Analytic description of atomic interaction at ultracold temperatures II: Scattering around a magnetic Feshbach resonance
aa r X i v : . [ phy s i c s . a t o m - ph ] M a y Analytic description of atomic interaction at ultracold temperatures II: Scatteringaround a magnetic Feshbach resonance
Bo Gao
1, 2, ∗ Institute for Theoretical Atomic, Molecular and Optical Physics (ITAMP),Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA Department of Physics and Astronomy, Mailstop 111, University of Toledo, Toledo, Ohio 43606, USA (Dated: August 15, 2018)Starting from a multichannel quantum-defect theory, we derive analytic descriptions of a magneticFeshbach resonance in an arbitrary partial wave l , and the atomic interactions around it. An analyticformula, applicable to both broad and narrow resonances of arbitrary l , is presented for ultracoldatomic scattering around a Feshbach resonance. Other related issues addressed include (a) theparametrization of a magnetic Feshbach resonance of arbitrary l , (b) rigorous definitions of “broad”and “narrow” resonances of arbitrary l and their different scattering characteristics, and (c) thetuning of the effective range and the generalized effective range by a magnetic field. PACS numbers: 34.10.+x,34.50.Cx,33.15.-e,03.75.Nt
I. INTRODUCTION
Analytic descriptions of two-body interactions arehighly desirable if any systematic understanding of quan-tum few-body and quantum many-body systems is to beexpected or achieved. The best-known example may bethe Gross-Pitaevskii theory of identical bosons [1], withits simplicity and generality depending intimately on ourability to parametrize the low-energy two-body interac-tion using the effective range theory (ERT) [2–4]. Thesame is true for quantum few-body theories in the uni-versal regime (see, e.g., Refs. [5, 6]).In a companion paper [7], referred to hereafter as paperI, we have discussed the limitations of the standard ERT[2–4] in describing atomic interactions at low tempera-tures, and how such limitations are overcome using ex-pansions derived from the quantum-defect theory (QDT)for − /r type of long-range potentials [8–10]. The focuswas on the case of a single channel, both out of the ne-cessity of theoretical development, but also to provide aset of single-channel universal behaviors that will serve asbenchmarks for understanding other types of behaviors.This article extends this discussion to atomic interac-tion around a magnetic Feshbach resonance [11–13] inan arbitrary partial wave l . It is a nontrivial extensionwith considerable new physics as a Feshbach resonance isnecessarily a multichannel phenomena [11–13], for whichfew analytic results have been derived in any general con-text. The theory includes the parametrization of the res-onance, the rigorous definitions of “broad” and “narrow”resonances [12–15], and an analytic description of theatomic scattering properties around them. Such under-standings, which have been mostly limited to the s wave[12, 13], are not only of interest by themselves, they arealso prerequisites for understanding atomic interactionin an optical lattice [16], and behaviors of quantum few- ∗ atom and many-atom systems around a Feshbach res-onance. For nonzero partial waves, the theory here isa necessity as ERT fails [7, 8, 17, 18]. Even for the s wave, it offers much improved analytic description, es-pecially for narrow resonances around which the energydependence of the scattering amplitude can become sosignificant that it has to be incorporated into the cor-responding few-body [19] and many-body theories (See,e.g., Refs. [20–23]).There are three main steps in developing an analyticdescription of a magnetic Feshbach resonance. The firstis the reduction of the underlying multichannel problem,as formulated in a multichannel quantum-defect theory(MQDT) of Ref. [24] to an effective single channel prob-lem. The second is an efficient parametrization of a mag-netic Feshbach resonance. The third is to apply the the-ory of paper I [7] to obtain the desired results such asthe scattering properties around the threshold, to be ad-dressed in this article.The paper is organized as follows. The reduction to aneffective single-channel problem is carried out in Sec. II.The parametrization of a magnetic Feshbach resonance isaddressed in Sec. III. In particular, we derive in Sec. III Bthe magnetic-field dependence of scattering lengths andthe generalized scattering lengths introduced in paper I[7]. We show in this section that regardless of l , thescattering length, or the generalized scattering length for l ≥
2, can be parametrized around a magnetic Feshbachresonance in a similar fashion as the s wave scatteringlength [12, 13, 25]. The parametrization is further devel-oped in Sec. III C in terms of scaled parameters. It leadsnot only to more concise analytic formulas, but moreimportantly, to rigorous definitions of “broad” and “nar-row” Feshbach resonances of arbitrary l . In Sec. IV, wepresent the QDT expansion [7] that provides an analyticdescription of ultracold scattering around a magnetic Fes-hbach resonance of arbitrary l . As sample applicationsof the QDT expansion, Sec. V presents and discussesthe generalized effective range expansion [7] for ultra-cold scattering around a magnetic Feshbach resonance.It includes a relationship between the (generalized) ef-fective range and the (generalized) scattering length thatis applicable to both broad and narrow resonances, andresonances of arbitrary l . It substantially extends a pre-vious relationship [7, 8, 20, 26] that is applicable only tobroad resonances. Two special cases of interest in cold-atom physics, the case of infinite scattering length (theunitarity limit) and the case of zero scattering length,are also discussed in this section as examples of the QDTexpansion. The conclusions are given in Sec. VI. II. REDUCTION OF A MULTICHANNELPROBLEM TO AN EFFECTIVESINGLE-CHANNEL PROBLEM
In cold-atom physics, most of the interest in atomicinteraction lies in a small range of energies around thelowest threshold (of a certain symmetry), below whichwe have true bound states. Ignoring weak couplings be-tween different partial waves due to the magnetic dipole-dipole [25, 27] and the second-order spin-orbit interaction[28–30], we can label the single channel of partial wave l that is associated with this lowest threshold “ a ”, and allthe other channels of partial wave l by “ c ”. Above thelowest threshold and below the energies at which the sec-ond or more channels becomes open, it is already clearfrom Ref. [24] that the MQDT for atom-atom interac-tion reduces to an effective single-channel problem withan effective short-range K-matrix, K c , given by K c eff = K caa + K cac ( χ ccc − K ccc ) − K cca . (1)Here K caa , K cac , K cca , K ccc , are submatrices of K c corre-sponding to the separation of all channels into a single“ a ” channel and N c closed “ c ” channels. χ ccc is an N c × N c diagonal matrix with elements χ cl ( ǫ si ), which is the uni-versal χ cl function, as given, e.g., by Eq. (54) in paper I,evaluated at properly scaled channel energies. This re-duction to an effective single-channel problem is a resultof the standard channel-closing procedure, and occursin similar fashion in any type of multichannel scatteringtheories. What is important, and maybe less well-known,is that the energies of the multichannel bound states be-low the threshold “ a ” can also be reduced to an effectivesingle-channel problem with the very same effective K c as given by Eq. (1). A proof is given in the Appendix A.With this reduction, the scattering below the secondthreshold and the multichannel bound states below thethreshold “ a ” are all described by an effective single-channel QDT [8–10] with an effective short-range K c ma-trix given by K c eff . Specifically K l = tan δ l = ( Z cgc K c eff − Z cfc )( Z cfs − Z cgs K c eff ) − , (2)gives the scattering K matrix between the lowest and thesecond thresholds, and the solutions of (see Appendix A) χ cl ( ǫ s ) = K c eff , (3) give the bound spectrum below the lowest threshold.Here Z cxy ( ǫ s , l ) are universal QDT functions for − /r potential, as given, e.g., by Eqs. (4)-(7) of paper I. They,and χ cl ( ǫ s ), are all evaluated at a scaled energy rela-tive to the lowest threshold of angular momentum l , ǫ s = ǫ/s E = ( E − E a ) /s E , with s E = (¯ h / µ )(1 /β ) being the energy scale, and β = (2 µC / ¯ h ) / being thelength scale associated with the − C /r van der Waalsinteraction in channel “ a ”. We note that other thanthe ignorance of weak interactions that couple different l states, there is no further approximation associated withthis reduction to a single channel.This effective single-channel problem differs from atrue single-channel problem in that the energy depen-dence of K c eff is generally not negligible, unlike the K c parameter for a single channel [9]. As will become clearthroughout this work, it is this energy dependence, whichoriginates from the energy dependence of χ ccc in Eq. (1),that leads to deviations from single-channel universal be-haviors of paper I [7], and makes the behaviors of a“narrow” Feshbach resonance to differ substantially fromthose of a “broad” Feshbach resonance. As another dif-ference from a true single-channel problem, the l depen-dence of χ ccc also makes K c eff l -dependent. The same for-malism applies to atomic interaction in an external mag-netic field [31], which has the additional effect of making K c eff to depend parametrically on B . We will use the no-tation of K c eff ( ǫ, l, B ), when necessary, to fully specify itsdependences.The equivalence, around the lowest threshold, of themultichannel atomic interaction in a B field to an effec-tive single channel problem with an effective short-range K c eff ( ǫ, l, B ) makes most results of paper I [7] immedi-ately applicable, except for a few that made explicit useof the energy- and/or l -insensitivity of K c . In particular,if we define a K c l parameter, which is more convenientfor descriptions of near-threshold properties [7, 10], as K c l ( ǫ, B ) = K c eff ( ǫ, l, B ) − tan( πν / πν / K c eff ( ǫ, l, B ) , (4)where ν = (2 l + 1) / − /r type of potential, the lo-cations of the zero-energy magnetic Feshbach resonancesin an arbitrary partial wave l , B l , namely the magneticfields corresponding to having a bound or quasiboundstate right at the threshold, can be conveniently foundas the roots of K c l ( ǫ = 0 , B ) [7, 32], namely as the solu-tions of K c l ( ǫ = 0 , B l ) = 0 . (5)The scattering length or the generalized scattering lengthfor an abitrary l is given by the zero-energy value of K c l through Eq. (48) of paper I [7], namely, e a l ( B ) = ¯ a l (cid:18) ( − l + 1 K c l ( ǫ = 0 , B ) (cid:19) , (6)where ¯ a l = ¯ a sl β l +16 is the mean scattering length (withscale included) for partial wave l that was defined in pa-per I [7], with¯ a sl = π l +1 [Γ( l/ / l + 3 / , (7)being the scaled mean scattering length. Recall that thegeneralized scattering length, e a l , reduces to the regularscattering lengths whenever they are well defined, namelyfor the s and p partial waves.Computationally, a similar theory based on MQDT[24] has been shown by Hanna et al. [31] to give an accu-rate description of magnetic Feshbach resonances over awide range of magnetic fields using only three parametersfor alkali-metal systems. Even better results can be ex-pected by incorporating the energy and/or partial-wavedependences of K cS and K cT [24] using a few more pa-rameters. Further calculations for specific systems andespecially nonzero partial waves will be presented else-where. Here we focus on the parametrization of one par-ticular resonance and the analytic description of atomicinteraction around it. III. PARAMETRIZATION OF A MAGNETICFESHBACH RESONANCEA. Derivation and general considerations
As the second step towards developing an analytic de-scription of a magnetic Feshbach resonance, we need asimple parametrization of K c eff or the corresponding K c l .For any isolated resonance, the second term in Eq. (1) hasa simple pole at ¯ ǫ l ( B ), determined by det( χ c − K ccc ) = 0.It represents the “bare” location of a Feshbach resonanceand depends on the magnetic field. (Here “bare” meansno coupling to the open channel “a”.) Around such asimple pole, the effective K c parameter, Eq. (1), can al-ways be parametrized as K c eff = K c bg l − Γ cl / ǫ − ¯ ǫ l ( B ) , (8)sufficiently close to the pole. Here Γ cl is a measure ofthe width of the resonance, ǫ and ¯ ǫ l are energies thatare conveniently chosen to be relative to channel “ a ”,e.g., ǫ = E − E a , and K c bg l is a background K c parame-ter, namely the K c for energies and magnetic fields awayfrom the resonance, such that | ǫ − ¯ ǫ l ( B ) | ≫ Γ cl . Using thefact that χ cl is a piecewise monotonically decreasing func-tions of energy [9], namely, dχ cl /dǫ s <
0, one can furthershow rigorously that Γ cl >
0, a property that will put im-portant constraints on other forms of parametrizations,all of which will be derived from Eq. (8).In writing Eq. (8), we have adopted a notation thatavoids unnecessary confusions without getting into thedetails of the MQDT [24] for atomic interaction in a mag-netic field [31]. Rigorously speaking, the K c matrix itself,and therefore the parameters K c bg l and Γ cl in Eq. (8), also depend on B . This dependence, however, is onlysignificant over a field range of the order of ∆ E hf /µ B ,where ∆ E hf is the atomic hyperfine splitting, and µ B is the Bohr magneton. Since our focus here is on theparametrization of an individual resonance, the width ofwhich is always much smaller than the hyperfine splitting[13], we adopt the notation of Eq. (8) to emphasize thatover the range of B field of interest here, the most rele-vant B field dependence is that of the “bare” Feshbachenergy, ¯ ǫ l ( B ).As discussed in paper I [7], analytic properties aroundthe threshold are more conveniently described using theshort-range parameter K c l . Substituting Eq. (8) intoEq. (4), we have K c l ( ǫ, B ) = K c l − Γ c l / ǫ − ¯ ǫ l ( B ) − f El , (9)where K c l is the background K c l parameter correspond-ing to K c bg l K c l = K c bg l − tan( πν / πν / K c bg l , (10)with a corresponding generalized background scatteringlength of [7] e a bg l = ¯ a l ( − l + 1 K c l ! , (11)and Γ c l = Γ cl ( πν / πν / K c bg l ] . (12)The f El in Eq. (9) is not an independent parameters. Itis related to K c l and Γ c l by f El = 12 Γ c l tan( πν / − tan( πν / K c l . (13)In describing a Feshbach resonance in terms of K c l , thefact that Γ cl > c l > s wave [12, 13, 25].For K c l = 0 ( e a bg l = ∞ ), namely in all cases whenthere is no background bound or quasi-bound state rightat the threshold [32], it is more convenient to rewriteEq. (9) as K c l ( ǫ, B ) = K c l ǫ − ǫ l ( B ) ǫ − ǫ l ( B ) − d El , (14)= K c l (cid:18) d El ǫ − ǫ l ( B ) − d El (cid:19) , (15)where ǫ l ( B ) = ¯ ǫ l ( B ) + Γ cl / K c bg l − tan( πν / , (16)and d El = (Γ cl /
2) 1 + tan ( πν / πν / − K c bg l ][1 + tan( πν / K c bg l ] . (17)In this form for K c l , the location of the zero-energy mag-netic Feshbach resonance, B l , determined by Eq. (5),translates into the solution of ǫ l ( B l ) = 0. And since weare interested here only in a range of B that covers a sin-gle Feshbach resonance, the ǫ l ( B ) in Eq. (15) can be ap-proximated, around B l , by ǫ l ( B ) ≈ δµ l ( B − B l ), where δµ l = dǫ l ( B ) /dB | B = B l is the difference of magnetic mo-ments between the molecular state and the separate-atomstate [13]. This approximation, together with Eq. (15),gives the following parameterization of the effective K c l around a magnetic Feshbach resonance, K c l ( ǫ, B ) = K c l (cid:18) d El ǫ − δµ l ( B − B l ) − d El (cid:19) . (18)It is a parametrization in terms of four parameters B l , K c l , δµ l , and d El , with the condition of K c l d El < c l >
0. These parameters, together with eitherthe C coefficient or the corresponding energy scale s E for a total of five parameters, provide a complete charac-terization of atomic interaction around a magnetic Fesh-bach resonance, through Eqs. (2) and (3). It is applica-ble for all partial waves l and for either broad or narrowFeshbach resonances (the precise definition of which willbe addressed in in Sec. III C), or anything in between.It fails only in the special case of having a background bound or quasibound state right at the threshold, whichcan happen only by pure coincidence. This special case,together with an alternative parametrization of magneticFeshbach resonances that is applicable for all cases, is dis-cussed in Appendix B. B. Tuning of the scattering lengths and generalizedscattering lengths
Contained in the parametrization of K c l is themagnetic-field dependence of the scattering length or thegeneralized scattering length for an abitrary l . Defining K c l ( B ) ≡ K c l ( ǫ = 0 , B ) to simplify the notation, wehave from Eq. (18) K c l ( B ) = K c l (cid:18) − d Bl B − B l + d Bl (cid:19) , (19)where d Bl = d El /δµ l . Upon substitution into Eq. (6), weobtain, for e a bg l = 0, e a l ( B ) = ¯ a l (cid:18) ( − l + 1 K c l ( B ) (cid:19) (20)= e a bg l (cid:18) − ∆ Bl B − B l (cid:19) . (21) Here e a bg l is the (generalized) background scatteringlength defined earlier by Eq. (11), and∆ Bl = − d Bl / [1 + ( − l K c bgl ] , (22)= − (cid:18) − ( − l e a bg l / ¯ a l (cid:19) d Bl . (23)For e a bg l = 0, we obtain e a l ( B ) = − ( − l ¯ a l d Bl B − B l . (24)Equation (21) shows that around a magnetic Feshbachresonance of arbitrary l , the (generalized) scatteringlength is tuned in a similar fashion by the magnetic fieldas around an s wave resonance, and can be parametrizedin a similar manner [12, 13, 25].The parametrization of the s wave scattering length inthe form of Eq. (21) has been popular for a good rea-son: every parameter in it has the simplest and the mostdirect experimental interpretation. It is worth pointingout, however, that theoretically it is not the most gen-eral parametrization possible as it fails for both e a bg l = ∞ and e a bg l = 0. As discussed in Appendix B, the failure ofEq. (21) at e a bg l = ∞ , and the corresponding failure ofEqs. (18) and (19) at K c l = 0, is a necessary sacrificefor using B l , which has a more direct physical inter-pretation than the ¯ B l parameter of Appendix B, butdoes not exist for e a bg l = ∞ . Its failure at e a bg l = 0 is theprice we pay for using the parameter ∆ Bl . An alternativeparametrization of the scattering length, which remainsapplicable for e a bg l = 0, is e a l ( B ) = e a bg l + e a bg l − ( − l ¯ a l ( B − B l ) /d Bl . (25)It can be obtained, e.g., by substituting Eq. (23) for ∆ Bl into Eq. (21). This parametrization is well defined, andreduces to Eq. (24) for e a bg l = 0. C. Parametrization in terms of scaled parametersand the definitions of “broad” and “narrow”resonances
Of the five parameters required to completely char-acterize the atomic interaction around a Feshbach reso-nance, such as B l , K c l , d El , δµ l , and s E (or C ), twoof them, K c l and d El can be replaced by e a bg l and ∆ Bl ,used in the parametrization of the (generalized) scatter-ing length. The resulting parametrization, in terms of B l , e a bg l , ∆ Bl , δµ l and s E (or C ), gives an alterna-tive that is the most direct generalization of the s waveparametrization [13] to other partial waves. Both sets,however, have the limitation that they are not fully trans-parent to the distinction between broad and narrow res-onances.The effective single channel K c l parameter for a mag-netic Feshbach resonance, as characterized, e.g., byEq. (18), is generally energy-dependent. Depending onthe relative importance of this energy variation, as com-pared to those due to the long-range van der Waals inter-action, a Feshbach resonance can be classified either as“broad” or “narrow”. For a broad Feshbach resonance,the energy dependence of K c l is insignificant comparedto those induced by the van der Waals interaction. Theatomic interaction around such a resonance follows, to alarge extent, the single-channel universal behavior of pa-per I [7] with a tunable (generalized) scattering length.A narrow Feshbach resonance corresponds to the oppo-site limit in which the energy dependence of K c l domi-nates. The atomic interaction around such a resonancecan differ completely from the single-channel universalbehavior.To better characterize the relative importance of theenergy dependence of K c l and therefore the defini-tion of broad and narrow resonances, we need to firstput it on the same energy scale as the other energy-dependent functions, namely on the energy scale s E =(¯ h / µ )(1 /β ) that is associated with the van der Waalsinteraction [7]. Defining g res = d El /s E , (26)and B s = ( B − B l ) /d Bl , (27)Eq. (18) can be written as K c l ( ǫ s , B s ) = K c l (cid:18) g res ǫ s − g res ( B s + 1) (cid:19) . (28)It describes K c l as a function of the scaled energy ǫ s and a scaled magnetic field B s using two dimensionlessparameters, K c l and g res , the meaning of which are il-lustrated in Fig. 1. K c l is the background K c l , namelyits value away from the resonance. g res is a measure ofthe width of the resonance. More specifically, K c l ( ǫ s , B )goes to infinity at ǫ s = g res ( B s + 1). It crosses zero ǫ s = g res B s . The distance between the two locationsis | g res | , which measures the width of resonance. Theparametrization of K c l using Eq. (28) divide the param-eters characterizing a magnetic Feshbach resonance intothree parameters, B l , d Bl , and s E , for location and scal-ing, and two dimensionless parameters, K c l and g res , forthe shape. Feshbach resonances with the same shape pa-rameters differ from each other only in scaling. The con-dition of Γ c l > K c l and g res are constrained by K c l g res < g res parameter, which measures thewidth of the resonance on the scale of s E , gives a rough,yet still imprecise, classification of “broad” ( | g res | ≫ | g res | ≪ s E is different for different partial waves. -6 -4 -2 0 2 4 6 ε s -2-1012 K c l g res (B s +1)K c0 l (B)g res B s K c0bg l FIG. 1. (Color online) Illustrations of the parameters de-scribing the energy dependence of K c l ( ǫ s , B s ) on the van derWaals energy scale. Γ c l > K c l g res <
0) implies that K c l is piecewise monotonically increasing function of energy. Thesignificance of its energy dependence is determined by com-paring it with that induced by the van der Waals interaction,the order-of-magnitude of which can be measured by the en-ergy dependence of the θ l ≈ − ǫ s / (2 l +3)(2 l −
1) function. Thedashed line illustrates the θ l function for l = 1. The energyvariation of θ l is less significant for higher partial waves. This is especially true for large l , for which the energyvariation around the threshold due to the van der Waalsinteraction is much less significant than that for the s wave.For a more precise definition of “broad” and “narrow”,we first recognize that the leading energy variation dueto the van der Waals interaction is characterized by the θ l ≈ − ǫ s / (2 l + 3)(2 l −
1) function defined by the Eq. (35)of paper I [7] (repeated as Eq. (43) in Sec. IV). Thisenergy variation, as measured by | ∂θ l /∂ǫ s ( ǫ s = 0) | = | / (2 l + 3)(2 l − | , is what should be compared with theenergy variation of the K c l at zero energy, as measuredby ∂K c l /∂ǫ s ( ǫ s = 0 , B s = 0) = − K c l /g res . This leadsto the definition of an auxiliary parameter ζ res ≡ g res (2 l + 3)(2 l − K c l , (29)which gives a precise characterization of “broad” and“narrow”. For a broad resonance with | ζ res | ≫
1, theenergy variation of the effective short-range parameter isinsignificant compared to that due to the van der Waalsinteraction, just like the case of a single channel [7]. Theatomic interaction around such a resonance can be ex-pected to follow the single channel universal behavior.For a narrow resonance with | ζ res | ≪
1, the energy vari-ation of the effective short-range parameter dominates.Within such a resonance, the atomic interaction deviatessubstantially from the single channel behavior. The con-straint K c l g res < ζ res is always positivefor l = 0, and always negative for l ≥
1. Specializing tothe s wave, the ζ res parameter is similar in spirit to theparameter s res of Chin et al. [13], which is equivalent tothe 1 /η parameter of K¨ohler et al. [12].To finish our discussion on parametrization, we sum-marize here the explicit relations between two sets of pa-rameters that we will use for the complete characteriza-tion of a Feshbach resonance. The first set is B l , e a bg l ,∆ Bl , δµ l and s E (or C ), which is more closely corre-lated with the parametrization of the (generalized) scat-tering length and the standard s wave parametrization[13]. The second set is B l , K c l , g res , d Bl , and s E (or C ), which is much more convenient with the QDT ex-pansion of Sec. IV, and correlates much more closely withthe distinction of broad and narrow resonances. Theydiffer in three parameters that are related by K c l = 1 e a bg l / ¯ a l − ( − l , (30) g res = − e a bg l / ¯ a l e a bg l / ¯ a l − ( − l (cid:18) δµ l ∆ Bl s E (cid:19) , (31) d Bl = − e a bg l / ¯ a l e a bg l / ¯ a l − ( − l ∆ Bl . (32)The condition of Γ cl > δµ l e a bg l ∆ Bl > K c l g res < s wave magnetic Fes-hbach resonances. The first set is taken from Table IVof Chin et al. [13]. The second set is calculated from thefirst using Eqs. (30)-(32). They are given here both forconvenient applications of the QDT expansion, and to il-lustrate the vast range of ζ res , from very narrow | ζ res | ≪ | ζ res | ≫
1. Since the parametrization is newfor nonzero partial waves, no parameters are yet availablefor them. Tentative theoretical predictions of resonancesand their parameters for nonzero partial waves, usingMQDT as briefly outlined in Sec. II, will be presentedelsewhere. It is hoped that they will stimulate furtherexperiment and theory for their more precise characteri-zation. Previous works on nonzero partial waves, such asthose in Ref. [13, 31, 40, 48–55], can also be re-analyzedto extract the parameters.The second set of parameters describes K c l throughEq. (28). A useful variation, which relates K c l explicitly to its value at zero energy, is given by K c l ( ǫ s , B s ) = K c l ( B s ) − K c l η ( B s ) ǫ s − η ( B s ) ǫ s , (33)where K c l ( B s ) = K c l ( ǫ s = 0 , B s ) = K c l B s B s + 1 , (34)is the value of K c l at zero energy, given earlier byEq. (19), expressed in terms of the scaled magnetic field,and we have defined η ( B s ) = 1 g res ( B s + 1) . (35)This representation of K c l makes it clear that K c l ( ǫ s , B s ) ∼ K c l ( B s ) in the broad-resonance limit of | g res | → ∞ . It also makes it easier, if ever desirable, torepresent K c l ( ǫ s , B s ) in term of (generalized) scatteringlength and (generalized) background scattering length,using Eqs. (30)-(32), and K c l ( B s ) = 1 e a l ( B ) / ¯ a l − ( − l , (36)which is a direct consequence of Eq. (20). IV. QDT EXPANSION FOR ULTRACOLDSCATTERING AROUND A MAGNETICFESHBACH RESONANCE
In deriving the QDT expansion for single-channel ul-tracold scattering of paper I [7], the only quantities ex-panded are the universal QDT functions, with no as-sumptions made about the behavior of the short-rangeparameter K c l , including its energy dependence. Thusthe same expansion is applicable to the effective single-channel problem that describes the magnetic Feshbachresonance. Specifically, we have for ǫ s > δ l ≈ K ( B ) l ( ǫ s ) + K ( D ) l ( ǫ s , B s ) , (37)where K ( B ) l ≈ − π ( ν − ν ) ≈ π (2 l + 5)(2 l + 3)(2 l + 1)(2 l − l − ǫ s , (38)is the Born term (see, e.g., Ref. [56]), and K ( D ) l ( ǫ s , B s ) = − e A sl ( ǫ s , B s ) k l +1 s , (39)describes the deviation from the Born term. Here e A sl ( ǫ s , B s ) = ¯ a sl (cid:20) ( − l + 1 + K c l ( ǫ s , B s ) θ l K c l ( ǫ s , B s ) − θ l − π ( ν − ν ) / (cid:21) , (40) TABLE I. Sample parameters for selective s wave magnetic Feshbach resonances, illustrating a vast range of ζ res values, fromvery narrow ( | ζ res | ≪
1) to very broad ( | ζ res | ≫ Li- Li and
Cs-
Cs systems have the bestresonances for the purpose of investigating universal behaviors. Here a is the Bohr radius and µ B is the Bohr magneton. Thedata sets of B l , ∆ Bl , a bg l , and δµ l are taken from Table IV of Chin et al. [13]. The channel identification also follows thesame reference. Note that there are many resonances that are not broad. The atomic interaction around them do not followsingle-channel universal behavior and is much better described using the QDT expansion presented here.system s E /k B ( µ K) ch. B l (G) ∆ Bl (G) a bg l /a δµ l /µ B K c l g res d Bl (G) ζ res references Li Li 7368 ab ac bc ab Li Li 5849 aa Na Na 933.1 cc aa
907 1 63 3.8 2.143 -0.8597 -3.143 0.1337 [13, 37, 38] aa
853 0.0025 63 3.8 2.143 -0.00215 -0.00786 0.00033 [13, 37, 38] K K 257.3 ab ac Rb Rb 75.58 ee Rb Rb 72.99 aa aa aa aa ae Cs Cs 31.97 aa -11.7 28.7 1720 2.30 0.05945 -146.9 -30.41 823.9 [13, 46, 47] aa
547 7.5 2500 1.79 0.04015 -29.34 -7.801 243.5 [13] aa
800 87.5 1940 1.75 0.05235 -338.6 -92.08 2156 [13] = ¯ a sl (cid:20) ( − l + (2 l + 3)(2 l − − K c l ( ǫ s , B s ) ǫ s (2 l + 3)(2 l − K c l ( ǫ s , B s ) + ǫ s + w l ǫ s (cid:21) , (41)with the w l in Eq. (41) being given by w l = 3 π l + 5)(2 l + 1)(2 l − , (42)and the θ l in Eq. (40) being given by θ l ≈ − l + 3)(2 l − ǫ s . (43)Equations (37)-(41), with a K c l that depends explic-itly on energy and parametrically on the magnetic field B , as described by either Eq. (28), Eq. (33), or Eq. (B1)of Appendix B, give the QDT expansion for scatteringaround a magnetic Feshbach resonance in an arbitrarypartial wave l . It is applicable to both broad and narrowresonances, or anything in between, and has the same en-ergy range of applicability as its single-channel counter-part, limited only by ǫ s being much less than the criticalscale energy ǫ scl , as discussed in more detail in paper I[7]. There is no restriction on the magnetic field exceptthat imposed by the validity of the isolated resonance.While the QDT expansion for scattering around amagnetic Feshbach resonance may be formally similar to QDT expansion for true single channel cases [7], itcontains considerable new physics beyond those of a sin-gle channel, including dramatically different behaviorsfor broad and narrow resonances. For broad resonanceswith | g res | ≫
1, or more precisely | ζ res | ≫
1, the en-ergy dependence of K c l ( ǫ s , B s ) is negligible. The QDTexpansion approaches the single-channel universal be-havior [7] defined by replacing K c l ( ǫ s , B s ) in Eqs. (40)and (41) with its zero-energy value of K c l ( B s ). In suchcases, multichannel scattering behaves the same as sin-gle channel with a tunable (generalized) scattering lengthnot only at the threshold, but over a range of energiesaround the threshold with an energy dependence deter-mined primary by the van der Waals interaction. Thisis illustrated in Figs. 2 and 3 using a broad resonanceof Li- Li. Narrow resonances behaves very differentlywith a much more complex energy dependence that isdetermined both by the properties of the resonance andby the van der Waals interaction. They change scatter-ing in a narrow range of energies around the resonance.Away from it, atomic interaction evolves towards a singlechannel universal behavior determined not by K c l ( B s ), ε s P a r ti a l C r o ss S ec ti on ( πβ ) s wave FIG. 2. (Color online) Comparison of near-threshold s wavescattering properties for narrow and broad Feshbach reso-nances. The solid line represents results for the Li ab chan-nel narrow resonance located at 543 G. The dash-dot linerepresents results for the Li ab channel broad resonance lo-cated at 834 G. In both cases, magnetic fields are chosento give the same s wave scattering length corresponding to a l =0 ( B ) / ¯ a l =0 = +10. The figure also shows that the Fesh-bach resonance at 834 G is sufficient broad that the scatter-ing properties around it is well approximated by the single-channel universal behavior (dashed line). but by K c l or equivalently the background (generalized)scattering length e a bg l , with K ( D ) l ∼ − ¯ a sl k l +1 s " ( − l + (2 l + 3)(2 l − − K c l ǫ s (2 l + 3)(2 l − K c l + ǫ s + w l ǫ s . (44)Figures 2 and 3 contain illustrations of narrow-resonancebehavior using an example from Li- Li scattering. Fur-ther conceptual understanding of the differences betweenbroad and narrow resonances can be found in the nextsection, in connections with the generalized effectiverange expansion and examples for infinite and zero (gen-eralized) scattering lengths.As an illustration of the breadth of the physics con-tained in the QDT expansion for a magnetic Feshbachresonance, Figure 4 shows its description of an avoidedcrossing between a narrow p wave Feshbach/shape reso-nance and a background p wave shape resonance in thethreshold region. It is an example of the coupling of abound state to a highly “structured” continuum, and isused here to emphasize that the only restriction on theapplicability of the QDT expansion is ǫ s ≪ ǫ scl [7].For nonzero partial waves, a Feshbach resonance abovethe threshold manifests itself as a Feshbach/shape res-onance. The qualitative characteristics of such a res-onance, such as its position and width, are containedwithin the QDT expansion, in a manner similar to thecase of a single channel [7]. We defer their discussions toa following paper, since they need to be combined withthe binding energy of a Feshbach molecule to give a com- ε s s i n ( δ l ) s wave FIG. 3. (Color online) The same as Fig. 2 except it is for a l =0 ( B ) / ¯ a l =0 = −
10. sin δ l is plotted, instead of the par-tial cross section to give better visibility to both sets of dataon the same figure. The single-channel universal behavior isindistinguishable from the broad-resonance results (dash-dotline) and is not plotted. Note that even though both set ofdata correspond to the same scattering length, the case ofnarrow Feshbach (solid line) has a resonance feature in thethreshold region that is absent for a broad Feshbach. (Seealso Ref. [13].)FIG. 4. (Color online) A plot of sin δ l vs ǫ s and B s for the p wave, with parameters K c l = − .
03 and g res = 0 .
02. Itillustrates the avoided crossing between a background shaperesonance located in the threshold region and a narrow Fesh-bach/shape resonance. plete picture of the evolution of a resonance across thethreshold.
V. SAMPLE APPLICATIONSA. The generalized effective range expansionaround a magnetic Feshbach resonance
One of the ways to understand some of the physicscontained in the QDT expansion for atomic interactionaround a magnetic Feshbach resonance is through thegeneralized effective range expansion contained withinit. As in paper I [7], the QDT expansion, given byEqs. (37)-(41), can be approximated by an generalized effective range expansion k l +1 cot( δ l − δ ( B ) l ) = − e a l + 12 e r el k + O ( k ln k ) , (45)where δ ( B ) l = − π ( ν − ν ) is approximated by Eq. (38).It reduces to the standard effective range expansion [2–4]for l = 0, and serves to define the generalized scatteringlength and the generalized effective range for other l [7].For scattering around a magnetic Feshbach resonance,both the (generalized) scattering length and the (gener-alized) effective range become magnetic-field dependent.The (generalized) scattering length, e a l , is tuned by themagnetic field according to Eq. (21). From the QDTexpansion, it is straightforward to show that the (gener-alized) effective range is given by e r el ( B ) = − a l β (2 l + 3)(2 l − e a l ( B )] " (cid:18) ( − l − e a l ( B )¯ a l (cid:19) − (cid:18) ¯ h µ e a bg l δµ l ∆ Bl (cid:19) (cid:18) ∆ Bl B − B l − ∆ Bl (cid:19) , (46)= − a l β (2 l + 3)(2 l − e a l ( B )] " (cid:18) ( − l − e a l ( B )¯ a l (cid:19) + (cid:18) ζ res (cid:19) β (2 l + 3)(2 l − a l (cid:18) ∆ Bl B − B l − ∆ Bl (cid:19) , (47)= − a l β (2 l + 3)(2 l − e a l ( B )] " (cid:18) ( − l − e a l ( B )¯ a l (cid:19) + (cid:18) ζ res (cid:19) a l β (2 l + 3)(2 l − e a l ( B )] (cid:20) e a l ( B ) − e a bg l ¯ a l (cid:21) . (48)It consists of two terms. The first corresponds to thesingle-channel universal behavior of paper I [7] and is dueto the long-range van der Waals interaction. The secondterm is due to the energy dependence of the effective K c l that comes from the coupling to a bound state in closedchannels. It is given here in three different forms withdistinctive insights.In the broad-resonance limit of | ζ res | → ∞ , the en-ergy dependence of the effective K c l is negligible, andthe the result reduces to the single-channel universalbehavior [7], in which the (generalized) effective rangeis uniquely determined by the (generalized) scatteringlength, e a l ( B ), independent of the other details of theresonance. Within a narrow resonance, the (generalized)effective is changed substantially from the universal be-havior, which is one way to understand its substantiallydifferent near-threshold behavior as illustrated in Figs. 2and 3. This change, as characterized by the second termsin Eqs. (46)-(48), depends sensitively both on the loca-tion within the resonance and on specific characteristics of the resonance, which are described, for instance, by the ζ res and e a bg l parameters in Eq. (48). Away from the res-onance, namely for | B − B l | ≫ | ∆ Bl | , the second termgoes away and the (generalized) effective range evolvesback towards a single-channel universal result [7] deter-mined by the (generalized) background scattering length, e a bg l .It is useful to note that the contribution to the (gen-eralized) effective range due to the coupling to a boundstate in closed channels, the second term, is always neg-ative. It comes from the constraint of Γ c > δµ l e a bg l ∆ Bl > ζ res > l = 0 and ζ res < e r el ( B ) is always negative for l > s wave,the two terms are always of opposite signs and the endresult can be either positive or negative.0Specializing to the case of s wave, the second term inEq. (46) is the expression for the effective range that isadopted by Zinner and Thogersen [23] in their investiga-tion of Bose-Einstein condensate (BEC) around a narrowFeshbach resonance. It comes from the work of Bruun etal. [57], which ignores the effect of van der Waals in-teraction. Having only the second term in their theoriesmeans that that their results are applicable only for nar-row resonances ( | ζ res | ≪ e a l ( B ) = 0,and has generally a much more limited range of applica-bility compared to the QDT expansion, from which it is derived. B. Examples of infinite and zero (generalized)scattering lengths
Both for the purpose of illustrating explicit energy de-pendences contained in the QDT expansion for magneticFeshbach resonances, and to facilitate future applica-tions, we give here explicit QDT expansion for two specialcases of interest in cold-atom physics. One is the case ofinfinite (generalized) scattering length, the so-called uni-tary limit. The other is the case of zero (generalized)scattering length.For infinite (generalized) scattering length, which oc-curs at B = B l , or equivalently, at B s = 0, the QDTexpansion for K ( D ) l , Eqs. (39) and (41), becomes K ( D ) l = ¯ a sl k l − s (2 l + 3)(2 l − g res − ǫ s h (2 l + 3)(2 l − − K c l ǫ s i (2 l + 3)(2 l − K c l + ǫ s + w l ǫ s − g res (1 + w l ǫ s ) − ( − l ¯ a sl k l +1 s . (49)In the broad resonance limit of | g res | → ∞ , it reducesto Eq. (51) of paper I [7]. For a narrow resonancewith | ζ res | ≪
1, scattering around the threshold followsthat of an infinite (generalized) scattering length, with K ( D ) l ∼ k l − s [7], only in a very small energy range of0 < ǫ s ≪ | g res | around the threshold. Outside this re-gion, namely for energies ǫ s ≫ | g res | , it evolves into thebackground scattering described by Eq. (44), as discussed in the previous section in a more general context.Another special case of interest, where the effectiverange expansion, including the generalized version ofSec. V A, fails completely, is that of zero (generalized)scattering length. It occurs at the magnetic field B = B l + ∆ Bl , or, equivalently at B s = − [1 + ( − l K c l ] − ,where Eqs. (39) and (41) becomes K ( D ) l = ¯ a sl k l +3 s K c l g res (2 + w l ǫ s ) − h ( − l + K c l i n (2 l + 3)(2 l − K c l + ǫ s + w l ǫ s + ( − l h (2 l + 3)(2 l − − K c l ǫ s io K c l g res h (2 l + 3)(2 l − K c l − ( − l ǫ s − ( − l w l ǫ s i + ( − l h ( − l + K c l i ǫ s h (2 l + 3)(2 l − K c l + ǫ s + w l ǫ s i , (50)In the broad resonance limit of | g res | → ∞ , it reducesto Eq. (50) of paper I. For a narrow resonance with | ζ res | ≪
1, it behaves as scattering of zero (generalized)scattering length, with K ( D ) l ∼ k l +3 s [7], only in a smallenergy range of 0 < ǫ s ≪ | g res | . Outside this range itbecomes that determined by the background scattering,as given by Eq. (44). Equations (49) and (50) illustratethe kind of energy dependences contained in the QDT ex-pansion for magnetic Feshbach resonances, and the com-plexity required to describe two types of behaviors, onedetermined by K c l ( B s ) or e a l ( B ) sufficiently close to thethreshold, and one by K c l or e a bg l away from the reso-nance, and the evolution between the two. VI. CONCLUSIONS
In conclusion, we have presented an analytic descrip-tion of a magnetic Feshbach resonance in an arbitrarypartial wave l and the atomic scattering around it at ul-tracold temperatures. It is derived by showing, in a verygeneral context, that a multichannel problem below thesecond threshold, all the way through the bound spec-trum, is equivalent to an effective single-channel prob-lem with a generally energy- and partial-wave-dependentshort range parameter. The relative significance of thisenergy dependence, in comparison with those induced bythe long-range interaction, leads to the classification ofFeshbach resonances of arbitrary l into broad and narrow1resonances, with vastly different scattering characteris-tics around the threshold.We have shown that, except for the special case of K c l = 0 (corresponding to e a bg l = ∞ , and discussedfurther in Appendix B), a magnetic Feshbach resonanceof arbitrary l can be parametrized in a similar fashionas an s wave Feshbach resonance [12, 13, 25], in termsof five parameters, which can be either B l , e a bg l , ∆ Bl , δµ l and s E (or C ), or B l , K c l , g res , d Bl , and s E (or C ). These parameters, together with the QDT expan-sion, give accurate analytic descriptions of atomic inter-actions around a magnetic Feshbach resonance, not onlyof the scattering properties presented here, but also of thebinding energies of a Feshbach molecule and of scatteringat negative energies [10], to be presented in a followingpublication. Such descriptions can now be incorporatedinto theories of atomic interaction in an optical lattice[16], using, e.g., the multiscale QDT of Ref. [17], andtheories of few-atom and many-atom systems around aFeshbach resonance, especially around a resonance thatis not broad (See, e.g., Refs. [14, 19–23]).Accurate determinations of Feshbach parameters willgenerally require a combination of theoretical and ex-perimental efforts, as have been done previously for the s wave [13]. The derivation of the parametrization, asgiven in Sec. II-III and Appendix B, also constitutes anoutline of a theory for these parameters and how they canbe computed from the MQDT formulation for atomic in-teraction in a magnetic field [24, 31]. The detailed imple-mentation of the theory and results for specific systemswill be presented elsewhere.It should be clear that while the focus of this articleis on magnetic Feshbach resonances, many of the con-cepts are much more generally applicable. In particular,the theoretical development followed here provides a verygeneral methodology on how analytic descriptions of cer-tain aspects of a multichannel problem may be developedand understood. The theory is also a necessary step to-wards resolving one remaining difficulty in the analyticdescription of ultracold atomic interaction, which is to ef-ficiently incorporate the weak magnetic dipole-dipole andsecond-order spin-orbit interactions [25, 27–30]. Beforeone can describe how such anistropic interactions can,e.g., couple a d wave resonance into the s wave [13], it isfirst necessary to efficiently characterize the resonanceswithout such coupling. ACKNOWLEDGMENTS
This work was partially supported by the NSF througha grant for ITAMP at Harvard University and Smithso-nian Astrophysical Observatory. The work at Toledo wassupported in part by the NSF under the Grant No. PHY-0758042.
Appendix A: Reduction of an N channel boundstate problem to an effective N a < N channel boundstate problem In MQDT for − /r n type of potentials with n > N channel problemis given generally by the solutions ofdet( χ c − K c ) = 0 , (A1)where K c is an N × N real and symmetric matrix, and χ c is an N × N diagonal matrix with elements χ c ( n i ) l ( ǫ si ).Separating the N channels into N a “ a ” channels and N c = N − N a “ c ” channels, the K c matrix can be writtenin a partitioned form as K c = K caa K cac K cca K ccc ! , (A2)where K caa is a N a × N a submatrix of K c , K ccc is a N c × N c submatrix, K cac is a N a × N c submatrix, and K cca is a N c × N a submatrix. From det( xy ) = det( x ) det( y ), wecan writedet( χ c − K c ) = det χ caa − K caa K cac K cca χ ccc − K ccc ! = det " A χ caa − K caa K cac K cca χ ccc − K ccc ! A − B = det( A ) det " χ caa − K caa K cac K cca χ ccc − K ccc ! A − B . (A3)Here A is an arbitrary nonsingular matrix, and B is anarbitrary matrix with det( B ) = 1. Choosing A = I χ ccc − K ccc ! , (A4)and B = I − K cca I ! , (A5)where I represents an unit matrix, we obtaindet( χ c − K c ) = det( χ ccc − K ccc ) det( χ caa − K c eff ) , (A6)where K c eff = K caa + K cac ( χ ccc − K ccc ) − K cca . (A7)Equation (A6) means that if the channels “ a ” andchannels “ c ” are not coupled, namely K cac = 0, thebound states separate into two sets, one for channels“ a ”, given by det( χ caa − K caa ) = 0, and one for chan-nels “ c ”, given by det( χ ccc − K ccc ) = 0. (Since K c , likeany other K matrix, is real and symmetric, K cac = 0also implies K cca = ( K cac ) T = 0.) If they are coupled,2which is the case that we are interested in, the solu-tions of det( χ ccc − K ccc ) = 0 are no longer solutions ofdet( χ c − K c ) = 0. This can be proven by taking thelimit of E → ¯ E in Eq. (A6), where ¯ E is one of the so-lutions of det( χ ccc − K ccc ) = 0, namely one of the bareresonance energies. Thus in the coupled case, all boundstate energies are given by the solutions ofdet( χ caa − K c eff ) = 0 , (A8)with the effective K c matrix being given by Eq. (A7).This procedure can in principle reduce an N channelbound state problem to an effective N a < N channelproblem with N a being an arbitrary number smaller than N . In reality, the choice of N a is of course determined bythe underlying physics. While we are using N a = 1 in thispaper, in which case Eq. (A8) reduces to Eq. (3), otherchoices are possible, and are likely to be important infuture treatments that incorporate the magnetic dipole-dipole and second-order spin-orbit interactions [25, 27–30]. Appendix B: An alternative parametrization ofmagnetic Feshbach resonances and the special caseof infinite (generalized) background scatteringlength
As stated in the main text, the parametrizationadopted, Eqs. (18) for the K c l ( ǫ, B ) and the correspond-ing Eq. (21) for the generalized scattering length, havethe limitation that they fail for K c l = 0, correspondingto an infinite (generalized) background scattering length e a bg l = ∞ . We show in this appendix that this is notan intrinsic difficulty of the theory, but due simply tothe desire of using parameters that have more direct ex-perimental interpretations. There are parametrizationsof K c l and e a l that would work for arbitrary backgroundscattering lengths, provided that we are willing to sacri-fice using B l .It is not surprising that any parametrization based on B l would have difficult at K c l = 0 ( e a bg l = ∞ ), corre-sponding to having a bound or quasibound backgroundstate right at the threshold. The interaction of this back-ground state and the “bare” Feshbach state is such that B l no longer exists, meaning that there can never be,in this case, a (coupled) bound state right at the thresh-old due to avoided crossing. This is reflected in the factthat the effective K c l , given by Eq. (9), does not havea solution for K c l ( ǫ = 0 , B l ) = 0 in the special case of K c l = 0.This difficulty can be overcome by expanding ¯ ǫ l ( B )in Eq. (9) around ¯ B l , determined by ¯ ǫ l ( ¯ B l ) = 0,which is the magnetic field at which the “bare” Fesh-bach resonance is crossing the threshold. This gives us¯ ǫ l ( B ) ≈ δ ¯ µ l ( B − ¯ B l ), where δ ¯ µ l = d ¯ ǫ l ( B ) /dB | B = ¯ B l .Equation (9) now becomes K c l ( ǫ, B ) = K c l − Γ c l / ǫ − δ ¯ µ l ( B − ¯ B l ) − f El . (B1)It is a parametrization with four parameters, K c l , ¯ B l , δ ¯ µ l , and Γ c l . [The f El is not an independent parame-ter and is still given by Eq. (13).] The correspondingparametrization of e a l is e a l ( B ) = ¯ a l [1 + ( − l K c l ]( B − ¯ B l + ¯ f Bl ) + ( − l Γ c Bl / K c l ( B − ¯ B l + ¯ f Bl ) + Γ c Bl / . (B2)where ¯ f Bl = f El /δ ¯ µ l and Γ c Bl = Γ c l /δ ¯ µ l . Equations (B1)and (B2) work for both infinite and zero (generalized)background scattering lengths. For example, in the caseof K c l = 0 ( e a bg l = ∞ ), Eq. (B2) becomes e a l ( B ) = ¯ a l (cid:20) ( − l + tan( πν /
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