Using a multiple-image reconstruction method applied to a harmonically trapped Bose gas, we determine the equation of state of uniform matter across the critical transition point, within the local density approximation. Our experimental results provide the canonical description of pressure as a function of the specific volume, emphasizing the dramatic deviations from the ideal Bose gas behavior caused by interactions. They also provide clear evidence for the nonmonotonic behavior with temperature of the chemical potential, which is a consequence of superfluidity and Bose-Einstein condensation. The measured thermodynamic quantities are compared to mean-field predictions available for the interacting Bose gas. The limits of applicability of the local density approximation near the critical point are also discussed, focusing on the behavior of the isothermal compressibility.
Using a multiple-image reconstruction method applied to a harmonically trapped Bose gas, we determine the equation of state of uniform matter across the critical transition point, within the local density approximation. Our experimental results provide the canonical description of pressure as a function of the specific volume, emphasizing the dramatic deviations from the ideal Bose gas behavior caused by interactions. They also provide clear evidence for the nonmonotonic behavior with temperature of the chemical potential, which is a consequence of superfluidity and Bose-Einstein condensation. The measured thermodynamic quantities are compared to mean-field predictions available for the interacting Bose gas. The limits of applicability of the local density approximation near the critical point are also discussed, focusing on the behavior of the isothermal compressibility.
The critical properties of the phase transition from a normal gas to a BEC (superfluid) of a harmonically confined Bose gas are analyzed using a proposed model for the equation of state of the underlying homogenous Bose fluid. The equation of state is given in terms of the homogenous particle density as a function of the chemical potential and temperature of the gas. On the normal side of the fluid it is assumed to be an ideal Bose gas while in the superfluid state we propose a function that, both, asymptotically reaches the Thomas-Fermi solution of a weakly interacting Bose gas at large densities and low temperatures and, at the transition, matches the critical properties of the ideal Bose gas. With this model we obtain the {\it global} thermodynamics of the harmonically confined gas which shows that, at the critical transition of the confined gas, the thermodynamic susceptibilities, such as the isothermal compressibility, do not diverge, in contrast to the behavior of the uniform gas. The singularity of the critical transition in the confined system appears now at the derivatives of the susceptibilities. We discuss how these properties can be experimentally tested.
Abstract The critical properties of the phase transition from a normal gas to a Bose–Einstein condensate (superfluid) of a harmonically confined Bose gas are addressed with the knowledge of an equation of state of the underlying homogeneous Bose fluid. It is shown that while the presence of the confinement trap arrests the usual divergences of the isothermal compressibility and heat capacities, the critical behavior manifests itself now in the divergence of derivatives of the mentioned susceptibilities. This result is illustrated with a mean-field like model of an equation of state for the homogeneous particle density as a function of the chemical potential and temperature of the gas. The model assumes the form of an ideal Bose gas in the normal fluid while in the superfluid state a function is proposed such that, both, asymptotically reaches the Thomas–Fermi solution of a weakly interacting Bose gas at large densities and low temperatures and, at the transition, matches the critical properties of the ideal Bose gas. With this model we obtain the global thermodynamics of the harmonically confined gas, from which we analyze its critical properties. We discuss how these properties can be experimentally tested.
Abstract. We describe a powerful method for determining the equation of state of an ultracold gas from in situ images. The method provides a measurement of the local pressure of an harmonically trapped gas and we give several applications to Bose and Fermi gases. We obtain the grand-canonical equation of state of a spin-balanced Fermi gas with resonant interactions as a function of temperature [1]. We compare our equation of state with an equation of state measured by the Tokyo group in [2], that reveals a significant difference in the high-temperature regime. The normal phase, at low temperature, is well described by a Landau Fermi liquid model, and we observe a clear thermodynamic signature of the superfluid transition. In a second part we apply the same procedure to Bose gases. From a single image of a quasi ideal Bose gas we determine the equation of state from the classical to the condensed regime. Finally the method is applied to a Bose gas in a 3D optical lattice in the Mott insulator regime. Our equation of state directly reveals the Mott insulator behavior and is suited to investigate finite-temperature effects. The equation of state of ultracold Bose and Fermi gases: a few examples 2
We describe a powerful method for determining the equation of state of an ultracold gas from in situ images. The method provides a measurement of the local pressure of a harmonically trapped gas and we give several applications to Bose and Fermi gases. We obtain the grand-canonical equation of state of a spin-balanced Fermi gas with resonant interactions as a function of temperature (Nascimbène et al 2010 Nature 463 1057). We compare our equation of state with an equation of state measured by the Tokyo group (Horikoshi et al 2010 Science 327 442), which reveals a significant difference in the high-temperature regime. The normal phase, at low temperature, is well described by a Landau Fermi liquid model, and we observe a clear thermodynamic signature of the superfluid transition. In a second part, we apply the same procedure to Bose gases. From a single image of a quasi-ideal Bose gas, we determine the equation of state from the classical to the condensed regime. Finally, the method is applied to a Bose gas in a three-dimensional optical lattice in the Mott insulator regime. Our equation of state directly reveals the Mott insulator behavior and is suited to investigate finite-temperature effects.
We apply the global thermodynamic variables approach to experimentally determine the isothermal compressibility parameter ${\ensuremath{\kappa}}_{T}$ of a trapped Bose gas across the phase transition. We demonstrate the behavior of ${\ensuremath{\kappa}}_{T}$ around the critical pressure, revealing the second-order nature of the phase transition. Compressibility is the most important susceptibility to characterize the system. The use of global variables shows advantages with respect to the usual local density approximation method and can be applied to a broad range of situations.
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 Bagnato 3, 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.