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arXiv:cond-mat/0304633 · Physical Review A

Unitarity-limited elastic collision rate in a harmonically trapped Fermi gas

Michael E. Gehm, S. L. Hemmer, Kenneth M. O’Hara, John E. Thomas

Published Jul 23, 2003• 38 citations
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Abstract

We derive the elastic collision rate for a harmonically trapped Fermi gas in the extreme unitarity limit where the s-wave scattering cross section is $\ensuremath{\sigma}(k)=4\ensuremath{\pi}{/k}^{2},$ with $\ensuremath{\Elzxh}k$ the relative momentum. The collision rate is given in the form $\ensuremath{\Gamma}=\ensuremath{\gamma}{I(T/T}_{F})$---the product of a universal collision rate $\ensuremath{\gamma}{=k}_{B}{T}_{F}/(6\ensuremath{\pi}\ensuremath{\Elzxh})$ and a dimensionless function of the ratio of the temperature T to the Fermi temperature ${T}_{F}.$ We find that I has a peak value of $\ensuremath{\simeq}4.6$ at ${T/T}_{F}\ensuremath{\simeq}0.4,$ $I\ensuremath{\simeq}82{(T/T}_{F}{)}^{2}$ for ${T/T}_{F}<~0.15,$ and $I\ensuremath{\simeq}{2(T}_{F}{/T)}^{2}$ for ${T/T}_{F}>1.5.$ We estimate the collision rate for recent experiments on a strongly-interacting degenerate Fermi gas of atoms.

DOI
10.1103/physreva.68.011603

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We derive the elastic collision rate for a harmonically trapped Fermi gas in the extreme unitarity limit where the s-wave scattering cross section is \ensuremathσ(k)=4\ensuremathπ/k2,\ensuremath{\sigma}(k)=4\ensuremath{\pi}{/k}^{2},\ensuremathσ(k)=4\ensuremathπ/k2, with \ensuremath\Elzxhk\ensuremath{\Elzxh}k\ensuremath\Elzxhk the relative momentum. The collision rate is given in the form \ensuremathΓ=\ensuremathγI(T/TF)\ensuremath{\Gamma}=\ensuremath{\gamma}{I(T/T}_{F})\ensuremathΓ=\ensuremathγI(T/TF​)---the product of a universal collision rate \ensuremathγ=kBTF/(6\ensuremathπ\ensuremath\Elzxh)\ensuremath{\gamma}{=k}_{B}{T}_{F}/(6\ensuremath{\pi}\ensuremath{\Elzxh})\ensuremathγ=kB​TF​/(6\ensuremathπ\ensuremath\Elzxh) and a dimensionless function of the ratio of the temperature T to the Fermi temperature TF.{T}_{F}.TF​. We find that I has a peak value of \ensuremath≃4.6\ensuremath{\simeq}4.6\ensuremath≃4.6 at T/TF\ensuremath≃0.4,{T/T}_{F}\ensuremath{\simeq}0.4,T/TF​\ensuremath≃0.4, I\ensuremath≃82(T/TF)2I\ensuremath{\simeq}82{(T/T}_{F}{)}^{2}I\ensuremath≃82(T/TF​)2 for T/TF< 0.15,{T/T}_{F}<~0.15,T/TF​< 0.15, and I\ensuremath≃2(TF/T)2I\ensuremath{\simeq}{2(T}_{F}{/T)}^{2}I\ensuremath≃2(TF​/T)2 for T/TF>1.5.{T/T}_{F}>1.5.T/TF​>1.5. We estimate the collision rate for recent experiments on a strongly-interacting degenerate Fermi gas of atoms.
Using a scattering length parametrization of the crossover from a BCS state to a Bose-Einstein condensate as well as the local density approximation for the density profile, we calculate the momentum distribution of a harmonically trapped atomic Fermi gas at zero temperature. Various interaction regimes are considered, including the BCS phase, the unitarity limit, and the molecular regime. We show that the relevant parameter which characterizes the crossover is given by the dimensionless combination N1/6a/aho,{N}^{1/6}{a/a}_{\mathrm{ho}},N1/6a/aho​, where N is the number of atoms, a is the scattering length, and aho{a}_{\mathrm{ho}}aho​ is the oscillator length. The width of the momentum distribution is shown to depend in a crucial way on the value and sign of this parameter. Our predictions can be relevant for experiments on ultracold atomic Fermi gases near a Feshbach resonance.
The viscous relaxation time of a trapped two-component gas of fermions in its normal phase is calculated as a function of temperature and scattering length, with the collision probability being determined by an energy-dependent sss-wave cross section. The result is used for calculating the temperature dependence of the frequency and damping of collective modes studied in recent experiments, starting from the kinetic equation for the fermion distribution function with mean-field effects included in the streaming terms.
We experimentally investigate the unitarity-limited behavior of the three-body loss near a ppp-wave Feshbach resonance in a single-component Fermi gas of 6Li^{6}\mathrm{Li}6Li atoms. At the unitarity limit, the three-body loss coefficient L3{L}_{3}L3​ exhibits universality in the sense that it is independent of the interaction strength and follows the predicted temperature scaling law of ${L}_{3}\ensuremath{\propto}{T}^{\ensuremath{-}2}$. When decreasing the interaction strength from the unitarity regime, the three-body loss coefficient as a function of the interaction strength and temperature can be described by the theory based on the association of an excited resonant quasibound state and its relaxation into a deep stable dimer by collision with a third atom in the framework of the standard Breit-Wigner theoretical approach. The results reported here are important to understand the properties of a resonant ppp-wave Fermi gas in the prospect of quantum few- and many-body physics.
The frequencies of the collective oscillations of a harmonically trapped Fermi gas interacting with large scattering lengths are calculated at zero temperature using hydrodynamic theory. Different regimes are considered, including the molecular Bose-Einstein condensate and the unitarity limit for collisions. We show that the frequency of the radial compressional mode in an elongated trap exhibits a pronounced non-monotonous dependence on the scattering length, reflecting the role of the interactions in the equation of state.
Abstract We revisit the properties of the two-component Fermi gas with short-range interactions in three dimensions, in the limit where the s -wave scattering length diverges. Such a unitary Fermi gas possesses universal thermodynamic and dynamical observables that are independent of any interaction length scale. Focusing on trapped systems of N fermions, where N ≤ 10 , we investigate how well we can determine the zero-temperature behavior of the many-body system from published few-body data on the ground-state energy and the contact. For the unpolarized case, we find that the Bertsch parameters extracted from trapped few-body systems all lie within 15% of the established value. Furthermore, the few-body values for the contact are well within the range of values determined in the literature for the many-body system. In the limit of large spin polarization, we obtain a similar accuracy for the polaron energy, and we estimate the polaron’s effective mass from the dependence of its energy on N . We also compute an upper bound for the squared wave-function overlap between the unitary Fermi system and the non-interacting ground state, both for the trapped and uniform cases. This allows us to prove that the trapped unpolarized ground state at unitarity has zero overlap with its non-interacting counterpart in the many-body limit N → ∞ .
Strongly interacting, dilute Fermi gases exhibit a scale-invariant, universal thermodynamic behavior. This is notoriously difficult to understand theoretically because of the absence of a small interaction parameter. Here, we present a systematic comparison of theoretical predictions from different quantum many-body theories with recent experimental data of Nascimbne et al (2010 Nature 463 1057). Our comparisons have no adjustable parameters, either theoretically or experimentally. All the model approximations seem to fluctuate rather than converge on the experimental data. It turns out that a simple Gaussian pair fluctuation theory gives the best quantitative agreement, except at the critical superfluid transition region. In the normal state, we also calculate the equation of state by using a quantum cluster expansion theory and explore in detail its applicability to low temperatures. Using the accurate experimental result for the thermodynamic function S ( T ), we determine the temperature T of a trapped Fermi gas at unitarity as a function of a non-interacting temperature T i , which can be obtained by an adiabatic sweep to the free gas limit. By analyzing the recent experimental data, we find a characteristic temperature ( T / T F ) 0 =0.19±0.02 or ( T i / T F ) 0 =0.16±0.02 in a harmonic trap, below which there are deviations from normal Fermi-liquid-like behavior that may be attributed to pairing effects. Here, T F is the Fermi temperature for a trapped ideal, non-interacting Fermi gas. Our thorough comparison may shed light on the further theoretical development of strongly interacting fermions.