Technical thermodynamics of an inhomogeneous gas around the Bose-Einstein transition using the global-variable method - Miotti, Marcos Paulo (2021)
Thermodynamics of non-uniform systems has always been a challenging topic in the physical sciences. In the field of quantum gases, that subject is mainly studied with ultracold samples trapped in a harmonic potential, the first kind of confinement ever made to atomic fluids.1 Nascimbène et al.2 have already described a method to measure locally the equation of state and heat capacity of a harmonically trapped quantum gas. Nonetheless, local measurements cannot provide the volume-dependent susceptibilities of an inhomogeneous system, namely the thermal expansion and compressibility, meaning that the thermodynamic description of harmonically trapped gases is still incomplete. To solve that issue, Romero and Bagnato3, 4 proposed canonical variables of work for a gas in a harmonic trap, which work like pressure and volume for a gas in a vessel and allow global measurements on the system. Therefore, we describe here our methodology to measure the thermodynamic susceptibilities of a harmonically trapped gas around the Bose-Einstein transition using the Romero-Bagnato formalism. Our experiments were divided in different phases, each having the harmonic trap with unique frequencies, related to the formalism´s extensive variable. In each phase, we prepared individual samples of rubidium-87 gas in the trap and imaged them using the time-of-flight technique. Next, we fitted our images with the bimodal model and found the in situ density profiles of the samples with the standard regression procedures. Then, we determined the phase´s equation of state as a function of temperature and number of atoms. For the thermodynamic analysis, we used an empirical model that we developed to fit the equation-of-state curves, allowing us to represent our data in terms of mathematical coefficients. In that way, we found the curves of internal energy, heat capacity, thermal expansion and compressibility, thus achieving a full thermodynamic description of a harmonically trapped gas. Our method and its results are unprecedented in the literature and might contribute to the further understanding of non-uniform systems, as well as the future development of quantum thermal engines.