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

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 $403G$ 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 $\Delta T_c/T_c^0=b_0^{\ast }+b_1^{\ast }(a/{\lambda }_T)+b_2^{\ast }(a/{\lambda }_T{)}^2$ with $b_0^{\ast }\simeq 0.0002$, $b_1^{\ast }\simeq -3.4$ and $b_2^{\ast }\simeq 47$, where $a$ is the s-wave scattering length, and ${\lambda }_T$ is the thermal wavelength at $T_c^0$. Generic expressions for the coefficients $b_0^{\ast }$, $b_1^{\ast }$ and $b_2^{\ast }$ are also obtained, which can be used to predict the temperature shift for other Feshbach resonances of ${}^{39}K$ or other atomic condensates.

4 pages