Brandon P. van Zyl

Simple analytical particle and kinetic energy densities for a dilute fermionic gas in a d-dimensional harmonic trap.

We derive simple analytical expressions for the particle density rho(r) and the kinetic energy density tau(r) for a system of noninteracting fermions in a d-dimensional isotropic harmonic oscillator potential. We test the Thomas-Fermi (TF, or local-density) approximation for the functional relation tau[rho] using the exact rho(r) and show that it locally reproduces the exact kinetic energy density tau(r), including the shell oscillations, surprisingly well everywhere except near the classical turning point. For the special case of two dimensions (2D), we obtain the unexpected analytical result that the integral of tau(TF)[rho(r)] yields the exact total kinetic energy.

Finite-temperature theory of the trapped two-dimensional Bose gas

We present a Hartree-Fock-Bogoliubov (HFB) theoretical treatment of the two-dimensional trapped Bose gas and indicate how semiclassical approximations to this and other formalisms have lead to confusion. We numerically obtain results for the quantum-mechanical HFB theory within the Popov approximation and show that the presence of the trap stabilizes the condensate against long wavelength fluctuations. These results are used to show where phase fluctuations lead to the formation of a quasicondensate.

The two-body problem of ultra-cold atoms in a harmonic trap

We consider two bosonic atoms interacting with a short-range potential and trapped in a spherically symmetric harmonic oscillator. The problem is exactly solvable and is relevant for the study of ultra-cold atoms. We show that the energy spectrum is universal, irrespective of the shape of the interaction potential, provided its range is much smaller than the oscillator length.

Some exact results for a trapped quantum gas at finite temperature

We present closed analytical expressions for the particle and kinetic-energy spatial densities at finite temperatures for a system of noninteracting fermions (bosons) trapped in a d-dimensional harmonic-oscillator potential. For d=2 and 3, exact expressions for the N-particle densities are used to calculate perturbatively the temperature dependence of the splittings of the energy levels in a given shell due to a very weak interparticle interaction in a dilute Fermi gas. In two dimensions, we obtain analytically the surprising result that the l degeneracy in a harmonic-oscillator shell is not lifted in the lowest order even when the exact, rather than the Thomas-Fermi expression for the particle density is used. We also demonstrate rigorously (in two dimensions) the reduction of the exact zero-temperature fermionic expressions to the Thomas-Fermi form in the large-N limit.

Analytical expression for the first-order density matrix of a d-dimensional harmonically confined Fermi gas at finite temperature

We present a closed-form expression for the finite temperature first-order density matrix of an isotropic harmonically trapped ideal Fermi gas in any dimension. This constitutes a much sought after generalization of the recent results in the literature, where analytical expressions have been limited to quantities derived from the diagonal first-order density matrix. We compare our results with the Thomas-Fermi approximation (TFA) and demonstrate numerically that the TFA provides an excellent description of the first-order density matrix in the large-N limit. As an interesting application, we derive a closed-form expression for the finite temperature Hartree-Fock exchange energy of a two-dimensional (2D) parabolically confined quantum dot. We numerically test this result against the 2D TF exchange functional, and comment on the applicability of the local-density approximation to the exchange energy of an inhomogeneous 2D Fermi gas.

An elementary exposition of the Efimov effect

Two particles that are just shy of binding may develop an infinite number of shallow bound states when a third particle is added. This counterintuitive effect was first predicted by Efimov for identical bosons interacting with a short-range pairwise potential. The Efimov effect persists for nonidentical particles if at least two of the three bonds are almost bound. The Efimov effect has recently been verified experimentally using ultracold atoms. We explain the origin of this effect using elementary quantum mechanics and summarize the experimental evidence for it.