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arXiv:0911.3312 · Physical Review A

Equation of state of a polarized Fermi gas in the Bose-Einstein-condensate limit

Florent Alzetto, X. Leyronas

Published Apr 8, 2010• 13 citations
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Abstract

We present a theoretical study of the BEC-BCS crossover in the Bose-Einstein-condensate (BEC) regime in the case of an unequal number of fermions of two species. We take full account of the composite nature of the dimers made of fermions. In the limit of low densities, we calculate the ground-state energy of the system, or equivalently the chemical potentials of each species, as well as the one-particle gap and the energy of an ``impurity'' immersed in a Fermi sea. For the chemical potentials we go up to order $(\mathrm{density}){}^{4/3}$. The results found involve the exact atom-dimer ${a}_{\mathrm{AD}}$ and dimer-dimer ${a}_{\mathrm{DD}}$ scattering lengths and therefore include the three- and four-body problems in the many-body problem. We briefly comment on the importance of the different mean-field corrections for recent experiments.

DOI
10.1103/physreva.81.043604

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We present a theoretical study of the BEC-BCS crossover in the Bose-Einstein-condensate (BEC) regime in the case of an unequal number of fermions of two species. We take full account of the composite nature of the dimers made of fermions. In the limit of low densities, we calculate the ground-state energy of the system, or equivalently the chemical potentials of each species, as well as the one-particle gap and the energy of an ``impurity'' immersed in a Fermi sea. For the chemical potentials we go up to order (density)4/3(\mathrm{density}){}^{4/3}(density)4/3. The results found involve the exact atom-dimer aAD{a}_{\mathrm{AD}}aAD​ and dimer-dimer aDD{a}_{\mathrm{DD}}aDD​ scattering lengths and therefore include the three- and four-body problems in the many-body problem. We briefly comment on the importance of the different mean-field corrections for recent experiments.
We present an exact many-body theory of ultracold fermionic gases for the Bose-Einstein condensation (BEC) regime of the BEC-BCS crossover. This is a purely fermionic approach which treats explicitly and systematically the dimers formed in the BEC regime as made of two fermions. We consider specifically the zero temperature case and calculate the first terms of the expansion of the chemical potential in powers of the density nnn. We derive first the mean-field contribution, which has the expected standard expression when it is written in terms of the dimer-dimer scattering length aM{a}_{M}aM​. We go next in the expansion to the Lee-Huang-Yang order, proportional to n3∕2{n}^{3∕2}n3∕2. We find the far less obvious result that it retains also the same expression in terms of aM{a}_{M}aM​ as for elementary bosons. The composite nature of the dimers appears only in the next term proportional to n2{n}^{2}n2.
We investigate the crossover from BCS pairing to molecular Bose-Einstein condensation (BEC) in an atomic gas with two fermion species with masses ${m}_{\ensuremath{\uparrow}}\ensuremath{\ne}{m}_{\ensuremath{\downarrow}}$ tuned through a Feshbach resonance. We present results for the T=0T=0T=0 equation of state as a function of the scattering length including the effects of Gaussian fluctuations about the mean field ground state. We compute the ground state energy as a function of ${m}_{\ensuremath{\uparrow}}/{m}_{\ensuremath{\downarrow}}$ at unitarity and find excellent agreement with the quantum Monte Carlo result for ${m}_{\ensuremath{\uparrow}}/{m}_{\ensuremath{\downarrow}}=6.67$ for a 40K^{40}\mathrm{K}40K-6Li^{6}\mathrm{Li}6Li mixture. We show that the dimer scattering length in the BEC limit as a function of ${m}_{\ensuremath{\uparrow}}/{m}_{\ensuremath{\downarrow}}$ compares well with the exact four-body results of Petrov et al. [J. Phys. B 38, S645 (2005)]. We also derive the condition for trapping frequencies to obtain an unpolarized gas in a harmonic trap.
We present an exact theory of the BEC-BCS crossover in the Bose-Einstein-condensate (BEC) regime, which treats explicitly dimers as made of two fermions. We apply our framework, at zero temperature, to the calculation of the equation of state. We find that, when expanding the chemical potential in powers of the density n up to the Lee-Huang-Yang order, proportional to n(3/2), the result is identical to the one of elementary bosons in terms of the dimer-dimer scattering length a(M), the composite nature of the dimers appearing only in the next order term proportional to n(2).
We determine the energetically lowest-lying states in the BEC-BCS crossover regime of sss-wave interacting two-component Fermi gases under harmonic confinement by solving the many-body Schr\"odinger equation using two distinct approaches. Essentially exact basis set expansion techniques are applied to determine the energy spectrum of systems with N=4N=4N=4 fermions. Fixed-node diffusion Monte Carlo methods are applied to systems with up to N=20N=20N=20 fermions, and a discussion of different guiding functions used in the Monte Carlo approach to impose the proper symmetry of the fermionic system is presented. The energies are calculated as a function of the sss-wave scattering length as{a}_{s}as​ for N=2−−20N=2--20N=2−−20 fermions and different mass ratios $\ensuremath{\kappa}$ of the two species. On the Bose-Einstein condensate (BEC) and BCS sides, our energies agree with analytically determined first-order correction terms. We extract the scattering length and the effective range of the dimer-dimer system up to $\ensuremath{\kappa}=20$. Our energies for the strongly interacting trapped system in the unitarity regime show essentially no shell structure, and are well described by a simple expression whose functional form can be derived using the local density approximation, with one or two parameters. The universal parameter $\ensuremath{\xi}$ for the trapped system for various $\ensuremath{\kappa}$ is determined, and comparisons with results for the homogeneous system are presented.
We show how the 3 and 4-body problems emerge in the BEC limit of the BEC-BCS crossover, where we treat explicitly dimers as made of two fermions. We give the argument leading, at zero temperature, to the calculation of the equation of state. We find that, when expanding the chemical potential in powers of the density n up to the Lee-Huang-Yang order, proportional to n 3/2, the result is identical to the one of elementary bosons in terms of the dimer-dimer scattering length a M , the composite nature of the dimers appearing only in the next order term proportional to n 2.
We present a theoretical study of the ground state of the BCS-BEC crossover in dilute two-dimensional Fermi gases. While the mean-field theory provides a simple and analytical equation of state, the pressure is equal to that of a noninteracting Fermi gas in the entire BCS-BEC crossover, which is not consistent with the features of a weakly interacting Bose condensate in the BEC limit and a weakly interacting Fermi liquid in the BCS limit. The inadequacy of the two-dimensional mean-field theory indicates that the quantum fluctuations are much more pronounced than those in three dimensions. In this work, we show that the inclusion of the Gaussian quantum fluctuations naturally recovers the above features in both the BEC and the BCS limits. In the BEC limit, the missing logarithmic dependence on the boson chemical potential is recovered by the quantum fluctuations. Near the quantum phase transition from the vacuum to the BEC phase, we compare our equation of state with the known grand canonical equation of state of two-dimensional Bose gases and determine the ratio of the composite boson scattering length aB{a}_{B}aB​ to the fermion scattering length a2D{a}_{2\mathrm{D}}a2D​. We find ${a}_{B}\ensuremath{\simeq}0.56{a}_{2\mathrm{D}}$, in good agreement with the exact four-body calculation. We compare our equation of state in the BCS-BEC crossover with recent results from the quantum Monte Carlo simulations and the experimental measurements and find good agreements.