Momentum Distribution and Condensate Fraction of a Fermion Gas in the BCS-BEC Crossover

Abstract

By using the diffusion Monte Carlo method we calculate the one-and two-body density matrix of an interacting Fermi gas at T = 0 in the BCS-BEC crossover. 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 fermionic pairs is calculated in the unitary limit and on both sides of the BCS-BEC crossover. PACS numbers:The physics of the crossover from Bardeen-CooperSchrieffer (BCS) superfluids to molecular Bose-Einstein condensates (BEC) in ultracold Fermi gases near a Feshbach resonance is a very exciting field that has recently attracted a lot of interest, both from the experimental [1,2] and the theoretical side [3]. An important experimental achievement is the observation of a condensate of pairs of fermionic atoms on the side of the Feshbach resonance where no stable molecules would exist in vacuum [4,5]. Although the interpretation of the experiment is not straightforward, as it involves an out-of-equilibrium projection technique of fermionic pairs onto bound molecules [6], it is believed that these results strongly support the existence of a superfluid order parameter in the strongly correlated regime on the BCS side of the resonance [5].The occurrence of off-diagonal long-range order (ODLRO) in interacting systems of bosons and fermions was investigated by C.N. Yang in terms of the asymptotic behavior of the one-and two-body density matrix [7]. In the case of a two-component Fermi gas with N ↑ spin-up and N ↓ spin-down particles, the one-body density matrix (OBDM) for spin-up particles, defined asdoes not possess any eigenvalue of order N ↑ . This behavior implies for homogeneous systems the asymptotic condition ρ 1 (rIn the above expression ψ † ↑ (r) (ψ ↑ (r)) denote the creation (annihilation) operator of spin-up particles. The same result holds for spin-down particles. ODLRO may occur instead in the two-body density matrix (TBDM), that is defined asFor an unpolarized gas with N ↑ = N ↓ = N/2, if ρ 2 has an eigenvalue of the order of the total number of particles N , the TBDM can be written as a spectral decomposition separating the largest eigenvalue,2 containing only eigenvalues of order one. The parameter α ≤ 1 in Eq. (3) is interpreted as the condensate fraction of pairs, in a similar way as the condensate fraction of single atoms is derived from the OBDM.The spectral decomposition (3) yields for homogeneous systems the following asymptotic behavior of the TBDMThe complex function ϕ is proportional to the order parameter ψ ↑ (r 1 )ψ ↓ (r 2 ) = αN/2ϕ(|r 1 − r 2 |), whose appearance distinguishes the superfluid state of the Fermi gas. Equation (4) should be contrasted with the behavior of Bose systems with ODLRO, where ρ 1 has an eigenvalue of order N [8], and consequently the largest eigenvalue of ρ 2 is of the order of N 2 . In thi...

Density-functional theory for fermions close to the unitary regime

“…The Monte Carlo results by Chang et al [31] and by Astrakharchik et al [22] yield c 1 ≈ −0.28 [32] and are very close to Steele's analytical result [24]. We are not aware of any estimate for the constant c 2 in Eq.…”

“…This is one of the main results of this work. We estimate that the systematic error of this coefficient is about 5%-15%, as this is the deviation by which the DFT estimates for ξ [27], and c 1 deviates from the Monte Carlo predictions [21,22]. The estimate for c 2 enables us to discuss a small systematic correction of the universal constant obtained from Monte Carlo calculations.…”

“…This makes it possible to study the system as it evolves from a dilute Fermi gas with weak attractive interactions to a bosonic gas of diatomic molecules. This transition from a superfluid BCS state to Bose Einstein condensation (BEC) has been the subject of many experimental [2,3,4,5,6,7,8,9,10,11,12] and theoretical works [13,14,15,16,17,18,19,20,21,22,23].…”

“…The estimate for c 2 enables us to discuss a small systematic correction of the universal constant obtained from Monte Carlo calculations. Recall that the Monte Carlo calculations [21] and [22] are based on potentials with a small effective range of about r 0 ρ 1/3 ≈ 0.05, and r 0 ρ 1/3 = 0.01, respectively. This suggests that their predictions for the universal constant ξ involve a small positive error of about c 2 r 0 ρ 1/3 ≈ 0.007 and c 2 r 0 ρ 1/3 ≈ 0.001, respectively, which is within the statistical error of these simulations.…”

Trapped two-component Fermi gases with up to six particles: Energetics, structural properties, and molecular condensate fraction

“…[81] for a cold-atom study). Instead, the analysis of finite-sized systems proceeds through the diagonalization of the two-body density matrix.…”