Collaborative Research: Joint NSF-BSF Proposal: Nonlinear Dynamics with Gross-Pitaevskii Breathers

能量守恒原理规定，任何封闭系统的总能量不随时间变化。对于大多数高度复杂的相互作用系统，能量是唯一这样的守恒量，但这一规则也有例外。在极少数情况下，即使是非常复杂的系统，也可能有额外的不变量，它们的存在限制了系统可能的物理配置。当系统具有一定程度的对称性时，通常会出现这些额外的不变量。由此产生的约束-所谓的守恒定律-例如，牛顿摇篮运动的钟状规律性，完美稳定的海洋岸波的存在，以及行星轨道形状的简单性。这个项目的目标是学习使用这种非标准守恒定律作为稳定原子干涉仪的工具；原子干涉仪是用于引力和其他领域的超灵敏探测器的量子力学装置。此外，该项目可能会揭示复杂系统在受到干扰后恢复平衡的方式，以及放松所需的时间尺度。这项工作涉及激发原子干涉仪的某些高度稳定的模式，即所谓的“原子呼吸”，然后研究这些模式在遇到势垒时的弛豫。干涉仪的高度灵敏度提供了比以前更仔细地研究非平衡过程的机会。该提案有三个主要部分：研究呼吸器的松弛动力学，包括对称性破坏的影响；制定创造高度稳定的呼吸器的实验方案；以及评估呼吸器的稳定性，防止解离和衰变。由四名主要研究人员组成的小组拥有非常适合这些任务的独特技能组合：它包括操纵玻色孤子(由玻色原子组成的呼吸子)、理论量子、平均场和经典非平衡动力学的实验技术、逆散射变换和非线性微扰理论，以及Bethe ansatz的应用。理论和实验的紧密结合是这个项目的一个关键方面，也是其成功的关键。

The principle of energy conservation dictates that the total energy of any closed system does not change with time. For most highly complex interacting systems, energy is the only such conserved quantity, but there are exceptions to this rule. In rare situations, even quite complex systems may have additional invariant quantities whose existence constrains the possible physical configurations of the system. These extra invariants generally arise when the system possesses some degree of symmetry. The resulting constraints---the so-called conservation laws---are responsible, for example, for the clock-like regularity of the motion of Newton's cradle, for the existence of perfectly straight stable ocean-shore waves, and for the simplicity of the shape of the planetary orbits. The goal of this project is to learn to employ such non-standard conservation laws as a tool to stabilize atom interferometers; quantum mechanical devices used in ultra-sensitive detectors of gravitational and other fields. In addition, the project may shed light on the manner by which complex systems, once disturbed, return to equilibrium, and the time-scale required for that relaxation. This work involves exciting certain highly stable modes, so-called "atomic breathers", of an atom interferometer, and then studying the relaxation of these modes as they encounter a potential barrier. The high degree of sensitivity of the interferometer provides an opportunity for much more careful scrutiny of non-equilibrium processes than has been possible previously. There are three major components of the proposal: a study of the relaxation dynamics of the breathers, including effects of symmetry-breaking; the development of an experimental protocol for the creation of highly-stable breathers; and an assessment of the stability of the breathers against dissociation and decay. The group of four principal investigators possess combine unique skill sets ideally suited for these tasks: it includes experimental techniques for manipulating bosonic solitons (breathers made of bosonic atoms), theoretical quantum, mean-field, and classical nonequilibrium dynamics, Inverse Scattering Transform and nonlinear perturbation theory, and applications of the Bethe ansatz. The tight coupling of theory and experiment is a crucial aspect of this project and will be essential for its success.