Analytical calculation of cold-atom scattering

Abstract

The interaction between atoms behaves as −α/r n at large distances, and, owing to the large reduced mass µ of the collision pair, allows semiclassical treatment within the potential well. As a result, the low-energy scattering is governed by two large parameters: the asymptotic parameter γ = √ 2µα/h ≫ a (n−2)/2 0 (a 0 is the Bohr radius), and the semiclassical zero-energy phase Φ ≫ 1. In our previous work [Phys. Rev. A 48, 546 (1993)] we obtained an analytical expression for the scattering length a, which showed that it has 75% preference for positive values for n = 6, characteristic of collisions between ground-state neutral atoms. In this paper we calculate the effective range and show that it is a function of a, r e = F n − G n /a + H n /a 2 , where F n , G n and H n depend only on γ. Thus, we know the s phase shift at low momenta k ≪ γ −2/(n−2) from the expansion k cot δ 0 ≃ −1/a +

Quantum-defect theory for−1/r4-type interactions

“…More generally for a −1/r n potential, there exist natural energy scales of ǫ (n−2)/2n s for positive energies and −(−ǫ s ) (n−2)/2n for negative energies. They are energy scales associated with the semiclassical behaviors away from the threshold [49][50][51][52]. In later figures covering both positive and negative energies, the natural energy scale for the −1/r 4 potential will be represented as sgn(ǫ s )|ǫ s | 1/4 , with sgn(ǫ s ) defined by…”

Low-energy universality and scaling of van der Waals forces

“…We conclude by underlining that, when suitably displayed, the analytical approach to Van der Waals forces pursued in previous works [10,11] acquires a quite universal character with indisputable phenomenological success; the leading 1/r 6 contribution suffices to accurately describe low energy atom-atom scattering with just two parameters in a wide energy range. We naturally expect new universality and scaling patterns to emerge from systems characterized by power law forces but less understood such as molecular interactions in the ultracold regime.…”

“…The line corresponds to Eq. (17) [10,11]. We analyze the robustness of this agreement by showing in Fig.…”

“…For the pure VdW case, V = −C 6 /r 6 , the effective range has been computed analytically [10,11] and in harmony with the general structure Eq. (16) reads…”

“…[10,11,12] (see also [13]),within a quantum-defect theoretical viewpoint. In this letter we systematically show that these simplified approaches work and analyze why they succeed.…”