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Trapped Atomic Fermi Gases

M. Ya. Amusia, ALFRED Z. MSEZANE

Published Jan 1, 2001• 4 citations
OpenAlex

Abstract

A many-body system of fermion atoms with a model interaction characterized by the scattering length a is considered. We treat both a and the density as parameters assuming that the system can be created artificially in a trap. If a is negative the system becomes strongly correlated at densities ρ ∼ |a | −3, provided the scattering length is the dominant parameter of the problem. It means that we consider |a | to be much bigger than the radius of the interaction or any other relevant parameter of the system. The density ρc1 at which the compressibility vanishes is defined by ρc1 ∼ |a | −3. Thus, a system composed of fermion atoms with the scattering length a → − ∞ is completely unstable at low densities, inevitably collapsing until the repulsive core stops the density growth. As a result, any Fermi system possesses the equilibrium density and energy if the bare particle-particle interaction is sufficiently strong to make a negative and to be the dominant parameter. This behavior can be realized in a trap. Our results show that a low density neutron matter can have the equilibrium density. PACS numbers: 05.30.Fk, 24.10.Cn, 71.10.Ca Key Words: trapped Fermi gases; effective interaction

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A many-body system of fermion atoms with a model interaction characterized by the scattering length a is considered. We treat both a and the density as parameters assuming that the system can be created artificially in a trap. If a is negative the system becomes strongly correlated at densities ρ ∼ |a | −3, provided the scattering length is the dominant parameter of the problem. It means that we consider |a | to be much bigger than the radius of the interaction or any other relevant parameter of the system. The density ρc1 at which the compressibility vanishes is defined by ρc1 ∼ |a | −3. Thus, a system composed of fermion atoms with the scattering length a → − ∞ is completely unstable at low densities, inevitably collapsing until the repulsive core stops the density growth. As a result, any Fermi system possesses the equilibrium density and energy if the bare particle-particle interaction is sufficiently strong to make a negative and to be the dominant parameter. This behavior can be realized in a trap. Our results show that a low density neutron matter can have the equilibrium density. PACS numbers: 05.30.Fk, 24.10.Cn, 71.10.Ca Key Words: trapped Fermi gases; effective interaction
Two-component equal-mass Fermi gases, in which unlike atoms interact through a short-range two-body potential and like atoms do not interact, are stable even when the interspecies sss-wave scattering length becomes infinitely large. Solving the many-body Schr\"odinger equation within a hyperspherical framework and by Monte Carlo techniques, this paper investigates how the properties of trapped two-component gases change if a third or fourth component are added. If all interspecies scattering lengths are equal and negative, our calculations suggest that both three- and four-component Fermi gases become unstable for a certain critical set of parameters. The relevant length scale associated with the collapse is set by the interspecies scattering length and we argue that the collapse is similar to the collapse of an attractive trapped Bose gas, a many-body phenomenon. Furthermore, we consider a three-component Fermi gas in which two interspecies scattering lengths are negative while the other interspecies scattering length is zero. In this case, the stability of the Fermi system is predicted to depend appreciably on the range of the underlying two-body potential. We find parameter combinations for which the system appears to become unstable for a finite negative scattering length and parameter combinations for which the system appears to be made up of weakly bound trimers that consist of one fermion of each species.
Dilute Fermi gases with large scattering lengths are intriguing physical systems which are just now becoming accessible in laboratory experiments with cold trapped atoms. Their properties are closely related to those of the dilute neutron‐rich matter which may be present in the surface of neutron stars. Fermi systems with such strong short‐range attractive forces lie between traditional BCS theories with small gaps and those where the fermions are tightly bound into pairs, yielding composite bosons. We discuss the Quantum Monte Carlo methods used to study such systems, and report results for the ground‐state energy, the superfluid gap, and other properties. In particular, we compare results with those of neutron matter at subnuclear densities.
Ground-state energies and superfluid gaps are calculated for degenerate Fermi systems interacting via long attractive scattering lengths such as cold atomic gases, neutron, and nuclear matter. In the intermediate region of densities, where the interparticle spacing $(\ensuremath{\sim}{1/k}_{F})$ is longer than the range of the interaction but shorter than the scattering length, the superfluid gaps and the energy per particle are found to be proportional to the Fermi energy and thus differ from the dilute and high-density limits. The attractive potential increase linearly with the spin-isospin or hyperspin statistical factor such that, e.g., symmetric nuclear matter undergoes spinodal decomposition and collapses whereas neutron matter and Fermionic atomic gases with two hyperspin states are mechanically stable in the intermediate density region. The regions of spinodal instabilities in the resulting phase diagram are reduced and do not prevent a superfluid transition.
We generalize the Noziéres-Schmitt-Rink method to study the repulsive Fermi gas in the absence of molecule formation, i.e., in the so-called "upper branch." We find that the system remains stable except close to resonance at sufficiently low temperatures. With increasing scattering length, the energy density of the system attains a maximum at a positive scattering length before resonance. This is shown to arise from Pauli blocking which causes the bound states of fermion pairs of different momenta to disappear at different scattering lengths. At the point of maximum energy, the compressibility of the system is substantially reduced, leading to a sizable uniform density core in a trapped gas. The change in spin susceptibility with increasing scattering length is moderate and does not indicate any magnetic instability. These features should also manifest in Fermi gases with unequal masses and/or spin populations.
We calculate the equation of state of a Fermi gas with resonant interactions when the effective range is appreciable. Using an effective field theory for a large scattering length and large effective range, we show how calculations in this regime become tractable. Our results are model independent, and as an application, we study the neutron matter equation of state at low densities of astrophysical interest 0.002 fm(-3) < rho < 0.02 fm(-3), for which the interparticle separation is comparable to the effective range. We compare our simple results with those of conventional many-body calculations.
We study the properties of strongly interacting Bose gases at the density and temperature regime when the three-body recombination rate is substantially reduced. In this regime, one can have a Bose gas with all particles in scattering states (i.e., the "upper branch") with little loss even at unitarity over the duration of the experiment. We show that because of bosonic enhancement, pair formation is shifted to the atomic side of the original resonance (where scattering length a(s)<0), opposite to the fermionic case. In a trap, a repulsive Bose gas remains mechanically stable when brought across resonance to the atomic side until it reaches a critical scattering length a(s)*<0. For a(s)<a(s)*, the density consists of a core of upper branch bosons surrounded by an outer layer of equilibrium branch. The conditions of low three-body recombination require that the particle number N<1.024(T/ω)(5/2) in a harmonic trap with frequency ω.