Virial relations for ultracold trapped Fermi gases with finite range interactions through the BCS-BEC crossover

In the absence of interaction, the virial theorem relates the energy per particle of a confined atomic gas with the trapping potential. If that potential is harmonic, the theorem states that the total energy per particle is twice the mean trapping potential energy E = 2E tr .

(

For strongly interacting two component Fermi gases, confined by a harmonic trap in the unitary limit, this relation was also shown to be valid experimentally and theoretically 3,4 . The first derivation of that theorem considered zerorange interactions and made use of the local density approximation. Further insight on the fundamental basis of this relation revealed several remarking features of the unitary gas such as its scaling properties 5 or a mapping, using group theory, between the trapped and the free space problem 6 . Recently, the Hellmann-Feynman theorem was used to prove Eq. (1) at the unitary limit 7,8 , and to generalize the virial relations for finite scattering lengths 1,2 . In fact, general confinement potentials and finite range interactions can directly be taken into account using a general virial theorem which can be stated as follows 2 : Consider a Hamiltonian for a system of N particles with arbitrary statistics:

where H ′ and its domain depend on p parameters with length dimensions ℓ 1 , …, ℓ p , on h and the mass of the particles. U ( r 1 , …, r N ) denotes a regular arbitrary function that allows the domains of H and H ′ to coincide, r i is the position vector for the i-th particle. Then,

with E the total energy. For N particles confined by a harmonic trap, U

)/2 and Eq. (3) becomes:

where E tr = U is the trapping potential energy.

In the present article, we study 2N fermionic atoms in two equally populated hyperfine states (N = N ↑ = N ↓ = 165) confined by an isotropic three-dimensional harmonic trap of frequency ω, and interacting through an attractive finite range potential V = -|V 0 |e -r/rv . This potential is characterized by two parameters, its strength V 0 and its range r v . When the kinetic energy of the atoms is low enough, the scattering length is a proper parameter to describe the interacting system. For a given number of s-wave bound states and a given r v , there is a one-to-one relationship between the strength of the potential V 0 and the scattering length a. We consider the case where at most one bound state is admitted by the potential and find the ground state of the many body Schrödinger equation approximately, via a variational Monte Carlo calculation, for several scattering lengths a and short potential ranges r v « r ho ≡ h/mω. We then study the behavior of the total, internal and trapping energy as a function of both length parameters a and r v to verify Eq. ( 4). The explicit expression of the Hamiltonian is

and the corresponding virial relation becomes

In the BEC side of the crossover the total energy E can become extremely large compared to the total energy E in the BCS side due to the contribution of the binding energy of the formed molecules. This fact increases the numerical errors in the evaluation of the derivatives in Eq. (7). In order to isolate this two-body effect from many-body effects, we have found convenient to take into account the behavior of the free space binding energy as follows. The two body problem,

is analytically solvable for s-states, so that the scattering length is explicitly given by

with ζ = (2r v |V 0 |m/h), C = 0.577215664901… is the Euler constant and J ν and N ν represent the Bessel function of the first and second kind of order ν, respectively. This problem has the following bound states

where N is a normalization factor and y = ζe -r/2rv . The boundary condition at the origin implies J xs (ζ) = 0, so the corresponding eigenenergies ε

That is, x s is determined by ζ and

We shall work with z 0 < ζ < z 1 with z 0 and z 1 the first two zeros of the Bessel function J 0 . Under these conditions, just one bound state is admitted for each positive scattering length a. Given a and r v and using Eq. ( 9),we can write

As a consequence, the ground state binding energy ε (rv) 0 of the two interacting particle system in otherwise free space satisfies the equation

Thus, if we define

and

the virial relation, Eq. ( 7), reads

This expression is easier to verify numerically than Eq. (7). Notice that, from a dimensional analysis, equations similar to Eq. ( 13) can be

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