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Communications in Theoretical Physics

Weak Nonlinear Matter Waves in a Trapped Spin-1 Condensates

Hong-Qiang Cai, Shu-Rong Yang, Ju-Kui Xue

Published Apr 1, 2011• 5 citations
Publisher

Abstract

The dynamics of the weak nonlinear matter solitary waves in a spin-1 condensates with harmonic external potential are investigated analytically by a perturbation method. It is shown that, in the small amplitude limit, the dynamics of the solitary waves are governed by a variable-coefficient Korteweg-de Vries (KdV) equation. The reduction to the (KdV) equation may be useful to understand the dynamics of nonlinear matter waves in spinor BECs. The analytical expressions for the evolution of soliton show that the small-amplitude vector solitons of the mixed types perform harmonic oscillations in the presence of the trap. Furthermore, the emitted radiation profiles and the soliton oscillation frequency are also obtained.

DOI
10.1088/0253-6102/55/4/10

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The dynamics of the weak nonlinear matter solitary waves in a spin-1 condensates with harmonic external potential are investigated analytically by a perturbation method. It is shown that, in the small amplitude limit, the dynamics of the solitary waves are governed by a variable-coefficient Korteweg-de Vries (KdV) equation. The reduction to the (KdV) equation may be useful to understand the dynamics of nonlinear matter waves in spinor BECs. The analytical expressions for the evolution of soliton show that the small-amplitude vector solitons of the mixed types perform harmonic oscillations in the presence of the trap. Furthermore, the emitted radiation profiles and the soliton oscillation frequency are also obtained.
The dynamics of the weak nonlinear matter solitary waves in two-component Bose–Einstein condensates (BEC) with cigar-shaped external potential are investigated analytically by a perturbation method. In the small amplitude limit, the two-components can be decoupled and the dynamics of solitary waves are governed by a variable-coefficient Korteweg–de Vries (KdV) equation. The reduction to the KdV equation may be useful to understand the dynamics of nonlinear matter waves in two-component BEC. The analytical expressions for the evolution of soliton, emitted radiation profiles and soliton oscillation frequency are also obtained.
弱非线性的事的动力学在 spin-1 的独居的波浪有泛音的冷凝物外部潜力被调查由一个不安方法的经分解。在小振幅限制，独居的波浪的动力学被一个可变系数的 Korteweg-de Vries (KdV ) 方程管理，这被显示出。减小到(KdV ) 方程可能是有用的在旋量 BEC 理解非线性的事波浪的动力学。为 soliton 的进化的分析表情证明混合类型的小振幅的向量 solitons 面对陷井执行泛音摆动。而且，射出的放射侧面和 soliton 摆动频率也被获得。
Waves with different symmetries exist in two-component Bose-Einstein condensates (BECs) whose dynamics is described by a system of coupled Gross-Pitaevskii (GP) equations. A first type of waves corresponds to excitations for which the motion of both components is locally in phase. In the second type of waves the two components have a counter-phase local motion. In the case of different values of inter- and intra-component interaction constants, the long wave-length behavior of these two modes corresponds to two types of sound with different velocities. In the limit of weak nonlinearity and small dispersion the first mode is described by the well-known Korteweg-de Vries (KdV) equation. We show that in the same limit the second mode can be described by the Gardner (modified KdV) equation, if the intra-component interaction constants have close enough values. This leads to a rich phenomenology of nonlinear excitations (solitons, kinks, algebraic solitons, breathers) which does not exist in the KdV description.
The dynamics of nonlinear solitary waves in studied by using the model of nonlinear Schrodinger equations(NSE) with an external harmonic potential. The model allows one to analyze on the general basis a variety of nonlinear phenomena appearing both in Bose-Einstein condensate in a magnetic trap, whose profile is describe by a quadratic function of coordinates, and an nonlinear optics, physics of lasers, and biophysics. It is shown that exact solutions for a quantum-mechanical particle in a harmonic potential and solutions obtained within the framework of the adiabatic perturbation theory of bright solitons in a parabolic trap are completely identical. This fact not only proves once more that solitons behave like particles but also that they can preserve such properties in different traps for the which the parabolic approximation is valid near potential energy minima. The conditions are found for formation of stable stationary states of antiphase soliton in a harmonic potential. The interaction dynamics of solitons in nonstationary potentials is studied and the possibility of the appearance of a soliton parametric resonance at with the amplitude of soliton oscillation in a trap exponentially increases with time is shown.
We investigate the moving matter-wave solitons in spin—orbit coupled Bose—Einstein condensates (BECs) by a perturbation method. Starting with the one-dimensional Gross—Pitaevskii equations, we derive a new KdV-like equation to which an approximate solution is obtained by assuming weak Raman coupling and strong spin—orbit coupling. The derivation of the KdV-like equation may be useful to understand the properties of solitons excitation in spin—orbit coupled BECs. We find different types of moving solitons: dark—bright, bright—bright and dark—dark solitons. Interestingly, moving dark—dark soliton for attractive intra- and inter-species interactions is found, which depends on the Raman coupling. The amplitude and velocity of the moving solitons strongly depend on the Raman coupling and spin—orbit coupling.
We develop a systematic analytical approach to consider the dynamics of linear and nonlinear excitations in trapped quasi-one-dimensional Bose-Einstein condensates with repulsive atom-atom interactions. We show that, for a condensate strongly confined in two transverse directions, the ground state of the system involves the high-order eigenmodes of the transverse confining potential in the transverse directions and effective high-order Thomas-Fermi wave functions in the axial direction. The linear excitations of the system have a Bogoliubov-type spectrum with the excitation frequency varying slowly along the axial direction. We find that, in a weak nonlinear approximation, the amplitude of a nonlinear excitation is governed by a variable coefficient Korteweg--de Vries equation with additional terms contributed from the transverse structure and the inhomogeneity in the axial direction of the condensate, which results in varying amplitude, width, and velocity for dark solitons. Because of the inhomogeneity the dark solitons undergo deformation and emit radiations when traveling along the axial direction. We finally demonstrate that a dark soliton will disintegrate into several ones plus a residual wave train when passing over a steplike potential.