BEC-BCS crossover of a trapped Fermi gas without using the local density approximation

Abstract

We perform a variational quantum Monte Carlo simulation of an interacting Fermi gas confined in a three dimensional harmonic potential. This gas is considered as the precursor system from which a molecular bosonic gas is formed. Based on the results of two-body calculations for trapped atoms, we propose a family of variational many-body wave functions that takes into account the qualitative different nature of the BCS and BEC regimes as a function of the scattering length. Energies, densities and correlation functions are calculated and compared with previous results for homogeneous gases. Universality tests at the unitarity limit are performed including the verification of the virial relation and the evaluation of the β parameter.PACS numbers: 03.75. Ss, 03.75.Hh, 05.30.Fk The crossover from a low interacting attractive Bardeen-Cooper-Schieffer (BCS) gas to a molecular BoseEinstein condensate (BEC) has been realized experimentally using a mixture of ultracold Fermi atoms in two hyperfine states [1,2,3,4,5]. In dilute Fermi gases, the atomic interactions have a range much smaller than the interparticle separation. Nevertheless, under proper conditions, a magnetic field can be used to tune the attractive potential so that the energy of a pair of scattering atoms is close to that of a molecular bound state (Feshbach resonance). In experiments where the resonance is broad, it can be represented by a single-channel model in which the s-wave scattering length a determines the general features of the ultracold atomic gas. At low enough temperatures, atoms in different hyperfine states pair into bound molecules for positive values of a, forming a molecular BEC that can be adiabatically converted into a degenerate Fermi gas by shifting a to negative values. At resonance (|a| → ∞), the gas acquires universal properties, i. e., they are independent of any feature of the atomic potentials [2,6,7]. This is the so called unitarity limit.The theoretical description of the BEC-BCS crossover is complicated because there is no ad hoc single parameter to control the relevant features in both regimes. A reasonable alternative to infer properties of the system at unitarity has been to consider an homogeneous Fermi gas, and to assume the universality hypothesis according to which the only dominant length is the interparticle separation. Then, the thermodynamic potentials have a universal form specified by only few universal numbers [7]. For instance, the interaction energy is proportional to the Fermi energy via a universal constant β.Quantum Monte Carlo (QMC) techniques support ab initio methods to theoretically test the universality hypothesis. They can be used to approximately solve the many-body Schrödinger equation for a given model of the interaction potential. In previous studies the BEC and BCS regimes have been explored using a fixed-node QMC technique, which is rigorous but also computationally demanding. In particular, the value of β has been predicted considering up to 66 particles [8,9]. Nevertheless, these QMC calcu...

Few-body physics with ultracold atomic and molecular systems in traps

“…These sets of energies have been chosen to illustrate our current understanding (see Refs. [194,208,209] for other calculations). The MC and DFT approaches are based on entirely different formulations and the relatively good agreement displayed in Fig.…”

“…Crossover studies for systems with N > 4 can be found in Refs. [189,194,196]. The energies of the (2, 2) system were obtained using the stochastic variational approach [197], a basis set expansion approach that results in stable and highly accurate results for trapped systems with up to N = 6 fermions throughout the entire crossover regime [167,183,187,188,189,196,198].…”

Energetics and structural properties of trapped two-component Fermi gases

“…The definition of the normalized energy crossover curve Λ (κ) N1,N2 introduced in Refs. [21,28] for even N can be extended to odd N ,…”

Pairing and molecule formation along the BCS-BEC crossover for finite range potentials

“…Since actual potentials are finite range, and do not depend explicitly on the scattering length, there has already been interest in analyzing the crossover and its properties taking those aspects into account. Previous analysis of the T = 0 BCS-BEC problem considering finite range potentials include both homogeneous and inhomogeneous cases and are based on Quantum Monte Carlo approaches [20][21][22], as well as on numerical solutions of the coupled equations of the BCS theory [23,24]. Here we show that the structure of the coupled equations for the energy gap and the density leads us to identify closed expressions for any finite range potential model with definite Fourier transform, thus, providing a route to address a variety of the physics of the crossover.…”