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Renormalization of Resonance Scattering Theory Resonance superfluidity : renormalization of resonance scattering theory Citation for published version (APA): Kokkelmans, S. J. J. M. F., Milstein, J. N., Chiofalo, M. L., Walser, R., & Holland, M. J. (2002). Resonance superfluidity : renormalization of resonance scattering theory. Physical Review A : Atomic, Molecular and Optical Physics, 65(5), 53617-1/14. [53617]. https://doi.org/10.1103/PhysRevA.65.053617 DOI: 10.1103/PhysRevA.65.053617 Document status and date: Published: 01/01/2002 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 27. Sep. 2021 PHYSICAL REVIEW A, VOLUME 65, 053617 Resonance superfluidity: Renormalization of resonance scattering theory S. J. J. M. F. Kokkelmans,1 J. N. Milstein,1 M. L. Chiofalo,2 R. Walser,1 and M. J. Holland1 1JILA, University of Colorado and National Institute of Standards and Technology, Boulder, Colorado 80309-0440 2INFM and Classe di Scienze, Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126, Pisa, Italy ͑Received 14 December 2001; published 14 May 2002͒ We derive a theory of superfluidity for a dilute Fermi gas that is valid when scattering resonances are present. The treatment of a resonance in many-body atomic physics requires a novel mean-field approach starting from an unconventional microscopic Hamiltonian. The mean-field equations incorporate the micro- scopic scattering physics, and the solutions to these equations reproduce the energy-dependent scattering properties. This theory describes the high-Tc behavior of the system, and predicts a value of Tc that is a significant fraction of the Fermi temperature. It is shown that this mean-field approach does not break down for typical experimental circumstances, even at detunings close to resonance. As an example of the application of our theory, we investigate the feasibility for achieving superfluidity in an ultracold gas of fermionic 6Li. DOI: 10.1103/PhysRevA.65.053617 PACS number͑s͒: 03.75.Fi, 67.60.Ϫg, 74.20.Ϫz I. INTRODUCTION treatment of strongly interacting fermionic systems where higher-order correlations must be treated explicitly. The remarkable accomplishment of reaching the regime In this paper, we show that an unconventional mean-field of quantum degeneracy ͓1͔ in a variety of ultracold atomic theory can still be appropriately exploited under the condi- gases enabled the examination of superfluid phenomena in a tion that the characteristic range R of the potential is such diverse range of novel quantum systems. Already many el- that nR3Ӷ1 ͑while na3տ1). The core issue is that around a ementary aspects of superfluid phenomena have been ob- resonance, the cross section becomes strongly dependent on served in bosonic systems including vortices ͓2͔. The chal- the scattering energy. This occurs when either a bound state lenge of achieving superfluidity in a Fermi gas remains, lies just below threshold, or when a quasibound state lies just however, although it appears possible that this situation may above the edge of the collision continuum. In both cases, the change in the near future. A number of candidate systems for scattering length—evaluated by considering the zero-energy realizing superfluidity in a fermionic gas appear very prom- limit of the scattering phase shift—does not characterize the ising and it is currently the goal of several experimental ef- full scattering physics over the complete energy range of forts to get into the required regime to observe the superfluid interest, even when in practice this may cover a range of only phase transition. So far both fermionic potassium ͓3͔ and a few microkelvin. lithium ͓4,5͔ have been cooled to the microkelvin regime and The paper is outlined as follows. In Sec. II, we present a are well below the Fermi temperature by now—a precursor systematic derivation of the renormalized potentials for an step for superfluidity. effective many-body Hamiltonian. This requires a detailed In order to make the superfluid phase transition experi- analysis of coupled-channels scattering. In Sec. III, we de- mentally accessible, it will likely be necessary to utilize the rive the resonance mean-field theory. In Sec. IV, we present rich internal hyperfine structure of atomic collisions. Scatter- the thermodynamic solutions allowing for resonance super- ing resonances, in particular, may prove to be extremely im- fluidity. We apply our theory to the specific case of 6Li and portant since they potentially allow a significant enhance- determine the critical temperature for the superfluid phase ment of the strength of the atomic interactions. It is transition. In Sec. V, we consider the validity of the mean- anticipated that by utilizing such a scattering resonance one field approach in the case of resonance coupling, and estab- may dramatically increase the critical temperature at which lish the equivalence with previous diagrammatic calculations the system becomes unstable towards the formation of Coo- of the crossover regime between fermionic and bosonic su- per pairs, thus bringing the critical temperature into the ex- perconductivity. perimentally accessible regime. In spite of its promise, this situation poses a number of II. TWO-BODY RESONANCE SCATTERING fundamental theoretical problems that must be addressed in order to provide an adequate minimal description of the criti- The position of the last bound state in the interatomic cal behavior. The scope of the complexities that arise in interaction potentials generally has a crucial effect on the treating a scattering resonance can be seen by examining the scattering properties. In a single-channel system, the scatter- convergence of the quantum kinetic perturbation theory of ing process becomes resonant when a bound state is close to the dilute gas. In this theory the small parameter is known as threshold. In a multichannel system the incoming channel the gaseous parameter, defined as ͱna3, where n is the par- ͑which is always open͒ may be coupled during the collision ticle density and a is the scattering length. Formally, when to other open or closed channels corresponding to different the scattering length is increased to the value at which na3 spin configurations. When a bound state in a closed channel Ϸ1, conventional perturbation theory breaks down ͓6,7͔. lies near the zero of the collision energy continuum, a Fesh- This situation is commonly associated with the theoretical bach resonance ͓8͔ may occur, giving rise to scattering prop- 1050-2947/2002/65͑5͒/053617͑14͒/$20.0065 053617-1 ©2002 The American Physical Society S. J. J. M. F. KOKKELMANS et al. PHYSICAL REVIEW A 65 053617 FIG. 1. Scattering length as a function of magnetic field, for the FIG. 2. Sequence of theoretical steps involved in formulating a ϭ Ϫ 6 ( f ,m f ) (1/2, 1/2) and (1/2,1/2) mixed spin channel of Li. renormalized scattering model of resonance physics for low-energy scattering. The starting point is a full coupled-channels ͑CC͒ calcu- erties that are tunable by an external magnetic field. The lation that leads us via an equivalent Feshbach theory, and an ana- tuning dependence arises from the magnetic moment differ- lytic coupled square-well theory, to a contact potential scattering ence ⌬␮mag between the open and closed channels ͓9͔. This theory that gives the renormalized equations for the resonance gives rise to a characteristic dispersive behavior of the system. s-wave scattering length at fields close to resonance given by Only the exact interatomic interaction will reproduce the ⌬B full T matrix over all energy scales. However, since only ϭ ͩ Ϫ ͪ ͑ ͒ a abg 1 Ϫ , 1 collision energies in the ultracold regime ͑of order mi- B B0 crokelvin͒ are relevant, a much simpler description is pos- sible. If the scattering length does not completely character- where abg is the background value that may itself depend weakly on magnetic field. The field width of the resonance is ize the low-energy scattering behavior in the presence of a given by ⌬B, and the bound state crosses threshold at a resonance, what is the minimal set of parameters that will field-value B . The field detuning can be converted into an do? 0 As illustrated in Fig. 2, we proceed to systematically re- energy detuning ¯␯ by the relation ¯␯ϭ(BϪB )⌬␮mag.An 0 solve this question by the following steps.
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