Bose and Einstein Meet Newton

TL;DR

This paper models the quantum dynamics of a Bose-Einstein condensate in a periodically excited optical lattice using a unitary operator, revealing eigenvalue behaviors linked to Newton's Theorem and analytic geometry.

Contribution

It introduces a novel mathematical framework for analyzing Bose-Einstein condensate dynamics with periodic excitation, connecting quantum eigenvalues to classical geometric theorems.

Findings

Eigenvalues are real analytic functions with period 1/q.

The characteristic polynomial factors into products involving these eigenvalues.

The phenomena are explained using Newton's Theorem and modern analytic geometry.

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 = L^2(R/Z,C^q) 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)) = Prod_j=1^q (z - lambda_j(t)) where each lambda_j 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.