On the role of hyperfine-interactions-mediated Zeeman effect in the condensation temperature shift of trapped atomic Bose-Einstein condensates

TL;DR

This paper investigates how hyperfine interactions mediated Zeeman effects influence the shift in condensation temperature of trapped atomic Bose-Einstein condensates, explaining experimental observations through a mean-field approach.

Contribution

It introduces a theoretical model incorporating hyperfine-mediated Zeeman interactions to predict temperature shifts in Bose-Einstein condensates, aligning with experimental data.

Findings

Predicted temperature shift coefficients match experimental values.

Derived general expressions for temperature shift coefficients.

Applicable to other Feshbach resonances and atomic species.

Abstract

We discuss the effect of interatomic interactions on the condensation temperature $T_c$ of a laboratory atomic Bose-Einstein condensate under the influence of an external trapping magnetic field. We predict that accounting for hyperfine interactions mediated Zeeman term in the mean-field approximation produces, in the case of the $403 \, G$ Feshbach resonance in the $|F,m_F> = |1,1>$ hyperfine state of a $^{39}K$ condensate, with $F$ the total spin of the atom, an experimentally observed (and not yet explained) shift in the condensation temperature $\Delta T_{c}/T_{c}^{0}=b^{*}_0+b^{*}_1 (a/\lambda_{T}) + b^{*}_2 (a/\lambda_{T})^2$ with $b^{*}_0 \simeq 0.0002$, $b^{*}_1 \simeq -3.4$ and $b^{*}_2 \simeq 47$, where $a$ is the s-wave scattering length, and $\lambda_T$ is the thermal wavelength at $T_{c}^{0}$. Generic expressions for the coefficients $b^*_0$, $b^*_1$ and $b^*_2$ are also obtained, which can be used to predict the temperature shift for other Feshbach resonances of $^{39}K$ or other atomic condensates.