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

这是数学，量子半导体物理学和GaN半导体材料科学接口的跨学科计划，致力于严格理解和控制无序系统，旨在授予设计的权力，不仅仅是观察，隐藏在无序半导体材料中的新量子物体。 将启动一项理论和实验研究计划，开创性地使用无序半导体合金中的局域态作为量子点，并在几个相关领域取得突破性成就，从无序系统的数学到量子物理学，再到纳米材料设计和表征。 该项目的宏伟目标是以精确、可量化的数学术语预测和操纵无序介质中电子物质波的局域化性质，并将其应用于半导体合金局域化区域中的新型行为良好的量子物体。具体而言，在数学方面，该项目的目标是第一个确定性理论，揭示波的精确几何结构和更普遍的解决方案，以偏微分方程的存在，disorder. 第一次处理Poisson-Schrodinger和类似的自洽系统中的局部化问题，第一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。 在实验材料科学方面，该项目旨在开创本地化工程领域，包括设计基于LED的InGaN材料系统的自发纳米结构，具有基于微观尺度上电子本地化控制的所需特性，优化其激子特性，控制本地化状态之间的传输，其弛豫和相干时间。 在物理学中，目标是无序系统中量子效应的新方法，重点是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，这是构建稳定纠缠态发生器的第一个强制性步骤。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.