The Road Less Traveled: Resonant Bose-Einstein Condensates via a Hyperspherical Lowest-Order Constrained Variational Approach

In this work, we study the ground state properties of a system of N harmonically trapped bosons of mass m interacting with two-body contact interactions, from small to large scattering lengths. This is accomplished in a hyperspherical coordinate system that is flexible enough to describe both the overall scale of the gas and two-body correlations. By adapting the lowestorder constrained variational (LOCV) method, we are able to semi-quantitatively attain BoseEinstein condensate ground state energies even for gases with infinite scattering length. In the large particle number limit, our method provides analytical estimates for the energy per particle E0/N ≈ 2.5N1/3~ω and two-body contact C2/N ≈ 16N1/6p mω/~ for a Bose gas on resonance, where ω is the trap frequency. Further, by considering only two-body correlations, we note that a sudden quench from small to large scattering lengths leads to out-of-equilibrium resonant BEC. As an alternative, we propose a two-step scheme that involves an intermediate scattering length, between 0 and ∞, which serves to maximize the transfer probability of N bosons in a harmonic trap with frequency ω to the resonant state. We find that the intermediate scattering length should be a ≈ 3.16N −2/3p ~/(mω), and that it produces an optimum transition probability of 1.03N −1/6
.