Dark soliton interaction of spinor Bose-Einstein condensates in an optical lattice

We study the magnetic soliton dynamics of spinor Bose-Einstein condensates in an optical lattice which results in an effective Hamiltonian of anisotropic pseudospin chain. An equation of nonlinear Schrödinger type is derived and exact magnetic soliton solutions are obtained analytically by means of Hirota method. Our results show that the critical external field is needed for creating the magnetic soliton in spinor Bose-Einstein condensates. The soliton size, velocity and shape frequency can be controlled in practical experiment by adjusting the magnetic field. Moreover, the elastic collision of two solitons is investigated in detail.     

玻色-爱因斯坦凝聚体干涉现象的数值模拟

采用含时非线性薛定谔方程的数值解法模拟了两种情况下玻色-爱因斯坦凝聚体的演化.一种是在初始状态中有两个包含粒子数差别很大、在空间上可区分的玻色-爱因斯坦凝聚体的情况;另一种是一个玻色-爱因斯坦凝聚体被外力一分为二后在自由空间及约束势阱中继续演化的情况. 关键词：     

The study of the stability of 1D confined Bose-Einstein condensates based on one novel orthogonal basis

One novel orthogonal basis of the 1D Gross-Pitaevskii equation with square well, describing the properties of the nonuniform Bose-Einstein condensates, was introduced. Based on this orthogonal basis, we investigated the dynamical stability of the ground state and the first excited state for the nonuniform condensates. We found that the ground state is stable even for large negative nonlinear interaction. But the first excited state performs dynamical instability when the nonlinear interaction is over the critical value. Our numerical results support our analytical analysis. Some underlying physics are discussed.     

Complete and orthonormal eigenfunctions of excitations to a one-dimensional dark soliton

We study linear excitations to a one-dimensional dark soliton described by a defocusing nonlinear Schödinger equation. By solving an eigenvalue problem for the excitations we obtain all eigenvalues and eigenfunctions and prove rigorously that these eigenfunctions are orthonormal and form a complete set. We then use the eigenfunctions to obtain the exact form of linear excitations for any given initial condition and to investigate the transverse stability of the dark soliton. The rigorous results reported in the present work can be applied to study the dynamics of dark solitons in various nonlinear optical media and Bose-Einstein condensates.     

Periodic spin domains of spinor Bose-Einstein condensates in an optical lattice

The periodic spin domains of spinor Bose-Einstein condensates confined in a one-dimensional optical lattice are studied in terms of the equation of motion of the spinor which is reduced to the nonlinear Schrödinger equation with the help of Holstein-Primakoff transformation. It is shown that the spin domains obtained analytically can be easily controlled by adjusting the light-induced dipole-dipole interaction, which is realizable in optical lattice created by red-detuned laser beams with modulating intensity. The dynamical stability of the spin domains is also demonstrated.     

Quantum field theoretical analysis on unstable behavior of Bose-Einstein condensates in optical lattices

We study the dynamics of Bose-Einstein condensates flowing in optical lattices on the basis of quantum field theory. For such a system, a Bose-Einstein condensate shows an unstable behavior which is called the dynamical instability. The unstable system is characterized by the appearance of modes with complex eigenvalues. Expanding the field operator in terms of excitation modes including complex ones, we attempt to diagonalize the unperturbative Hamiltonian and to find its eigenstates. It turns out that although the unperturbed Hamiltonian is not diagonalizable in the conventional bosonic representation the appropriate choice of physical states leads to a consistent formulation. Then we analyze the dynamics of the system in the regime of the linear response theory. Its numerical results are consistent with those given by the discrete nonlinear Schrödinger equation.     

Modulation of breathers in cigar-shaped Bose-Einstein condensates

We present new solutions to the nonautonomous nonlinear Schrödinger equation that may be realized through convenient manipulation of Bose-Einstein condensates. The procedure is based on the modulation of breathers through an analytical study of the one-dimensional Gross-Pitaevskii equation, which is known to offer a good theoretical model to describe quasi-one-dimensional cigar-shaped condensates. Using a specific ansatz, we transform the nonautonomous nonlinear equation into an autonomous one, which engenders composed states corresponding to solutions localized in space, with an oscillating behavior in time. Numerical simulations confirm stability of the modulated breathers against random perturbation on the input profile of the solutions.     

Energy transfer in weakly coupled nonlinear oscillator chains: Transition from a wandering breather to nonlinear self-trapping

We present analytical and numerical studies of phase-coherent dynamics of intrinsically localized excitations (breathers) in a system of two weakly coupled nonlinear oscillator chains. We show that there are two qualitatively different dynamical regimes of the coupled breathers, either immovable or slowly moving: the periodic wandering of the low-amplitude breather between the chains, and the one-chain-localization of the high-amplitude breather. These two modes of coupled breathers can be mapped exactly onto two solutions of a pendulum equation, detached by a separatrix mode. We also show that these two regimes of the coupled breathers are similar, and are described by a similar pair of equations, to the two regimes in the nonlinear tunneling dynamics of two weakly coupled Bose-Einstein condensates. On the basis of this analogy, we predict a new tunneling mode of two weakly coupled Bose-Einstein condensates in which their relative phase oscillates around π/2 modulo π.     

Collapse in coupled nonlinear Schrödinger equations: Sufficient conditions and applications

In this paper we study blow-up phenomena in general coupled nonlinear Schrödinger equations with different dispersion coefficients. We find sufficient conditions for blow-up and for the existence of global solutions. We discuss several applications of our results to heteronuclear multispecies Bose-Einstein condensates and to degenerate boson-fermion mixtures.     

Matter-wave interference in Bose-Einstein condensates: A dispersive hydrodynamic perspective

The interference pattern generated by the merging interaction of two Bose-Einstein condensates reveals the coherent, quantum wave nature of matter. An asymptotic analysis of the nonlinear Schrödinger equation in the small dispersion (semiclassical) limit, experimental results, and three-dimensional numerical simulations show that this interference pattern can be interpreted as a modulated soliton train generated by the interaction of two rarefaction waves propagating through the vacuum. The soliton train is shown to emerge from a linear, trigonometric interference pattern and is found by use of the Whitham modulation theory for nonlinear waves. This dispersive hydrodynamic perspective offers a new viewpoint on the mechanism driving matter-wave interference.     

推广的Gross-Pitaevskii理论中的涡旋问题(英文)

利用推广 Gross- Pitaevskii方程 ,分别研究了 (2 +1 )维时空和 3维空间的 Bose- Einstein凝聚体中涡旋的拓扑结构 .这一推广的方程能够被用于非均匀并且高度非线形的 Bose- Einstein凝聚系统 .利用Φ映射拓扑流理论 ,给出了基于序参数的涡旋速度场,以及该速度场的拓扑结构 .最后 ,仔细地探讨了这两种 Bose- Einstein系统中涡旋的各种分支条件.We studied the topological structure of vortex in the Bose-Einstein condensation with a generalized Gross-Pitaevskii equation in (2+1)-dimensional space-time and 3-dimensional space, respectively. Such equation can be used in discussing Bose-Einstein condensates in heterogeneous and highly nonlinear systems. An explicit expression for the vortex velocity field as a function of the order parameter field is derived in terms of the Φ -mapping theory, and the topological structure of ...     

Nonautonomous “rogons” in the inhomogeneous nonlinear Schrödinger equation with variable coefficients

The analytical nonautonomous rogons are reported for the inhomogeneous nonlinear Schrödinger equation with variable coefficients in terms of rational-like functions by using the similarity transformation and direct ansatz. These obtained solutions can be used to describe the possible formation mechanisms for optical, oceanic, and matter rogue wave phenomenon in optical fibres, the deep ocean, and Bose-Einstein condensates, respectively. Moreover, the snake propagation traces and the fascinating interactions of two nonautonomous rogons are generated for the chosen different parameters. The obtained nonautonomous rogons may excite the possibility of relative experiments and potential applications for the rogue wave phenomenon in the field of nonlinear science.     

Averaging of nonlinearity-managed pulses

We consider the nonlinear Schrodinger equation with the nonlinearity management which describes Bose-Einstein condensates under Feshbach resonance. By using an averaging theory, we derive the Hamiltonian averaged equation and compare it with other averaging methods developed for this problem. The averaged equation is used for analytical approximations of nonlinearity-managed solitons.     

Non-quantum liquefaction of coherent gases

We show that a gas-to-liquid phase transition at zero temperature may occur in a coherent gas of bosons in the presence of competing nonlinear effects. This situation can take place in atomic systems like Bose-Einstein condensates in alkali gases with two-body and three-body interactions of opposite signs, as well as in laser beams which propagate through optical media with Kerr (focusing) and higher order (defocusing) nonlinear responses. The liquefaction process takes place in the absence of any quantum effect and can be formulated in the framework of a mean field theory, in terms of the minimization of a thermodynamic potential. We study from a thermodynamic point of view all the stationary solutions of the cubic-quintic nonlinear Schrödinger equation which describes our system. We show that solitonic localized solutions connect the gaseous and liquid phases. Furthermore, we also perform a numerical simulation in the presence of linear gain and three-body recombination where a rich dynamics, including the emergence of self-organization behavior, is found.     

Dynamics of Bose-Einstein condensates near Feshbach resonance in external potential

We review our recent theoretical advances in the dynamics of Bose-Einstein condensates with tunable interactions using Feshbach resonance and external potential. A set of analytic and numerical methods for Gross-Pitaevskii equations are developed to study the nonlinear dynamics of Bose-Einstein condensates. Analytically, we present the integrable conditions for the Gross-Pitaevskii equations with tunable interactions and external potential, and obtain a family of exact analytical solutions for one- and two-component Bose-Einstein condensates in one and two-dimensional cases. Then we apply these models to investigate the dynamics of solitons and collisions between two solitons. Numerically, the stability of the analytic exact solutions are checked and the phenomena, such as the dynamics and modulation of the ring dark soliton and vector-soliton, soliton conversion via Feshbach resonance, quantized soliton and vortex in quasi-two-dimensional are also investigated. Both the exact and numerical solutions show that the dynamics of Bose-Einstein condensates can be effectively controlled by the Feshbach resonance and external potential, which offer a good opportunity for manipulation of atomic matter waves and nonlinear excitations in Bose-Einstein condensates.     

Coherent control of the self-trapping transition

We discuss the dynamics of two weakly coupled Bose-Einstein condensates in a double-well potential, contrasting the mean-field picture to the exact N-particle evolution. On the mean-field level, a self-trapping transition occurs when the scaled interaction strength exceeds a critical value; this transition essentially persists in small condensates comprising about 1000 atoms. When the double-well is modulated periodically in time, Floquet-type solutions to the nonlinear Schr?dinger equation take over the role of the stationary mean-field states. These nonlinear Floquet states can be classified as “unbalanced” or “balanced”, depending on whether or not they entail long-time confinement of most particles to one well. Since the emergence of unbalanced Floquet states depends on the amplitude and frequency of the modulating force, we predict that the onset of self-trapping can efficiently be controlled by varying these parameters. This prediction is verified numerically by both mean-field and N-particle calculations. Received 5 November 2000 and Received in final form 16 February 2001     

Splitting of coupled bright solitons in two-component Bose-Einstein condensates under parametric perturbation

We analyze the dynamics of bright-bright solitons in two-component Bose-Einstein condensates (BECs) subject to parametric perturbations using the variational approach and direct numerical simulations. The system is described by a vector nonlinear Schrödinger equation (NLSE) appropriate to coupled multi-component BECs. A periodic variation of the inter-component coupling coefficient is used to explore nonlinear resonances and splitting of the coupled bright solitons. The analytical predictions are confirmed by direct numerical simulations of the vector NLSE.     

Integrability of the Gross-Pitaevskii equation with Feshbach resonance management

In this Letter we study the integrability of a class of Gross-Pitaevskii equations managed by Feshbach resonance in an expulsive parabolic external potential. By using WTC test, we find a condition under which the Gross-Pitaevskii equation is completely integrable. Under the present model, this integrability condition is completely consistent with that proposed by Serkin, Hasegawa, and Belyaeva [V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. 98 (2007) 074102]. Furthermore, this integrability can also be explicitly shown by a transformation, which can convert the Gross-Pitaevskii equation into the well-known standard nonlinear Schrödinger equation. By this transformation, each exact solution of the standard nonlinear Schrödinger equation can be converted into that of the Gross-Pitaevskii equation, which builds a systematical connection between the canonical solitons and the so-called nonautonomous ones. The finding of this transformation has a significant contribution to understanding the essential properties of the nonautonomous solitons and the dynamics of the Bose-Einstein condensates by using the Feshbach resonance technique.     

Dynamics of Bright Solitons in Bose-Einstein Condensates with Complicated Potential

We present analytical solutions of the one-dimensional nonlinear Schrodinger equations of Bose-Einstein condensates in an expulsive parabolic background with a complex potential and gravitational field, by performing the Darboux transformation from a trivial seed solution. It is shown that under a safe range of parameter, the shape of bright soliton can be controlled well by adjusting the experimental parameter of the ratio of axial oscillation to radial oscillation and feeding condensates from a thermal cloud. The gravitational field can change the contrail of the bright soliton trains without changing their peak and width.     

One-dimensional delocalizing transitions of matter waves in optical lattices

We investigate, both analytically and numerically, the conditions for the occurrence of the delocalizing transition phenomenon of one-dimensional localized modes of several nonlinear continuous periodic and discrete systems of the nonlinear Schrödinger type. We show that either non-existence of solitons in the small amplitude limit or the loss of stability along existence branches can lead to delocalizing transitions, which occur following different scenarios. Examples of delocalizing transitions of both types are provided for a class of equations which describe single component and binary mixtures of Bose-Einstein condensates trapped in linear and nonlinear optical lattices.