Number-Theory-Inspired Effects in Cold Atoms

最近在为单个原子创造量身定制的陷阱方面的实验进展，使得完全控制这些陷阱中的量子能级成为可能。这些技术的潜在应用之一是“实验数论”。在这样的应用中，量子系统的物理效应是基于特定猜想或数论特定定理的有效性来设计的。在这笔资助下，PI将研究一个与整数平方和有关的数学恒等式，以及一个数学猜想，即任何偶数都是两个素数的和。在这两种情况下，可以预见到以前未知的物理效应的出现，例如违反量子-经典对应的新例子。该项目的更广泛影响包括与汉斯科姆空军基地的国防部STARBASE学院青年拓展计划合作，并制定一项教育计划，该计划介绍了未来QIS学习者的9个NSF关键概念中的5个，目标是马萨诸塞州东北部的第一学区。下面讨论与数论有关的量子系统。(a)其单粒子谱是所有自然数的对数的势。这一潜力有望证明共振跃迁级联，在其经典对应物中不存在。共振级联中的居群动力学对所涉及能级的质因数分解特性很敏感。(b)一种势，其单粒子谱由自然数的两个平方和的对数给出。这组也支持共振级联通过所谓的Diophantus-Brahmagupta-Fibonacci身份。(c)谱为素数的外势中的两个弱相互作用原子，加上频率为2的小周期性扰动。该系统中共振级联的存在是基于哥德巴赫猜想的有效性，而哥德巴赫猜想是出了名的尚未被证明。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。

Recent experimental advances in creating tailored traps for individual atoms have allowed for complete control of the quantum energy levels in those traps. One of the potential applications of these techniques is "experimental number theory". In such applications, the physical effects of a quantum system are engineered based on the validity of a particular conjecture or a particular theorem of the number theory. Under this grant, the PI will address a mathematical identity that is related to sums of squares of integers and also a mathematical conjecture which states that any even number is a sum of two prime numbers. In both cases, an appearance of previously unknown physical effects is foreseen, such as a new example of a violation of the quantum-classical correspondence. Broader impacts of the project include collaboration with the DoD STARBASE Academy youth outreach program at the Hanscom Air Force Base and the development of an educational program which introduces five out of the nine NSF Key Concepts for Future QIS Learners, aimed at Title I school districts in northeastern Massachusetts. The following quantum systems relevant to the number theory are addressed. (a) A potential whose one-particle spectrum is the logarithms of all natural numbers. This potential is expected to demonstrate a resonant transition cascade, absent in its classical counterpart. Population dynamics in a resonant cascade will be sensitive to the properties of the prime factorization of the energy levels involved. (b) A potential whose one particle spectrum is given by the logarithms of sums of two squares of the naturals. This set also supports resonant cascades via the so-called Diophantus-Brahmagupta–Fibonacci identity. (c) Two weakly interacting atoms in an external potential whose spectrum is the primes, plus a small periodic perturbation whose frequency is 2. The existence of a resonant cascade in this system is predicated on the validity of Goldbach’s conjecture, which is famously still unproven.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.