Fermion - like description of condensed Bosons in a trap

A Bose-Einstein condensate of atoms, trapped in an axially symmetric harmonic potential, is considered. By averaging the spatial density along the symmetry direction over a length that preserves the aspect ratio, the system may be mapped on to a zero temperature noninteracting Fermi-like gas. The “mock fermions” have a state occupancy factor (>> 1) proportional to the ratio of the coherance length to the “hard-core” radius of the atom. The mapping reproduces the ground state properties of the condensate, and is used to estimate the vortex excitation energy analytically. The “mock-fermion” description predicts some novel collective excitation in the condensed phase. PACS:03.75.Fi, 05.30.Jp, 32.80.Pj Typeset using REVTEX Permanent Address: The Institute of Mathematical Sciences, Madras 600 113, India 1 Recently there has been a renewed interest in the Bose-Einstein condensation(BEC) of a gas after its experimental demonstration [1] with rubidium vapour in a trap at a temperature of 170 nanokelvin and at a number density ρ = 2.5 × 10 atoms per cc. This experiment has been followed by others using alkali atoms [2]. At these temperatures the atoms form a weakly interacting metastable gas of Bosons. For a non-technical account see the review by Burnett [3]. In a typical device, atoms are trapped in a potential which is well described by an axially symmetric parabolic confinement. The oscillator frequency in the symmetry direction is larger than the frequency in the plane perpendicular to it. The experimental situation of interest to us is the one with rubidium vapour, where the s-wave scattering length between two atoms is known to be positive. The effect of the interatomic interaction may be mocked up by a repulsive pseudo-potential [4]. The interaction energy is propotional to aρ, where a is the s-wave scattering length and ρ is the number density of the atoms. The properties of the condensate have been studied by constructing the density functional involving this replusive interaction energy, and the potential energy of the atoms in the trap [5–7]. In this paper, we first note that by averaging the spatial density of the condensed bosons along one direction, it may be reduced to the same form as the density of a non-interacting Fermi gas. We chose the averaging direction to be the symmetry axis (the z-direction ), along which the harmonic confinement is steeper. This enables us to use the Fermi gas model to compute the low-lying planar excitations of the condensate in the shallow well. Moreover, the averaging distance is chosen to preserve the aspect ratio (the ratio of the length scales in the planar to perpendicular directions ) of the original trap. The Bose-condensate is now described by a three-dimensional noninteracting “Fermi” gas, trapped in the same planar parabolic potential as the original system, but free to move along the z-direction within the averaging distance. One peculiarity of these mock-fermions, as we call them, is their occupancy factor per state. Instead of being one (or zero ) at T = 0, it is multiplied by a large factor proportional to λF/2a, where λF = h/pF , and 2a the apparent size of the atom. In fact, the Fermi momentum h̄kF is such that k −1 F = ξ, where ξ = (8πρa) −1/2 is just the coherance length in the bose condensate. It is remarkable that the kinetic energy of these 2 mock fermions exactly reproduces the condensate energy in the large-N limit. The latter is in fact calculated by neglecting the kinetic energy of the bosons. After having shown this equivalence, we go on to use this model to calculate some other properties. These include the velocity of sound, and the vortex excitation energy. The sound velocity in the mock-fermion ideal gas is the same as in the Bose-condensate. A simple estimate of the vortex excitation energy is made by “digging” a hole in the central density, i.e., promoting all the s-state mock-fermions out of the Fermi sea to states of non-zero angular momentum. This reproduces the numerical results of Dalfovo and Stringari [6] satisfactorily. The latter calculation involved solving a nonlinear Schrödinger equation that was obtained from the density-functional formalism. Finally, our description also predicts some novel collective excited states with zero angular momentum involving a large number of mock-fermions. We begin with the ground state energy for condensed bosons given by the GinzbergGross-Pitaevskii [8] energy functional, E[ψ] = ∫ dr [ h̄ 2m |∇ψ| + m 2 (ω ⊥ r ⊥ + ω 3 z)|ψ(r)| + 2πh̄ a m |ψ(r)| ] , (1) where m is the mass of the atom, ω⊥, ω3 denote the oscillator frequencies in the transverse direction and in the direction of the symmetry axis( z-axis), and a is the s-wave scattering length which defines the strength of the interaction in the pseudo-potential method. The condensate wave function is usually denoted as ψ(r) = √