Number-Theory-Inspired Effects in Cold Atoms

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.