Effective theory for the propagation of a wave packet in a disordered and nonlinear medium
Effective theory for the propagation of a wave packet in a disordered and nonlinear medium
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波包在无序非线性介质中传播的有效理论
DOI：
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发表时间：
2013
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通讯作者：
A. Finkelstein
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作者：
G. Schwiete;A. Finkelstein
波包在非线性无序介质中的传播表现出有趣的动力学。在这里，我们提出了一个分析的基础上的非线性薛定谔方程（Gross-Pitaevskii方程）。这个问题直接关系到玻色气体的膨胀实验和非线性光学介质中横向局域化的研究。在非线性介质中，波包的能量储存在动能部分和势能部分中，其传播的细节在很大程度上取决于从一种形式的能量到另一种形式的能量的转移。描述波包演化的理论在[G. Schwiete和A. Finkelstein，Phys. Rev. Lett. 104，103904（2010）]的非线性动力学方程。在本文中，我们提出的动力学方程的推导和分析的细节。作为一个重要的新的组成部分，我们研究了由非线性引起的粒子间碰撞，并导出了相应的碰撞积分。我们限制自己的弱非线性限制，无序散射是占主导地位的散射机制。我们发现，在特殊的白色噪声杂质势的情况下，二维系统的均方半径与t成线性关系。这个结果以前在无碰撞极限下得到过，但在存在碰撞的情况下也成立。最后，我们提到不同的机制，通过这些机制，非线性可能会影响本地化的扩展波包。
The propagation of a wave-packet in a nonlinear disordered medium exhibits interesting dynamics. Here, we present an analysis based on the nonlinear Schr"odinger equation (Gross-Pitaevskii equation). This problem is directly connected to experiments on expanding Bose gases and to studies of transverse localization in nonlinear optical media. In a nonlinear medium the energy of the wave-packet is stored both in the kinetic and potential parts, and details of its propagation are to a large extent determined by the transfer from one form of energy to the other. A theory describing the evolution of the wave-packet has been formulated in [G. Schwiete and A. Finkelstein, Phys. Rev. Lett. 104, 103904 (2010)] in terms of a nonlinear kinetic equation. In this paper, we present details of the derivation of the kinetic equation and of its analysis. As an important new ingredient we study interparticle-collisions induced by the nonlinearity and derive the corresponding collision integral. We restrict ourselves to the weakly nonlinear limit, for which disorder scattering is the dominant scattering mechanism. We find that in the special case of a white noise impurity potential the mean squared radius in a two-dimensional system scales linearly with t. This result has previously been obtained in the collisionless limit, but it also holds in the presence of collisions. Finally, we mention different mechanisms through which the nonlinearity may influence localization of the expanding wave-packet.