RAISE-TAQS: The Hidden Structure of the Disorder in Quantum Systems

这是数学、量子半导体物理和GaN半导体材料科学的交叉学科，致力于对无序系统的严格理解和控制，目的是赋予设计的能力，而不仅仅是观察，隐藏在无序半导体材料中的新量子物体。将启动一个理论和实验研究项目，该项目将率先在无序半导体合金中使用局域态作为量子点，并在几个相关领域提供突破性成果，这些领域包括无序系统的数学、量子物理、纳米级材料的设计和表征。这个项目的宏伟目标是以精确的、可量化的、数学术语来预测和操纵无序介质中电子物质波的局域性，并将其应用于半导体合金局域区域的新型行为良好的量子物体。具体来说，在数学方面，该项目旨在建立第一个确定性理论，揭示波的精确几何结构，并在无序存在下更普遍地解决偏微分方程。首次处理泊松-薛定谔及类似自洽系统中的局部化问题，首次用几何方法处理局部化问题，并完全解决了椭圆测度的绝对连续性问题。在实验材料科学领域，该项目旨在开启局域化工程领域，包括设计基于led的InGaN材料系统的自发生纳米结构，其特性基于微观尺度上的电子局域化控制，激子特性的优化，局域态之间的输运控制，弛豫和相干时间。在物理学方面，目标是在无序系统中寻找量子效应的新方法，重点是在量子化半导体结构中退相干的突出开放问题。关键是证明无序可以极大地改善量子参数，如相干时间，这是建立稳定纠缠态发生器的第一步。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This is an interdisciplinary program at the interface of mathematics, quantum semiconductor physics, and the material science of GaN semiconductors, devoted to the rigorous understanding and control of the disordered systems with the aim to grant the power to design, beyond just observing, new quantum objects hidden in disordered semiconductor materials. A program of theoretical and experimental research will be launched which pioneers the use of localized states in disordered semiconductor alloys as quantum dots and offers breakthrough achievements across several related areas which range from mathematics of the disordered systems, to quantum physics, to nanoscale materials design and characterization. The grand goal of this project is to predict and manipulate localization properties of electron matter waves in disordered media in precise, quantifiable, mathematical terms, with the applications to novel well-behaved quantum objects in the localization regions of semiconductor alloys.Specifically, in mathematics, the project aims at the first deterministic theory revealing the precise geometric structure of waves and more general solutions to PDEs in the presence of disorder. The first treatment of localization in Poisson-Schrodinger and similar self-consistent systems, and the first treatment of localization by geometry, and a complete resolution of the problem of absolute continuity of elliptic measure. In experimental material science, the project aims at the inauguration of the field of localization engineering, including designing self-occurring nanostructures, based on the InGaN materials system of LEDs, with desired properties based on the control of electron localization at the microscopic scale, optimization of their exciton properties, control of the transport between the localized states, their relaxation and coherence times. In physics, the goal is a new approach to quantum effects in disordered systems, with the emphasis on the outstanding open problems of decoherence in quantized semiconductor structures. At stake is the demonstration that disorder can greatly improve quantum parameters such as the coherence time, which is the first mandatory step for building stable entangled state generators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.