Tuning of Efimov states in non-integer dimensions

Abstract: The purpose of this paper is to show that, by combining Feshbach resonances with external confining potentials, the energy scale factor of neighboring Efimov states can be tremendously reduced. The Efimov conditions can be reached for systems made of three different particles. For the case of two identical light particles and a heavy particle the energy factor can be reduced by many orders of magnitude, and the Efimov states are in this way more easily reachable experimentally. The equivalence between external potentials and the formulation in terms of non-integer dimensions, $d$, is exploited. The technically simpler $d$-method is used to derive analytic expressions for two-component relative wave functions describing two short-range square-well interacting particles. The two components express one open and one closed channel. The scattering length is obtained after phase shift expansion, providing an analytic form for the Efimov condition. We illustrate the results by means of systems made of $^7$Li, $^{39}$K, and $^{87}$Rb, with realistic parameters. The related pairs of dimension and magnetic field are shown and discussed. The results are universal as they only rely on large-distance properties.