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arXiv:1702.01583 · Physical Review A

Effective-range dependence of resonant Fermi gases

Lars Schonenberg, G. J. Conduit

Published Jan 27, 2017• 22 citations
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Abstract

A Fermi gas of cold atoms allows precise control over the dimensionless effective range, ${k}_{\mathrm{F}}{R}_{\mathrm{eff}}$, of the Feshbach resonance. Our pseudopotential formalism allows us to create smooth potentials with effective range, $\ensuremath{-}2\ensuremath{\le}{k}_{\mathrm{F}}{R}_{\mathrm{eff}}\ensuremath{\le}2$, which we use for a variational and diffusion Monte Carlo study of the ground state of a unitary Fermi gas. We report values for the universal constants of $\ensuremath{\xi}=0.388(1)$ and $\ensuremath{\zeta}=0.087(1)$, and compute the condensate fraction, momentum distribution, and pair correlations functions. Finally, we show that a gas with ${k}_{\mathrm{F}}{R}_{\mathrm{eff}}\ensuremath{\gtrsim}1.9$ is thermodynamically unstable.

DOI
10.1103/physreva.95.013633

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A Fermi gas of cold atoms allows precise control over the dimensionless effective range, kFReff{k}_{\mathrm{F}}{R}_{\mathrm{eff}}kF​Reff​, of the Feshbach resonance. Our pseudopotential formalism allows us to create smooth potentials with effective range, \ensuremath−2\ensuremath≤kFReff\ensuremath≤2\ensuremath{-}2\ensuremath{\le}{k}_{\mathrm{F}}{R}_{\mathrm{eff}}\ensuremath{\le}2\ensuremath−2\ensuremath≤kF​Reff​\ensuremath≤2, which we use for a variational and diffusion Monte Carlo study of the ground state of a unitary Fermi gas. We report values for the universal constants of \ensuremathξ=0.388(1)\ensuremath{\xi}=0.388(1)\ensuremathξ=0.388(1) and \ensuremathζ=0.087(1)\ensuremath{\zeta}=0.087(1)\ensuremathζ=0.087(1), and compute the condensate fraction, momentum distribution, and pair correlations functions. Finally, we show that a gas with kFReff\ensuremath≳1.9{k}_{\mathrm{F}}{R}_{\mathrm{eff}}\ensuremath{\gtrsim}1.9kF​Reff​\ensuremath≳1.9 is thermodynamically unstable.
The Feshbach resonance provides precise control over the scattering length and effective range of interactions between ultracold atoms. We propose the ultratransferable pseudopotential to model effective interaction ranges $\ensuremath{-}1.5\ensuremath{\le}{k}_{\mathrm{F}}^{2}{R}_{\mathrm{eff}}^{2}\ensuremath{\le}0$, where Reff{R}_{\mathrm{eff}}Reff​ is the effective range and kF{k}_{\text{F}}kF​ is the Fermi wave vector, describing narrow to broad Feshbach resonances. We develop a mean-field treatment and exploit the pseudopotential to perform a variational and diffusion Monte Carlo study of the ground state of the two-dimensional Fermi gas, reporting on the ground-state energy, contact, condensate fraction, momentum distribution, and pair-correlation functions as a function of the effective interaction range across the BEC-BCS crossover. The limit ${k}_{\mathrm{F}}^{2}{R}_{\mathrm{eff}}^{2}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ is a gas of bosons with zero binding energy, whereas $ln({k}_{\mathrm{F}}a)\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ corresponds to noninteracting bosons with infinite binding energy.
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.
We calculate the ground-state properties of an unpolarized two-component Fermi gas with the aid of the diffusion quantum Monte Carlo (DMC) methods. Using an extrapolation to the zero effective range of the attractive two-particle interaction, we find E/EfreeE/{E}_{\mathrm{free}}E/Efree​ in the unitary limit to be 0.212(2), 0.407(2), 0.409(3), and 0.398(3) for 4, 14, 38, and 66 atoms, respectively. Our calculations indicate that the dependence of the total energy on the effective range of the interaction Reff{R}_{\mathrm{eff}}Reff​ is sizable and the extrapolation to Reff=0{R}_{\mathrm{eff}}=0Reff​=0 is therefore important for reaching the true unitary limit. To test the quality of nodal surfaces and to estimate the impact of the fixed-node approximation, we perform released-node DMC calculations for 4 and 14 atoms. Analysis of the released-node and the fixed-node results suggests that the main sources of the fixed-node errors are long-range correlations, which are difficult to sample in the released-node approaches due to the fast growth of the bosonic noise. Besides energies, we evaluate the two-body density matrix and the condensate fraction. We find that the condensate fraction for the 66-atom system converges to 0.56(1) after the extrapolation to the zero interaction range.
We develop a simple, mean-field-like theory for the normal phase of a unitary Fermi gas by deriving a self-consistent equation for its self-energy via a momentum-dependent coupling constant for both attractive and repulsive universal fermions. For attractive universal fermions in the lower branch of a Feshbach resonance, we use zero-temperature Monte Carlo results as a starting point for one-step iteration in order to derive an analytical expression for the momentum-dependent self-energy. For repulsive universal fermions in the upper branch of a Feshbach resonance, we iteratively calculate the momentum-dependent self-energy via our self-consistent equation. Lastly, for the case of population imbalance, we propose an ansatz for higher-order virial expansion coefficents. Overall, we find that our theory is in good agreement with currently available, high-temperature experimental data.
We use quantum Monte Carlo, including backflow corrections, to investigate a two-component Fermi gas on the upper branch of a Feshbach resonance and contrast it with the hard sphere gas. We find that, in both cases, the Fermi liquid becomes unstable to ferromagnetism at a kFak_F akF​a smaller than the mean field result, where kFk_FkF​ is the Fermi wavevector and aaa the scattering length. Even though the total energies E(kFa)E(k_F a)E(kF​a) are similar in the two cases, their pair correlations and kinetic energies are completely different, reflecting the underlying potentials. We discuss the extent to which our calculations shed light on recent experiments.
Properties of the ground state of an unpolarized ultracold Fermi gas along the BCS-BEC crossover are investigated by the variational and diffusion Monte Carlo methods. We apply the Wigner-Bethe-Peierls boundary condition in our calculations to avoid any bias from using an interatomic potential with finite effective range. Properties for several values of the scattering length are studied in the range $\ensuremath{-}8\ensuremath{\le}1/a{k}_{F}\ensuremath{\le}4$, including the unitary limit. The contact parameters as a function of scattering length are obtained by fitting the pair distribution functions for particles with different spins. The energies and contact parameters are in very good agreement with experimental data reported in the literature.