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arXiv:cond-mat/0306187 · Physical Review Letters

High Temperature Expansion Applied to Fermions near Feshbach Resonance

Tin-Lun Ho, Erich J. Mueller

Published Apr 20, 2004• 154 citations
arXivHTMLPDF

Abstract

We show that, apart from a difference in scale, all of the surprising recently observed properties of a degenerate Fermi gas near a Feshbach resonance persist in the high temperature Boltzmann regime. In this regime, the Feshbach resonance is unshifted. By sweeping across the resonance, a thermal distribution of bound states (molecules) can be reversibly generated. Throughout this process, the interaction energy is negative and continuous. We also show that this behavior must persist at lower temperatures unless there is a phase transition as the temperature is lowered. We rigorously demonstrate universal behavior near the resonance.

DOI
10.1103/physrevlett.92.160404

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We show that, apart from a difference in scale, all of the surprising recently observed properties of a degenerate Fermi gas near a Feshbach resonance persist in the high temperature Boltzmann regime. In this regime, the Feshbach resonance is unshifted. By sweeping across the resonance, a thermal distribution of bound states (molecules) can be reversibly generated. Throughout this process, the interaction energy is negative and continuous. We also show that this behavior must persist at lower temperatures unless there is a phase transition as the temperature is lowered. We rigorously demonstrate universal behavior near the resonance.
Near Feshbach resonances, n|a|3 ≫ 1, systems of Bose and Fermi particles become strongly interacting/dense. In this unitary limit both bosons and fermions have very different properties than in a dilute gas, e.g., the energy per particle approaches a value ℏ2n2/3/m times a universal many-body constant. Calculations based upon an approximate Jastrow wavefunction can quantitatively describe recent measurements of trapped Bose and Fermi atoms near Feshbach resonances. The pairing gap between attractive fermions also scales as Δ ∼ ℏ2n2/3/m near Feshbach resonances and is a large fraction of the Fermi energy—promising for observing BCS superfluidity in traps. Pairing undergoes several transitions depending on interaction strength and the number of particles in the trap and can also be compared with pairing in nuclei.
Near Feshbach resonances, n | a | 3 ≫ 1, systems of Bose and Fermi particles become strongly interacting/dense. In this unitary limit both bosons and fermions have very different properties than in a dilute gas, e.g., the energy per particle approaches a value ℏ 2 n 2/3 / m times a universal many-body constant. Calculations based upon an approximate Jastrow wavefunction can quantitatively describe recent measurements of trapped Bose and Fermi atoms near Feshbach resonances. The pairing gap between attractive fermions also scales as Δ ∼ ℏ 2 n 2/3 / m near Feshbach resonances and is a large fraction of the Fermi energy—promising for observing BCS superfluidity in traps. Pairing undergoes several transitions depending on interaction strength and the number of particles in the trap and can also be compared with pairing in nuclei.
Using high-temperature virial expansion, we study the quench dynamics of the thermal Bose gases near wide, narrow, and intermediate Feshbach resonances. Our results show that the shallow bound state near Feshbach resonances leads to oscillation of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{\mathbf{k}}$. Near the wide Feshbach resonance, the long-time momentum distribution ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{\mathbf{k}}$ oscillates when the scattering length as{a}_{s}as​ is quenched to large but with finite positive values. The oscillation frequency is determined by the bound-state energy. When as{a}_{s}as​ is quenched to infinity or negative value, the oscillation vanishes. Near the narrow Feshbach resonance, when the background scattering length abg{a}_{\mathrm{bg}}abg​ is larger than de Broglie wave length $\ensuremath{\lambda}$, there is an oscillation, and the frequency is determined by the energy of the shallow bound state in the open channel. When abg<0{a}_{\mathrm{bg}}<0abg​<0 or $0<{a}_{\mathrm{bg}}\ensuremath{\ll}\ensuremath{\lambda}$, there is no shallow bound state in the open channel, and hence no oscillation. We check our conclusion using some realistic systems, and the results are consistent with our conclusion.
We consider the superfluid phase transition that arises when a Feshbach resonance pairing occurs in a dilute Fermi gas. We apply our theory to consider a specific resonance in potassium ((40)K), and find that for achievable experimental conditions, the transition to a superfluid phase is possible at the high critical temperature of about 0.5T(F). Observation of superfluidity in this regime would provide the opportunity to experimentally study the crossover from the superfluid phase of weakly coupled fermions to the Bose-Einstein condensation of strongly bound composite bosons.
The entropy and kinetic, potential, and interaction energies of an atomic Fermi gas in a trap are studied under the assumption of thermal equilibrium for finite temperature. A Feshbach resonance can cause the fermions to pair into diatomic molecules. The entropy and energies of mixtures of such molecules with unpaired atoms are calculated, in relation to recent experiments on molecular Bose-Einstein condensates produced in this manner. It is shown that, starting with a Fermi gas of temperature T=0.1TF0T=0.1{T}_{F}^{0}T=0.1TF0​, where TF0{T}_{F}^{0}TF0​ is the noninteracting Fermi temperature, an extremely cold degenerate Fermi gas of temperature $T\ensuremath{\lesssim}0.01{T}_{F}^{0}$ may be produced without further evaporative cooling. This requires adiabatic passage of the resonance, subsequent sudden removal of unpaired atoms, and adiabatic return. We also calculate the ratio of the interaction energy to the kinetic energy, a straightforward experimental signal which may be used to determine the temperature of the atoms and indicate condensation of the molecules.
In summary, we use all-optical methods with evaporative cooling near a Feshbach resonance to produce a strongly interacting degenerate Fermi gas. We observe hydrodynamic behavior in the expansion dynamics. At low temperatures, collisions may not explain the expansion dynamics. We observe hydrodynamics in the trapped gas. Our observations include collisionally-damped excitation spectra at high temperature which were not discussed above. In addition, we observe weakly damped breathing modes at low temperature. The observed temperature dependence of the damping time and hydrodynamic frequency are not consistent with collisional dynamics nor with collisionless mean field interactions. These observations constitute the first evidence for superfluid hydrodynamics in a Fermi gas.