One-dimensional Discrete Lattice Research Articles

Topological multiband photonic superlattices

A one-dimensional discrete lattice of dimers is known to possess topologically protected edge states when interdimer coupling is stronger than intradimer coupling. Here, we address richer topological properties of photonic superlattices having arbitrary number of elements in each unit cell. It is shown that the superlattice provides tunable number of topologically protected edge and interface states depending on certain restrictions on intra- and intercell couplings maintaining inversion symmetry of the lattice. Simultaneous and stable propagation of multiple topological interface states, their interference pattern, and stable oscillation are reported. The superlattice configuration can be relevant for topologically protected mode-division multiplexing through a narrow route in photonic devices.

Matter-wave soliton buffer realized by a tailored one-dimensional lattice

By introducing the attractive dipole–dipole interaction, we provide a proposal to buffer the matter-wave soliton in a one-dimensional discrete lattice, which is divided into three parts with different local potentials produced by the external field modulation. Our study shows that, for sufficiently small initial phase tilt of the matter wave and deep enough potential well, the soliton will be trapped in the lattice. Otherwise, the soliton will propagate through the lattice. It appears that under certain conditions such a condensate system acts as a buffer, which can be used to temporarily store the matter wave. Meanwhile, in this buffer, the matter-wave soliton performs an unidirectional propagation, which makes it serve as a matter wave diode. The norm distribution in the lattice with respect to the initial phase tilt of the input wave is discussed, and it suggests there exists a critical phase tilt, below which the soliton will be trapped in the lattice.

Scattering of a composite quasiparticle by an impurity on a lattice

We study scattering of a composite quasiparticle, which possesses a degree of freedom corresponding to relative separation between two bound excitations, by a delta-like impurity potential on a one-dimensional discrete lattice. Firstly, we show that, due to specific properties of their dispersion, lattice excitations bind to impurities with both negative and positive potentials. We demonstrate that the finite size of the composite excitation leads to formation of multiple excitation-impurity bound states. The number and the degree of localization of these bound states depend on the signs and relative magnitudes of the impurity potential and the binding strength of two quasiparticles. We also report the existence of excitation-impurity bound states whose energies are located in the continuum band. Secondly, we study a change in the entanglement between the centre of mass and relative coordinate degrees of freedom of a biexciton wave packet during single impurity scattering and decoherence caused by it. For a composite quasiparticle on a lattice, the entanglement between its relative and centre of mass coordinate degrees of freedom arises naturally due to inseparability of the two-particle Hamiltonian. One of the main focuses of our study is to investigate how this inseparability affects the creation of the biexciton-impurity bound states and the entanglement dynamics.

Quantum entanglement and drifting generated by an ac field resonant with frequency-doubled Bloch oscillations of correlated particles

We study the dynamics of initially localized and uncorrelated two-particle quantum wave packets evolving in a one-dimensional discrete lattice. In particular, we show that the particles become strongly entangled when directed by a harmonic ac field which is resonant with frequency-doubled Bloch oscillations promoted by a static dc field. Some degree of entanglement is also achieved when the ac field is resonant with the single-particle Bloch oscillations. However, in this case, entanglement is strongly limited by the survival of anticorrelated unbounded states. We further show that the drift velocity of the correlated-particle wave-packet centroid depends on the phase of the ac field. This dependence is similar to the semiclassical prediction for single-particle motion. The drift velocity vanishes in the limit of uncorrelated particles, as well as for Fock-like initial states, which have a null expectation value of the kinetic operator. We reveal that the interparticle interaction influences unbounded- and bounded-state components differently. This leads to a nontrivial nonmonotonic dependence of the drift velocity on the interaction strength.

Antisymmetry breaking of discrete dipole gap solitons induced by a phase-slip defect

In this paper, we study the antisymmetry breaking of discrete dipole soliton induced by a phase-slip one-dimensional discrete lattice, which contains on-site self-repulsive nonlinearity. This system can be realized in waveguide arrays system in optics or Bose–Einstein condensate in optical lattice. Different from the symmetry breaking occurring in the ground-state, antisymmetry breaking occurs in the first excited state, which contains antisymmetry. For this system, stable antisymmetric dipole soliton and anti-asymmetric are found, the symmetry transition between them is supercritical type. It is found that, increasing the total norm of the soliton or decreasing the coupled strength of the defect waveguide can lead to the antisymmetry breaking. Such kind of symmetry breaking can lead to the change of the tendency of some characters of the soliton.

Beyond kinetic relations

We introduce the concept of kinetic equations representing a natural extension of the more conventional notion of a kinetic relation. Algebraic kinetic relations, widely used to model dynamics of dislocations, cracks and phase boundaries, link the instantaneous value of the velocity of a defect with an instantaneous value of the driving force. The new approach generalizes kinetic relations by implying a relation between the velocity and the driving force which is nonlocal in time. To make this relations explicit one needs to integrate the system of kinetic equations. We illustrate the difference between kinetic relation and kinetic equations by working out in full detail a prototypical model of an overdamped defect in a one-dimensional discrete lattice. We show that the minimal nonlocal kinetic description containing now an internal time scale is furnished by a system of two ordinary differential equations coupling the spatial location of defect with another internal parameter that describes configuration of the core region.

Connection between matrix-product states and superposition of Bernoulli shock measures

We consider a generalized coagulation-decoagulation system on a one-dimensional discrete lattice with reflecting boundaries. It is known that a Bernoulli shock measure with two shock fronts might have a simple random-walk dynamics, provided that some constraints on the microscopic reaction rates of this system are fulfilled. Under these constraints the steady state of the system can be written as a linear superposition of such shock measures. We show that the coefficients of this expansion can be calculated using the finite-dimensional representation of the quadratic algebra of the system obtained from a matrix-product approach.

Interparticle gap distributions on one-dimensional lattices

We analyse the successive binding of two species of particles on a one-dimensional discrete lattice, where the second variety is deposited only after complete adsorption of the first. We consider the two extreme cases of a perfectly irreversible initial deposition, with non-sliding particles, and that of a fully equilibrated one. For the latter we construct the exact gap distribution from the Tonks gas partition function. This distribution is contrasted with that obtained from the random sequential adsorption process. We discuss implications for the kinetics of adsorption of the second species, as well as experimental relevance of our results.

One-dimensional generalized branching annihilating random walker process with stochastic generation of offsprings

We introduce a branching annihilating random walker process with two species, particles A and B, which diffuse creating new particles and annihilating instantaneously ( A + B → 0) when they meet. Each kind of particle branches stochastically having offsprings of the same or different type. The model is defined and studied by means of epidemic simulations in a one-dimensional discrete lattice. The phase diagram of the model exhibits two states, the vacuum and the active one, separated by a critical line. Along that line the system undergoes irreversible second order phase transitions. Monte Carlo results show that the transitions belong to the same universality class as directed percolation. In the limiting case when the generation of offsprings is forbidden, the model is mapped into the standard diffusion-limited reaction A+ B→0 which asymptotically evolves towards the vacuum state. The transition between the stationary regime and such vacuum states is also studied.

Relaxation and trapping of excitons in J-aggregates of a thiacarbocynine dye

Abstract Exciton relaxation processes in J-aggregates of 3,3′,9-triethyl-5,5′-dichlorothiacarbocyanine iodide (TDC) dye have been studied by means of site-selective, steady-state and time-resolved spectroscopy. We found that TDC forms three different types of J-aggregates (J1,J2, and J3) in frozen solutions. Fluorescence polarization measurements showed that exciton diffusion between randomly oriented segments of aggregates plays a minor role in exciton relaxation. Optical properties of J3-aggregate fluorescent states are well described by the theoretical model of barrierless self-trapping of an excitation in a strictly one-dimensional discrete lattice. Upon optical excitation J3-aggregates also show thermally activated transformation to J2-aggregates. The height of the related potential barrier is approximately equal to 30 cm−1. Hence, two different exciton relaxation processes are proposed to take place in the system under study: barrierless self-trapping in J3-aggregates and thermally activated J3 → J2 photorearrangement.

Spatial patterns in a simple reaction-diffusion system

Bifurcations of the steady homogeneous solution of a simpel reaction-diffusion system, distributed over a one-dimensional discrete lattice, are examined, and the different types of steady spatially inhomogeneous solutions that can appear are indicated. Bifurcations in the infinite-dimensional system are related to branchings (see below) of a two-dimensional area-preserving map, and the result is applied to establish the appearance and stability of wavelength-two solutions. Looking into bifurcations of these wavelength-two solutions, we show that no futher wavelength doubling takes place. The possibility of appearance of spatially chaotic time-invariant structures, and of more complex spatio-temporal structures including temporal intermittency, is briefly speculated upon.

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

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.

Shock profiles caused by different end conditions in one-dimensional quiescent lattices

Shock propagation in a one-dimensional discrete lattice is generated by accelerating the end-most particle from zero to its final velocity in a finite rise time after which the end particle is maintained at that velocity. The wave profiles for various rise times are compared to the zero-time case in a quiescent lattice. For the anharmonic lattice the classical equations of motion of the atoms are solved numerically on the computer. A Morse-type potential is assumed. For a finite rise time the amplitude of the wave passing through the surface atoms is diminished when compared with he zero-time case. For the anharmonic lattice the head of the wave develops into a solitary wave train with an oscillatory tail, and for certain rise times and anharmonicity parameters an apparent envelope soliton forms behind the shock front. This envelope soliton travels much slower than the shock wave.