Quantum scattering states in a nonlinear coherent medium
Quantum scattering states in a nonlinear coherent medium
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DOI：
10.1103/physreva.108.023314
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发表时间：
2023-01
影响因子：
2.9
通讯作者：
Allison Brattley;Hongyi Huang;K. Das
中科院分区：
文献类型：
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作者：
Allison Brattley;Hongyi Huang;K. Das
我们提出了一个全面的研究在一个相干介质中的定态与二次或克尔非线性的存在下，在一个维度（1D）的非线性项的正负符号，以及障碍和威尔斯本地化的潜力。该描述是在非线性薛定谔方程（NLSE），因此适用于各种系统，包括相互作用的超冷原子在平均场制度和光在光纤中的传播。我们确定解决方案的完整景观，在一个潜在的步骤和建立解决方案的矩形障碍和势阱。结果表明，所有的解决方案可以表示在一个雅可比椭圆函数与包含一个复值相移。我们的解决方案的方法依赖于一个三次多项式的根与流体动力学的图片，它提供了一个简单的分类所有的解决方案，有界和无界的，而边界条件直观地可视化为相空间曲线的交叉点。我们比较了开放边界条件的解决方案与那些在一个环上的势垒，也表明，数值计算的解决方案，光滑的障碍同意定性与矩形障碍的解析解。波动的Bogoliubov方程的基础上的解决方案的稳定性分析表明，持久的不稳定性是本地化的尖锐的边界，并预测的边界上的平均密度变化的值的导数的密度在边缘的关系。我们研究的散射波包的势垒势，并表明，在任何时刻的散射状态很好地描述了我们得到的固定的解决方案，表明我们的结果和方法的应用程序的非线性散射问题。
We present a comprehensive study of stationary states in a coherent medium with a quadratic or Kerr nonlinearity in the presence of localized potentials in one dimension (1D) for both positive and negative signs of the nonlinear term, as well as for barriers and wells. The description is in terms of the nonlinear Schr\"odinger equation (NLSE) and hence applicable to a variety of systems, including interacting ultracold atoms in the mean field regime and light propagation in optical fibers. We determine the full landscape of solutions, in terms of a potential step and build solutions for rectangular barrier and well potentials. It is shown that all the solutions can be expressed in terms of a Jacobi elliptic function with the inclusion of a complex-valued phase shift. Our solution method relies on the roots of a cubic polynomial associated with a hydrodynamic picture, which provides a simple classification of all the solutions, both bounded and unbounded, while the boundary conditions are intuitively visualized as intersections of phase space curves. We compare solutions for open boundary conditions with those for a barrier potential on a ring, and also show that numerically computed solutions for smooth barriers agree qualitatively with analytical solutions for rectangular barriers. A stability analysis of solutions based on the Bogoliubov equations for fluctuations show that persistent instabilities are localized at sharp boundaries, and are predicated by the relation of the mean density change across the boundary to the value of the derivative of the density at the edge. We examine the scattering of a wavepacket by a barrier potential and show that at any instant the scattered states are well described by the stationary solutions we obtain, indicating applications of our results and methods to nonlinear scattering problems.