The pseudopotential in resonant regimes Ludovic Pricoupenko To cite this version: Ludovic Pricoupenko. The pseudopotential in resonant regimes. 2005. hal-00007776v1 HAL Id: hal-00007776 https://hal.archives-ouvertes.fr/hal-00007776v1 Preprint submitted on 3 Aug 2005 (v1), last revised 10 Jan 2006 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The pseudopotential in resonant regimes Ludovic Pricoupenko Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, Universit´ePierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France. (Dated: August 4, 2005) The zero-range approach is extended for describing situations where two-body scattering is res- onant in arbitrary partial waves. The formalism generalizes then the Fermi pseudopotential which can be used only for s-wave broad resonances. In a given channel, the interaction is described either in term of a boundary condition on the wave function or with a family of pseudopotentials. Scattering states being not mutually orthogonal in the zero-range scheme (except for the Fermi pseudopotential), the inconsistency is solved by introducing a regularized scalar product adapted to these particular Hilbert spaces. PACS numbers: 03.65.Nk,03.75.Ss,05.30.Fk,34.50.-s I. INTRODUCTION change the exact treatment of the two-body problem, while the Born approximation can be adjusted to the A. Zero-range approach in ultra-cold physics exact zero-range solution for arbitrary colliding energy. The λ-potential has been applied for ultra-cold bosons to take into account self-consistently the back action of the The roots of the zero range approach lie in the works on excitations on the condensate using the Hartree-Fock- the neutron-proton interaction in Nuclear Physics. First, Bogolubov formalism in three dimensional systems [7] H. Bethe and R. Peierls [1] shown that it was possible to and also in quasi two-dimensional traps [8]. A powerful get numerous scattering properties on the Deuteron by application of zero-range approaches and pseudopoten- replacing the neutron-proton force with a suitable bound- tials in ultra-cold gases concerns also low dimensional ary condition on the wave function in the s-wave chan- configurations [9, 10, 11] where scattering properties of nel. One year after, E. Fermi [2] introduced the idea of a confined particles can be deduced from the Bethe-Peierls zero-range effective potential allowing for a Born approxi- boundary condition on the 3D wave function. This is by mation in the computation of scattering cross-section be- this way that the concept of tunable interaction in quasi tween a slow neutron and a proton. Equivalence between one-dimensional and quasi two-dimensional systems have these two approaches was developed further by G. Breit been discovered. The few body problem is another im- [3] and led to the final formulation of the so-called Fermi portant direction of research in which the zero-range ap- pseudopotential [4]. The fact that the range of the nu- − proach has proven to be fruitful. In this case, it appears clear force ( 2 10 15 m) is much less than the s-wave − clearly that the zero-range pseudopotential is not only a neutron-proton≃ triplet scattering length ( 4.3 10 15 m) parameterization of the two-body low energy scattering was a key ingredient for the success of this≃ way of mod- properties but has to be understood as a tool for obtain- eling low energy processes. ing exact results in the few body problem [12, 13, 14, 15] Interestingly, more than one half century after these pi- and for the unitary quantum gas [16, 17]. These studies oneer works, zero-range pseudopotentials appear as ref- are relied on the resonant regime in the s-wave channel erence tools in the physics of ultra-cold atoms. There which can be achieved in ultra-cold systems by use of are several reasons for this renewed interest. First, the Feshbach resonances [18] tuned by an external magnetic full two-body potential contains information on “high en- field. ergy” processes which are of no use for studying prop- erties of the metastable dilute gas phase: this is the More generally, study of high correlations regimes ob- case for example of the deep bound-states and of the tained through scattering resonances appears actually hard core repulsion. As will be shown below, the zero- as one of the more challenging directions in the field range approach permits to introduce only the relevant of ultra-cold atoms. The experimental achievement of parameters for describing the low energy processes which the BCS-BEC crossover in two-components Fermi gases ccsd-00007776, version 1 - 3 Aug 2005 are dominant in ultra-cold atomic gases. A second key [19, 20, 21, 22, 23] is a spectacular example of such sit- property at the origin of the pseudopotential is that it uation where the system is not only interesting in itself can be used in the Born approximation while this is but also serves as a model for studying accurately ques- not possible with the exact two-body potential. As a tions raised in the quantum many body problem [24, 25]. consequence, mean field treatments like Gross-Pitaevskii Actually, the domain enlarges to resonant scattering in and Bogolubov approaches [5] for bosons or BCS [6] for p-wave [26, 27, 28, 29, 30] and d-wave channels [31], and fermions can be implemented using the Fermi pseudopo- possible studies of the BCS-BEC crossover in channels tential. This idea has been generalized by introducing of higher angular momentum are now at hand. Major a family of equivalent zero-range pseudopotentials: the interest in these systems comes from the fact that the λ-potentials [7]. Varying the free parameter λ doesn’t strength of correlations can be tuned arbitrarily while 2 the mean inter-particle distance remains large with re- However, it is of interest when dealing with wave func- spect to the potential radius R (for example, the s-wave tions of arbitrary symmetry (for example in presence of scattering length can be adjusted to several order of mag- an anisotropic external potential) to have the full expres- nitude larger than R). As a consequence, one can ex- sion of the pseudopotential, that is an expression which pect that short range physics is not directly involved in contains implicitly the projection operator over the in- a large class of many body properties and can be de- teracting channel (noted Πl for the channel of angular scribed only in terms of the low energy two-body behav- quantum number l). ior. It is worth pointing out that this fundamental feature has been verified in the BCS-BEC crossover of the two- component Fermi gas. For this reason, the zero-range C. Organization of the Paper approach which is parameterized only by the two-body low energy physics is a very appealing tool for study- ing these regimes. Actually, the state of the art is as The paper is organized as follow: in the first part, the follow: broad s-wave resonances (like the one in 6Li at regime of interest (two-body resonant scattering) is intro- 835 Gauss [21]) can be accurately parameterized with the duced and characterized in each partial wave channel by scattering length by use of the Fermi pseudopotential, a set of two parameters in the phase-shift. In the second and for narrow s-wave resonance (for example in 23Na part, the different tools of the zero-range approach in this at 907 Gauss [32]) where the effective range is large and regime are defined. We show that the problem can be de- negative, the method has been generalized by modifying fined in terms of boundary conditions on the wave func- the Bethe-Peierls boundary conditions in Ref.[14]. Re- tion which generalize the Bethe-Peierls approach. We cently, a pseudopotential has been introduced for p-wave express these boundary conditions in spherical coordi- resonances [33] and the aim of this paper is to extend this nates and also in Cartesian coordinates by use of the approach for resonant regimes in arbitrary partial waves. symmetric trace free tensors appearing in the usual mul- tipolar expansion. This way of defining the zero-range approach permits to include quite easily both the projec- tion operator Πl and also the l-th derivatives of the delta B. Which generalization of actual pseudopotentials distribution in the pseudopotential. It appears that in is needed ? each interacting channel there exists a family of pseu- dopotentials generated by a free parameter, thus gen- The idea of replacing a true finite range two-body po- eralizing the λ-potential approach [7, 33]. This degree tential by a zero-range pseudopotential acting on each of freedom makes possible to use the pseudopotential in partial wave has been first developed by Huang and Yang the Born approximation for mean-field approaches. The [34] in the context of the hard-sphere model in view of third and last part, illustrates the formalism. We show applications to superfluid 4He. However, due to the im- how to recover two-body scattering states by use of the portance of the short range behavior in these systems, Green’s method.