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

Abstract

We investigate small equal-mass two-component Fermi gases under external spherically symmetric confinement in which atoms with opposite spins interact through a short-range two-body model potential. We employ a non-perturbative microscopic framework, the stochastic variational approach, and determine the system properties as functions of the interspecies s-wave scattering length as, the orbital angular momentum L of the system, and the numbers N1 and N2 of spin-up and spin-down atoms (with N1 − N2 = 0 or 1 and N ≤ 6, where N = N1 + N2). At unitarity, we determine the energies of the five-and six-particle systems for various ranges r0 of the underlying two-body model potential and extrapolate to the zero-range limit. These energies serve as benchmark results that can be used to validate and assess other numerical approaches. We also present structural properties such as the pair distribution function and the radial density. Furthermore, we analyze the one-body and two-body density matrices. A measure for the molecular condensate fraction is proposed and applied. Our calculations show explicitly that the natural orbitals and the momentum distributions of atomic Fermi gases approach those characteristic for a molecular Bose gas if the s-wave scattering length as, as > 0, is sufficiently small.

Lattice Monte Carlo calculations for unitary fermions in a harmonic trap

“…Our final fit results for N ≤ 6 are indicated in Fig. 7, and presented along with a comparison to an exact result for N = 3 [15] and high-precision Hamiltonian results of [57] for N = 4 − 6 in Table IV. For N > 6, it is likely that both discretization and finite volume errors will grow, since we expect the wave function to spread out in both position and momentum space when more particles are added to the system. Numerical evidence suggests that extrapolations become necessary for N 20.…”

“…Not known exactly but determined to high numerical accuracy are (iv) the few lowest energy levels for three unitary fermions in a box, extrapolated from a lattice Hamiltonian diagonalization very close to the continuum limit, with lattice size up to L = 50 [29]; and (v) the ground state energies for 4, 5, 6 unitary fermions in a harmonic trap, obtained by solving the Schrödinger equation [57]. The ground state energy for N = 4 fermions in a box has also recently been precisely studied by several methods in Ref.…”

Natural and unnatural parity states of small trapped equal-mass two-component Fermi gases at unitarity and fourth-order virial coefficient

“…This implies that the determination of the eigenenergies amounts to the diagonalization of a generalized eigenvalue problem defined by the Hamiltonian and overlap matrices [32]. While one might think, at first sight, that the non-orthogonality of the basis functions could introduce numerical instabilities, it has been shown in previous work [5,6,9,12,14] that numerical instabilities due to linear dependence issues can be avoided completely for the systems of interest in this work if the basis sets are chosen carefully.…”

“…From a theoretical point of view, small harmonically trapped Fermi gases with central short-range interactions are particularly appealing since they can be treated with comparatively high accuracy by a variety of methods, including techniques that have been developed in the context of atomic physics, nuclear physics and quantum chemistry problems [4][5][6][7][8][9][10][11][12][13][14][15][16]. For the harmonically trapped equal-mass system, the center of mass degrees of freedom separate.…”

Universal noise and Efimov physics

“…At unitarity, the energies of N-body bosonic systems have been shown to be tied to the spectrum of Efimov trimers [9,10], whose binding energies are known to obey a discrete scaling symmetry, for at least N 16 [11][12][13][14][15]. One method for showing this is to study the ratio of some N-body…”