Nonlinear Hamiltonian Research Articles

Observer Design for a Class of Nonlinear Hamiltonian Systems

In this paper, the observer design problem for Port-Controlled Hamiltonian systems is approached. It is considered a particular class of these systems and a full-order order observer is proposed which belongs to the Structure Preserving approach since it is a copy of the original system with an output corrective term. Concerning the class of systems, it corresponds to nonlinear systems that exhibit nonlinearities given by the products between components of the state vector. The fact that this kind of behavior corresponds to a special property of the interconnection matrix is exploited to obtain a representation for the estimation error dynamics that is suitable to formally state the convergence properties of the observer. The usefulness of the contribution is illustrated by solving the speed observation problem for a Permanent Magnet Synchronous Motor.

Thermalization and its breakdown for a large nonlinear spin

By developing a semi-classical analysis based on the Eigenstate Thermalization Hypothesis, we determine the long time behavior of a large spin evolving with a nonlinear Hamiltonian. Despite integrable classical dynamics, we find the Eigenstate Thermalization Hypothesis for the diagonal matrix elements of observables is satisfied in the majority of eigenstates, and thermalization of long time averaged observables is generic. The exception is a novel mechanism for the breakdown of thermalization based on an unstable fixed point in the classical dynamics. Using the semi-classical analysis we derive how the equilibrium values of observables encode properties of the initial state. This analysis shows an unusual memory effect in which the remembered initial state property is not conserved in the integrable classical dynamics. We conclude with a discussion of relevant experiments and the potential generality of this mechanism for long time memory and the breakdown of thermalization.

Sharp DNA denaturation in a helicoidal mesoscopic model

AbstractThe Peyrard-Bishop DNA model describes the molecular interactions with simple potentials which allow efficient calculations of melting temperatures. However, it is based on a Hamiltonian that does not consider the helical twist or any other relevant molecular dimensions. Here, we start from a more realistic 3D model and work out several approximations to arrive at a new non-linear 1D Hamiltonian with a twist angle dependence. Our approximations were numerically compared to full 3D calculations, and established its validity in the regime of small angles. For long DNA sequences we obtain sharp, first-order-like melting, transitions.

On the Reducibility of a Class of Nonlinear Periodic Hamiltonian Systems with Degenerate Equilibrium

In this paper, we prove the existence of periodic solutions of a class of Hamiltonian systems with degenerate equilibriums under small nonlinear periodic perturbations. Actually we prove that the periodic Hamiltonian systems with small perturbations can be reducible to a periodic Hamiltonian system with an equilibrium by a periodic symplectic mapping. This result is a reformulation of the result in Lu and Xu (Nonlinear Differ Equ Appl 21:361–370, 2014) in the case of Hamiltonian systems.

A Kernel-Based explicit unconditionally stable scheme for Hamilton-Jacobi equations on nonuniform meshes

In [11], the authors developed a class of high-order numerical schemes for the Hamilton-Jacobi (H-J) equations, which are unconditionally stable, yet take the form of an explicit scheme. This paper extends such schemes, so that they are more effective at capturing sharp gradients, especially on nonuniform meshes. In particular, we modify the weighted essentially non-oscillatory (WENO) methodology in the previously developed schemes by incorporating an exponential basis and adapting the previously developed nonlinear filters used to control oscillations. The main advantages of the proposed schemes are their effectiveness and simplicity, since they can be easily implemented on higher-dimensional nonuniform meshes. We perform numerical experiments on a collection of examples, including H-J equations with linear, nonlinear, convex and non-convex Hamiltonians. To demonstrate the flexibility of the proposed schemes, we also include test problems defined on non-trivial geometry.

The Emergence of Chaos in Quantum Mechanics

Nonlinearity in Quantum Mechanics may have extrinsic or intrinsic origins and is a liable route to a chaotic behaviour that can be of difficult observations. In this paper, we propose two forms of nonlinear Hamiltonian, which explicitly depend upon the phase of the wave function and produce chaotic behaviour. To speed up the slow manifestation of chaotic effects, a resonant laser field assisting the time evolution of the systems causes cumulative effects that might be revealed, at least in principle. The nonlinear Schrödinger equation is solved within the two-state approximation; the solution displays features with characteristics similar to those found in chaotic Classical Mechanics: sensitivity on the initial state, dense power spectrum, irregular filling of the Poincaré map and exponential separation of the trajectories of the Bloch vector σ in the Bloch sphere.

Statistical Mechanics of Freely Fluctuating Two-Dimensional Elastic Crystals

In this paper, some facts related Joel L. Lebowitz are mentioned. In addition, some of the known theory concerning the statistical physics of freely fluctuating two-dimensional crystals subject to a non-linear elastic hamiltonian is described. Particular topics that are discussed include the existence of two-dimensional crystals in relation to the Hohenberg–Mermin–Wagner theorem, the crumpling transition for freely suspended crystalline membranes and the renormalization of elastic moduli. Although much of what is known has been uncovered since the mid-80s, the topic has become of interest once again due to the discovery of graphene and other two-dimensional crystals. The field is vast so it is the aim of this note to describe some of its fundamental properties.

Optimal estimation with quantum optomechanical systems in the nonlinear regime

We study the fundamental bounds on precision measurements of parameters contained in a time-dependent nonlinear optomechanical Hamiltonian, which includes the nonlinear light-matter coupling, a mechanical displacement term, and a single-mode mechanical squeezing term. By using a recently developed method to solve the dynamics of this system, we derive a general expression for the quantum Fisher information and demonstrate its applicability through three concrete examples: estimation of the strength of a nonlinear light-matter coupling, the strength of a time-modulated mechanical displacement, and a single-mode mechanical squeezing parameter, all of which are modulated at resonance. Our results can be used to compute the sensitivity of a nonlinear optomechanical system to a number of external and internal effects, such as forces acting on the system or modulations of the light--matter coupling.

Time evolution of two harmonic oscillators with cross-Kerr interactions

We study the time evolution of two coupled quantum harmonic oscillators interacting through nonlinear optomechanical-like Hamiltonians that include cross-Kerr interactions. We employ techniques developed to decouple the time-evolution operator and obtain the analytical solution for the time evolution of the system. We apply these results to obtain explicit expressions of a few quantities of interest. Our results do not require approximations and therefore allow us to study the nature and implications of the full nonlinearity of the system. As a potential application, we show that it is possible to greatly increase the population of phonons using a suitable combination of cubic and cross-Kerr interactions.

Time-evolution of nonlinear optomechanical systems: interplay of mechanical squeezing and non-Gaussianity

We solve the time evolution of a nonlinear optomechanical Hamiltonian with arbitrary time-dependent mechanical displacement, mechanical single-mode squeezing and a time-dependent optomechanical coupling up to the solution of two second-order differential equations. The solution is based on identifying a minimal and finite Lie algebra that generates the time-evolution of the system. This reduces the problem to considering a finite set of coupled ordinary differential equations of real functions. To demonstrate the applicability of our method, we compute the degree of non-Gaussianity of the time-evolved state of the system by means of a measure based on the relative entropy of the non-Gaussian state and its closest Gaussian reference state. We find that the addition of a constant mechanical squeezing term to the standard optomechanical Hamiltonian generally decreases the overall non-Gaussian character of the state. For sinusoidally modulated squeezing, the two second-order differential equations mentioned above take the form of the Mathieu equation. We derive perturbative solutions for a small squeezing amplitude at parametric resonance and show that they correspond to the rotating-wave approximation at times larger than the scale set by the mechanical frequency. We find that the non-Gaussianity of the state increases with both time and the squeezing parameter in this specific regime.

Finite-time robust control for transformerless H-bridge cascaded STATCOM with star configuration

This article presents a finite-time robust control (FTRC) of a transformerless STATCOM based on a cascaded multilevel H-bridge converter (CMHC) with star configuration. The FTRC is first proposed for the current loop control of a CMHC-based transformerless STATCOM by using the finite time robust control theory. Taking the parameters, perturbations and external disturbances into account and using coordinate transformation method, the nonlinear dynamic model of the CMHC-based transformerless STATCOM is transformed into a standard nonlinear port-controlled dissipative Hamiltonian (PCDH) structure. Based on the PCDH structure, an FTRC is designed for the CMHC-based transformerless STATCOM to improve the transient stability and oscillation damping of power system. Finally, the simulation results demonstrate that the FTRC has better dynamic performance and strong robustness in comparison with the passivity-based control of the CMHC-based transformerless STATCOM.

Photon gas at the Planck scale within the doubly special relativity

Within the approach to doubly special relativity (DSR) suggested by Magueijo and Smolin, a new algebraically justified rule of so-called κ-addition for the energies of identical particles is proposed. This rule permits to introduce the nonlinear κ-dependent Hamiltonian for one-mode multi-photon (sub)system. On its base, with different modes treated as independent, the thermodynamics of black-body radiation is explored within DSR, and main thermodynamic quantities are obtained. In their derivation, we use both the analytical tools within mean field approximation (MFA) and numerical evaluations based on exact formulas. The entropy of one-mode subsystem turns out to be finite (bounded). Another unusual result is the existence of threshold temperature above which radiation is present. Specific features of the obtained results are explained and illustrated with a number of plots. Comparison with some works of relevance is given.

Finite-time stability and stabilization of nonlinear singular time-delay systems via Hamiltonian method

This paper investigates the finite-time stability (FTS) and finite-time stabilization for a class of nonlinear singular time-delay Hamiltonian systems, and proposes a number of new results on these issues. Firstly, an equivalent form is obtained for the nonlinear singular time-delay Hamiltonian systems by the singular matrix decomposition method, based on which some delay-independent and delay-dependent conditions on the FTS are derived for the systems by constructing a kind of novel Lyapunov function. Secondly, we use the equivalent form as well as the energy shaping plus damping injection technique to investigate the finite-time stabilization problem for a class of nonlinear singular port-controlled Hamiltonian (PCH) systems with time delay, and present a specific control design procedure for the systems. Finally, we give several illustrative examples to show the effectiveness of the results obtained in this paper.

Robust $${{\cal H}_\infty}$$ Control for Switched Nonlinear Port-controlled Hamiltonian Systems

In the paper, the robust $${{\cal H}_\infty}$$ control for switched nonlinear port-controlled Hamiltonian (SNPCH) systems with external disturbances is investigated via the multiple Lyapunov functions method. Under two cases that the SNPCH system is subject to actuator saturation (AS) and the SNPCH system is without AS, a design approach for the switching robust $${{\cal H}_\infty}$$ feedback controller is developed to attenuate the external disturbances. The sufficient conditions are established for $${{\cal H}_\infty}$$ control of the corresponding closed-loop systems based on Hamilton-Jacobi inequality techniques. Furthermore, via the Hamiltonian realization method, the obtained results are applied to solve the $${{\cal H}_\infty}$$ control for switched nonlinear affine systems with external disturbances. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed $${{\cal H}_\infty}$$ control methods.

Enhanced continuous generation of non-Gaussianity through optomechanical modulation

We study the non-Gaussian character of quantum optomechanical systems evolving under the fully nonlinear optomechanical Hamiltonian. By using a measure of non-Gaussianity based on the relative entropy of an initially Gaussian state, we quantify the amount of non-Gaussianity induced by both a constant and time-dependent cubic light–matter coupling and study its general and asymptotic behaviour. We find analytical approximate expressions for the measure of non-Gaussianity and show that initial thermal phonon occupation of the mechanical element does not significantly impact the non-Gaussianity. More importantly, we also show that it is possible to continuously increase the amount of non-Gassuianity of the state by driving the light–matter coupling at the frequency of mechanical resonance, suggesting a viable mechanism for increasing the non-Gaussianity of optomechanical systems even in the presence of noise.

Many-body effects in quantum metrology

We study the impact of many-body effects on the fundamental precision limits in quantum metrology. On the one hand such effects may lead to nonlinear Hamiltonians, studied in the field of nonlinear quantum metrology, while on the other hand they may result in decoherence processes that cannot be described using single-body noise models. We provide a general reasoning that allows to predict the fundamental scaling of precision in such models as a function of the number of atoms present in the system. Moreover, we describe a computationally efficient approach that allows for a simple derivation of quantitative bounds. We illustrate these general considerations by a detailed analysis of fundamental precision bounds in a paradigmatic atomic interferometry experiment with standard linear Hamiltonian but with both single and two-body losses taken into account—a model which is motivated by the most recent Bose–Einstein condensate magnetometry experiments. Using this example we also highlight the impact of the atom number super-selection rule on the possibility of protecting interferometric protocols against decoherence.

Recurrence network analysis in a model tripartite quantum system

In a novel approach to quantum dynamics, we apply the tools of the recurrence network analysis to the dynamics of the quantum-mechanical expectation values of observables. We construct and analyse ϵ-recurrence networks from the time-series data of the mean photon number in a model tripartite quantum system governed by a nonlinear Hamiltonian. The role played by the intensity-dependent field-atom coupling in the dynamics is investigated. Interesting features emerge as a function of a parameter characterising this intensity-dependent coupling in both the short-time and the long-time dynamics. In particular, we examine the manner in which standard measures of the network theory such as the average path length, the link density and the clustering coefficient, depend on this parameter.

Quantum Tomography of Time-Dependent Nonlinear Hamiltonian Systems

We discuss several models of nonlinear oscillators within the framework of a tomographic probability representation of quantum mechanics. Using the connection between the Green's function and the integrals of motion of quantum systems with the time-dependent Schrodinger equation with variable nonlinear Hamiltonians, the explicit tomograms for such systems are found. The case of quadratic and quasi-quadratic Hamiltonians are studied in detail.

Sensitivity bounds for interferometry with Ising Hamiltonians

Sensitivity bounds for a generic interferometric phase estimation problem are the shot noise and the Heisenberg limits. The shot noise is the highest sensitivity that can be reached with separable states, while the Heisenberg limit is the ultimate bound in sensitivity which can be saturated with entangled states. The scaling of these bounds with the number of particles $N$ entering the interferometer depends on the specific Hamiltonian governing the same interferometer. In typical cases, the Hamiltonian is linear, and the shot-noise and Heisenberg limits scale with ${N}^{\ensuremath{-}1/2}$ and with ${N}^{\ensuremath{-}1}$, respectively. With interferometers described by generic, nonlinear Hamiltonian, the scalings with the number of particles can be rather different. Here we study the shot-noise and Heisenberg limits of Ising-like Hamiltonians in the presence of longitudinal and transverse fields, both in the nearest-neighbor and in the fully connected spin interaction cases. We provide the explicit forms of the states saturating the shot-noise and Heisenberg limits. These results can be relevant not only for precision measurement purposes but also to characterize quantum phase transitions and, more generally, the witnessing of multipartite entanglement.

Breaking of Goldstone modes in a two-component Bose-Einstein condensate

We study the decay rate $\Gamma(k)$ of density excitations of two-component Bose-Einstein condensates at zero temperature. Those excitations, where the two components oscillate in phase, include the Goldstone mode resulting from condensation. While within Bogoliubov approximation the density sector and the spin (out-of-phase) sector are independent, they couple at the three-phonon level. For a Bose-Bose mixture we find that the Belyaev decay is slightly modified due to the coupling with the gapless spin mode. At the phase separation point the decay rate changes instead from the standard $k^5$ to a $k^{5/2}$ behaviour due to the parabolic nature of the spin mode. In presence of coherent coupling between the two components the spin sector is gapped and, away from the ferromagnetic-like phase transition point, the decay of density mode is not affected. On the other hand at the transition point, when the spin fluctuations become critical, the Goldstone mode is not well defined anymore since $\Gamma(k)\propto k$. As a consequence, we show that the friction induced by a moving impurity is enahnced -- a feature which could be experimentally tested. Our results apply to every non-linear 2-component quantum hydrodynamic Hamiltonian which is time-reversal invariant, and possesses an $U(1)\times {\mathbf Z}_2$ symmetry.