Discrete breathers

Nonlinear classical Hamiltonian lattices exhibit generic solutions in the form of discrete breathers. These solutions are time-periodic and (typically exponentially) localized in space. The lattices exhibit discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. Necessary ingredients for their occurrence are the existence of upper bounds on the phonon spectrum (of small fluctuations around the groundstate) of the system as well as the nonlinearity in the differential equations. We will present existence proofs, formulate necessary existence conditions, and discuss structural stability of discrete breathers. The following results will be also discussed: the creation of breathers through tangent bifurcation of band edge plane waves; dynamical stability; details of the spatial decay; numerical methods of obtaining breathers; interaction of breathers with phonons and electrons; movability; influence of the lattice dimension on discrete breather properties; quantum lattices — quantum breathers.Finally we will formulate a new conceptual approach capable of predicting whether discrete breathers exist for a given system or not, without actually solving for the breather. We discuss potential applications in lattice dynamics of solids (especially molecular crystals), selective bond excitations in large molecules, dynamical properties of coupled arrays of Josephson junctions, and localization of electromagnetic waves in photonic crystals with nonlinear response.     

Discrete moving breather collisions in a Klein-Gordon chain of oscillators

We study the collisions of moving breathers with the same frequency, traveling with opposite directions within a Klein-Gordon chain of oscillators. Two types of collisions have been analyzed: symmetric and non-symmetric, head-on collisions. For low enough frequency the outcome is strongly dependent of the dynamical states of the two colliding breathers just before the collision. For symmetric collisions, several results can be observed: breather generation, with the formation of a trapped breather and two new moving breathers; breather reflection; generation of two new moving breathers; and breather fusion bringing about a trapped breather. For non-symmetric collisions some possible results are: breather generation, with the formation of three new moving breathers; breather fusion, originating a new moving breather; breather trapping with breather reflection; generation of two new moving breathers; and two new moving breathers traveling as a bound state. Breather annihilation has never been observed.     

Semiclassical Ehrenfest paths

Trajectories are a central concept in our understanding of classical phenomena and also in rationalizing quantum mechanical effects. In this work we provide a way to determine semiclassical paths, approximations to quantum averages in phase space, directly from classical trajectories. We avoid the need of intermediate steps, like particular solutions to the Schroedinger equation or numerical integration in phase space by considering the system to be initially in a coherent state and by assuming that its early dynamics is governed by the Heller semiclassical approximation. Our result is valid for short propagation times only, but gives non-trivial information on the quantum-classical transition.     

Quantum to classical correspondence for the Fermi-acceleration model

A quantum particle which is confined to the interior of a box with infinitely high but periodically oscillating walls can have an unusual semiclassical limit: For the special case of a one-dimensional linear wall motion we show that the semiclassical domain corresponds to a classical motion in phase space where the initial momentum depends on the particle's position in the box. Another result is that quantum states which correspond to classical cycle-1 fixed points have maximum stability against the boundary induced perturbation (caused by the moving wall). Higher cycle-n fixed points are calculated by numerical bookkeeping up to n = 20. The classical motion is marginally stable. We show how a slight change in the boundary condition will lead to chaotic motion.     

Quantum Breathers in Anisotropy Ferromagnetic Chains with Second-Order Coupling

Under considering the next-nearest-neighbor interaction, quantum breathers in one-dimensional anisotropy ferromagnetic chains are theortically studied. By introducing the Dyson-Maleev transformation for spin operators, a map to a Heisenberg ferromagnetic spin lattice into an extended Bose–Hubbard model can be established. In the case of a small number of bosons, by means of the numerical diagonalization technique, the energy spectrum of the corresponding extended Bose–Hubbard model containing two bosons is calculated. When the strength of the single-ion anisotropy is enough large, a isolated single band appears. This isolated single band corresponds to two-boson bound state, which is the simplest quantum breather state. It is shown that the introduction of the next-nearest-neighbor interaction will lead to interesting band structures. In the case of a large number of bosons, by applying the time-dependent Hartree approximation, quantum breather states for the system is constructed. In this case, the effect of the next-nearest-neighbor interaction on quantum breathers is also analyzed.     

Quantum freeze of fidelity decay for chaotic dynamics

We show that the mechanism of quantum freeze of fidelity decay for perturbations with a zero time average, recently discovered for a specific case of integrable dynamics [New J. Phys. 5, 109 (2003)], can be generalized to arbitrary quantum dynamics. We work out explicitly the case of a chaotic classical counterpart, for which we find semiclassical expressions for the value and the range of the plateau of fidelity. After the plateau ends, we find explicit expressions for the asymptotic decay, which can be exponential or Gaussian depending on the ratio of the Heisenberg time to the decay time. Arbitrary initial states can be considered; e.g., we discuss coherent states and random states.     

Bound states of breathers in the Frenkel-Kontorova model

In the dissipative, driven standard Frenkel-Kontorova model propagating breathers exist as attractors of the dynamics. In collisions, these excitations interact through the phonons they emit. A possible result of a two-breather collision is a bound state of two breathers. After looking at phonons and breather collisions, we present phenomenological results on breather bound states obtained from lattice dynamics simulations. In particular, we find that bound states can be characterised by the distance between the two breathers they comprise and their propagation velocity. Contrary to the single breather case, several values of the propagation velocity are easily accessible to bound states at fixed model parameters. The results are interpreted on the basis of the observed phonon spectra. The latter can easily be explained as Doppler-shifted combination frequencies of breather harmonics and a discreteness-induced perturbation frequency.Received: 18 December 2003, Published online: 15 March 2004PACS: 63.20.Ry Anharmonic lattice modes - 63.20.Pw Localized modes     

Moving breather collisions in Klein-Gordon chains of oscillators

We investigate the collisions of moving breathers, with the same frequency, in three different Klein-Gordon chains of oscillators. The on-site potentials are: the asymmetric and soft Morse potential, the symmetric and soft sine-Gordon potential and the symmetric and hard φ⁴ potential. The simulation of a collision begins generating two identical moving breathers traveling with opposite velocities, they are obtained after perturbing two identical stationary breathers which centers are separated by a fixed number of particles. If this number is odd we obtain an on-site collision, but if this number is even we obtain an inter-site collision. Apart from this distinction, we have considered symmetric collisions, if the colliding moving breathers are vibrating in phase, and anti-symmetric collisions, if the colliding moving breathers are vibrating in anti-phase. The simulations show that the collision properties of the three chains are different. The main observed phenomena are: breather generation with trapping, with the appearance of two new moving breathers with opposite velocities, and a stationary breather trapped at the collision region; breather generation without trapping, with the appearance of new moving breathers with opposite velocities; breather trapping at the collision region, without the appearance of new moving breathers; and breather reflection. For each Klein-Gordon chain, the collision outcomes depend on the lattice parameters, the frequency of the perturbed stationary breathers, the internal structure of the moving breathers and the number of particles that initially separates the stationary breathers when they are perturbed.     

Wave scattering by discrete breathers

We present a theoretical study of linear wave scattering in one-dimensional nonlinear lattices by intrinsic spatially localized dynamic excitations or discrete breathers. These states appear in various nonlinear systems and present a time-periodic localized scattering potential for plane waves. We consider the case of elastic one-channel scattering, when the frequencies of incoming and transmitted waves coincide, but the breather provides with additional spatially localized ac channels whose presence may lead to various interference patterns. The dependence of the transmission coefficient on the wave number q and the breather frequency Omega(b) is studied for different types of breathers: acoustic and optical breathers, and rotobreathers. We identify several typical scattering setups where the internal time dependence of the breather is of crucial importance for the observed transmission properties.     

Modulational instability,quantum breathers and two-breathers in a frustrated ferromagnetic spin lattice under an external magnetic field

The modulational instability, quantum breathers and two-breathers in a frustrated easy-axis ferromagnetic zig-zag chain under an external magnetic field are investigated within the Hartree approximation. By means of a linear stability analysis, we analytically study the discrete modulational instability and analyze the effect of the frustration strength on the discrete modulational instability region. Using the results from the discrete modulational instability analysis, the presence conditions of those stationary bright type localized solutions are presented. On the other hand, we obtain the analytical expressions for the stationary bright localized solutions and analyze the effect of the frustration on their emergence conditions. By taking advantage of these bright type single-magnon bound wave functions obtained, quantum breather states in the present frustrated ferromagnetic zig-zag lattice are constructed. What is more, the analytical forms for quantum two-breather states are also obtained. In particular, the energy level formulas of quantum breathers and two-breathers are derived.     

Coherent States and Modified de Broglie-Bohm Complex Quantum Trajectories

This paper examines the nature of classical correspondence in the case of coherent states at the level of quantum trajectories. We first show that for a harmonic oscillator, the coherent state complex quantum trajectories and the complex classical trajectories are identical to each other. This congruence in the complex plane, not restricted to high quantum numbers alone, illustrates that the harmonic oscillator in a coherent state executes classical motion. The quantum trajectories we consider are those conceived in a modified de Broglie-Bohm scheme. Though quantum trajectory representations are widely discussed in recent years, identical classical and quantum trajectories for coherent states are obtained only in the present approach. We may note that this result for standard harmonic oscillator coherent states is not totally unexpected because of their holomorphic nature. The study is extended to coherent states of a particle in an infinite potential well and that in a symmetric Poschl-Teller potential by solving for the trajectories numerically. For the Gazeau-Klauder coherent state of the infinite potential well, almost identical classical and quantum trajectories are obtained whereas for the Poschl-Teller potential, though classical trajectories are not regained, a periodic motion results as t→∞. Similar features were found for the SUSY quantum mechanics-based coherent states of the Poschl-Teller potential too, but this time the pattern of complex trajectories is quite different from that of the previous case. Thus we find that the method is a potential tool in analyzing the properties of generalized coherent states.     

Vector breather of surface plasmon polaritons in left-handed material

A theory of an optical vector breather of self-induced transparency of surface plasmon polaritons is constructed. The nonlinear surface TM-modes propagating along the interface separating an isotropic medium and anisotropic left-handed material are investigated. A transition layer sandwiched between the connected media is described using a model of a two-dimensional gas of semiconductor quantum dots. Explicit analytical expressions for the shape and parameters of the surface vector breathers are obtained as well as simulations of the space-time dynamics of two-dimensional vector breathers presented with realistic parameters which can be reached in current experiments. It is shown that properties of a surface vector breather with phase modulation depends on the parameters of the quantum dots, the connected media and the transverse structure of the surface wave.     

Semiclassical theory of spin-orbit interactions using spin coherent states

We formulate a semiclassical theory for systems with spin-orbit interactions. Using spin coherent states, we start from the path integral in an extended phase space, formulate the classical dynamics of the coupled orbital and spin degrees of freedom, and calculate the ingredients of Gutzwiller's trace formula for the density of states. For a two-dimensional quantum dot with a spin-orbit interaction of Rashba type, we obtain satisfactory agreement with fully quantum-mechanical calculations. The mode-conversion problem, which arose in an earlier semiclassical approach, has hereby been overcome.     

Two-Dimensional Breather Lattice Solutions and Compact-Like Discrete Breathers and Their Stability in Discrete Two-Dimensional Monatomic β-FPU Lattice

We restrict our attention to the discrete two-dimensional monatomic β-FPU lattice. We look for two- dimensional breather lattice solutions and two-dimensional compact-like discrete breathers by using trying method and analyze their stability by using Aubry＇s linearly stable theory. We obtain the conditions of existence and stability of two-dimensional breather lattice solutions and two-dimensional compact-like discrete breathers in the discrete two- dimensional monatomic β-FPU lattice.     

Discrete nonlinear Schrödinger breathers in a phonon bath

We study the dynamics of the discrete nonlinear Schr?dinger lattice initialized such that a very long transitory period of time in which standard Boltzmann statistics is insufficient is reached. Our study of the nonlinear system locked in this non-Gibbsian state focuses on the dynamics of discrete breathers (also called intrinsic localized modes). It is found that part of the energy spontaneously condenses into several discrete breathers. Although these discrete breathers are extremely long lived, their total number is found to decrease as the evolution progresses. Even though the total number of discrete breathers decreases we report the surprising observation that the energy content in the discrete breather population increases. We interpret these observations in the perspective of discrete breather creation and annihilation and find that the death of a discrete breather cause effective energy transfer to a spatially nearby discrete breather. It is found that the concepts of a multi-frequency discrete breather and of internal modes is crucial for this process. Finally, we find that the existence of a discrete breather tends to soften the lattice in its immediate neighborhood, resulting in high amplitude thermal fluctuation close to an existing discrete breather. This in turn nucleates discrete breather creation close to a already existing discrete breather. Received 21 January 1999 and Received in final form 20 September 1999     

Quantum coherent operators: A generalization of coherent states

We introduce a technique to compare different, but related, quantum systems, thereby generalizing the way that coherent states are used to compare quantum systems to classical systems in semiclassical analysis. We then use this technique to estimate the dependence of the free energy of the quantum Heisenberg model on the spin value, and to estimate the relation between the ferromagnetic and antiferromagnetic free energies.Work supported in part by the U.S. National Science Foundation grant PHY-9019433.Work supported in part by the U.S. National Science Foundation grant DMS-9002416.     

Magnetic order in the Shastry-Sutherland model

We investigate the influence of energetic disorder, viscous damping and an external field on the electron transfer (ET) in DNA. The double helix structure of the λ-form of DNA is modeled by a steric oscillator network. In the context of the base-pair picture two different kinds of modes representing twist motions of the base pairs and H-bond distortions are coupled to the electron amplitude. Through the nonlinear interaction between the electronic and the vibrational degrees of freedom localized stationary states in the form of standing electron-vibron breathers are produced which we derive with a stationary map method. We show that in the presence of additional energetic disorder the degree of localization of such breathers is enhanced. It is demonstrated how an applied electric field initiates the long-range coherent motion of breathers along the bases of a DNA strand. These moving electron-vibron breathers, absorbing energy from the applied field, sustain energetic losses due to the viscous friction caused by the aqueous solvent as well as the impact of a moderate amount of energetic disorder. Moreover, it is illustrated that with the choice of the amplitude and frequency of the external field, the breather can be steered to a desired lattice position achieving control of the ET. Received 5 July 2002 Published online 29 November 2002