We discuss the thermal gas to Bose-Einstein Condensation (BEC) transition of an ultracold Bose gas in a linear quadrupolar trap with contact interatomic interactions within the Hartree-Fock (HF) approximation. We briefly review the theoretical framework of the thermodynamics of a gas confined by a non-uniform potential to show how mechanical equilibria must be formulated in term of generalized volume and pressure variables V and P that replace the usual volume and hydrostatic pressure of a uniform system. We solve for the density profile within HF as a function of temperature T and molar volume v = V/N. With this information, we are able to calculate all the thermodynamic properties of the system and analyze the behavior of the gas through the BEC (superfluid) transition. We find that the transition is completely smooth, showing no sign of critical behavior. We emphasize how these predictions can be readily measured with the current experimental setups.
We discuss the thermal gas to Bose-Einstein Condensation (BEC) transition of an ultracold Bose gas in a linear quadrupolar trap with contact interatomic interactions within the Hartree-Fock (HF) approximation. We briefly review the theoretical framework of the thermodynamics of a gas confined by a non-uniform potential to show how mechanical equilibria must be formulated in term of generalized volume and pressure variables V and P that replace the usual volume and hydrostatic pressure of a uniform system. We solve for the density profile within HF as a function of temperature T and molar volume v = V/N. With this information, we are able to calculate all the thermodynamic properties of the system and analyze the behavior of the gas through the BEC (superfluid) transition. We find that the transition is completely smooth, showing no sign of critical behavior. We emphasize how these predictions can be readily measured with the current experimental setups.
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.
Predictions from Hartree–Fock (HF), Popov (P), Yukalov–Yukalova (YY) and t -matrix approximations regarding the thermodynamics from the normal to the Bose–Einstein condensation (BEC) phase in weakly interacting Bose gases are considered. By analysing the dependence of the chemical potential μ on the temperature T and particle density ρ, we show that none of them predicts a second-order phase transition as required by symmetry-breaking general considerations. In this work, we find that the isothermal compressibility κ T predicted by these theories does not diverge at criticality as expected in a true second-order phase transition. Moreover, the isotherms μ = μ(ρ, T ) typically exhibit non-single-valued behaviour in the vicinity of the BEC transition, a feature forbidden by general thermodynamic principles. This behaviour can be avoided if a first-order phase transition is appealed. The facts described above show that although these mean field approximations give correct results near zero temperature, they are endowed with thermodynamic anomalies in the vicinity of the BEC transition. We address the implications of these results in the interpretation of current experiments with ultracold trapped alkali gases.
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.
We present the full thermodynamics of an interacting fluid confined by an arbitrary external potential. We show that for each confining potential, there emerge "generalized" volume and pressure variables V and P , that replace the usual volume and hydrostatic pressure of a uniform system. This scheme is validated with the derivation of the virial expansion of the grand potential. We discuss how this approach yields experimentally amenable procedures to find the equation of state of the fluid, P=P(VN,T) with N the number of atoms, as well as its heat capacity at constant generalized volume C_{V}=C_{V}(V,N,T) . With these two functions, all the thermodynamics properties of the system may be found. As specific examples we study weakly interacting Bose gases trapped by harmonic and by linear quadrupolar potentials within the Hartree-Fock approximation. We claim that this route provides an additional and useful tool to analyze both the thermodynamic variables of an ultracold trapped gas as well as its elementary excitations.
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∊/ k , where Δ∊ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest-energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermodynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced. From this we obtain the density profile and the velocity distribution of particles, and use these to determine the shape of an anisotropic cloud after free expansion. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. Finally, Sec. 2.5 treats corrections to the transition temperature due to a finite particle number.
Abstract This chapter is devoted to the thermodynamic behaviour of an interacting trapped Bose gas. It developes the Hartree–Fock theory for nonuniform interacting Bose gases at low temperatures, and discusses the problem of the shift of the critical temperature caused by interactions and also the thermodynamic behaviour of the trapped gas below the critical temperature. The chapter outlines general procedures, which do not only hold for Bose gases, to determine the equation of state of uniform matter, starting from the knowledge of the density profile in the trap. Finally, the chapter discusses some properties of the collective oscillations of a trapped Bose gas at a finite temperature.