Thermodynamics of inhomogenous imperfect quantum gases in harmonic traps

Abstract

We discuss thermodynamic properties of harmonically trapped imperfect quantum gases. The spatial inhomogeneity of these systems imposes a redefinition of the mean-field interparticle potential energy as compared to the homogeneous case. In our approach, it takes the form $\frac{a}{2}{N}^2{\omega }^d$, where $N$ is the number of particles, $\omega$ - the harmonic trap frequency, $d$ - system's dimensionality, and $a$ is a parameter characterizing the interparticle interaction. We provide arguments that this model corresponds to the limiting case of a long-ranged interparticle potential of vanishingly small amplitude. This conclusion is drawn from a computation similar to the well-known Kac scaling procedure, which is presented here in a form adapted to the case of an isotropic harmonic trap. We show that within our model, the imperfect gas of trapped repulsive bosons undergoes the Bose-Einstein condensation provided $d>1$. The main result of our analysis is that in $d=1$ the gas of attractive imperfect fermions with $a=-a_F<0$ is thermodynamically equivalent to the gas of repulsive bosons with $a=a_B>0$ provided the parameters $a_F$ and $a_B$ fulfill the relation $a_B+a_F=\hslash$. This result supplements similar recent conclusion about thermodynamic equivalence of two-dimensional uniform imperfect repulsive Bose and attractive Fermi gases.

arXiv commentRevised version, comments added

References in corpus (20)

• Theory of Bose-Einstein condensation in trapped gases
• Berezinskii-Kosterlitz-Thouless Crossover in a Trapped Atomic Gas
• Regimes of quantum degeneracy in trapped 1D gases
• Bose-Einstein condensation in quasi2D trapped gases
• Fermi gases in one dimension: From Bethe Ansatz to experiments
• Spin-Imbalance in a One-Dimensional Fermi Gas
• Proof of Bose-Einstein Condensation for Dilute Trapped Gases
• Two-dimensional Bose-Einstein condensate in an optical surface trap
• Observation of the Berezinskii-Kosterlitz-Thouless Phase Transition in an Ultracold Fermi Gas
• Weakly Interacting, Dilute Bose Gases in 2D
• Bose-Einstein Condensation in a Harmonic Potential
• Finite temperature phase diagram of a spin-polarized ultracold Fermi gas in a highly elongated harmonic trap
• Measurement of the Dynamical Structure Factor of a 1D Interacting Fermi Gas
• Existence of Long-Range Order for Trapped Interacting Bosons
• Casimir force induced by imperfect Bose gas
• The imperfect Bose gas in d dimensions: critical behavior and Casimir forces
• Bose-Einstein Condensation in a Dilute, Trapped Gas at Positive Temperature
• Mean-Field Interacting Boson Random Point Fields in Weak Harmonic Traps
• Thermodynamic equivalence of two-dimensional imperfect attractive Fermi and repulsive Bose gases
• Bose-Einstein Condensation in Geometrically Deformed Tubes

Cited by in corpus (1)

• Dimensional crossovers and Casimir forces for the Bose gas in anisotropic optical lattices