Thermodynamics of ultracold trapped gases. Generalized mechanical variables, equation of state and heat capacity

Abstract

The thermodynamics framework of an interacting quantum gas trapped by an arbitrary external potential is reviewed. We show that for each confining potential, in the thermodynamic limit, there emerge "generalized" volume and pressure variables $\mathcal{V}$ and $\mathcal{P}$, 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 show that this approach yields experimentally amenable procedures to find the equation of state of the fluid, $\mathcal{P}=\mathcal{P}(\mathcal{V}/N,T)$ with $N$ the number of atoms, as well as its heat capacity at constant generalized volume $C_{\mathcal{V}}=C_{\mathcal{V}}(\mathcal{V},N,T)$. 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. Comparisons with experimental results of a ${}^{23}$Na ultracold gas are also presented. We claim that this route should provide an additional and useful tool to analyze both the thermodynamic variables of a trapped gas as well as its elementary excitations.

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• Making statistics work: a quantum engine in the BEC-BCS crossover
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• Thermodynamic engine with a quantum degenerate working fluid
• Isothermal compressibility determination across Bose-Einstein condensation
• Validity of Gross-Pitaevskii solutions of harmonically confined BEC gases in reduced dimensions
• Semiclassical Hartree-Fock theory of a rotating Bose-Einstein condensation
• Thermodynamics of low-dimensional trapped Fermi gases
• Scale invariant thermodynamics of a toroidally trapped Bose gas
• Critical properties of weakly interacting Bose gases as modified by a harmonic confinement
• BEC phase diagram of a ${}^{87}$Rb trapped gas in terms of macroscopic thermodynamic parameters
• Thermal Global Expansion Coefficient Measurement for a Harmonic Trapped Gas Across Bose-Einstein Condensation