Modern nanotechnology allows one to scale down various important devices (sensors, chips, fibers, etc.) and thus opens up new horizons for their applications. The efficiency of most of them is based on fundamental physical phenomena, such as transport of wave excitations and resonances. Short propagation distances make phase-coherent processes of waves important. Often the scattering of waves involves propagation along different paths and, as a consequence, results in interference phenomena, where constructive interference corresponds to resonant enhancement and destructive interference to resonant suppression of the transmission. Recently, a variety of experimental and theoretical work has revealed such patterns in different physical settings. The purpose of this review is to relate resonant scattering to Fano resonances, known from atomic physics. One of the main features of the Fano resonance is its asymmetric line profile. The asymmetry originates from a close coexistence of resonant transmission and resonant reflection and can be reduced to the interaction of a discrete (localized) state with a continuum of propagation modes. The basic concepts of Fano resonances are introduced, their geometrical and/or dynamical origin are explained, and theoretical and experimental studies of light propagation in photonic devices, charge transport through quantum dots, plasmon scattering in Josephson-junction networks, and matter-wave scattering in ultracold atom systems, among others are reviewed.

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Fano resonances in nanoscale structures

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 occurence 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 aproach capable of predicting whether discrete breather 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 Breathers

Existence of 'breathers', that is, time-periodic, spatially localized solutions, is proved for a broad range of time-reversible or Hamiltonian networks of weakly coupled oscillators. Some of their properties are discussed, some generalizations suggested, and several open questions raised.

https://iopscience.iop.org/article/10.1088/0951-7715/7/6/006/pdf

Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators

Discrete breathers — Advances in theory and applications

Breathers in nonlinear lattices: existence, linear stability and quantization

In this paper we derive master equations for two or more systems coupled to each other, perhaps strongly, by using a generalization of the usual projection-operator technique to include time-dependent projection operators. The coupled systems may be either similar or dissimilar and classical or quantum mechanical. Whereas the customary approaches to coupled systems are best able to treat situations in which some of the systems are "baths" with a specified density operator or phase-space probability density, our approach allows us to treat situations where it is necessary or convenient to treat the coupled systems on an equal footing. In our scheme the "relevant" part of the full density operator is considered to be the uncorrelated part of the full density operator and is a symmetric functional of the reduced density operators of each of the coupled subsystems. The "irrelevant" part of the density operator is then the part describing correlations between the coupled systems. Our formalism is particularly useful where systems are coupled to one another predominantly in a self-consistent fashion. First, we develop exact master equations for two coupled systems, taking as our prototype the dynamical problem of quantum optics, where a spatially extended collection of two-level atoms interact with a multimode optical field. We then generalize our results to $N$ coupled systems, taking as our prototype the kinetics of a classical nonideal gas interacting through two-body forces, and derive exact master equations for the system. We then consider as examples several approximate theories resulting from our exact equations. In the case of the imperfect gas we investigate the low-density limit and show how Bogoliubov's form of the Boltzmann equation emerges from our formalism, as well as corrections due to Klimontovich. We consider as special cases of our exact quantum-optical equations the equations in the first Born approximation, with and without memory, and show how several existing quantum-optical master equations are contained in our general results. As a second example in quantum optics, we consider the case where the predominant behavior of the system is described by the self-consistent-field or coupled Bloch and Maxwell equations and derive a first-order perturbation description for deviations from self-consistent-field behavior.

Time-dependent projection-operator approach to master equations for coupled systems

We use a collective-variable approach to study the emission of phonon radiation from a discrete sine-Gordon kink. We find that a kink, trapped and oscillating in the nonlinear Peierls-Nabarro potential well not only radiates phonons smoothly but emits large and sudden bursts of phonon radiation when the frequency of oscillation reaches certain critical values. We show that the bursts occur whenever a harmonic of the kink's oscillation frequency crosses the lower phonon band edge and thus begins to resonate with phonon modes in the high-density-of-states region of the phonon spectrum. We show that the mechanism that explains the radiation bursts and kink motion in the trapped case also accounts for the behavior of a radiating untrapped kink. We introduce into a general discrete kink system the collective variable X, which is the position of the center of the kink, and we derive, using a projection-operator approach we have recently developed, the exact equations of motion, valid for any kink size, for X, and coupled field variables, which dynamically dress the kink and describe the radiation. The equations of motion are valid for any discrete nonlinear Klein-Gordon wave equation that admits a single kink solution. We apply the equations of motion to the discrete sine-Gordon system and obtain excellent agreement with simulation. We also comment on the problems of treating discreteness as a perturbation.

Spontaneous emission of radiation from a discrete sine-Gordon kink.

We derive a complete Hamiltonian formalism for a kink on a one-dimensional discrete lattice in which the position of the center of the kink appears as one of the canonical variables. Our method is a generalization to the discrete lattice of the method used in field theory to introduce the soliton as a canonical degree of freedom. The derivation is valid for a particle chain in a periodic potential when there exists a solitary-wave solution in the continuum limit. We show that the discrete lattice is responsible for an adiabatic dressing of the kink and for spontaneous emission of phonons. In the limit where the effective length of the kink is much larger than the interparticle spacing the kink experiences the well-known periodic Peierls-Nabarro potential. In the case of a short kink, the discrete lattice causes the continuum kink configuration to be adiabatically dressed, leading to a renormalization of the Peierls-Nabarro potential and in turn to an enhancement of corresponding small-amplitude oscillatory frequency. In addition, we formally derive an equation that describes the radiation of phonons by the moving kink, an effect of the lattice discreteness.

Sine-Gordon kinks on a discrete lattice. I. Hamiltonian formalism.

We consider classical Hamiltonian systems in which there exist collective modes where the motion associated with each collective mode is describable by a collective coordinate. The formalism we develop is applicable to both continuous and discrete systems where the aim is to investigate the dynamics of kink or solitonlike solutions to nonlinear Klein-Gordon equations which arise in field theory and condensed-matter theory. We present a new calculational procedure for obtaining the equations of motion for the collective coordinates and coupled fields based on Dirac's treatment of constrained Hamiltonian systems. The virtue of this new (projection-operator) procedure is the ease with which the equations of motion for the collective variables and coupled fields are derived relative to the amount of work needed to calculate them from the Dirac brackets directly. Introducing collective coordinates as dynamical variables into a system enlarges the phase space accessible to the possible trajectories describing the system's evolution. This introduces extra solutions to the new equations of motion which do not satisfy the original equations of motion. It is therefore necessary to introduce constraints in order to conserve the number of degrees of freedom of the original system. We show that the constraints have the effect of projecting out the motion in the enlarged phase space onto the appropriate submanifold corresponding to the available phase space of the original system. We show that the Dirac bracket accomplishes this projection, and we give an explicit formula for this projection operator. We use the Dirac brackets to construct a family of canonical transformations to the system of new coordinates (which contains the collective variables) and to construct a Hamiltonian in this new system of variables. We show the equations of motion that are derived through the lengthy Dirac bracket prescription are obtainable through the simple projection-operator procedure. We provide examples that illustrate the ease of this projection-operator method for the single- and multiple-collective-variable cases. We also discuss advantages of particular forms of the Ansatz used for introducing the collective variables into the original system.

C. R. Willis

Papers

Discrete Breathers

Time-dependent projection-operator approach to master equations for coupled systems

Spontaneous emission of radiation from a discrete sine-Gordon kink.

Sine-Gordon kinks on a discrete lattice. I. Hamiltonian formalism.

Hamiltonian equations for multiple-collective-variable theories of nonlinear Klein-Gordon equations: A projection-operator approach.