Tunneling dynamics of Bose-Einstein condensates with higher-order interactions in optical lattice

The nonlinear Landau-Zener tunneling and nonlinear Rabi oscillations of Bose-Einstein condensate (BEC) with higher-order atomic interaction between the Bloch bands in an accelerating optical lattice are discussed. Within the two-level model, the tunneling probability of BEC with higher-order atomic interaction between Bloch bands is obtained. We finds that the tunneling rate is closely related to the higher-order atomic interaction. Furthermore, the nonlinear Rabi oscillations of BEC with higher-order atomic interaction between the bands are discussed by imposing a periodic modulation on the level bias. Analytical expressions of the critical higher-order atomic interaction for suppressing/enhancing the Rabi oscillations are obtained. It is shown that the critical value strongly depends on the modulation parameters (i.e., the modulation amplitude and frequency) and the strength of periodic potential.     

Observation of nonlinear self-trapping of broad beams in defocusing waveguide arrays

We demonstrate experimentally the localization of broad optical beams in periodic arrays of optical waveguides with defocusing nonlinearity. This observation in optics is linked to nonlinear self-trapping of Bose-Einstein-condensed atoms in stationary periodic potentials being associated with the generation of truncated nonlinear Bloch states, existing in the gaps of the linear transmission spectrum. We reveal that unlike gap solitons, these novel localized states can have an arbitrary width defined solely by the size of the input beam while independent of nonlinearity.     

THE NONLINEAR SCHRÖDINGER EQUATION FOR THE DELTA-COMB POTENTIAL: QUASI-CLASSICAL CHAOS AND BIFURCATIONS OF PERIODIC STATIONARY SOLUTIONS

The nonlinear Schrödinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schrödinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight into the features of nonlinear stationary states of periodic potentials. Phenomena well-known from classical chaos are found, such as a bifurcation of periodic stationary states and a transition to spatial chaos. The relation to new features of nonlinear Bloch bands, such as looped and period doubled bands, are analyzed in detail. An analytic expression for the critical nonlinearity for the emergence of looped bands is derived. The results for the delta-comb are generalized to a more realistic potential consisting of a periodic sequence of narrow Gaussian peaks and the dynamical stability of periodic solutions in a Gaussian comb is discussed.     

Soliton complexes and flat-top nonlinear modes in optical lattices

We describe a continuous analog of the quasirectangular flat-top nonlinear modes earlier found for discrete nonlinear models. We show that these novel nonlinear modes can be understood as multi-soliton complexes with either in-phase or out-of-phase neighboring solitons trapped by the periodic potential of the lattice. We demonstrate a link between the flat-top states and the truncated nonlinear Bloch waves, and discuss how these nonlinear localized modes can be monitored experimentally in photonics and Bose–Einstein condensates. PACS 42.65.Tg; 42.65.Jx; 03.75.Lm     

周期性磁谐振材料本构参数的理论分析

将薄的磁谐振介质板等效为面磁流, 利用周期性边界条件, 给出了面磁流的指数形式. 通过计算无穷个面磁流在不同空间位置上产生的总电场和总磁场, 推导出了周期性磁谐振人工材料的色散关系和布洛赫阻抗, 进而获取了布洛赫本构参数的理论计算公式. 由于考虑了磁谐振人工材料中的电反谐振对布洛赫介电常数和磁导率的影响, 所以基于仿真实验的布洛赫本构参数的提取值和布洛赫本构参数理论预测值之间的误差很小, 这说明本文推导的布洛赫本构参数的理论计算公式在描述周期性磁谐振材料的电磁特性方面是十分有效的. 这些理论公式将在解释磁谐振现象、设计和优化周期性磁谐振材料等方面提供重要的理论依据. 关键词： 周期性结构 磁谐振 布洛赫本构参数 面磁流     

Justification of the Lattice Equation for a Nonlinear Elliptic Problem with a Periodic Potential

We justify the use of the lattice equation (the discrete nonlinear Schrödinger equation) for the tight-binding approximation of stationary localized solutions in the context of a continuous nonlinear elliptic problem with a periodic potential. We rely on properties of the Floquet band-gap spectrum and the Fourier–Bloch decomposition for a linear Schrödinger operator with a periodic potential. Solutions of the nonlinear elliptic problem are represented in terms of Wannier functions and the problem is reduced, using elliptic theory, to a set of nonlinear algebraic equations solvable with the Implicit Function Theorem. Our analysis is developed for a class of piecewise-constant periodic potentials with disjoint spectral bands, which reduce, in a singular limit, to a periodic sequence of infinite walls of a non-zero width. The discrete nonlinear Schrödinger equation is applied to classify localized solutions of the Gross–Pitaevskii equation with a periodic potential.     

Vanishing integral relations and expectation values for Bloch functions in finite domains

Integral identities for particular Bloch functions in finite periodic systems are derived. All following statements are proven for a finite domain consisting of an integer number of unit cells. It is shown that matrix elements of particular Bloch functions with respect to periodic differential operators vanish identically. The real valuedness, the time-independence and a summation property of the expectation values of periodic differential operators applied to superpositions of specific Bloch functions are derived.     

非线性Kronig-Penney超晶格的二维实映射分析

通过二维实映射研究非线性Kronig-Penney超晶格中的波输运特性，数值计算得到波矢一定情况下不同非线性系数的映射图、以及非线性系数一定情况下不同波矢的映射图.非线性Kronig-Penney超晶格中的非线性，对超晶格中波函数的Bloch波矢有明显的调节作用.随着非线性系数的增大，映射图由周期函数的有限个分立点变为准周期函数的一条闭合轨道、以及有规则或无规则的点分布. 关键词： 非线性Kronig-Penney超晶格 非线性效应 二维实映射 波矢     

Optical zoomeron as a result of beatings of the internal modes of a Bragg soliton

A new solution of two-wave Maxwell-Bloch equations has been obtained analytically and numerically. It describes the propagation of an oscillating nonlinear optical solitary wave, or optical zoomeron, in a one-dimensional periodic resonant Bragg structure. It has been shown that the appearance of large oscillations in the velocity and total amplitude of Bloch modes of the pulse is caused by beating of internal modes of the perturbed Bragg soliton.     

Self-trapped spatially localized states in combined linear-nonlinear periodic potentials

We analyze the existence and stability of two kinds of self-trapped spatially localized gap modes,gap solitons and truncated nonlinear Bloch waves,in one-and two-dimensional optical or matter-wave media with self-focusing nonlinearity,supported by a combination of linear and nonlinear periodic lattice potentials.The former is found to be stable once placed inside a single well of the nonlinear lattice,it is unstable otherwise.Contrary to the case with constant self-focusing nonlinearity,where the latter solution is always unstable,here,we demonstrate that it nevertheless can be stabilized by the nonlinear lattice since the model under consideration combines the unique properties of both the linear and nonlinear lattices.The practical possibilities for experimental realization of the predicted solutions are also discussed.     

Two-dimensional array of room-temperature nanophotonic logic gates using InAs quantum dots in mesa structures

We present a detailed study of the dynamics of light in passive nonlinear resonators with shallow and deep intracavity periodic modulation of the refractive index in both longitudinal and transverse directions of the resonator. We investigate solutions localized in the transverse direction (so-called Bloch cavity solitons) by means of envelope equations for underlying linear Bloch modes and solving Maxwell’s equations directly. Using a round-trip model for forward and backward propagating waves we review different types of Bloch cavity solitons supported by both focusing (at normal diffraction) and defocussing (at anomalous diffraction) nonlinearities in a cavity with a weak-contrast modulation of the refractive index. Moreover, we identify Bloch cavity solitons in a Kerr-nonlinear all-photonic crystal resonator solving Maxwell’s equations directly. In order to analyze the properties of Bloch cavity solitons and to obtain analytical access we develop a modified mean-field model and prove its validity. In particular, we demonstrate a substantial narrowing of Bloch cavity solitons near the zero-diffraction regime. Adjusting the quality factor and resonance frequencies of the resonator optimal Bloch cavity solitons in terms of width and pump energy are identified.     

Nonlinear photonic crystals: I. Quadratic nonlinearity

Abstract

We develop a consistent mathematical theory of weakly nonlinear periodic dielectric media for the dimensions one, two and three. The theory is based on the Maxwell equations with classical quadratic and cubic constitutive relations. In particular, we give a complete classification of different nonlinear interactions between Floquet–Bloch modes based on indices which measure the strength of the interactions. The indices take on a small number of prescribed values which are collected in a table. The theory rests on the asymptotic analysis of oscillatory integrals describing the nonlinear interactions.     

Nonlinear photonic crystals: II. Interaction classification for quadratic nonlinearities

Weakly nonlinear interactions between wavepackets in lossless periodic dielectric media are studied based on the classical nonlinear Maxwell equations. We consider nonlinear processes such that: (i) the amplitude of the wave component due to the nonlinearity does not exceed the amplitude of its linear component; (ii) the spatial range of a probing wavepacket is much smaller than the dimension of the medium sample, and it is not too small compared with the dimension of the primitive cell. These nonlinear processes are naturally described in terms of the Bloch modes and the dispersion relations of the underlying linear periodic medium. It turns out that only a few triads of modes have significant nonlinear interactions. They are singled out by the frequency and phase matching conditions and, as we show, by an additional selection rule: the group velocity matching condition. The latter condition is the most important selection rule for the nonlinear regimes. We give a complete quantitative classification of all possible significant interactions for quadratic nonlinearities. The classification is based on a universal system of indices reflecting the intensity of nonlinear interactions. The obtained classification points to the second harmonic generation as being one of the stronger nonlinear interactions, and often the strongest one.     

Long-living BLOCH oscillations of matter waves in periodic potentials

The dynamics of matter waves in linear and nonlinear optical lattices subject to a spatially uniform linear force is studied both analytically and numerically. It is shown that by properly designing the spatial dependence of the scattering length it is possible to induce long-living Bloch oscillations of gap-soliton matter waves in optical lattices. This occurs when the effective nonlinearity and the effective mass of the soliton have opposite signs for all values of the crystal momentum in the Brillouin zone. The results apply to all systems modeled by the periodic nonlinear Schr?dinger equation, including propagation of light in photonic and photorefractive crystals with tilted band structures.     

Discrete solitons and breathers with dilute Bose-Einstein condensates

We study the dynamical phase diagram of a dilute Bose-Einstein condensate (BEC) trapped in a periodic potential. The dynamics is governed by a discrete nonlinear Schr?dinger equation: intrinsically localized excitations, including discrete solitons and breathers, can be created even if the BEC's interatomic potential is repulsive. Furthermore, we analyze the Anderson-Kasevich experiment [Science 282, 1686 (1998)], pointing out that mean field effects lead to a coherent destruction of the interwell Bloch oscillations.     

Exact wave-based Bloch analysis procedure for investigating wave propagation in two-dimensional periodic lattices

This paper presents an exact, wave-based approach for determining Bloch waves in two-dimensional periodic lattices. This is in contrast to existing methods which employ approximate approaches (e.g., finite difference, Ritz, finite element, or plane wave expansion methods) to compute Bloch waves in general two-dimensional lattices. The analysis combines the recently introduced wave-based vibration analysis technique with specialized Bloch boundary conditions developed herein. Timoshenko beams with axial extension are used in modeling the lattice members. The Bloch boundary conditions incorporate a propagation constant capturing Bloch wave propagation in a single direction, but applied to all wave directions propagating in the lattice members. This results in a unique and properly posed Bloch analysis. Results are generated for the simple problem of a periodic bi-material beam, and then for the more complex examples of square, diamond, and hexagonal honeycomb lattices. The bi-material beam clearly introduces the concepts, but also allows the Bloch wave mode to be explored using insight from the technique. The square, diamond, and hexagonal honeycomb lattices illustrate application of the developed technique to two-dimensional periodic lattices, and allow comparison to a finite element approach. Differences are noted in the predicted dispersion curves, and therefore band gaps, which are attributed to the exact procedure more-faithfully modeling the finite nature of lattice connection points. The exact method also differs from approximate methods in that the same number of solution degrees of freedom is needed to resolve low frequency, and arbitrarily high frequency, dispersion branches. These advantageous features may make the method attractive to researchers studying dispersion characteristics, band gap behavior, and energy propagation in two-dimensional periodic lattices.     

Nonlinear photonic crystals: II. Interaction classification for quadratic nonlinearities

Abstract

Weakly nonlinear interactions between wavepackets in lossless periodic dielectric media are studied based on the classical nonlinear Maxwell equations. We consider nonlinear processes such that: (i) the amplitude of the wave component due to the nonlinearity does not exceed the amplitude of its linear component; (ii) the spatial range of a probing wavepacket is much smaller than the dimension of the medium sample, and it is not too small compared with the dimension of the primitive cell. These nonlinear processes are naturally described in terms of the Bloch modes and the dispersion relations of the underlying linear periodic medium. It turns out that only a few triads of modes have significant nonlinear interactions. They are singled out by the frequency and phase matching conditions and, as we show, by an additional selection rule: the group velocity matching condition. The latter condition is the most important selection rule for the nonlinear regimes. We give a complete quantitative classification of all possible significant interactions for quadratic nonlinearities. The classification is based on a universal system of indices reflecting the intensity of nonlinear interactions. The obtained classification points to the second harmonic generation as being one of the stronger nonlinear interactions, and often the strongest one.     

Nonlinear tunneling and chaos between two Bose–Einstein condensates trapped in time-dependent potential

In this Letter, we discuss the macroscopic quantum self-trapping (MQST) and coherent atomic tunneling between two weakly coupled Bose–Einstein condensates confined in a time-dependent double-well trap. It is shown that the nonlinear interaction and the time-periodic external field can dramatically affect the MQST, lead to the high-order Bloch harmonics of the population imbalance, and periodic or chaotic behavior. We give out the onset point of chaos by Lyapunov exponent and the phase plots. It is found that the sudden change on the quasi-energy corresponds to the transition from libration to rotation.     

Spectral properties of reduced Bloch Hamiltonians

Bloch Hamiltonians are defined, and the existence of bands is proven for a large class of periodic operators. The results are strong enough to include most of the reasonable physical models of a single electron in crystals. A notable exception is the Dirac Bloch Hamiltonian for a Coulombic crystal with high charge. Properties of the Bloch waves are briefly described and it is shown that “simple” Bloch Hamiltonians do not have Bloch waves with a finite number of Fourier coefficients. The asymptotic distribution of the bands is determined, and it is shown that for a large class of Hamiltonians, it is determined by the kinetic energy alone.     

Influence of parameters on light propagation dynamics in optically induced planar waveguide arrays

The diffraction and refraction of light beam in optical periodic structures can be determined by the photonic band-gap structures of spatial frequency. In this paper, by employing the equation governing the nonlinear light propagations in photorefractive crystals, we study the photonic band-gap structures, Bloch modes, and light transmission properties of optically induced planar waveguide arrays. The relationship between the photonic band-gap structures and the light diffraction characteristics is discussed in detail. Then the influence of the parameters of planar waveguide arrays on the band-gaps structures, Bloch modes, and linear light transmissions is analyzed. It is revealed that the linear light transmission properties of waveguide arrays are tightly related to the diffraction relationships determined by band-gap structures. And the Bloch modes corresponding to different transmission bands can be excited by different excitation schemes. Both the increases of the intensity and the period of the array writing beam will lead to the broadening of the forbidden gaps and the concentration of the energy of the Bloch modes to the high-index regions. Furthermore, the broadening of the forbidden gaps will lead to separation and transition between the Bloch modes of neighboring bands around the Bragg angle. Additionally, with the increase of the intensity of the array writing beams, the influences from light intensity will tend to be steady due to the saturation of the photorefractive effect. Supported by the Youth for Northwestern Polytechnical University (NPU) Teachers Scientific and Technological Innovation Foundation, the NPU Foundation for Fundamental Research, and the Doctorate Foundation of NPU (Grant No. CX200514)