Energy-dependent effective interactions for dilute many-body systems

Abstract

We address the issue of determining an effective two-body interaction for mean-field calculations of energies of many-body systems. We show that the effective interaction is proportional to the phase shift, and demonstrate this result in the quasiclassical approximation when there is a trapping potential in addition to the short-range interaction between a pair of particles. We calculate numerically energy levels for the case of an interaction with a short-range square-well and a harmonic trapping potential and show that the numerical results agree well with the analytical expression. We derive a generalized Gross--Pitaevskii equation which includes effective range corrections and discuss the form of the electron--atom effective interaction to be used in calculations of Rydberg atoms and molecules.Comment: 6 pages, 2 figure

Simple model for interactions and corrections to the Gross-Pitaevskii equation

“…For λ = 0 one finds the standard GPE (1). The contribution due to the nonvanishing size of the particles r out was found by Collin, Massignan, and Pethick [6] (for earlier work see [7]). In our case it takes the form…”

“…Here, we add an extra term of the order of const×a 3 k 2 to the standard GPE [see for example (6) where r out is of order a and λ is of the same magnitude as g which is proportional to a]. Note that taking into account contributions from components of higher angular momentum in the partial waves expansion will also add extra terms to the GPE.…”

“…According to [6], it is possible to formulate a modified GPE which takes the range of the pair interaction into account, as follows. For λ = 0 one finds the standard GPE (1).…”

Finite-range corrections to the thermodynamics of the one-dimensional Bose gas

“…Moreover, while a universal GPE-based approach is reliable only when r e ≪ a s ≪ d, with d being the average distance between atoms, it has been shown [17,19,21] that one can greatly enhance finite-range effects by means of Fano-Feshbach resonances. In regimes where r s ≈ a s , deviation from universalities are relevant, and every dynamical picture of the system has to take into account not only quantum fluctuations, but also finite-range corrections modelled around Eq.…”

“…The resulting thermodynamics is universal, since the only dependence from the inter-atomic potential is via the swave scattering length. However, deviations from universality due to the finite-range of interaction potential have been shown to be quite important for a better understanding of the 3D Bose gas [13][14][15][16][17][18][19][20][21][22][23]. Finite-range effects naturally arise by modelling the two-body interactions beyond the unphysical zero-range approximation.…”