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

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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 microscopic 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 super 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-

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 possible. 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. We start from a example of such a resonance is given in Fig. 1, where a numerical solution of the complete coupled-channels scatter- coupled-channels calculation is shown of the scattering 6 ing problem for a given real physical system. In Sec. II A we length of Li for collisions between atoms in the ( f ,m f ) ϭ Ϫ ͓ ͔ demonstrate that the results of these full numerical calcula- (1/2, 1/2) and (1/2,1/2) states 10 . The background scat- tions can be adequately replicated by giving an analytic de- tering length changes slowly as a function of magnetic field scription of resonance scattering provided by Feshbach’s due to a field-dependent mixing of a second resonance that resonance theory. The point of this connection is to demon- comes from the triplet potential. This full coupled-channels strate that only a few parameters are necessary to account for calculation includes the state-of-the-art interatomic potentials ͓ ͔ ͓ ͔ all the collision properties. This implies that the scattering 11 and the complete internal hyperfine structure 13 . model is not unique. There are many microscopic models The scattering length is often used in many-body theory that could be described by the same Feshbach theory. In Sec. to describe interactions in the s-wave regime. That the scat- II B we show this explicitly by presenting a simple double- tering length completely encapsulates the collision physics well model for which analytic solutions are accessible. over relevant energy scales is implicitly assumed in the deri- Thereby we derive a limiting model in which the range of the vation of the conventional Bardeen-Cooper-Schrieffer ͑BCS͒ ͓ ͔ square well potentials and coupling matrix elements are theory for degenerate gases 14,15 , as well as the Gross- taken to zero. This leads in Sec. II C to a scattering model of Pitaevskii description of Bose-Einstein condensates. How- contact potentials. We show that such a scattering solution is ever, the scattering length is only a useful concept in the ␦ able to reproduce well the results of the intricate full numeri- energy regime where the s-wave scattering phase shift 0 ␦ ϭϪ cal model we began with. The utility of this result is that, as depends linearly on the wave number k, i.e., 0 ka. For a will be apparent later, it greatly simplifies the many-body Feshbach resonance system at a finite temperature there will theoretic description. always be a magnetic field value where this approximation breaks down and the scattering properties become strongly energy dependent. In close proximity to a resonance, the A. Feshbach resonance theory scattering process then has to be treated by means of the Here we briefly describe the Feshbach resonance formal- energy-dependent T matrix. ism and derive the elastic S matrices and T matrices for two-

053617-2 RESONANCE SUPERFLUIDITY: RENORMALIZATION . . . PHYSICAL REVIEW A 65 053617 body scattering. These matrices are related to the transition hand side becomes the T matrix for the total scattering pro- probabilities for scattering from an initial channel ␣ to a final cess. The unscattered state is related to the scattering wave ␤ PϪ channel . A more detailed treatment of this formalism can function ͉␺␤ ͘ with incoming boundary conditions via be found in the literature ͓8͔. In Feshbach resonance theory two projection operators P V P PϪ and Q are introduced, which project onto the subspaces ͉␺ ͘ϭ͉␹␤͘ϩ ͉␹␤͘. ͑8͒ ␤ ϪϪ and Q. These subspaces are two orthogonal components that E H PP together span the full Hilbert space of both scattering and bound wave functions. The open and closed channels are The T matrix giving the transition amplitude is then contained in P and Q, respectively. The operators P and Q Ϫ split the Schro¨dinger equation for the two-body problem into ͗␺ P ͉H ͉␾ ͗͘␾ ͉H ͉␺ P͘ T ϭT P ϩ ␤ PQ i i QP ͑ ͒ two parts: ␤␣ ␤␣ ͚ Ϫ⑀ , 9 i E i ͑EϪH ͉͒␺ P͘ϭH ͉␺Q͘, ͑2͒ PP PQ P where T ␤␣ is the amplitude for the direct ͑nonresonant͒ pro- ͑EϪH ͉͒␺Q͘ϭH ͉␺ P͘, ͑3͒ cess. From the T matrix we can easily go to the S matrix that QQ QP Ϫ ϩ is defined as S␤␣ϭ͗␺␤ ͉␺␣ ͘. Since we consider s-wave scat- ϭ ϭ ␺ where H PP PHP, H PQ PHQ, etc., and is the total tering only, in our case there exists a simple relation between scattering wave function. The projections on the two sub- the S matrix and T matrix: S␤␣ϭ1Ϫ2␲iT␤␣ ͓16͔, and this spaces are indicated by P͉␺͘ϭ͉␺ P͘ and Q͉␺͘ϭ͉␺Q͘. The allows us to rewrite Eq. ͑9͒ as ϭ ϩ Hamiltonian H H0 V consists of the sum of the single- ϩ P particle interactions H0 and the two-body interaction V. 2␲i͗␺␥ ͉H ͉␾ ͗͘␾ ͉H ͉␺ ͘ ͑ ͒ ϭ P Ϫ P PQ i i QP Equation 3 can be formally solved S␤␣ S␤␣ ͚ S␤␥͚ Ϫ⑀ . ␥ i E i ͑10͒ 1 ͉␺Q͘ϭ H ͉␺ P͘, ͑4͒ ϩϪ QP P E HQQ The nonresonant factors S␤␥ describe the direct scattering process from an open channel ␥ to the outgoing channel ␤. ϩϭ ϩ ␦ ␦ where E E i with approaching zero from positive Returning to Eq. ͑7͒, we can solve for the component ͑ ͒ values. Substituting this result into Eq. 2 , the open channels ͗␾ ͉H ͉␺ P͘ by multiplying both sides with ͗␾ ͉H . Ϫ ͉␺ P ϭ i QP i QP equation can be written as (E Heff) ͘ 0, where 1. Single resonance 1 H ϭH ϩH H . ͑5͒ For the case of only one resonant bound state and only eff PP PQ ϩϪ QP E HQQ one open channel, the solution of Eq. ͑7͒ gives rise to the following elastic S-matrix element ͑we will omit now the The resolvant operator is now expanded in the discrete and incoming channel label ␣͒: continuum eigenstates of HQQ : ␲ ͉ ␺ Pϩ͉ ͉␾ ͉2 H ͉␾ ͗͘␾ ͉H 2 i ͗ H PQ 1͘ ϭ ϩ PQ i i QP SϭS P 1Ϫ . Heff H PP ͚ Ϫ⑀ i E i ͫ 1 ͬ EϪ⑀ Ϫ͗␾ ͉H H ͉␾ ͘ 1 1 QP ϩϪ PQ 1 ͉␾͑⑀͒͗͘␾͑⑀͉͒ E H PP H PQ HQP ͑ ͒ ϩ ͵ d⑀. ͑6͒ 11 EϩϪ⑀ The nonresonant S matrix is related to the background scat- ⑀ Pϭ Ϫ Here the i’s are the uncoupled bound-state eigenvalues. In tering length via S exp͓ 2ikabg͔. The term in the numera- practice, only a few bound states will significantly affect the tor gives rise to the energy width of the resonance, ⌫ open-channel properties. In this paper, we will consider ei- ϭ ␲͉ ␺ Pϩ͉ ͉␾ ͉2 2 ͗ H PQ 1͘ , which is proportional to the incom- ther one or two bound states and neglect the continuum ex- ing wave number k and coupling constant ¯g ͓17͔. The ͑ ͒ ͉␺ P͘ 1 pansion in Eq. 6 . Then the formal solution for is given bracket in the denominator gives rise to a shift of the bound- by state energy, and to an additional width term i⌫/2. When we denote the energy shift between the collision continuum and 1 H ͉␾ ͗͘␾ ͉H ͉␺ P͘ P Pϩ PQ i i QP ¯␯ ͉␺ ͘ϭ͉␺␣ ͘ϩ ͚ , the bound state by 1, and represent the kinetic energy sim- ϩϪ EϪ⑀ 2 2 E H PP i i ply by ប k /m, the S-matrix element can be rewritten as ͑7͒

ϩ ͉¯ ͉2 ͉␺ P ͘ Ϫ 2ik g1 where ␣ is the eigenstate of the direct interaction H PP S͑k͒ϭe 2ikabg 1Ϫ . ͫ 4␲ប2 ប2k2 ͬ that satisﬁes the outgoing wave boundary condition in chan- Ϫ ͩ¯␯ Ϫ ͪ ϩ ͉¯ ͉2 1 ik g1 nel ␣. By multiplying from the left with ͗␹␤͉V, where ͉␹␤͘ m m is an unscattered state in the outgoing channel ␤, the left- ͑12͒

053617-3 S. J. J. M. F. KOKKELMANS et al. PHYSICAL REVIEW A 65 053617

The resulting total scattering length has exactly the same nition for T in the many-body theory will be slightly different dispersive line shape for the resonant scattering length as we in order to give it the conventional dimensions of energy per have presented originally as Eq. ͑1͒. unit density:

2. Double resonance 2␲ប2i T͑k͒ϭ ͓S͑k͒Ϫ1͔. ͑15͒ Often more than one resonance may need to be consid- mk ered. For example, the scattering properties for the (1/2, Ϫ ϩ 6 1/2) (1/2,1/2) channel of Li are dominated by a combi- B. Coupled square-well scattering nation of two resonances: a triplet potential resonance and a Feshbach resonance. This can be clearly seen from Fig. 1, In this subsection we describe the coupled-channels ex- where the residual scattering length, which would arise in the tension of a textbook single-channel square-well scattering absence of the Feshbach resonance coupling, would be very problem. One reason that this model is interesting to study is → large and negative and vary with magnetic ﬁeld. This can be because we can take the limit of the potential range R 0, ␦ compared with the value of the nonresonant background thus giving an explicit representation of a set of coupled scattering length for the triplet potential for Li that is only function potentials that simpliﬁes the description in the many-body problem to follow. 31a0, which is an accurate measure of the characteristic range of this potential. An adequate scattering model for this The scattering equations for such a coupled system are system therefore requires inclusion of both bound-state reso- written as 6 nances. Since for Li the coupling between these two bound ប2 ͒ ͑ ͒ states is small, it will be neglected in the double-resonance ␺ P͑ ͒ϭͫϪ ٌ2ϩ P͑ ͒ͬ␺ P͑ ͒ϩ ͑ ͒␺Q͑ E r r V r r g r r , 16 model presented here. The double-resonance S matrix, with m again only one open channel, follows then from Eq. ͑10͒ and includes a summation over two bound states. After solving ប2 ͒ ␺Q͑ ͒ϭͫϪ ٌ2ϩ Q͑ ͒ϩ⑀ͬ␺Q͑ ͒ϩ ͑ ͒␺ P͑ ␾ ͉ ͉␺ P E r r V r r g* r r , for the two components ͗ i HQP ͘ of wave function m ͉␺ P͘, the S matrix can be written as ͑17͒

͉͑¯ ͉2⌬ ϩ͉¯ ͉2⌬ ͒ with ⑀ being the energy shift of the closed channel with Ϫ 2ik g1 2 g2 1 S͑k͒ϭe 2ikabgͫ 1Ϫ ͬ ϭប2 2 ͉͑¯ ͉2⌬ ϩ͉¯ ͉2⌬ ͒Ϫ⌬ ⌬ respect to the collision continuum and E k /m the rela- ik g1 2 g2 1 1 2 tive kinetic energy of the two colliding particles in the ͑13͒ center-of-mass frame. The coupled square-well model encapsulates the general properties of two-body alkali interactions. ⌬ ϭ ¯␯ Ϫប2 2 ␲ប2 ¯␯ ¯ with 1 ( 1 k /m)4 /m, where 1 and g1 are the There we can divide the internuclear separation into two re- detuning and coupling strengths for state 1. Equivalent defi- gions: the inner region where the exchange interaction ͑the nitions are used for state 2. Later we will show that this difference between the singlet and triplet potentials͒ is much simple analytic Feshbach scattering model mimics the larger than the hyperfine splitting, and the outer region where 6 coupled-channels calculation of Li. The parameters of this the hyperfine interaction dominates. Here we make a similar model, which are related to the positions and widths of the distinction for the coupled square wells. In analogy to the last bound states, can be directly found from a plot of the real singlet and triplet potentials, we use for the inner region scattering length versus magnetic field as given, for example, two artificial square-well potentials labeled as V1 and V2.We by Fig. 1. The scattering length behavior should be repro- take the coupling g(r) to be constant over the range of the duced by the analytic expression for the scattering length square-well potentials rϽR, and to be zero outside this range ͑ ͒ following from Eq. 13 : ͑see Fig. 3͒. Then the problem can be simply solved by means of basis rotations at the boundary R giving rise to ͉¯ ͉2 ͉¯ ͉2 Ͼ m g1 g2 simple analytic expressions. For r R, we therefore consider aϭa Ϫ ͩ ϩ ͪ . ͑14͒ bg ␲ប2 ¯␯ ¯␯ one open channel and one closed channel, with wave num- 4 1 2 bers k P and kQ . In analogy with a real physical system, we can refer to the inner range channels (rϽR) as a molecular The advantage of a double pole over a single-pole S-matrix basis, and the channel wave functions are just linear combi- parametrization is that we can account for the interplay be- nations of the u and u wave functions. At the boundary R, tween a potential resonance and a Feshbach resonance, 1 2 these wave functions have accumulated a phase ␾ ϭk R which in principle can radically change the scattering prop- 1 1 and ␾ ϭk R. The coupling strength is effectively given by erties. This interplay is not only important for the description 2 2 the basis-rotation angle ␪ for the scattering wave functions: of 6Li interactions, but also for other atomic systems that have an almost resonant triplet potential, such as bosonic u ͑R͒ cos ␪ Ϫsin ␪ u ͑R͒ 133Cs ͓18,19͔ and 85Rb ͓20͔. ͩ P ͪ ϭͩ ͪͩ 1 ͪ ͑ ͒ ͒ ␪ ␪ ͒ , 18 In the many-body part of this paper, Sec. III, the scatter- uQ͑R sin cos u2͑R ing properties are represented by a T matrix instead of an S matrix. We have shown in the above that in our case there allowing for an analytic solution of the scattering model. exists a simple relation between the two, however, the defi- This leads to the following expression for the S matrix:

053617-4 RESONANCE SUPERFLUIDITY: RENORMALIZATION . . . PHYSICAL REVIEW A 65 053617

FIG. 3. Illustration of the coupled square-well system. Outer region rϾR: the solid line corresponds to the open channel potential P, and the dotted line to the closed channel potential Q. The ϳ wave functions are given by u P(r) sin͗kPr͘ and uQ(r) ϳ Ϫ Ͻ exp( kQr), respectively. Inner region r R: the solid and dotted FIG. 4. Comparison of the real part of the T matrix for coupled ͑ ͒ ϭ lines correspond to the molecular potentials V1 and V2, respec- square-well scattering solid line with a potential range R 1a0,to ϳ ͑ ͒ tively. The wave functions are given by u1(r) sin k1r and u2(r) Feshbach scattering dashed line , for a detuning that yields a scat- ϳ Ϫ sin(k2r). The dashed line corresponds to the kinetic energy E in tering length of about 2750a0. A similar, good agreement is found ϭͱ ប the open channel. The wave vectors are defined as k P mE/ , for all detunings. ϭͱ ⑀Ϫ ប ϭͱ ϩ ប kQ m( E)/ , k1 m(E V1)/ , and k2 ϭͱ ϩ Ϫ⑀ ប ⑀ m(E V2 )/ . The detuning can be chosen such that a strong energy dependence of the T matrix, the two scattering bound state of square-well potential V2 enters the collision con- representations agree very well. tinuum, causing a Feshbach resonance in the open channel. C. Contact potential scattering and renormalization Ϫ ϭ 2ikPR Ϫ Ϫ ␾ 2␪ϩ S e ͓1 ͕ 2ikP͑k2cot 2cos kQ In this section the Lippmann-Schwinger scattering equa- ϩ ␾ 2␪͖͒ ͕ ϩ ␾ ͑ 2␪Ϫ 2␪͒ tion is solved for a resonance system with contact potentials. k1cot 1sin / k PkQ k1cot 1 k Psin kQcos As in the preceding section, we make use of an open sub- ϩ ␾ ␾ ϩ 2␪ϩ 2␪͒ ͑ ͒ space that is coupled to a closed subspace. The contact po- ik2cot 2͑k1cot 1 k Pcos kQsin ͖͔. 19 tentials are defined by P͑ ͒ϭ P␦͑ ͒ ͑ ͒ An extension to treat more than two coupled potentials, V r V r , 20 which would be required to model more than one resonance, Q͑ ͒ϭ Q␦͑ ͒ is also straightforward. V r V r , The parameters of the two wells have to be chosen such ͑ ͒ϭ ␦͑ ͒ ͑ ͒ that the results of a real scattering calculation are reproduced g r g r , 21 for a given physical system. In fact, all the parameters are where ␦(r) is the three-dimensional Dirac ␦ function. Here completely determined from the field dependence of the scat- P P tering length, and all other scattering properties, such as the V (r) is the open channel potential with strength V . The function VQ(r) is a closed-channel potential with strength energy dependence of the scattering phase shift, can then be Q derived. First we choose a range R, typically of the order of V , and g(r) is a coupling between the closed and open channel with strength g. The procedure of renormalization an interatomic potential range (100a0) or less. Now we have ␪ relates the physical units (a , ¯g , and ¯␯ ͒ from Sec. II A to only to determine the set of parameters V1 , V2, and . The bg i i these parameters of the contact potential scattering model for potential depth V1 is chosen such that the scattering length is a given momentum cutoff; a relationship for which we will equal to the background scattering length abg , while keeping ␪ϭ now obtain explicit expressions. The first step is to solve 0. Also, V1 should be large enough that the wave number ␪ again the scattering Eqs. ͑16͒ and ͑17͒ for these contact po- k1 depends weakly on the scattering energy. Then, we set to be nonzero, and change the detuning until a bound state tentials. As we have seen in Sec. II A, we can formally solve ͑ ͒ crosses threshold, giving rise to a Feshbach resonance. The the bound-state equations, and make use of Eq. 6 to expand the Green’s function in bound-state solutions. In this case it value of V2 is more or less arbitrary, but we typically choose ␪ can be written as it to be larger than V1. Finally, we change the value of to give the Feshbach resonance the desired width. We will later show that the resulting scattering properties ␾Q͑ ͒ ͵ 3 Ј␾Q ͑ Ј͒ ͑ Ј͒␺ P͑ Ј͒ i r d r i * r g* r r converge for R→0. In Fig. 4 the coupled square-well system ␺Q͑ ͒ϭ͚ 40 r Ϫ⑀ , is compared with the Feshbach scattering theory, for K i E i scattering parameters. Even despite the fact that there is a ͑22͒

053617-5 S. J. J. M. F. KOKKELMANS et al. PHYSICAL REVIEW A 65 053617

␾Q ⑀ with i (r) a bound-state solution and i its eigenenergy. We V Pm k i␲ now define an amplitude for the system to be in this bound T͑k͒ϭV PϪ T͑k͒ͫKϪarctanh ϩ kͬϩ͚ g ␾ . ␲2ប2 K 2 i i state that will later be useful in the mean-field equations: 2 i ␾ ϭ͗␾Q͉␺Q͘ ͑28͒ i i , and together with the open channel equation ϭ ␾Q and the definition gi(r) g(r) i (r), we get a new set of This is a variant of the Lippmann-Schwinger equation. The scattering equations, closed-channel scattering solutions are now used to eliminate ␾ ͑ ͒ the amplitude functions i . In Fourier space, Eq. 24 has ប2k2 ប2 ␺ P͑ ͒ϭͫϪ ٌ2ϩ P͑ ͒ͬ␺ P͑ ͒ϩ ͑ ͒␾ the form r r V r r ͚ gi r i , m m i ͑23͒ ប2k2 1 ␾ ϭ␯ ␾ ϩ ͵ ␺ P͑ ͒ 3 ͑ ͒ i i i gi* p d p. 29 m ͑2␲͒3 ប2k2 ␾ ϭ␯ ␾ ϩ ͵ 3 Ј ͑ Ј͒␺ P͑ Ј͒ ͑ ͒ i i i d r g* r r . 24 ͑ ͒ ␾ m i After substitution of Eq. 26 the expression for i is linked to the T matrix: The energy difference between the bound-state energy and ␯ m k i␲ the threshold of the collision continuum is given by i . The *ͩ Ϫ ͑ ͒ͫ Ϫ ϩ ͬͪ gi 1 T k K arctanh k open channel solution for Eq. ͑23͒ can be formulated as 2␲2ប2 K 2 ␾ ϭ . i ប2k2 ik͉rÀrЈ͉ Ϫ␯ m e m i ␺ P͑r͒ϭ␹͑r͒Ϫ ͵ d3rЈ ͫ V P͑rЈ͒␺ P͑rЈ͒ 4␲ប2 ͉rÀrЈ͉ ͑30͒ ␾ ͑ ͒ Eliminating i from Eq. 28 gives a complete expression for ϩ͚ g ͑rЈ͒␾ ͬ i i the Lippmann-Schwinger equation i eikr V Pm k i␲ ϭ␹͑r͒ϩ f ͑␪͒ ,asr→ϱ. ͑25͒ T͑k͒ϭV PϪ T͑k͒ͫKϪarctanh ϩ kͬ r 2␲2ប2 K 2

␹ 1 m k i␲ Here (r) is the unscattered wave function, and in the other ͉ ͉2ͩ Ϫ ͑ ͒ͫ Ϫ ϩ ͬͪ gi 1 T k K arctanh k term we recognize the scattered part that is usually formu- 2␲2 ប2 K 2 lated in terms of the scattering amplitude f (␪). The momen- ϩ͚ . ប2k2 tum representation of this last line is ͓7͔ i Ϫ␯ m i 4␲ f ͑k,p͒ ͑31͒ ␺ P͑p͒ϭ͑2␲͒3␦͑kÀp͒Ϫ . ͑26͒ 2 2 k Ϫp ϩi␦ Similar to the Feshbach and coupled square-well problems, the k→0 behavior of T(k) should reproduce the scat- Combining Eq. ͑26͒ with our expression for the scattering tering length, and, the result should not depend on the arbi- amplitude we find trary momentum cutoff K. For an analytic expression of the scattering length, we conveniently use the Feshbach repre- 4␲ប2 sentation. A comparison between the latter and the expres- Ϫ f ͑k,p͒ 4␲ប2 1 m sion for the scattering length a that results from solving Eq. Ϫ ͑ Ј͒ϭ Pϩ P ͵ 3 ͑ ͒ f k,k V V d p 2 2 2 2 31 , tells us how to relate the coupling constants for contact m ͑2␲͒3 ប k ប p Ϫ ϩi␦ scattering to the Feshbach coupling constants. By making m m use of the definitions ⌫ϭ(1Ϫ␣U)Ϫ1, ␣ϭmK/(2␲2ប2), ϭ ␲ប2 and U 4 abg /m, we find the very concise relations ϩ ␾ ͑ ͒ ͚ gi i . 27 i V Pϭ⌫U, ͑32͒

The typical temperature range of a system we are interested which is valid also in the case where no resonance is present, in will only allow for elastic s-wave scattering, therefore the and in addition, scattering amplitude has no angular dependence, and incom- ϭ⌫¯ ͑ ͒ ing and outgoing wave numbers are the same, i.e., kϭkЈ. g1 g1 , 33 The scattering amplitude can then be simply linked to the T ϭϪ ␲ប2 ␯ ϭ¯␯ ϩ␣ ¯ ͑ ͒ matrix via the relation T(k) (4 /m) f (k). The integral 1 1 g1g1 34 has a principal-value part, and the integration ranges from zero to a momentum cutoff K. Equation ͑27͒ then has as for the open-channel potential and the ﬁrst resonance. For solution, the second resonance, if present, we ﬁnd

053617-6 RESONANCE SUPERFLUIDITY: RENORMALIZATION . . . PHYSICAL REVIEW A 65 053617

FIG. 5. Comparison of the real part of the T matrix for coupled square-well scattering for three different values of the potential ϭ ͑ ͒ ϭ ͑ ͒ range: R 100a0 dash-dotted line , R 30a0 dashed line , and R ϭ ͑ ͒ 40 1a0 solid line . The interaction parameters for K have been used here, and the magnetic ﬁeld is chosen such that a scattering ϭ length of a 300a0 is obtained. Also plotted is the T matrix for contact scattering, which clearly agrees very well as it coincides with the solid line of the double-well scattering.

¯ g2 g ϭ , ͑35͒ 2 ␣¯ 2 ¯␯ ϩ⌫Ϫ1 g1/ 1

␯ ϭ¯␯ ϩ␣ ¯ ͑ ͒ 2 2 g2g2 . 36

Obviously, our approach can be systematically extended further, order by order, to give an arbitrarily accurate representation of the microscopic scattering physics. These expressions we refer to as the renormalizing equa- FIG. 6. ͑a͒ Real part of the T matrix as a function of collision tions of the resonance theory since they remove the ultravio- energy, for the Feshbach model and the cutoff model ͑overlapping let divergence that would otherwise appear in the field equa- solid lines͒, and for a coupled-channels calculation ͑dashed line͒. tions. Any many-body theory based on contact scattering The atomic species considered is 6Li, for atoms colliding in the around a Feshbach resonance will need to apply these ex- (1/2,Ϫ1/2)ϩ(1/2,1/2) channel. ͑b͒ Same as ͑a͒ for the imaginary pressions in order to renormalize the theory. These equations part. ͑32͒–͑36͒ therefore represent one of the major results of this 6 paper. imaginary parts of the T matrix applied to the case of Li, In Fig. 5 the T matrix as a function of energy is shown for and compare the cutoff and Feshbach scattering representa- contact scattering, in comparison with the square-well scat- tions to a full coupled-channels calculation. The agreement is tering for different values of the potential range. The contact- surprisingly good, and holds basically for all magnetic fields scattering model is demonstrated to be the limiting case of ͑i.e., similar agreement is found at all detunings͒. the coupled square-well system when R→0. In this section we have discovered a remarkable fact that even a complex system including internal structure and resonances can be simply described with contact potentials and a D. Discussion of different models few coupling parameters. This was known for off resonance In Sec. II C it has been shown that the resonance contact scattering where only a single parameter ͑the scattering scattering representation is the limiting case of the coupled length͒ is required to encapsulate the collision physics at a square-well system, when the range of the potentials is taken very low temperature. However, to our knowledge this has to zero. Also, in Sec. II B it has been shown that the double- not been pointed out before for the resonance system, where well system is in good agreement with the Feshbach scatter- an analogous parameter set is required to describe a system ing theory. Now we will show how well these scattering where the scattering length may even pass through infinity. representations agree with the full numerical coupled- We have shown in a very concise set of formulas on how to channels calculation ͓10͔. In Fig. 6 we show the real and derive the resonance parameters associated with contact po-

053617-7 S. J. J. M. F. KOKKELMANS et al. PHYSICAL REVIEW A 65 053617

tentials. This result is important for the incorporation of the kinetic and potential energies in the presence of external two-body scattering in a many-body system, as we will show traps, which is measured relative to the energy ␮ of a coro- later in this paper. tating reference system. Thus, we define Other papers have also proposed a simple scattering ͓ ͔ ប2 model to reproduce coupled-channels calculations 21,22 .In 2 H␴͑x͒ϭϪ ٌ ϩV␴͑x͒Ϫ␮, ͑39͒ these papers real potentials are used, and they give a fair 2m agreement. Here, however, we use models that need input from a coupled-channels calculation to give information ប2 about the positions of the bound states and the coupling to Hm͑x͒ϭϪ ٌ2ϩVm͑x͒ϩ␯ Ϫ␮ . ͑40͒ i 2M i i m the closed channels. All this information can be extracted from a plot of the scattering length as a function of magnetic Here, m denotes the atomic mass as used previously, M field. ϭ ␮ ϭ ␮ 2m is the molecular mass, m 2 is the energy offset of the molecules with respect to the reference system, V␴(x) III. MANY-BODY RESONANCE SCATTERING are external spin-dependent atomic trapping potentials, and m We will now proceed to a many-body description of reso- Vi (x) are the external molecular trapping potentials. The nance superfluidity and connect it to our theory of the two- molecular single-particle energy has an additional energy ␯ body scattering problem described earlier. This section ex- term i that accounts for the detuning of the molecular state plains in detail the similar approach in our papers devoted to i relative to the threshold of the collision continuum. P Ϫ resonance superfluidity in potassium ͓23,24͔. The general The binary interaction potential V (x1 x2) accounts for methods of nonequilibrium dynamics has been described in the nonresonant interaction of spin-up and spin-down fermi- Ϫ Ref. ͓25͔ and we have applied them in the context of con- ons, and coupling potentials gi(x1 x2) convert free fermi- densed bosonic fields ͓26,27͔. onic particles into bound bosonic molecular excitations. In the language of second quantization, we describe the Thus, we find for the total system Hamiltonian of the atomic and molecular fields, many-body system with fermionic fields ␺ˆ ␴(x) that remove a single fermionic particle from position x in internal elec- ˆ ϭ ˆ ϩ ˆ ͑ ͒ ␴ ␾ˆ H H0 H1 , 41 tronic state , and molecular bosonic fields i(x) that anni- hilate a composite-bound two-particle excitation from space where the free Hˆ and interaction contributions Hˆ are de- point x in internal configuration i. These field operators and 0 1 fined as their adjoints satisfy the usual fermionic anticommutation rules ˆ ϭ ͵ ␺ˆ † ͑ ͒ ͑ ͒␺ˆ ͑ ͒ H0 d1͚ ␴ 1 H␴ 1 ␴ 1 † ␴ ͕␺ˆ ␴ ͑x ͒,␺ˆ ␴ ͑x ͖͒ϭ␦͑x Ϫx ͒ ␦␴ ␴ ϵ␦ , 1 1 2 2 1 2 1 2 12 ϩ ͵ ␾ˆ †͑ ͒ m͑ ͒␾ˆ ͑ ͒ ͑ ͒ ͕␺ˆ ␴ ͑x ͒,␺ˆ ␴ ͑x ͖͒ϭ0, ͑37͒ d1͚ i 1 Hi 1 i 1 , 42 1 1 2 2 i and bosonic commutation rules ˆ ϭ ͵ ␺ˆ †͑ ͒␺ˆ †͑ ͒ P͑ Ϫ ͒␺ˆ ͑ ͒␺ˆ ͑ ͒ H d1d2ͭ ↑ 1 ↓ 2 V 1 2 ↓ 2 ↑ 1 1 ͓␾ˆ ͑x ͒,␾ˆ † ͑x ͔͒ϭ␦͑x Ϫx ͒ ␦ ϵ␦ , i1 1 i2 2 1 2 i1i2 12 1ϩ2 ͓␾ˆ ͑ ͒ ␾ˆ ͑ ͔͒ϭ ͑ ͒ ϩ ͫ␾ˆ †ͩ ͪ ͑ Ϫ ͒␺ˆ ͑ ͒␺ˆ ͑ ͒ϩ ͬ ͚ g* 1 2 ↓ 2 ↑ 1 H.c. ͮ . i x1 , i x2 0, 38 i i 1 2 i 2 respectively. Here and in the following discussion, we will ͑43͒ also try to simplify the notational complexity by adopting the notation convention of many-particle physics. This means, Here, H.c. denotes the Hermitian conjugate. In the present we will identify the complete set of quantum numbers picture, we deliberately neglect the interactions among the ␴ ϵ molecules. Several other papers have treated a Feshbach uniquely by its subscript index, i.e., ͕x1 , 1͖ 1. If only the ϵ resonance in a related way ͓28–31͔. position coordinate is involved, we will use boldface x2 2. In the double-resonance case of lithium, we have to distin- In order to derive dynamical Hartree-Fock-Bogoliubov guish only two internal atomic configurations for the free ͑HFB͒ equations from this Hamiltonian, we also need to de- fermionic single-particle states ␴ϭ͕↑,↓͖ and we need at fine a generalized density matrix to describe the state of the most two indices iϭ͕1,2͖ to differentiate between the fermionic system ͓32͔ and an expectation value for the bosonic molecular resonances. bosonic molecular field. The elements of the 4ϫ4 density The dynamics of the multicomponent gas is governed by matrix G are given by a total system Hamiltonian Hˆ ϭHˆ ϩHˆ , which consists of 0 1 G ͑ ͒ϭ͗ ˆ †͑ ͒ ˆ ͑ ͒͘ ͑ ͒ ˆ ˆ pq 12 Aq x2 A p x1 , 44 the free-evolution Hamiltonian H0 and the interactions H1 between atoms and molecules. We assume that the free dy- ˆ ͑ ͒ϭ͓␺ˆ ͑ ͒ ␺ˆ ͑ ͒ ␺ˆ †͑ ͒ ␺ˆ †͑ ͔͒T ͑ ͒ namics of the atoms and molecules is determined by their A x ↑ x , ↓ x , ↑ x , ↓ x , 45

053617-8 RESONANCE SUPERFLUIDITY: RENORMALIZATION . . . PHYSICAL REVIEW A 65 053617 and symmetry-broken molecular fields are defined as d 1ϩ2 iប ␾ ͑3͒ϭHm͑3͒ ␾ ͑3͒ϩ ͵ d1d2 ␦ͩ Ϫ3ͪ dt i i i 2 ␾ ͒ϭ ␾ˆ ͒ ͑ ͒ i͑1 ͗ i͑x1 ͘. 46 ϫ ͑ Ϫ ͒G ͑ ͒ ͑ ͒ gi* 1 2 p 12 . 52 As usual, we define the quantum averages of an arbitrary First, the free-evolution ⌺0 is obviously related to the single- operator Oˆ with respect to a many-body density matrix ␳ by particle Hamiltonians of Eq. ͑42͒. In complete analogy to the ͗Oˆ ͘ϭ ͓Oˆ ␳͔ Tr , and we calculate higher-order correlation generalized density matrix, it has a simple 4ϫ4 structure functions by a Gaussian factorization approximation known ͓ ͔ ϫ 0 as Wick’s theorem 32 . The structure of the 4 4 density ⌺N͑12͒ 0 ⌺0͑ ͒ϭͩ ͪ ͑ ͒ matrix, 12 0 , 53 0 Ϫ⌺N͑12͒* G ͑12͒ G ͑12͒ G͑ ͒ϭͩ N A ͪ ͑ ͒ which can be factorized into 2ϫ2 submatrices as 12 ϪG ͑ ͒ ϫ␦ ϪG ͑ ͒ , 47 A 12 * 1 12 N 12 * H↑͑1͒ 0 ⌺0 ͑ ͒ϭ␦ ͩ ͪ ͑ ͒ N 12 12 . 54 is very simple, if one recognizes that it is formed out of a 0 H↓͑1͒ ϫ G 2 2 single-particle density matrix N , a pair correlation G ␦ matrix A and obviously the vacuum fluctuations 12 . The Second, one obtains from the interaction Hamiltonian of Eq. single-particle submatrix is given by ͑43͒ the first-order self-energy ⌺1 as

G ͒ G ͒ ⌺1 ͑ ͒ ⌺1 ͑ ͒ n↑͑12 m͑12 N 12 A 12 G ͑ ͒ϭͩ ͪ ͑ ͒ ⌺1͑12͒ϭͩ ͪ . ͑55͒ N 12 G ͒ G ͒ , 48 Ϫ⌺1 ͑ ͒ Ϫ⌺1 ͑ ͒ m͑21 * n↓͑12 A 12 * N 12 *

1 † The normal potential matrix ⌺N has the usual structure of where G ␴(12)ϭ͗␺ˆ (x )␺ˆ ␴(x )͘ is the density of spin-up n ␴ 2 1 ͓ G ϭ͗␺ˆ † ␺ˆ ͘ direct contributions i.e., local Hartree potentials propor- and spin-down particles and m(12) ↓(x2) ↑(x1) de- ␦ ͔ ͓ tional to 12 and exchange terms i.e., nonlocal Fock poten- notes a cross-level coherence, or ‘‘magnetization’’ between P Ϫ ͔ G tials proportional to V (1 2) : the states. The pair-correlation submatrix A is defined analogously as ␦ G ↓͑44͒ Ϫ␦ G ͑12͒ ⌺1 ͑ ͒ϭ ͵ P͑ Ϫ ͒ͩ 12 n 14 m ͪ N 12 d4V 2 4 Ϫ␦ G ͒ ␦ G ͒ . 14 m͑21 * 12 n↑͑44 G ↑͑12͒ G ͑12͒ G ͑ ͒ϭͩ a p ͪ ͑ ͒ ͑56͒ A 12 ϪG ͒ G ͒ , 49 p͑21 a↓͑12 The zeros that appear in the diagonal of the anomalous cou- G ϭ ␺ˆ ␺ˆ pling matrix where a␴(12) ͗ ␴(x2) ␴(x1)͘ is an anomalous pairing field within the same level and the usual cross-level pairing 0 ⌬͑12͒ field of BCS theory is defined here as G (12) ⌺1 ͑ ͒ϭͩ ͪ ͑ ͒ p A 12 Ϫ⌬͑ ͒ 57 ϭ ␺ˆ ␺ˆ 21 0 ͗ ↓(x2) ↑(x1)͘. reflect the fact that there is no low-energy (s-wave͒ interac- A. General dynamic Hartree-Fock-Bogoliubov equations tion between same spin particles due to the Pauli exclusion of motion principle. The off-diagonal element defines a gap function as From these physical assumptions about the system’s 1ϩ2 Hamiltonian Eq. ͑41͒ and the postulated mean fields ␾ of ⌬͑ ͒ϭ P͑ Ϫ ͒ G ͑ ͒ϩ ͑ Ϫ ͒ ␾ ͩ ͪ i 12 V 1 2 p 12 ͚ gi 1 2 i . Eq. ͑46͒ and G of Eq. ͑47͒, one can now derive kinetic equa- i 2 tions for the expectation values ͗O͘ for an operator O by a ͑58͒ systematic application of Heisenberg’s equation B. The homogeneous limit and the contact potential d iប Oˆ ϭ͓Oˆ ,Hˆ ͔, ͑50͒ approximation dt In this section, we will apply the general HFB equations of motion ͓Eq. ͑51͔͒ to the case of a spatially homogeneous and Wick’s theorem. isotropic system. Furthermore, we will approximate the The first-order kinetic equation for the Hermitian density P Ϫ finite-range interaction potentials V (x1 x2) and gi(x1 matrix G has the general form of a commutator and the time- Ϫ ͑ ͒ x2) by the contact approximation as introduced in Eq. 20 , evolution is determined by a Hermitian self-energy matrix and assume equal populations for spin-up and spin-down ⌺ϭ⌺0ϩ⌺1 . In general, one finds atoms. Spatial homogeneity implies that a physical system is d iប G͑13͒ϭ ͵ d2͓⌺͑12͒ G͑23͒ϪG͑12͒⌺͑23͔͒, ͑51͒ translationally invariant. Thus, any single-particle field must dt be constant in space and any two-particle quantity or pair-

053617-9 S. J. J. M. F. KOKKELMANS et al. PHYSICAL REVIEW A 65 053617 correlation function can depend on the coordinate difference d iប G ͑r͒ϭ0, ͑65͒ only: dt n ␾ ͒ϭ␾ ͒ϵ␾ ͑ ͒ i͑x i͑0 i , 59 d ប2 ប G ͑ ͒ϭͫϪ ٌ2Ϫ ␮ϩ P͑ ͒ͬG ͑ ͒ϩ ͑ ͒ ␾ G͑x ,x ͒ϭG͑x Ϫx ͒ϭG͑r͒. ͑60͒ i p r r 2 V r p r ͚ gi r i , 1 2 1 2 dt m i ͑ ͒ This assumption implies also that there can be no external 66 ϭ m ϭ trapping potentials present, i.e., V␴(x) Vi (x) 0, as this would break the translational symmetry. d iប ␾ ϭ͑␯ Ϫ␮ ͒ ␾ ϩg* G ͑0͒. ͑67͒ Furthermore, we want to consider a special situation dt i i m i i p where there is no population difference in spin-up and spin- G ϭ͉ ͉ ϭG down particles n(r r ) n␴(r), there exists no cross- The scattering solution of Eqs. ͑66͒ and ͑67͒ is ‘‘summa- G ϭ level coherence or ‘‘magnetization’’ m(r) 0, and the rized’’ by the energy-dependent two-body T matrix, which G ϭ anomalous pairing field a(r) 0. It is important to note that we have discussed in the preceeding sections. In order to this special scenario is consistent with the full evolution incorporate the full energy dependence of the scattering equation and, on the other hand, leads to a greatly simplified physics, we propose to upgrade the direct energy shift PG Re G Re sparse density matrix, V n(r)to͗T (k)͘ n(r), where ͗T (k)͘ represents the ͗ ͘ G ͑ ͒ G ͑ ͒ real part of the two-body T matrix, and ••• denotes two- n r 00 p r particle thermal averaging over a Fermi distribution. A de- G ͒ ϪG ͒ 0 n͑r p͑r 0 tailed calculation of the proper upgrade procedure will be G͑12͒ϭ , ͩ 0 ϪG*͑r͒ ␦͑r͒ϪG ͑r͒ 0 ͪ presented in a forthcoming publication. p n The translationally invariant HFB Equations ͑63͒ and ͑64͒ G ͑ ͒ ␦͑ ͒ϪG ͑ ͒ *p r 00r n r are best analyzed in momentum-space. Thus, we will intro- ˆ ͑61͒ duce the Fourier transformed field-operators ak␴ by where rϭ͉r͉ϭ͉1Ϫ2͉. Similarly, one finds a translationally Ϫ invariant self-energy ⌺(12)ϭ⌺(1Ϫ2) with e ikx ␺ˆ ␴͑x͒ϭ͚ aˆ ␴ , ͑68͒ ͱ⍀ k ⌺͑1͒ 00 ⌬ k 0 ⌺͑1͒ Ϫ⌬ 0 ⍀ ⌺͑12͒ϭ␦ , ͑62͒ where is the quantization volume. If we define the Fourier 12ͩ Ϫ⌬ Ϫ⌺͑ ͒ ͪ 0 * 1 0 components of the translationally invariant mean fields as ⌬* 00Ϫ⌺͑1͒ Ϫ Ϫ G͑ ͒ϭG͑ Ϫ ͒ϭ ik(x1 x2) G͑ ͒ ͑ ͒ ⌺ ϭϪប2 ٌ2Ϫ␮ϩ PG r x1 x2 ͚ e k , 69 and (x) /(2m) x V n(0) and a complex en- k ⌬ϭ PG ϩ͚ ␾ ergy gap V p(0) igi i . These assumptions lead to a significant simplification of the HFB equations. we obtain the following relations between the real-space den- The structure of the HFB equations can be elucidated fur- sity of particles n ͑the same for both spins͒ and the real-space ther by separating out the bare two-particle interactions from density of particle pairs p: the many-body contributions. One can achieve this by splitting the self-energy into the kinetic energy and mean-field 1 shifts ⌺ϭ⌺0ϩ⌺1, and by separating the density matrix into ϭG ͑ ϭ ͒ϭ G ͑ ͒ϭ ͗ ˆ † ˆ ͘ ͑ ͒ n n␴ r 0 ͚ n␴ k ⍀ ͚ ak␴ak␴ , 70 the vacuum contribution G 0 ͓proportional to ␦(r)͔ and the k k remaining mean fields GϭG 0ϩG 1: 1 d pϭG ͑rϭ0͒ϭ͚ G ͑k͒ϭ ͚ ͗aˆ Ϫ ↓aˆ ↑͘. ͑71͒ iប G 1Ϫ͓⌺0,G 1͔Ϫ͓⌺1,G 0͔ϭ͓⌺1,G 1͔, ͑63͒ p p ⍀ k k dt k k

d The Fourier-transformed HFB equations are now local in iប ␾ ϭ͑␯ Ϫ␮ ͒ ␾ ϩg* G ͑0͒. ͑64͒ momentum space, dt i i m i i p

In this fashion, we can now identify the physics of resonance d iប G(k)ϭ͓⌺͑k͒,G͑k͔͒, ͑72͒ scattering of two particles in vacuo ͓left-hand side of Eq. dt ͑63͔͒ from the many-body corrections due to the presence of a medium ͓right-hand side of Eq. ͑63͔͒. d In the limit of very low densities, we can ignore many- ប ␾ ϭ͑␯ Ϫ␮ ͒ ␾ ϩ ͑ ͒ i i i m i gi* p, 73 body effects and rediscover Eqs. ͑23͒ and ͑24͒ of Sec. II C, dt but given here in a time-dependent form. They describe the scattering problem that we have solved already: and the self-energy is given by

053617-10 RESONANCE SUPERFLUIDITY: RENORMALIZATION . . . PHYSICAL REVIEW A 65 053617

⌺ ⌬ k 00 0 ⌺ Ϫ⌬ 0 ⌺͑ ͒ϭ k ͑ ͒ k ͩ Ϫ⌬ Ϫ⌺ ͪ . 74 0 * k 0 ⌬ Ϫ⌺ * 00 k

Here, the upgraded single-particle excitation energy is now ⌺ ϭ⑀ Ϫ␮ϩ͗ Re͘ ⑀ ϭប2 2 k k Tk n, k k /2m denotes the kinetic energy and the gap energy is still deﬁned as ⌬ϭV Pp ϩ͚ ␾ igi i .

IV. THERMODYNAMICS In this paper we focus on the properties of thermodynamic equilibrium. Thermodynamic equilibrium can be ⌽ ϭ reached by demanding that the grand potential G Ϫ ⌶ kbTln at a fixed temperature has a minimal value. In this ⌶ FIG. 7. Two overlapping quasiparticle energy spectra as a func- definition kb is Boltzmann’s constant, and the partition tion of single-particle kinetic energy Eϭប2k2/(2m). The total den- function ⌶ϭTr͓exp(ϪHˆ /k T)͔. The exponent containing Ϫ diag b sity of the gas is nϭ1014 cm 3, the magnetic field is Bϭ900 G, the diagonalized Hamiltonian reads ϭ and the temperature is T 0.01TF . The energy spectrum is calcu- ϭ lated for two different values of the cutoff; i.e., K 32kF and K ϭ64k . The fact that the two lines are overlapping to the extent ˆ ϭ ␯ Ϫ␮ ͉͒␾ ͉2 F Hdiag ͚ ͑ i m i i that the difference is difficult to see shows the renormalization in practice, demonstrating the validity of Eqs. ͑32͒–͑36͒. ϩ ͓⌺ ϩ ͑␣ˆ † ␣ˆ ϩ␣ˆ † ␣ˆ Ϫ ͔͒ ͑ ͒ ͚ k Ek k↑ k↑ k↓ k↓ 1 , 75 ¯ k gip ␾ ϭϪ , ͑77͒ i ¯␯ Ϫ␮ i m which is a quadratic approximation to the original Hamil- ͑ ͒ tonian. The energy spectrum Ek results from a local diago- which is also the stationary solution of Eq. 67 . This equal- nalization by the Bogoliubov transformation of the self- ity is very useful because we can effectively eliminate the ⌺ energy matrix k at each k, where the obtained quasiparticle molecular field from the equations. The quasiparticle states ϭͱ⌺2ϩ⌬2 are now populated according to the Fermi-Dirac distribution spectrum is Ek k . Note that the first summation n ϭ͓exp(E /k T)ϩ1͔Ϫ1. The mean fields are then deter- term in Hˆ results from a contribution from Q space, and k k b diag mined by integrating the equilibrium single-particle density the second summation term from P space of Sec. II A. The matrix elements given by rotation to Bogoliubov quasiparticles is given by the general canonical transformation 1 K ϭ ͵ ͓͑ Ϫ ͒ ␪ ϩ ͔ ͑ ͒ n dk 2nk 1 cos 2 k 1 , 78 ͑ ␲͒2 0 i␥ 2 ␣ˆ ↑ cos ␪ Ϫe sin␪ aˆ ↑ ͩ k ͪ ϭͩ ͪͩ k ͪ ͑ ͒ i␥ , 76 ␣ˆ † e sin ␪ cos ␪ aˆ † 1 K Ϫk↓ Ϫk↓ ϭ ͵ ͑ Ϫ ͒ ␪ ͑ ͒ p dk 2nk 1 sin 2 k . 79 ͑2␲͒2 0 ␪ ϭ͉⌬͉ ⌺ where tan 2 k / k is the Bogoliubov transformation ␪ angle. The quasiparticle annihilation and creation operators Since k depends on n and p, these equations require self- ␣ˆ ␣ˆ † consistent solutions that are found from a numerical iterative are indicated by k and k . In Fig. 7 we show a typical quasiparticle energy spectrum for 6Li versus the single- method. ϭ In Fig. 8 we show a plot of the chemical potential as a particle kinetic energy, at a magnetic field of B 900 G and 6 a temperature of Tϭ0.01T . The figure demonstrates how function of temperature, for the case of Li in a homoge- F ϭ well the renormalizing equations ͑32͒–͑36͒ work in obtain- neous gas, at a magnetic field of B 900 G. Figure 9 shows ing a cutoff-independent energy spectrum. This is important the ratio of the critical temperature Tc to the Fermi tempera- because it implies that all the thermodynamics that follow ture TF as a function of detuning. It clearly shows that there will also be cutoff independent. is a limiting value of Tc of about 0.5TF , similar to the value 40 ͓ ͔ For the stationary solution the grand potential, or equiva- that has been predicted for K in Ref. 23 . The BCS result lently, the free energy, has indeed a minimum. This follows for the critical temperature, given by the formula easily from setting the partial derivative of the grand poten- T ␲ ͒ ͑ ␾ ϭ0. This gives the c ϳ Ϫץ/ ⌽ץ :tial with respect to ␾ to zero i G i expͫ ͉ ͉ ͬ, 80 solution TF 2 a kF

053617-11 S. J. J. M. F. KOKKELMANS et al. PHYSICAL REVIEW A 65 053617

6 ͑ ͒ FIG. 8. Chemical potential as a function of temperature, for a FIG. 10. Density profile for a gas of Li atoms solid line , magnetic field of Bϭ900 G for 6Li. evenly distributed among the two lowest hyperfine states. The tem- ϭ ϭ perature is T 0.2TF at a magnetic field of B 900 G. The trap ␻ϭ ␲ϫ Ϫ1 ϭ has been plotted in the same graph for comparison. The BCS constant is 2 500 s , and we have a total number N 5 ϫ 5 line gives a curve for T higher than the resonance theory, 10 atoms. We compare this with a profile resulting from the c same ␮ ͑but for different total number N), where artificially no since it does not contain the energy dependence of the T superfluid is present by setting the pairing field p equal to zero matrix. The absolute value of the scattering length in this ͑dashed line͒. magnetic field range is always larger than 2000a0, which ͉ ͉Ͼ implies that kF a 1—a clear indication that the BCS theory breaks down in this regime. nonsuperfluid system, this gives the well-known Thomas- So far, our calculation has been done for a homogeneous Fermi solution. For a resonance system, however, a density gas. We will also present results for a trapped lithium gas in bulge appears in the center of the trap, which is caused by a a harmonic oscillator potential V(r) with a total number of change in compressibility when a superfluid is present. This Nϭ5ϫ105 atoms, similar to what we presented for 40Kin is shown in Fig. 10, for a spherical trap with a trap constant ␻ϭ ␲ϫ Ϫ1 Ref. ͓24͔. We treat the inhomogeneity by making use of the of 2 500 s . This bulge is a signature of superflu- semiclassical local-density approximation, which involves idity and could experimentally be seen by fitting the density mainly the replacement of the chemical potential by a spa- distribution in the outer wings to a nonresonant system, and tially dependent version ␮(r)ϭ␮ϪV(r). The thermody- thus obtaining an excess density in the middle of the trap. For a discussion of the abrupt change in the compressibility, namic equations for the homogeneous system are then solved ͓ ͔ at each point in space ͓24͔. As a result, we obtain a spatially see Ref. 24 . dependent density distribution. At zero temperature, for a V. FLUCTUATIONS IN THE MEAN FIELDS AND CROSSOVER MODEL In this section we make some comments on the connection between the resonance superfluidity theory we have presented and related mean-field approaches to discuss the crossover of superconductivity from weak to strong coupling. In the mean-field theory of BEC, most often reflected in the literature by the Gross-Pitaevskii equation or finite- temperature derivatives, a small parameter is derived to jus- tify the application of the theory. This parameter, ͱna3 ͓6͔, may be obtained from a study of higher-order corrections to the quasiparticle energy spectrum. It has been suggested that for a fermi system that exhibits superfluidity the small parameter is given by a power of kFa, and that the BCS theory breaks down when this parameter approaches unity. How- ever, the small parameter in the theory of resonance superfluidity cannot be simply a function of the scattering length FIG. 9. Dependence of critical temperature on magnetic field for for detunings close to resonance. This can already be seen 6Li, for a total density of nϭ1014 cmϪ3 ͑solid line͒. The dashed from the energy dependence of the T matrix, which shows line is, for comparison, the prediction of the regular BCS theory. that around the Fermi energy, the T matrix may have an

053617-12 RESONANCE SUPERFLUIDITY: RENORMALIZATION . . . PHYSICAL REVIEW A 65 053617

absolute value much smaller than at zero energy where the adds significantly to the total density equation in both the scattering length is defined. Moreover, even right on reso- crossover models of superconductivity and in the theory we nance when ␯ϭ0 and the scattering length passes through have presented here. Moreover, the inclusion of the molecu- infinity, the T matrix remains well behaved. lar term allows for a smooth interpolation between the BCS Instead of calculating the small parameter of this system, and BEC limits. This is clearly a substantial topic in its own we choose a different approach based on crossover models right, and will be addressed further in a future publication between BCS and BEC, formulated by Nozie`res and ͓35͔. Schmitt-Rink ͓33͔, and later expanded upon by Randeria ͓ ͔ 34 . In the regular BCS theory for weakly coupled systems VI. CONCLUSIONS the value of the critical temperature is given by the exponen- tial dependence in Eq. ͑80͒, but for strongly coupled systems We have shown that it is possible to derive a mean-field this model results in a logarithmically divergent prediction theory of resonance superfluidity, which can be applied to Ϫ1 6 40 for Tc . The parameter (kFa) is usually taken to describe ultracold Fermi gases such as Li and K. The Hamiltonian Ϫ1 the crossover from the weak-coupling Bose limit ͓(kFa) we use treats the resonant states explicitly, and automatically →Ϫϱ Ϫ1→ϩϱ ͔ to the strong-coupling ͓(kFa) ͔ BEC limit. builds the coupled scattering equations into the many-body The unphysical divergence in Tc occurs because the process theory. With a study of analytical scattering we have shown that dominates the transition in the weak-coupling regime is that these scattering equations can completely reproduce a the dissociation of pairs of fermions. For a strongly coupled full coupled-channels calculation for the relevant energy re- system, however, the fermions are so tightly bound that the gime. The energy dependence of the s-wave phase shifts can wave functions of pairs of atoms begin to overlap, and the be described by a small set of parameters that correspond to onset of coherence is signaled by excitations of the con- physical properties, such as the nonresonant background densed state, which occurs at a temperature well below the value of the scattering length, and the widths and detunings dissociation temperature of the Cooper pairs. Thus, when of the Feshbach resonances. Close to resonance, we predict a moving from weak to strong coupling, the nature of the tran- large relative value of 0.5TF for the critical temperature. The sition changes from a BCS- to a BEC-type mechanism. An particular resonance under study for 6Li occurs in the explicit inclusion of the process of molecule formation, char- (1/2,1/2)ϩ(1/2,Ϫ1/2) collision channel, and has its peak at ϭ ⌬ Ϸ ͓ ͔ acterized by the detuning, resonance width, and resonance B0 844 G, and a width of about B 185 G 10,11 . This position, will allow us to move from one regime to the other. large width translates into a large magnetic field range where The lowest-order correction that connects between BCS- the critical temperature is within a factor of 2 from its peak and BEC-type superconductivities can be made by augment- value. This range is, for comparison, much larger than for ing the density equation to account for the formation of pairs 40K. For 6Li there are also two other Feshbach resonances, of atoms. This is done by using the thermodynamic number one in the (1/2,1/2)ϩ(3/2,Ϫ3/2) state, as also noted by ␮ ⌽ ͓ ͔ Ϫ ϩץ ⌽ץϭϪ equation N G / , with G the total thermodynamic O’Hara et al. 36 , and another in the (1/2, 1/2) (3/2, grand potential Ϫ3/2) state. They result from coupling to the same singlet ϭ bound state, and occur at field values of about B0 823 G ⌽ ϭ⌽0 Ϫk T⌺ ln ⌫͑q,iq ͒. ͑81͒ ϭ G G b q,iql l and B0 705 G, and have a similar width to the (1/2,1/2) ϩ(1/2,Ϫ1/2) resonance. The disadvantage of these reso- ⌽0 The term G is a grand potential that does not include the nances, however, is that the atoms in these channels suffer quasibound molecules and results from regular BCS theory. from dipolar losses, which are also resonantly enhanced. Retaining only this term yields a theory that can only ac- Three-body interactions will be largely suppressed, as count for the free and scattered fermionic atoms that contrib- asymptotic p-wave collisions will give very little contribu- ute to the fermion density, therefore the theory breaks down tion in the temperature regime considered ͑an s-wave colli- if a sizeable number of bound states are formed. In the ex- sion is always forbidden for at least one of the pairs͒. From a ⌽0 treme limit of strong coupling, G becomes negligible and study of crossover models between BCS and BEC we find no Eq. ͑81͒ just reduces to the thermodynamic potential of an indication of breakdown effects of the applied mean-field ideal Bose gas. In this regime, the theory predicts the forma- theory. tion of a condensate of molecules below the BEC transition temperature. ⌫ ACKNOWLEDGMENTS The function (q,iql), which is a function of momentum q and thermal frequencies iql , is mostly negligible for a We thank J. Cooper, E. Cornell, D. Jin, C. Wieman, B. J. weakly coupled system and has little effect on the value of Verhaar, and B. DeMarco for very stimulating discussions. Tc in this regime. It allows for the inclusion of the lowest Support is acknowledged for S.J.J.M.F.K. and J.N.M. from contributing order of quantum fluctuations ͓33,34͔ by means the U.S. Department of Energy, Office of Basic Energy Sci- of a general inclusion of mechanisms for molecular pair for- ences via the Chemical Sciences, Geosciences and Bio- mation. In the resonance superfluidity model, a similar term sciences Division, and for M.L.C. from SNS, Pisa ͑Italy͒. is present due to the formation of bosonic molecular bound Support is acknowledged for M.J.H. and M.L.C. from the ␾ states i , and prevents the critical temperature from diverg- National Science Foundation, and for R.W. from the Austrian ing. When the coupling increases, the formation of molecules Academy of Sciences.

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