Analytical results for energy spectrum and eigenstates of a Bose-Einstein condensate in a Mott insulator state

Abstract

We present a method to solve the model describing either a Bose-Einstein condensate (BEC) in a Mott-insulator state or a double-well BEC. We show that all the energy eigenvalues and eigenstates for an arbitrary (small or large) total atom number N can be explicitly expressed analytically in terms of a parameter $\ensuremath{\lambda}$ whose values are determined by the roots of the polynomials of the order of at most $1+\mathrm{int}(N/2),$ with $\mathrm{int}(x)$ denoting x's integer part. We also show that \ensuremath{\lambda}'s explicit analytical expressions for $N<~7$ can be readily obtained by a simple MATHEMATICA code. Besides, finding the roots of the polynomials of the order of at most $1+\mathrm{int}(N/2)$ to give explicitly all the energy eigenvalues and eigenstates greatly simplifies the corresponding calculations, particularly when the total atom number N is large.