively and quantitatively very different from the one-dimensional spatial case.

Analytical Solution for Formation Flying Problem near Equatorial-Circular Reference Orbit

The relative motion between multiple satellites is a developed technique with many applications. Formation-flying missions use the relative motion dynamics in their design. In this work, the motion in invariant relative orbits is considered under the effects of second-order zonal harmonics in an equatorial orbit. The Hamiltonian framework is used to formulate the problem. All the possible conditions of the invariant relative motion are obtained with different inclinations of the follower satellite orbits. These second-order conditions warrantee the drift rates keeping two, or more, neighboring orbits from drifting apart. The conditions have been modeled. All the possibilities of choosing mean elements of the leader satellite orbit and differences in momenta between leader and follower satellites’ orbits are presented.

Stochastic electron heating in the laser and quasi-static electric and magnetic fields

The dynamics of relativistic electrons in the intense laser radiation and quasi-static electromagnetic fields both along and across the laser propagating direction are studied in the 3/2 dimensional (3/2D) Hamiltonian framework. It is shown that the unperturbed oscillations of the relativistic electron in these electric fields could exhibit a long tail of the amplitude of harmonics which makes an onset of stochastic electron motion be a primary candidate for electron heating. Chirikov-like mappings which describe the recurrence relations of electron energy and time passing through zero canonical momentum plane are derived, and then, the criteria for instability are obtained. It follows that for both transverse and longitudinal electric fields, there exist upper limits of the stochastic electron energy depending on the laser intensity and electric field strength. These maximum energies could be increased by a weak electric field. However, the maximum energy is reduced for the superluminal phase velocity in both cases. The impacts of the magnetic fields on the electron dynamics are different for these two cases and discussed qualitatively. These analytic results are confirmed by the numerical simulations of solving the 3/2D Hamiltonian equations directly.

Time-Varying Impedance Control of Port Hamiltonian System with a New Energy-Storing Tank

In order to guarantee the passivity of a kind of conservative system, the port Hamiltonian framework combined with a new energy tank is proposed in this paper. A time-varying impedance controller is designed based on this new framework. The time-varying impedance control method is an extension of conventional impedance control and overcomes the singularity problem that existed in the traditional form of energy tank. The validity of the controller designed in this paper is shown by numerical examples. The simulation results show that the proposed controller can not only eliminate the singularity problem but can also improve the control performance.

Symplectic Analysis of Wrinkles in Elastic Layers With Graded Stiffnesses

Wrinkles in layered neo-Hookean structures were recently formulated as a Hamiltonian system by taking the thickness direction as a pseudo-time variable. This enabled an efficient and accurate numerical method to solve the eigenvalue problem for onset wrinkles. Here, we show that wrinkles in graded elastic layers can also be described as a time-varying Hamiltonian system. The connection between wrinkles and the Hamiltonian system is established through an energy method. Within the Hamiltonian framework, the eigenvalue problem of predicting wrinkles is defined by a series of ordinary differential equations with varying coefficients. By modifying the boundary conditions at the top surface, the eigenvalue problem can be efficiently and accurately solved with numerical solvers of boundary value problems. We demonstrated the accuracy of the symplectic analysis by comparing the theoretically predicted displacement eigenfunctions, critical strains, and wavelengths of wrinkles in two typical graded structures with finite element simulations.

Nonlinear static response of low-aspect-ratio inverted flags subjected to a steady flow

The nonlinear static response of inverted flags, otherwise known as cantilevered plates in a reverse steady axial flow, is investigated. Recent studies have shown that the stability of low-aspect-ratio inverted flags in fluid flow is different to that of wide inverted flags. The latter undergo a supercritical divergence with increasing flow velocity, while a low-aspect-ratio inverted flag buckles abruptly within a wide bistable region. This paper aims to understand the effect of aspect ratio on the stability and bifurcation characteristics of the multiple equilibria of low-aspect-ratio inverted flags. To this end, the inverted flag is idealized as a thin Euler–Bernoulli beam coupled with the time-invariant form of the fluid forces obtained from a potential flow theory. A Hamiltonian framework is employed to derive the nonlinear integro-differential equation governing the motion expressed in terms of the mid-line rotation angle of the flag along its longitudinal axis. This allows for reliable predictions even at very large deflection amplitudes. The presence of an initial curvature along the length of the flag is also accounted for in the modelling. The Galerkin modal decomposition technique along with a Newton method is employed and the fixed-points of the governing equation are found. Bifurcation diagrams of the response with respect to the flow velocity are constructed. It is shown that the system with small aspect ratio undergoes a static divergence via a subcritical pitchfork bifurcation followed by a saddle–node bifurcation. Supercritical pitchfork bifurcations are expected for sufficiently large aspect ratios. In addition, numerical simulations are performed to investigate the effect of initial curvature amplitude on the nonlinear response of the system. The numerical predictions stemming from the proposed model compare well with existing experimental measurements available in the literature. However, it should be noted that some of the reported static responses of the inverted flag might violate the assumption of potential flow theory.

Hamiltonian structure and connection dynamics of Weyl gravity

A crucial property of Weyl gravity is its conformal invariance. It is shown how this gauge symmetry is exactly reflected by the two constraints in the Hamiltonian framework. Since the spatial 3-metric is one of the configuration variables, the phase space of Weyl gravity can be extended to include internal gauge freedom by triad formalism. Moreover, by canonical transformations, we obtain two new Hamiltonian formulations of Weyl gravity with an SU(2) connection as one of its configuration variables. The connection-dynamical formalisms lay the foundation to quantize Weyl gravity nonperturbatively by applying the method of loop quantum gravity. In one of the formulations, the so-called Immirzi parameter ambiguity in loop quantum gravity is avoided by the conformal invariance.

Vortex sheets and diffeomorphism groupoids

In 1966 V. Arnold suggested a group-theoretic approach to ideal hydrodynamics in which the motion of an inviscid incompressible fluid is described as the geodesic flow of the right-invariant L2-metric on the group of volume-preserving diffeomorphisms of the flow domain. Here we propose geodesic, group-theoretic, and Hamiltonian frameworks to include fluid flows with vortex sheets. It turns out that the corresponding dynamics is related to a certain groupoid of pairs of volume-preserving diffeomorphisms with common interface. We also develop a general framework for Euler–Arnold equations for geodesics on groupoids equipped with one-sided invariant metrics.

Gauge invariant variables for cosmological perturbation theory using geometrical clocks

Using the extended ADM-phase space formulation in the canonical framework we analyze the relationship between various gauge choices made in cosmological perturbation theory and the choice of geometrical clocks in the relational formalism. We show that various gauge invariant variables obtained in the conventional analysis of cosmological perturbation theory correspond to Dirac observables tied to a specific choice of geometrical clocks. As examples, we show that the Bardeen potentials and the Mukhanov–Sasaki variable emerge naturally in our analysis as observables when gauge fixing conditions are determined via clocks in the Hamiltonian framework. Similarly other gauge invariant variables for various gauges can be systematically obtained. We demonstrate this by analyzing five common gauge choices: longitudinal, spatially flat, uniform field, synchronous and comoving gauge. For all these, we apply the observable map in the context of the relational formalism and obtain the corresponding Dirac observables associated with these choices of clocks. At the linear order, our analysis generalizes the existing results in canonical cosmological perturbation theory twofold. On the one hand we can include also gauges that can only be analyzed in the context of the extended ADM-phase space and furthermore, we obtain a set of natural gauge invariant variables, namely the Dirac observables, for each considered choice of gauge conditions. Our analysis provides insights on which clocks should be used to extract the relevant natural physical observables both at the classical and quantum level. We also discuss how to generalize our analysis in a straightforward way to higher orders in the perturbation theory to understand gauge conditions and the construction of gauge invariant quantities beyond linear order.

Vibration analysis of shear-deformable AFG imperfect beams

The vibrations of axially functionally graded (AFG) beams is examined incorporating shear deformation and an imperfection. A coupled nonlinear model is developed in the Hamiltonian framework for an AFG imperfect beam. Different sources of displacement nonlinearities are involved in this problem, namely, the geometrical nonlinearities due to the large deformations, and the nonlinearities due to the presence of a geometrical imperfection along the length of the beam. Expressions for the virtual work of external excitation and damping as well as formulas for the motion and elastic energies of the system are inserted into Hamilton’s energy/work principle. The equations are truncated employing Galerkin’s technique while incorporating symmetric as well as asymmetric modes. It is shown that the nonuniform material properties along with the tapered geometry of the beam as well as the presence of geometric imperfections results in the contribution of both asymmetric and symmetric vibration modes. Hence, a well-optimised scheme on the basis of a continuation method is developed to solve the high DOF nonlinear problem. Numerical simulations are performed to investigate the influences of the gradient index, the imperfection and the taper values.

Extended Quantum Field Theory, Index Theory, and the Parity Anomaly

We use techniques from functorial quantum field theory to provide a geometric description of the parity anomaly in fermionic systems coupled to background gauge and gravitational fields on odd-dimensional spacetimes. We give an explicit construction of a geometric cobordism bicategory which incorporates general background fields in a stack, and together with the theory of symmetric monoidal bicategories we use it to provide the concrete forms of invertible extended quantum field theories which capture anomalies in both the path integral and Hamiltonian frameworks. Specialising this situation by using the extension of the Atiyah-Patodi-Singer index theorem to manifolds with corners due to Loya and Melrose, we obtain a new Hamiltonian perspective on the parity anomaly. We compute explicitly the 2-cocycle of the projective representation of the gauge symmetry on the quantum state space, which is defined in a parity-symmetric way by suitably augmenting the standard chiral fermionic Fock spaces with Lagrangian subspaces of zero modes of the Dirac Hamiltonian that naturally appear in the index theorem. We describe the significance of our constructions for the bulk-boundary correspondence in a large class of time-reversal invariant gauge-gravity symmetry-protected topological phases of quantum matter with gapless charged boundary fermions, including the standard topological insulator in 3+1 dimensions.

Stability and nonlinear dynamical analysis of functionally graded microplates

This paper conducts an investigation into the nonlinear dynamical analysis of a functionally graded microplate resting on an elastic foundation. The first part of the paper is devoted to formulate the forced nonlinear mathematical model of the functionally graded microsystem in the Hamiltonian framework. To this end, the von Karman plate theory together with the modified couple stress theory is used to obtain the in-plane and transverse motions of the microplate; the Mori–Tanaka homogenisation scheme is employed to take into account the variable material properties of the microplate through its thickness. A Winkler-type elastic foundation is considered as the bed of the microplate; a series of linear and nonlinear springs are attached to the microplate so as to approximate the elastic characteristics of the foundation. In the second part of this study, the resonant behaviour of the functionally graded microplate is obtained by numerical simulations via use of a parameter-continuation technique together with direct time-integration scheme. Moreover, the influences of the key parameters of the system (such as the material index and foundation’s linear and nonlinear coefficients) affecting the bifurcations on the forced resonant behaviour of the functionally graded microplate are explored through extensive numerical simulations. The outcome of this paper, which is the size-dependent nonlinear resonant oscillations, can readily be used in MEMS industry, where functionally graded microplates are used as deformable electrodes.

A Nonlinear Bike Model for Purposes of Controller and Observer Design

This paper introduces a nonlinear model for a bike including self-stability behaviour and nonholonomic constraints. The model is derived in the Lagrangian and Hamiltonian framework yielding a pseudo-Hamiltonian system, that is suitable for a variety of systematic synthesis, such as passivity-based controller and observer. The model is, furthermore, simplified in a structure preserving manner to be described by constant tensors. The model validity is investigated by different simulations and comparison to a benchmark model from the literature.

Hamiltonian systems discrete-time approximation: Losslessness, passivity and composability

In this paper a passive integrator dedicated to input/output Hamiltonian systems approximation is presented. In a first step, a discrete Hamiltonian framework endowed with a Lie derivative-like formula is introduced. It is shown that the discrete dynamics encodes energy conservation and passivity. Additionally, the characterization of the discrete dynamics in terms of Dirac structure is shown to be invariant by interconnection. The class is thus composable: networked systems belong to the class. In a second step, the discrete dynamics is considered as a one-step integration method. The method is shown to be convergent and provides a discrete-time approximation of an input/output Hamiltonian system. Accordingly, the discrete dynamics inherits intrinsic energetic characteristics (storage function and dissipation rate) from the original system. The method is thus tagged as passive integrator. As an illustration, the closed-loop behavior of interconnected subsystems and the stabilization of a rigid body spinning around its center of mass are presented.

Effective Floquet Hamiltonian theory of multiple-quantum NMR in anisotropic solids involving quadrupolar spins: Challenges and Perspectives.

The response of a quadrupolar nucleus (nuclear spin with I >12) to an oscillating radio-frequency pulse/field is delicately dependent on the ratio of the quadrupolar coupling constant to the amplitude of the pulse in addition to its duration and oscillating frequency. Consequently, analytic description of the excitation process in the density operator formalism has remained less transparent within existing theoretical frameworks. As an alternative, the utility of the "concept of effective Floquet Hamiltonians" is explored in the present study to explicate the nuances of the excitation process in multilevel systems. Employing spin I =32 as a case study, a unified theoretical framework for describing the excitation of multiple-quantum transitions in static isotropic and anisotropic solids is proposed within the framework of perturbation theory. The challenges resulting from the anisotropic nature of the quadrupolar interactions are addressed within the effective Hamiltonian framework. The possible role of the various interaction frames on the convergence of the perturbation corrections is discussed along with a proposal for a "hybrid method" for describing the excitation process in anisotropic solids. Employing suitable model systems, the validity of the proposed hybrid method is substantiated through a rigorous comparison between simulations emerging from exact numerical and analytic methods.

Hamiltonian analysis of the cuscuton

The cuscuton was introduced in the context of cosmology as a field with infinite speed of propagation. It has been claimed to resemble Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity in a certain limit, and it is a good candidate for an ether theory in which a time-dependent cosmological constant appears naturally. The analysis of its properties is usually performed in the Lagrangian framework, which makes issues like the counting of its dynamical degrees of freedom less clear-cut. Here we perform a Hamiltonian analysis of the theory. We show that the homogeneous limit with local degrees of freedom has singular behavior in the Hamiltonian framework. In other frames, it has an extra scalar degree of freedom. The homogeneous field has regular behavior only if defined a priori as a spatially constant field in a constant mean curvature foliation and contributing with a single global degree of freedom. Lastly, we find conditions on the cuscuton potential for the resulting lapse function to be nonzero throughout evolution.

Structure-preserving properties of Störmer-Verlet scheme for mathematical pendulum

The structure-preserving property, in both the time domain and the frequency domain, is an important index for evaluating validity of a numerical method. Even in the known structure-preserving methods such as the symplectic method, the inherent conservation law in the frequency domain is hardly conserved. By considering a mathematical pendulum model, a Stormer-Verlet scheme is first constructed in a Hamiltonian framework. The conservation law of the Stormer-Verlet scheme is derived, including the total energy expressed in the time domain and periodicity in the frequency domain. To track the structure-preserving properties of the Stormer-Verlet scheme associated with the conservation law, the motion of the mathematical pendulum is simulated with different time step lengths. The numerical results illustrate that the Stormer-Verlet scheme can preserve the total energy of the model but cannot preserve periodicity at all. A phase correction is performed for the Stormer-Verlet scheme. The results imply that the phase correction can improve the conservative property of periodicity of the Stormer-Verlet scheme.