Higher order terms in the condensate fraction of a homogeneous and dilute Bose gas

TL;DR

This paper analytically derives higher order coefficients in the power series expansion of the condensate fraction for a homogeneous dilute Bose gas, extending known results to include terms up to order $(n a^3)^{3/2}$.

Contribution

The authors develop a method to analytically calculate higher order terms in the condensate fraction expansion, which were previously unknown.

Findings

Coefficients for higher order terms are explicitly derived.

The method involves a double application of canonical transformations.

Results include specific values for $c_2$ and $c_3$.

Abstract

The condensate fraction of a homogeneous and dilute Bose gas is expanded as a power series of $\sqrt{n a^3}$ as $N_0/N = 1 -c_1 (n a^3)^{1/2} -c_2 (n a^3) - c_3 (n a^3)^{3/2}\hdots.$ The coefficient $c_1$ is well-known as $8/3 \sqrt{\pi}$, but the others are unknown yet. Considering two-body contact interactions and applying a canonical transformation method twice we developed the method to obtain the higher order coefficients analytically. An iteration method is applied to make up a cutoff in a fluctuation term. The coefficients ares $c_2=2(\pi - 8/3)$ and $c_3=(4/\sqrt{\pi}) (\pi -8/3)(10/3-\pi)$.