Bose and Einstein Meet Newton
Abstract
We model the time evolution of a Bose-Einstein condensate, subject to a special periodically excited optical lattice, by a unitary quantum operator U on a Hilbert space H. If a certain parameter alpha = p/q, where p and q are coprime positive integers, then H = L2(R/Z,Cq) and U is represented by a q x q matrix-valued function M on R/Z that acts pointwise on functions in H. The dynamics of the quantum system is described by the eigenvalues of M. Numerical computations show that the characteristic polynomial det(zI - M(t)) = Prodj=1q (z - lambdaj(t)) where each lambdaj is a real analytic functions that has period 1/q. We discuss this phenomena using Newton's Theorem, published in Geometria analytica in 1660, and modern concepts from analytic geometry.