非线性薛定谔方程的波包动力学
Wavepacket Dynamics in Nonlinear Schrödinger Equations
摘要：
相干态由于其在时间演化下的独特性质，在量子力学中扮演着重要的角色。在这里，我们探讨这个概念的一维排斥非线性薛定谔方程，它描述弱相互作用的玻色-爱因斯坦凝聚体或光在非线性介质中的传播。结果表明，谐波势的基态的相空间平移的动力学是相当简单的：中心遵循经典的轨迹，而它的形状不随时间变化。抛物势是唯一满足这一性质的势。我们研究了这些非线性相干态的形状扰动下的时间演化，或限制势。各种各样的影响出现。特别是，在存在非谐性的情况下，我们观察到数据包分裂成两个不同的组件。一小部分的冷凝物被转移到非相干的高能量模式，而剩余的相干分量的振荡幅度被阻尼朝向井底。
英文摘要：
Coherent states play an important role in quantum mechanics because of their unique properties under time evolution. Here we explore this concept for one-dimensional repulsive nonlinear Schrödinger equations, which describe weakly interacting Bose-Einstein condensates or light propagation in a nonlinear medium. It is shown that the dynamics of phase-space translations of the ground state of a harmonic potential is quite simple: the centre follows a classical trajectory whereas its shape does not vary in time. The parabolic potential is the only one that satisfies this property. We study the time evolution of these nonlinear coherent states under perturbations of their shape, or of the confining potential. A rich variety of effects emerges. In particular, in the presence of anharmonicities, we observe that the packet splits into two distinct components. A fraction of the condensate is transferred towards uncoherent high-energy modes, while the amplitude of oscillation of the remaining coherent component is damped towards the bottom of the well.
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