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arXiv:0909.2344 · Laser Physics Letters

Surface effects in the unitary Fermi gas

Luca Salasnich, Francesco Ancilotto, Flavio Toigo

Published Sep 24, 2009• 19 citations
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Abstract

We study the extended Thomas-Fermi (ETF) density functional of the superfluid unitary Fermi gas. This functional includes a gradient term which is essential to describe accurately the surface effects of the system, in particular with a small number of atoms, where the Thomas-Fermi (local density) approximation fails. We find that our ETF functional gives density profiles which are in good agreement with recent Monte Carlo results and also with a more sophisticated superfluid density functional based on Bogoliubov-de Gennes equations. In addition, by using extended hydrodynamics equations of superfluids, we calculate the frequencies of collective surface oscillations of the unitary Fermi gas, showing that quadrupole and octupole modes strongly depend on the number of trapped atoms.

DOI
10.1002/lapl.200910107

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We study the extended Thomas-Fermi (ETF) density functional of the superfluid unitary Fermi gas. This functional includes a gradient term which is essential to describe accurately the surface effects of the system, in particular with a small number of atoms, where the Thomas-Fermi (local density) approximation fails. We find that our ETF functional gives density profiles which are in good agreement with recent Monte Carlo results and also with a more sophisticated superfluid density functional based on Bogoliubov-de Gennes equations. In addition, by using extended hydrodynamics equations of superfluids, we calculate the frequencies of collective surface oscillations of the unitary Fermi gas, showing that quadrupole and octupole modes strongly depend on the number of trapped atoms.
We determine the energy density $\ensuremath{\xi}(3∕5)n{\ensuremath{\epsilon}}_{F}$ and the gradient correction $\ensuremath{\lambda}{\ensuremath{\hbar}}^{2}{(\mathbf{\ensuremath{\nabla}}n)}^{2}∕(8mn)$ of the extended Thomas-Fermi (ETF) density functional, where nnn is the number density and ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy, for a trapped two-component Fermi gas with infinite scattering length (unitary Fermi gas) on the basis of recent diffusion Monte Carlo (DMC) calculations [Phys. Rev. Lett. 99, 233201 (2007)]. In particular we find that $\ensuremath{\xi}=0.455$ and $\ensuremath{\lambda}=0.13$ give the best fit of the DMC data with an even number NNN of particles. We also study the odd-even splitting $\ensuremath{\gamma}{N}^{1∕9}\ensuremath{\hbar}\ensuremath{\omega}$ of the ground-state energy for the unitary gas in a harmonic trap of frequency $\ensuremath{\omega}$ determining the constant $\ensuremath{\gamma}$. Finally we investigate the effect of the gradient term in the time-dependent ETF model by introducing generalized Galilei-invariant hydrodynamics equations.
We study a spin-balanced unitary Fermi superfluid bound by an attractive 1/r1/r1/r interaction within an extended Thomas--Fermi density functional. Its large-particle-number limit is the Lane--Emden polytrope of index 3/23/23/2. We prove the leading gradient correction to the ground-state energy and strong convergence of the density amplitude. A rescaled surface problem predicts the next energy correction and the leading integrated density error. Matching a vortex core to the polytropic bulk gives the energetic threshold of a singly quantized central vortex, including the constant beyond the leading logarithm. Linearizing the same surface problem predicts a local energetic threshold for collective surface excitations that, in the natural TF frequency units, decreases more slowly with particle number than the central-vortex threshold. Radial stationary, spherical Bogoliubov--Poisson, and axisymmetric real-time calculations test the predicted energy corrections and vortex matching, and resolve finite-size surface-mode minima whose approach to the continuum surface coefficient is followed to N≃4×106N\simeq4\times10^6N≃4×106. For the reference gradient coefficient the vortex constant is c1=−0.535c_1=-0.535c1​=−0.535, and the surface Landau scale is 1.417 ωK(ℏeff2/2RTF)1/6∝N−1/91.417\,ω_K(\hbar_{\rm eff}^2/2R_{\rm TF})^{1/6}\propto N^{-1/9}1.417ωK​(ℏeff2​/2RTF​)1/6∝N−1/9; sector energies at N≃4.6×105N\simeq4.6\times10^5N≃4.6×105, computed after the fit, agree with the extrapolated vortex threshold to 10−410^{-4}10−4. These results distinguish the accuracy of the Thomas--Fermi bulk from the gradient physics that controls the surface and vortex core, and separate vortex energetics from the open problem of nonlinear vortex entry.
In this thesis, following L. Salasnich and F. Toigo [Phys. Rev. A 78, 053626 (2008)] we use an extended Thomas-Fermi (TF) density functional to describe a gas of interacting fermions composed of an even number N of particle in the unitary regime confined by a spherically symmetric harmonic trap. The great advantage in taking such a functional, relative to functionals based on single-fermion wave functions, is to depend only on the density number n and on its derivatives, so that the numerical load is independent of the number of particles. By minimizing the ETF functional with respect to the density we obtain a time-independent non-linear Schrodinger equation which we solve using the imaginary time propagation method, by integrating numerically the corresponding time-dependent Schrodinger equation. We determine the two characteristic parameters of the functionals ξ and λ by fitting the Monte Carlo total energy data obtained by D. Blume et al. [Phys. Rev. Lett. 99, 233201 (2007)]. We obtain the values ξ = 0.469 and λ = 0.082 leading to a χ2 = 42. These values are to be compared with ξ = 0.455 and λ = 0.13, leading to a χ2 = 282 as proposed by Salasnich and Toigo. We also study the dependence of the ground-state density profile on the values of the two parameters, finding that ξ enhances the width of the profile, while increasing λ smooths out the outer boundary of the fermionic cloud. Advisor: Dr. Nicola Manini Co-Advisor: Dr. Luca Salasnich
We present a comparison between simulated dynamics of the unitary fermion gas using the superfluid local density approximation (SLDA) and a simplified bosonic model, the extended Thomas-Fermi (ETF) with a unitary equation of state. Small-amplitude fluctuations have similar dynamics in both theories for frequencies far below the pair-breaking threshold and wave vectors much smaller than the Fermi momentum. The low-frequency linear responses in both match well for surprisingly large wave vectors, even up to the Fermi momentum. For nonlinear dynamics such as vortex generation, the ETF provides a semiquantitative description of SLDA dynamics as long as the fluctuations do not have significant power near the pair-breaking threshold; otherwise the dynamics of the ETF cannot be trusted. Nonlinearities in the ETF tend to generate high-frequency fluctuations, and with no normal component to remove this energy from the superfluid, features such as vortex lattices cannot relax and crystallize as they do in the SLDA.
Two species superfluid Fermi gas is investigated on the BCS side up to the Feshbach resonance. Using the Greens's function technique gradient corrections are calculated to the generalized Thomas-Fermi theory including Cooper pairing. Their relative magnitude is found to be measured by the small parameter (d/RTF)4(d/{R}_{\mathrm{TF}}){}^{4}(d/RTF​)4, where ddd is the oscillator length of the trap potential and RTF{R}_{\mathrm{TF}}RTF​ is the radial extension of the density nnn in the Thomas-Fermi approximation. In particular, at the Feshbach resonance the universal corrections to the local density approximation are calculated and a universal prefactor ${\ensuremath{\kappa}}_{W}=7/27$ is derived for the von Weizs\"acker--type correction ${\ensuremath{\kappa}}_{W}(\ensuremath{\hbar}{}^{2}/2m)({\ensuremath{\nabla}}^{2}{n}^{1/2}/{n}^{1/2})$.
We discuss collective monopole and quadrupole excitations of a collisionless gas of trapped Fermionic atoms in the superfluid BCS phase, comparing the fully microscopic Bogoliubov--de Gennes and quasiparticle random-phase approximation method with widely used semiclassical methods. In particular, the microscopic treatment allows us to address the questions of temperature dependence and nontrivial dependence on the trap parameters, which cannot be answered within the semiclassical approach. The main result concerning the temperature dependence is a strong Landau damping at intermediate temperature, which disappears in the limits of zero and critical temperature. However, even at zero temperature, considerable deviations from superfluid hydrodynamics are found if the trap frequency is of the same order of magnitude as the pairing gap.