Measuring the heat capacity in a Bose-Einstein condensation using global variables

Abstract

Phase transitions are well understood and generally followed by the behavior of the associated thermodynamic quantities, such as in the case of the $\ensuremath{\lambda}$-point superfluid transition of liquid He, which is observed in its heat capacity. In the case of a trapped Bose-Einstein condensate, the heat capacity cannot be directly measured. In this work, we present a technique capable of determining the global heat capacity from the density distribution of a weakly interacting gas trapped in an inhomogeneous potential. This approach represents an alternative to models based on the local density approximation. By defining a pair of global conjugate variables, we determine the total internal energy and its temperature derivative, the heat capacity. We then apply the technique to a trapped $^{87}\mathrm{Rb}$ BEC, and a $\ensuremath{\lambda}$-type transition dependent on the atom number is observed, and the deviations from the noninteracting, ideal gas case are discussed. Finally, we discuss the chances of using this method to study the heat capacity at $T\ensuremath{\rightarrow}0$.