Showing papers on "Floquet theory published in 2005"

Superfluid-Insulator Transition in a Periodically Driven Optical Lattice

Abstract: We demonstrate that the transition from a superfluid to a Mott insulator in the Bose-Hubbard model can be induced by an oscillating force through an effective renormalization of the tunneling matrix element. The mechanism involves adiabatic following of Floquet states, and can be tested experimentally with Bose-Einstein condensates in periodically driven optical lattices. Its extension from small to very large systems yields nontrivial information on the condensate dynamics.

Symmetry breaking of two-dimensional time-periodic wakes

Abstract: A number of two-dimensional time-periodic flows, for example the Karman street wake of a symmetrical bluff body such as a circular cylinder, possess a spatio-temporal symmetry: a combination of evolution by half a period in time and a spatial reflection leaves the solution invariant. Floquet analyses for the stability of these flows to three-dimensional perturbations have in the past been based on the Poincare map, without attempting to exploit the spatio-temporal symmetry. Here, Floquet analysis based on the half-period- flip map provides a comprehensive interpretation of the symmetry breaking bifurcations.

A Floquet-Bloch Decomposition of Maxwell's Equations Applied to Homogenization

Abstract: Using Bloch waves to represent the full solution of Maxwell's equations in periodic media, we study the limit where the material's period becomes much smaller than the wavelength. It is seen that for steady state fields, only a few of the Bloch waves contribute to the full solution. Effective material parameters can be explicitly represented in terms of dyadic products of the mean values of the nonvanishing Bloch waves, providing a new means of homogenization. The representation is valid for an arbitrary wave vector in the first Brillouin zone.

Spectral Stability of Periodic Solutions of Viscous Conservation Laws: Large Wavelength Analysis

Abstract: We complete and unify the works by Oh & Zumbrun [13] and by the author [16], about the spectral stability of traveling waves that are spatially periodic, in systems of n conservation laws. Our context is one-dimensional. These systems are of order larger than one, in general. For instance, they could be viscous approximation of first-order systems that are not everywhere hyperbolic. However, modelling considerations often lead to higher-order terms, like capillarity in fluid dynamics ; our framework remains valid in this more general setting. We make generic assumptions, saying in particular that the set of periodic traveling waves is a manifold of maximal dimension, under the restrictions given by the conserved quantities. The spectral stability of a periodic traveling wave is studied through Floquet’s theory. Following Gardner [4], we introduce an Evans function D(‚;µ), with ‚ the Laplace frequency and µ the phase shift. The large wavelength analysis is the description of the zero set of D around the origin. Our main result tells that this zero set is described, at the leading order, by a characteristic equation det(‚IN i iµ@F(U)=@U) = 0: This formula involves a flux F, which enters in a first-order system of conservation laws @tU + @xF(U) = 0; describing the slow modulation of the periodic traveling waves. Its size N is in practice larger than n. The important consequence is that hyperbolicity of the latter system is a necessary condition for spectral stability of periodic traveling waves.

Linear response at the 4-component relativistic density-functional level: application to the frequency-dependent dipole polarizability of Hg, AuH and PtH2

Abstract: We report the implementation and application of linear response density-functional theory (DFT) based on the 4-component relativistic Dirac–Coulomb Hamiltonian. The theory is cast in the language of second quantization and is based on the quasienergy formalism (Floquet theory), replacing the initial state dependence of the Runge–Gross theorem by periodic boundary conditions. Contradictions in causality and symmetry of the time arguments are thereby avoided and the exchange-correlation potential and kernel can be expressed as functional derivatives of the quasienergy. We critically review the derivation of the quasienergy analogues of the Hohenberg–Kohn theorem and the Kohn–Sham formalism and discuss the nature of the quasienergy exchange-correlation functional. Structure is imposed on the response equations in terms of Hermiticity and time-reversal symmetry. It is observed that functionals of spin and current densities, corresponding to time-antisymmetric operators, contribute to frequency-dependent and not static electric properties. Physically, this follows from the fact that only a time-dependent electric field creates a magnetic field. It is furthermore observed that hybrid functionals enhance spin polarization since only exact exchange contributes to anti-Hermitian trial vectors. We apply 4-component relativistic linear response DFT to the calculation of the frequency-dependent polarizability of the isoelectronic series Hg, AuH and PtH2. Unlike for the molecules, the effect of electron correlation on the polarizability of the mercury atom is very large, about 25%. We observe a remarkable performance of the local-density approximation (LDA) functional in reproducing the experimental frequency-dependent polarizability of this atom, clearly superior to that of the BLYP and B3LYP functionals. This allows us to extract Cauchy moments (S(−4) = 382.82 and S(−6) = 6090.89 a.u.) that we believe are superior to experiment since we go to higher order in the Cauchy moment expansion.

Nonadiabatic Electron Pumping: Maximal Current with Minimal Noise

Abstract: The noise properties of pump currents through an open double-quantum-dot setup with nonadiabatic ac driving are investigated. Driving frequencies close to the internal resonances of the double-dot system mark the optimal working points at which the pump current assumes a maximum while its noise power possesses a remarkably low minimum. A rotating-wave approximation provides analytical expressions for the current and its noise power and allows to optimize the noise characteristics. The analytical results are compared to numerical results from a Floquet transport theory.

Kubo formula for Floquet states and photoconductivity oscillations in a two-dimensional electron gas

Abstract: high mobility 2DES under the influence of microwave radiation of frequency v at moderate values of the magnetic field exhibits strong oscillations with zero-resistance states sZRSd governed by the ratio v / vc, where vc is the cyclotron frequency. In this work we present a model for the photoconductivity of a 2DES subjected to a magnetic field. The model includes the microwave and Landau contributions in a nonperturbative, exact way, while impurity-scattering effects are treated perturbatively. In our model, the Landau-Floquet states act coherently with respect to the oscillating field of the impurities that in turn induces transitions between these levels. Based on this formalism, we provide a Kubo-like formula that takes into account the oscillatory Floquet structure of the problem. We study the effects of both short-range and long-range disorder on the photoconductivity. Our calculation yields a magnetoresistance oscillatory behavior with the correct period and phase. It is found that, in agreement with experiment, negative dissipation can only be induced in very high mobility samples. We analyze the dependence of the results on the microwave power and polarization. For highintensity radiation, multiphoton processes take place predicting negative-resistance states centered at v / vc = 1 2 and v / vc = 3 2 .

Decoupling and recoupling using continuous-wave irradiation in magic-angle-spinning solid-state NMR: a unified description using bimodal Floquet theory.

Abstract: The application of two or more different time-dependent coherent perturbations with, in general, incommensurable frequencies occurs quite commonly in NMR experiments. Here we develop a unified description of the entire class of experiments using bimodal Floquet theory and van Vleck–Primas perturbation theory. This treatment leads to a time-independent effective Hamiltonian in Hilbert space and can be looked at as a generalization of average Hamiltonian theory to several incommensurate time dependencies. As a prototype experiment we treat the application of continuous-wave (cw) radio-frequency irradiation in combination with magic-angle sample spinning. Practically relevant examples of this type of experiments are heteronuclear spin decoupling and recoupling experiments using cw irradiation, e.g., rotary-resonance recoupling. Perturbations up to the third order must be taken into account to explain all experimentally observed resonance conditions.

Modal properties of surface and leaky waves propagating at arbitrary angles along a metal strip grating on a grounded slab

Abstract: A full-wave analysis is presented of the dispersion properties of modes supported by a grounded dielectric slab periodically loaded with metal strips, which represents a canonical configuration employed in planar microwave antennas and arrays and in the realization of artificially hard and soft surfaces. Propagation of surface and leaky modes at arbitrary angles is considered here, without any restrictive assumption on the values of the involved physical and geometrical parameters. Spectral properties of modes are studied, by deriving generalized conditions for establishing the proper or improper nature of the spatial harmonics in the Floquet representation of the fields. The proposed approach, based on a full-wave moment-method discretization of the relevant electric-field integral equation in the spectral domain, is validated through comparisons with the available data in the literature. Novel results are presented which illustrate the continuous evolution of modes as a function of the propagation angle along the grating, both in surface and leaky propagation regimes.

A three-dimensional time-domain finite-element formulation for periodic structures

Abstract: A formulation is presented for a three-dimensional time-domain finite-element method that can be used to analyze the scattering of a plane wave obliquely incident on a (doubly) infinite periodic structure using one unit cell. A broadband frequency response can be obtained in a single execution. The specifics of the method are shown for scattering problems, but it should be straightforward to extend it to radiation problems. The method solves for a transformed field variable (instead of solving directly for the electric field) in order to easily enable periodic boundary conditions in the time domain. The accuracy and stability of the method is demonstrated by a series of examples where the new formulation is compared with reference solutions. Accurate results are obtained when the excitation (frequency range) and the geometry are such that no higher order propagating Floquet modes are present. The accuracy is degraded in the presence of higher order propagating modes due to the rather simple absorbing boundary condition that is used with the present formulation. The method is found to be stable even for angles of incidence close to grazing.

A Holling II functional response food chain model with impulsive perturbations

Abstract: In this paper, we investigate a three trophic level food chain system with Holling II functional responses and periodic constant impulsive perturbations of top predator. Conditions for extinction of predator as a pest are given. By using the Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability of predator eradication periodic solution. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows the rich dynamics (for example: period doubling, period halfing, chaos crisis) in the positive octant. The dynamics behavior is found to be very sensitive to the parameter values and initial value.

Pattern reconfigurable leaky-wave antenna design by FDTD method and Floquet's Theorem

Abstract: A pattern reconfigurable millimeter-wave coplanar waveguide leaky-wave antenna is designed by using the finite-difference time-domain (FDTD) method combined with Floquet's theorem and the simple linear interpolation technique. The concept of equivalent period is presented for mixed-periodic structures. The novel design process results in a reduction in the design time compared with that of the traditional FDTD method.

Ordinary differential equations with applications

Abstract: # Fundamental Theory # Linear Systems # Stability of Nonlinear Systems # Method of Lyapunov Functions # Two-Dimensional Systems # Second Order Linear Equations # The Index Theory and Brouwer Degree # Introduction to Perturbation Methods

Adiabatic theorem for non-Hermitian time-dependent open systems

Abstract: In the conventional quantum mechanics i.e., Hermitian quantum mechanics the adiabatic theorem for systems subjected to time-periodic fields holds only for bound systems and not for open ones where ionization and dissociation take placeD. W. Hone, R. Ketzmerik, and W. Kohn, Phys. Rev. A 56, 4045 1997. Here with the help of the t, t formalism combined with the complex scaling method we derive an adiabatic theorem for open systems and provide an analytical criterion for the validity of the adiabatic limit. The use of the complex scaling transformation plays a key role in our derivation. As a numerical example we apply the adiabatic theorem we derived to a one-dimensional model Hamiltonian of Xe atom which interacts with strong, monochromatic sine-square laser pulses. We show that the generation of odd-order harmonics and the absence of hyper-Raman lines, even when the pulses are extremely short, can be explained with the help of the adiabatic theorem we derived. DOI: 10.1103/PhysRevA.72.032103 I. MOTIVATION When matter is exposed to intense laser fields, high harmonics HH’s of the incident radiation may be produced. Usually, only odd harmonics are obtained even when the laser pulses are short for theoretical and experimental work which demonstrates this see 1,2, respectively. Since the duration of the pulse in time is inversely proportional to its width in energy space, one may find this result surprising, as one may expect to obtain also a large distribution of frequencies in the scattered field. Why are only odd harmonics obtained even when the laser pulses are short? For cw lasers and symmetric field-free potential using the non-Hermitian Floquet theory it was proved that only odd harmonics are obtained when the dynamics is controlled by a single-resonance Floquet quasienergy QE state 3,4. When laser pulses are used it was argued that this proof still holds since usually the populated resonance states are associated with very different lifetimes and the dynamics is controlled by the resonance state which has the longest lifetime. However, this argument may hold only when the duration of the laser pulses is large enough to enable the decay of the short-lived resonances. Indeed numerical simulations showed that the harmonic generation spectra HGS as obtained from a single non-Hermitian complex-scaled resonance Floquet state is in remarkable agreement with the results obtained from conventional i.e., Hermitian timedependent simulations 5. The question that is addressed in this work is whether an analytical criterion for the shape and duration of the laser pulse for which the system is controlled by a singleresonance Floquet state can be given. It is obvious that the question regarding the possibility of the population of a single-resonance state is connected with the question regarding the degree of adiabaticity of the process. The question is therefore under which conditions can a short laser pulse be defined as an adiabatic one. The answer to this question is important not only to harmonic generation HG studies but also for other, more general studies where lasers are used to

Chaos in three species food chain system with impulsive perturbations

Abstract: In this paper, we investigate three species food chain system with periodic constant impulsive perturbations of mid-level predator. Conditions for extinction of lowest-level prey and top predator are given. By using the Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability of lowest-level prey and top predator eradication periodic solution. Further, influences of the impulsive perturbations on the inherent oscillation are studied numerically, which shows the rich dynamics (for example: period doubling, period halfing, non-unique dynamics) in the positive octant. The dynamics behavior is found to be very sensitive to the parameter values and initial value.

A food chain model with impulsive perturbations and Holling IV functional response

Abstract: In this paper, we investigate a three trophic level food chain system with Holling IV functional responses and periodic constant impulsive perturbations of top predator. Conditions for extinction of predator are given. By using the Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability of predator eradication periodic solution. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows the rich dynamics in the positive octant.

Magnetic-field symmetry of pump currents of adiabatically driven mesoscopic structures

Abstract: We examine the scattering properties of a slowly and periodically driven mesoscopic sample using the Floquet function approach. One might expect that at sufficiently low driving frequencies it is only the frozen scattering matrix which is important. The frozen scattering matrix reflects the properties of the sample at a given instant of time. Indeed many aspects of adiabatic scattering can be described in terms of the frozen scattering matrix. However, we demonstrate that the Floquet scattering matrix, to first order in the driving frequency, is determined by an additional matrix which reflects the fact that the scatterer is time dependent. This low-frequency irreducible part of the Floquet matrix has symmetry properties with respect to time and/or a magnetic field direction reversal opposite to that of the frozen scattering matrix. Using the adiabatic decomposition of the Floquet scattering matrix we split the dc current flowing through the pump into several parts with well defined properties with respect to a magnetic field inversion.

Multipole-multimode Floquet theory in nuclear magnetic resonance.

Abstract: In this paper, we present a new analytical approach for describing the spin dynamics of synchronous and asynchronous time-dependent modulations in solid-state nuclear magnetic resonance experiments. The approach, based on multimode Floquet theory, employs the multipole operator basis of Sanctuary for spin description and illustrates the time evolution in the Floquet–Liouville space using the effective Hamiltonians obtained from the contact (or van Vleck) transformation procedure. Since the Hamiltonian and the density operator are expressed in terms of irreducible tensor operators, extensions to higher spin magnitudes (I>1∕2) and multiple spins are quite straightforward and permit analytical treatments for many problems. We outline the general underlying principles involved in this approach with a brief mention of its potential application in other branches of spectroscopy.

The Linear Periodic Output Regulation Problem

Abstract: The problem of asymptotic output regulation for linear systems driven by time-varying, T-periodic exosystems is considered in this paper. Necessary and sufficient condition for its solvability based on the existence of periodic solutions of differential Sylvester equations are derived. These conditions constitute a generalization to the periodic case of the celebrated algebraic regulator equations of Francis. A general algorithm for the synthesis of an error-feedback regulator is given. For the special case of minimum-phase systems, it is shown that the regulator design can be carried out without the knowledge of the Floquet decomposition of the exosystem. The issue of robust regulation by error feedback is also briefly addressed.

Floquet theory and stability of nonlinear integro-differential equations

Abstract: One of the classical topics in the qualitative theory of differential equations is the Floquet theory. It provides a means to represent solutions and helps in particular for stability analysis. In this paper first we shall study Floquet theory for integro-differential equations (IDE), and then employ it to address stability problems for linear and nonlinear equations.

Computation of period sensitivity functions for the simulation of phase noise in oscillators

Abstract: Accurate phase noise simulation of circuits for radio frequency applications is of great importance during the design and development of wireless communication systems. In this paper, we present an approach based on the Floquet theory for the analysis and numerical computation of phase noise that solves some drawbacks implicitly present in previously proposed algorithms. In particular, we present an approach that computes the perturbation projection vector directly from the Jacobian matrix of the shooting method adopted to compute the steady-state solution. Further, we address some problems that arise when dealing with circuits whose modeling equations do not satisfy the Lipschitz condition at least from the numerical point of view. Frequency-domain aspects of phase noise analysis are also considered and, finally, simulation results for some benchmark circuits are presented.

A High Order Perturbation Analysis of the Sitnikov Problem

Abstract: The Sitnikov problem is one of the most simple cases of the elliptic restricted three body system. A massless body oscillates along a line (z) perpendicular to a plane (x,y) in which two equally massive bodies, called primary masses, perform Keplerian orbits around their common barycentre with a given eccentricity e. The crossing point of the line of motion of the third mass with the plane is equal to the centre of gravity of the entire system. In spite of its simple geometrical structure, the system is nonlinear and explicitly time dependent. It is globally non integrable and therefore represents an interesting application for advanced perturbative methods. In the present work a high order perturbation approach to the problem was performed, by using symbolic algorithms written in Mathematica. Floquet theory was used to derive solutions of the linearized equation up to 17th order in e. In this way precise analytical expressions for the stability of the system were obtained. Then, applying the Courant and Snyder transformation to the nonlinear equation, algebraic solutions of seventh order in z and e were derived using the method of Poincare–Lindstedt. The enormous amount of necessary computations were performed by extensive use of symbolic programming. We developed automated and highly modularized algorithms in order to master the problem of ordering an increasing number of algebraic terms originating from high order perturbation theory.

Three-dimensional Floquet stability analysis of the flow produced by an oscillating circular cylinder in quiescent fluid

Abstract: We study the two-dimensional symmetry breaking transitions in the time-periodic flow generated in quiescent fluid by a rigid cylinder with simple harmonic rectilinear translation in a direction normal to its axis. The base flow possesses two symmetries: a spatio-temporal symmetry and a spatial reflection symmetry about the axis of oscillation. Two distinct regimes of marginal stability are identified through two-dimensional Floquet analysis. These correspond to (I) a pair of real Floquet multipliers simultaneously crossing the unit circle at µ =+ 1 and (II) a pair of complex-conjugate multipliers crossing the unit circle, µ = e ±iθ (a Neimark–Sacker bifurcation). In both transitions the spatial reflection symmetry of the base flow is broken, but for type I transitions, the spatio–temporal symmetry of the base flow is retained.  2003 Elsevier SAS. All rights reserved.

Stability Analysis of Complex Multibody Systems

Abstract: The linearized stability analysis of dynamical systems modeled using finite element based multibody formulations is addressed in this paper. The use of classical methods for stability analysis of these system, such as the characteristic exponent method or Floquet theory, results in computationally prohibitive costs. Since comprehensive multibody models are “virtual prototypes” of actual systems, the applicability to numerical models of the stability analysis tools that are used in experimental settings is investigated in this work. Various experimental tools for stability analysis are reviewed. It is proved that Prony’s method, generally regarded as a curve fitting method, is equivalent, and sometimes identical, to Floquet theory and to the partial Floquet method. This observation gives Prony’s method a sound theoretical, footing, and considerably improves the robustness of its predictions when applied to comprehensive models of complex multibody system. Numerical applications are presented to demonstrate the efficiency of the proposed procedure.Copyright © 2005 by ASME