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arXiv:cond-mat/0402383 · Physical Review A

Particle density distributions in Fermi gas superfluids: Differences between one- and two-channel models in the Bose-Einstein-condensation limit

Jelena Stajic, Qijin Chen, Kathryn Levin

Published Mar 2, 2005• 6 citations
arXivHTMLPDF

Abstract

We discuss the differences between one- and two-channel descriptions of fermionic gases with arbitrarily tunable attractive interactions; these two cases correspond to whether molecular bosonic degrees of freedom are omitted or included. We adopt the standard ground state wave function for the fermionic component associated with the BCS to BEC crossover problem: for weak attraction the system is in the BCS state while it crosses over continuously to a Bose-Einstein-condensed (BEC) state as the interaction strength is increased. Our analysis focuses on the BEC and near-BEC limit where the differences between the one- and two-channel descriptions are most notable, and where analytical calculations are most tractable. Among the differences we elucidate are the equations of state at general $T$ below ${T}_{c}$ and related particle density profiles. We find a narrowing of the density profile in the two-channel problem relative to the one-channel analog. Importantly, we infer that the ratio between bosonic and fermionic scattering lengths depends on the magnetic detuning and is generally smaller than its one-channel counterpart. Future experiments will be required to determine to what extent this ratio varies with magnetic fields, as predicted here.

DOI
10.1103/physreva.71.033601

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We discuss the differences between one- and two-channel descriptions of fermionic gases with arbitrarily tunable attractive interactions; these two cases correspond to whether molecular bosonic degrees of freedom are omitted or included. We adopt the standard ground state wave function for the fermionic component associated with the BCS to BEC crossover problem: for weak attraction the system is in the BCS state while it crosses over continuously to a Bose-Einstein-condensed (BEC) state as the interaction strength is increased. Our analysis focuses on the BEC and near-BEC limit where the differences between the one- and two-channel descriptions are most notable, and where analytical calculations are most tractable. Among the differences we elucidate are the equations of state at general TTT below Tc{T}_{c}Tc​ and related particle density profiles. We find a narrowing of the density profile in the two-channel problem relative to the one-channel analog. Importantly, we infer that the ratio between bosonic and fermionic scattering lengths depends on the magnetic detuning and is generally smaller than its one-channel counterpart. Future experiments will be required to determine to what extent this ratio varies with magnetic fields, as predicted here.
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We report results of quantum Monte Carlo calculations of the ground state of dilute Fermi gases with attractive short-range two-body interactions. The strength of the interaction is varied to study different pairing regimes which are characterized by the product of the sss-wave scattering length and the Fermi wave vector, akFa{k}_{F}akF​. We report results for the ground-state energy, the pairing gap $\ensuremath{\Delta}$, and the quasiparticle spectrum. In the weak-coupling regime, $1∕a{k}_{F}<\ensuremath{-}1$, we obtain Bardeen-Cooper-Schrieffer (BCS) superfluid and the energy gap $\ensuremath{\Delta}$ is much smaller than the Fermi gas energy EFG{E}_{\mathrm{FG}}EFG​. When a>0a>0a>0, the interaction is strong enough to form bound molecules with energy Emol{E}_{\mathrm{mol}}Emol​. For $1∕a{k}_{F}\ensuremath{\gtrsim}0.5$, we find that weakly interacting composite bosons are formed in the superfluid gas with $\ensuremath{\Delta}$ and gas energy per particle approaching $\ensuremath{\mid}{E}_{\mathrm{mol}}\ensuremath{\mid}∕2$. In this region, we seem to have Bose-Einstein condensation (BEC) of molecules. The behavior of the energy and the gap in the BCS-to-BEC transition region, $\ensuremath{-}0.5<1∕a{k}_{F}<0.5$, is discussed.
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BCS-Bose-Einstein condensation (BEC) crossover is effected by increasing pairing strength between fermions from weak to strong in the particle-particle channel, and has attracted a lot of attention since the experimental realization of quantum degenerate atomic Fermi gases. Here we study the effect of the (often dropped) particle-hole channel on the zero T gap Δ(0), superfluid transition temperature Tc, the pseudogap at Tc, and the mean-field ratio 2Δ(0)/, from BCS through BEC regimes, using a pairing fluctuation theory which includes self-consistently the contributions of finite-momentum pairs and features a pseudogap in single particle excitation spectrum. Summing over the infinite particle-hole ladder diagrams, we find a complex dynamical structure for the particle-hole susceptibility χph, and conclude that neglecting the self-energy feedback causes a serious over-estimate of χph. While our result in the BCS limit agrees with Gor'kov et al., the particle-hole channel effect becomes more complex and pronounced in the crossover regime, where χph is reduced by both a smaller Fermi surface and a big (pseudo)gap. Deep in the BEC regime, the particle-hole channel contributions drop to zero. We predict a density dependence of the magnetic field at the Feshbach resonance, which can be used to quantify χph and test different theories.
The physics of quantum degenerate atomic Fermi gases in uniform as well as in harmonically trapped configurations is reviewed from a theoretical perspective. Emphasis is given to the effect of interactions that play a crucial role, bringing the gas into a superfluid phase at low temperature. In these dilute systems, interactions are characterized by a single parameter, the sss-wave scattering length, whose value can be tuned using an external magnetic field near a broad Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the Bose-Einstein condensation (BEC) of dimers, and the unitary limit of large scattering length are important regimes exhibited by interacting Fermi gases. In particular, the BEC and the unitary regimes are characterized by a high value of the superfluid critical temperature, on the order of the Fermi temperature. Different physical properties are discussed, including the density profiles and the energy of the ground-state configurations, the momentum distribution, the fraction of condensed pairs, collective oscillations and pair-breaking effects, the expansion of the gas, the main thermodynamic properties, the behavior in the presence of optical lattices, and the signatures of superfluidity, such as the existence of quantized vortices, the quenching of the moment of inertia, and the consequences of spin polarization. Various theoretical approaches are considered, ranging from the mean-field description of the BCS-BEC crossover to nonperturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare theoretical predictions with available experimental results.
The ground-state properties of a two-component Fermi gas with attractive short-range interactions are calculated using the fixed-node diffusion Monte Carlo method. The interaction strength is varied over a wide range by tuning the value a of the s-wave scattering length of the two-body potential. We calculate the energy per particle, the one- and two-body density matrix as a function of the interaction strength. Results for the momentum distribution of the atoms, as obtained from the Fourier transform of the one-body density matrix, are reported as a function of the interaction strength. Off-diagonal long-range order in the system is investigated through the asymptotic behavior of the two-body density matrix. The condensate fraction of pairs is calculated in the unitary limit and on both sides of the BCS-BEC crossover.