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arXiv:cond-mat/0406020 · Comptes Rendus Physique

Exact scaling transform for a unitary quantum gas in a time dependent harmonic potential

Yvan Castin

Published Apr 1, 2004• 90 citations
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Abstract

A unitary quantum gas is a gas of quantum particles with a binary interaction of infinite scattering length and negligible range. It has been produced in recent experiments with gases of fermionic atoms by means of a Feshbach resonance. Using the Fermi pseudo-potential model for the atomic interaction, we show that the time evolution of such a gas in an isotropic three-dimensional time dependent harmonic trap is exactly given by a gauge and scaling transform.

DOI
10.1016/j.crhy.2004.03.017

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A unitary quantum gas is a gas of quantum particles with a binary interaction of infinite scattering length and negligible range. It has been produced in recent experiments with gases of fermionic atoms by means of a Feshbach resonance. Using the Fermi pseudo-potential model for the atomic interaction, we show that the time evolution of such a gas in an isotropic three-dimensional time dependent harmonic trap is exactly given by a gauge and scaling transform.
Dynamics of strongly interacting Fermi gases, consisting of a 50-50 mixture of two different fermionic species, is investigated. For the equation of state we consider a Padé [2/2] approximations, which gives the weak-coupling perturbative formula (up to 4th order) in the low density regime, the unitary-limit Monte Carlo result in the high density regime, and reproduces the 4-fermion prediction for dimer-dimer scattering length in the BEC region. We use a time-dependent LDA to derive various properties of the Fermi gas under a harmonic confinement and compare them with the data of very recent experiments of 6Li atoms across a Feshbach resonance.
A scale-invariant two-component Fermi gas in a time-dependent isotropic harmonic potential is investigated. The exact time evolution of the density distribution in position space in any spatial dimension is obtained. Two experimentally relevant examples, an abrupt change and a periodic modulation of the trapping frequency, are solved. Small deviations from scale invariance are addressed within first-order perturbation theory. The consequences for experiments with ultracold quantum gases, such as the excitation of a tower of undamped breathing modes and an alternative for measuring the Tan contact, are discussed.
The probability amplitude for NNN particles in a quantum gas with negligible range of interparticle interaction potentials to come to a small region of size rrr scales like rγr^γrγ. It is shown that γγγ is quantitatively related to the ground state energy of these NNN fermions in the unitarity limit, confined by an isotropic harmonic potential. For large NNN, the short range density distribution of these NNN particles is predominantly the same as the Thomas-Fermi profile of the gas in the unitarity limit confined by such a harmonic potential. These results may shed light on strongly interacting ultracold atomic Fermi gases, in a trap or on an optical lattice.
We consider NNN atoms trapped in an isotropic harmonic potential, with sss-wave interactions of infinite scattering length. In the zero-range limit, we obtain several exact analytical results: mapping between the trapped problem and the free-space zero-energy problem, separability in hyperspherical coordinates, SO(2,1) hidden symmetry, existence of a decoupled bosonic degree of freedom, and relations between the moments of the trapping potential energy and the moments of the total energy.
Single-component quantum gas confined in a harmonic potential, but otherwise isolated, is considered. From the invariance of the system of the gas under a displacement-type transformation, it is shown that the center of mass oscillates along a classical trajectory of a harmonic oscillator. It is also shown that this harmonic motion of the center has, in fact, been implied by Kohn's theorem. If there is no interaction between the atoms of the gas, the system in a time-independent isotropic potential of frequency ${\ensuremath{\nu}}_{c}$ is invariant under a squeeze-type unitary transformation, which gives collective radial breathing motion of frequency $2{\ensuremath{\nu}}_{c}$ to the gas. The amplitudes of the oscillating and breathing motions from the exact invariances could be arbitrarily large. For a Fermi system, appearance of $2{\ensuremath{\nu}}_{c}$ mode of the large breathing motion indicates that there is no interaction between the atoms, except for a possible long-range interaction through the inverse-square-type potential.