Number-Theory-Inspired Effects in Cold Atoms

最近在为单个原子制造量身定做的陷阱方面的实验进展使人们能够完全控制这些陷阱中的量子能级。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的特定定理的有效性而设计的。根据这项拨款，PI将解决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，都可以预见到以前未知的物理效应的出现，例如一个违反量子经典对应的新例子。该项目的更广泛影响包括与美国国防部STARBASE学院在汉斯康姆空军基地的青年外展计划合作，以及开发一项教育计划，该计划针对马萨诸塞州东北部的TITLI学区，介绍了NSF面向未来QIS学习者的九个关键概念中的五个。以下是与数论相关的量子系统。(A)其单粒子谱是所有自然数的对数的势。这种势能有望显示出一种共振的跃迁级联，而经典的势能是不存在的。共振级联中的布居动力学将对所涉及的能级的素因式分解的性质很敏感。(B)一个粒子谱由自然数的两个平方的和的对数给出的势。这套装置还支持通过所谓的丢番图-婆罗马古塔-斐波纳契恒等式的共振级联。(C)外势中两个弱相互作用的原子，其频谱是素数，加上一个频率为2的小周期微扰。该系统中共振级联的存在是基于哥德巴赫猜想的有效性，这一猜想是出了名的仍未得到证实。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。

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.