Three-body recombination near the d-wave resonance in ultracold ^{85}Rb\,-^{87}Rb mixtures
TThree-body recombination near the d-wave resonance in ultracold Rb - Rb mixtures
Cai-Yun Zhao,
1, 2
Hui-Li Han, ∗ and Ting-Yun Shi State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics,Chinese Academy of Sciences, Wuhan 430071, P. R. China University of Chinese Academy of Sciences, 100049, Beijing, P. R. China (Dated: February 26, 2021)
Abstract
We have studied the three-body recombination rates on both sides of the interspecies d-waveFeshbach resonance in the Rb - Rb- Rb system using the R -matrix propagation method in thehyperspherical coordinate frame. Two different mechanisms of recombination rate enhancementfor positive and negative Rb - Rb d-wave scattering lengths are analyzed. On the positivescattering length side, the recombination rate enhancement occurs due to the existence of three-body shape resonance, while on the negative scattering length side, the coupling between thelowest entrance channel and the highest recombination channel is crucial to the appearance of theenhancement. In addition, our study shows that the intraspecies interaction plays a significantrole in determining the emergence of recombination rate enhancements. Compared to the case inwhich the three pairwise interactions are all in d-wave resonance, when the Rb- Rb interactionis near the d-wave resonance, the values of the interspecies scattering length that produce therecombination enhancement shift. In particular, when the Rb- Rb interaction is away from thed-wave resonance, the enhancement disappears on the negative interspecies scattering length side. ∗ [email protected] a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b . INTRODUCTION Three-body recombination (TBR) occurs in ultracold atomic gases as a result of a three-body collision in which the atoms gain kinetic energy due to the formation of a two-bodybound state. This process is vitally important in a number of physical and chemical contextsand has been recognized as one of the most important scattering observables [1–5]. As anexothermic reaction, TBR is one of the main loss mechanisms in systems with ultracoldatoms, limiting the density and lifetime of a Bose-Einstein condensate [6–8]. Additionally,the recombination process is used as a way to form weakly bound diatoms in ultracolddegenerate Fermi gases [9–11].Particles with resonant s-wave interactions will exhibit the Efimov effect, i.e., an infinitesequence of universal bound states characterized by discrete scale invariance [12, 13]. Theuniversal properties of the Efimov effect have been studied both theoretically [14–23] andexperimentally [24–47] in ultracold atomic gases. Due to the novel character of the Efimoveffect in few-body physics, whether the Efimov effect is possible for p-wave or higher partial-wave interaction is fundamentally important [48, 49]. Nisida [50] noted that Efimov statescannot be realized in physical situations for non-s-wave interactions. The studies of Efremov et al. [51] indicated that the effective potential is attractive and decreases as the third powerof the interatomic distance for a heavy-heavy-light system with a p-wave resonant interactionwhen employing the Born-Oppenheimer approximation method. With the same method,Zhu and Tan [52] provided a more general discussion of the universal properties for atomsnear higher partial-wave Feshbach resonances and found that the effective potential behavedas 1 /ρ L +1 when the distance ρ between two heavy atoms was large. These two works alsodemonstrated that the Efimov effect does not occur in few-body systems interacting viahigher partial-wave resonant interactions.For d-wave dimers, Gao [53] predicted that the bound states always appear at a universalvalue of the s-wave scattering length of a s ≈ . r vdW . r vdW = (2 µ b C ) / / − C /r with two-body reduced mass µ b . For a systemof three identical bosons interacting via a d-wave resonant interaction, the study of Wang et al. [54] showed that a universal three-body state associated with the d-wave dimer isformed at a s ≈ . r vdW . Despite this progress, universal properties of heteronuclear systems2nteracting via resonant higher partial-wave interactions are less well understood [55].Experimentally, great efforts have been devoted to studying strongly interacting atomicBose gases with s-wave resonances [24–27, 32–47, 56–60], and only a few studies have inves-tigated the many-body properties of interacting fermions near a p-wave resonance [61–66].For higher partial-wave resonances, challenges arise from their short lifetime and narrowresonance width [67]. Recently, two broad d-wave resonances were observed via atom lossin a Rb - Rb mixture [67]. Very recently, a d-wave shape resonance and a Feshbach reso-nance were observed in degenerate K and K gases, respectively [68, 69]. Moreover, bothp-wave and d-wave Feshbach resonances were observed in the
Cs - Li system [70]. Thisexperimental progress has provided a platform to study the universal properties of few-bodyphysics with d-wave resonant interactions.In this paper, we investigate the TBR process in a heteronuclear system with d-waveresonant interactions. In real ultracold atomic systems, inter- and intraspecies interactionsare generally not controlled independently. Thus, complications arise in heteronuclear three-body systems due to the two different scattering lengths. Near the interspecies Feshbachresonance, two identical atoms interact with each other with a smaller scattering length.As a result, ultracold gases of heteronuclear systems are expected to show rich few-bodyphysics compared to the homonuclear case. Since two broad d-wave Feshbach resonanceshave been experimentally observed in Rb - Rb mixtures [67], we choose the Rb - Rb - Rb system as an example to explore the TBR process near d-wave resonances. We alsofocus on the more realistic case in which two heteronuclear atoms are in d-wave resonancewhile homonuclear atoms interact via a finite scattering length.The TBR rates are obtained using quantum calculations based on a combination of theslow variable discretization (SVD) method, traditional hyperspherical adiabatic method and R -matrix propagation method [71–76]. Following the method of Ref [74], first, we solve theSchr¨odinger equation with the hyperradius divided into two regimes. At short distances,the SVD method is employed to overcome the numerical difficulties at sharp nonadiabaticavoided crossings, and at large distances, the traditional adiabatic hyperspherical method isutilized to avoid the large memory and CPU time needed in SVD. Second, by propagatingthe R matrix from short to large distances, we can obtain scattering properties through the S matrix by matching the R matrix with asymptotic functions and boundary conditions.The Lennard-Jones potential, which has been shown to be an excellent model potential, is3tilized to mimic the interaction between atoms [14, 54, 77].This paper is organized as follows. In Sec. II, our calculation method and all necessaryformulas for calculations are presented. In Sec. III, we discuss the results and emphasizethe significant role of intraspecies interactions in heteronuclear systems. Finally, we give abrief summary. Atomic units are used throughout unless stated otherwise. II. THEORETICAL FORMALISM
The present numerical study focuses on the heteronuclear system with total angularmomentum J = 0, with parameters adjusted to represent the Rb - Rb- Rb system. Weuse m i (i=1,2,3) to represent the mass of three atoms, and r ij represents their distance. Inthe center-of-mass frame, six coordinates are needed to describe the three-particle system.Three of these are taken to be the Euler angles α , β , and γ , which specify the orientation ofthe body-fixed frame relative to the space-fixed frame. The remaining degrees of freedom canbe represented by hyperradius R and two hyperangles θ and φ . In our method, we employDelves’ hyperspherical coordinates. We first introduce the mass-scaled Jacobi coordinates. (cid:126)ρ is the vector from atom 1 to atom 2, with the reduced mass denoted by µ ; the secondJacobi (cid:126)ρ is measured from the diatom center of mass to the third atom, with reduced mass µ . θ is the angle between (cid:126)ρ and (cid:126)ρ . The hyperradius R and hyperangle φ are defined as µR = µ ρ + µ ρ (1)and tan φ = (cid:114) µ µ ρ ρ , ≤ φ ≤ π , (2)respectively, where µ is an arbitrary scaling factor and chosen as µ = √ µ µ in our calcu-lations. R is the only coordinate with the dimension of length, which represents the overallsize of the three-body system. θ , φ and the three Euler angles ( α, β, γ ) can be collectivelyrepresented by Ω [Ω ≡ ( θ, φ, α, β, γ )], which describe the rotation of the plane containingthe three particles.In hyperspherical coordinates, the Schr¨odinger equation can be written in terms of therescaled wave function ψ υ (cid:48) ( R ; θ, φ ) = Ψ υ (cid:48) ( R ; θ, φ ) R / sin φ cos φ : (cid:20) − µ d dR + (cid:18) Λ − µR + V ( R ; θ, φ ) (cid:19)(cid:21) ψ υ (cid:48) ( R ; Ω) = Eψ υ (cid:48) ( R ; Ω) , (3)4here Λ is the squared “grand angular momentum operator”, whose expression is givenin Ref [78]. The volume element relevant to integrals over | ψ υ (cid:48) ( R ; Ω) | then becomes dR sin θdθdφdα sin βdβdγ . The index υ (cid:48) labels the different independent solutions. Thethree-body interaction V ( R ; θ, φ ) in Eq. (3) is taken to be a sum of the three pairwisetwo-body interactions υ ( r ij ): V ( R ; θ, φ ) = υ ( r ) + υ ( r ) + υ ( r ) . (4)The interparticle distances r ij can be described in terms of the internal coordinates as follows: r = R (cid:114) µµ cos φ, (5) r = R (cid:16) µµ sin φ + 14 µµ cos φ −
12 sin 2 φ cos θ (cid:17) / , (6) r = R (cid:16) µµ sin φ + 14 µµ cos φ + 12 sin 2 φ cos θ (cid:17) / . (7)The wave function ψ υ (cid:48) can then be expanded with the complete, orthonormal adiabaticchannel functions Φ υ as ψ υ (cid:48) ( R ; Ω) = ∞ (cid:88) ν =0 F νυ (cid:48) ( R )Φ ν ( R ; Ω) . (8)We first determine the adiabatic potentials U ν ( R ) and the corresponding channel functionsΦ ν ( R ; Ω) at fixed R by solving the following adiabatic eigenvalue equation: (cid:18) Λ − µR + V ( R ; θ, φ ) (cid:19) Φ ν ( R ; Ω) = U ν ( R )Φ ν ( R ; Ω) . (9)The channel function is further expanded on Wigner rotation matrices D JKM asΦ J Π Mν ( R ; Ω) = J (cid:88) K =0 u νK ( R ; θ, φ ) D J Π KM ( α, β, γ ) , (10) D J Π KM = 14 π √ J + 1 (cid:2) D JKM + ( − K + J Π D J − KM (cid:3) , (11)where J is the total nuclear orbital angular momentum, M is its projection onto thelaboratory-fixed axis, and Π is the parity with respect to the inversion of the nuclear coordi-nates. The quantum number K denotes the projection of J onto the body-frame z axis andtakes the values K = J, J − , . . . , − ( J − , − J for the “parity-favored” case, Π = ( − J ,and K = J − , J − , . . . , − ( J − , − ( J −
1) for the “parity-unfavored” case, Π = ( − J +1 .5or the Rb - Rb - Rb system, the wave function is symmetric with respect to theexchange of the two Rb atoms, and thus, this exchange symmetry can be built into theboundary conditions of the body-frame components as follows: P D J Π KM = Π( − K D J Π KM , (12) P θ = π − θ. (13)For even parity, u νK should be symmetric about π/ K and antisymmetric withodd K . For odd parity, u νK should be antisymmetric for even K and symmetric for odd K .To satisfy the permutation requirements, u νK is expanded with symmetric B-spline basissets, u νK ( R ; θ, φ ) = N φ (cid:88) i N θ / (cid:88) j c i,j B i ( φ ) ( B j ( θ ) + B N θ +1 − j ( π − θ )) , (14)or antisymmetric B-spline basis sets, u νK ( R ; θ, φ ) = N φ (cid:88) i N θ / (cid:88) j c i,j B i ( φ ) ( B j ( θ ) − B N θ +1 − j ( π − θ )) , (15)where N θ and N φ are the sizes of the basis sets in the θ and φ directions, respectively. Theconstructed symmetric B-spline basis sets used in the θ direction reduce the number of basisfunctions to N θ / S from the solutionsof Eq. (3). We first calculate the R matrix, which is defined as R ( R ) = F ( R )[ (cid:101) F ( R )] − , (16)where matrices F and (cid:101) F can be calculated from the solution of Eq. (3) and Eq. (9) by F ν,υ (cid:48) ( R ) = (cid:90) d ΩΦ ν ( R ; Ω) ∗ ψ υ (cid:48) ( R ; Ω) , (17) (cid:101) F ν,υ (cid:48) ( R ) = (cid:90) d ΩΦ ν ( R ; Ω) ∗ ∂∂R ψ υ (cid:48) ( R ; Ω) . (18)Following the method of Ref. [74], we divide the hyperradius into ( N −
1) intervals witha set of grid points R < R < · · · R N . At a short distance, we use the SVD method to solveEq. (3) in the interval [ R i , R i + 1]. In the SVD method, the total wave function ψ υ (cid:48) ( R ; Ω)6s expanded in terms of the discrete variable representation (DVR) basis π i and the channelfunctions Φ ν ( R ; Ω) as ψ υ (cid:48) ( R ; Ω) = N DV R (cid:88) i N chan (cid:88) ν C υ (cid:48) iν π i ( R )Φ ν ( R i ; Ω) , (19)where N DV R is the number of DVR basis functions and N chan is the number of includedchannel functions. Inserting ψ υ (cid:48) ( R ; Ω) into the three-body Schrodinger equation results inthe standard algebraic problem for the coefficients C υ (cid:48) iν : N DV R (cid:88) j N chan (cid:88) µ T ij O iν,jµ C υ (cid:48) iν + U ν ( R i ) C υ (cid:48) iν = E υ (cid:48) C υ (cid:48) iν , (20)where T ij = 12 µ (cid:90) R i +1 R i ddR π i ( R ) ddR π j ( R ) dR (21)are the kinetic energy matrix elements, with R i and R i +1 being the boundaries of the cal-culation box, and O iν,jµ = (cid:104) Φ ν ( R i ; Ω) | Φ µ ( R j ; Ω) (cid:105) (22)are the overlap matrix elements between the adiabatic channels defined at different quadra-ture points.At large distances, the traditional adiabatic hyperspherical method is used to solveEq. (3). When substituting the wave functions ψ ( R ; Ω) into Eq. (3), one obtains a setof coupled ordinary differential equations:[ − µ d dR + U ν ( R ) − E ] F ν,υ (cid:48) ( R ) − µ (cid:88) µ [2 P µν ( R ) ddR + Q µν ( R )] F µυ (cid:48) ( R ) = 0 , (23)where P µν ( R ) = (cid:90) d ΩΦ µ ( R ; Ω) ∗ ∂∂R Φ ν ( R ; Ω) (24)and Q µν ( R ) = (cid:90) d ΩΦ µ ( R ; Ω) ∗ ∂ ∂R Φ ν ( R ; Ω) (25)7re the nonadiabatic couplings that control the inelastic transitions as well as the width ofthe resonance supported by adiabatic potential U ν ( R ) . In our calculations, the relationbetween P and Q is ddR P = − P + Q , where P νµ ( R ) = − (cid:90) d Ω ∂∂R Φ ν ( R ; Ω) ∗ ∂∂R Φ µ ( R ; Ω) . (26)The coupling matrices have the following properties: P νµ = − P µν and P νµ = P µν , whichleads to P νν = 0 , and Q νν = − P νν . The effective hyperradial potentials that includehyperradial kinetic energy contributions with the P νν term are more physical than adiabatichyperpotentials and are defined as W νν ( R ) = U ν ( R ) − ¯ h µ P νν ( R ) . (27)Next, the R -matrix propagation method is used. Over an interval [ R , R ], for a given R matrix Eq. (16), one uses the R -matrix propagation method to calculate the corresponding R matrix at another point R = R as follows: R ( R ) = R − R [ R + R ( R )] − R . (28)The K matrix can be expressed in the following matrix equation: K = ( f − f (cid:48) R )( g − g (cid:48) R ) − , (29)where f νν (cid:48) = (cid:113) µk ν π Rj l ν ( k ν R ) δ νν (cid:48) and g νν (cid:48) = (cid:113) µk ν π Rn l ν ( k ν R ) δ νν (cid:48) are the diagonal matricesof energy-normalized spherical Bessel and Neumann functions. For the recombination chan-nel, l ν is the angular momentum of the third atom relative to the dimer, and k ν is givenby k ν = (cid:112) µ ( E − E b ). For the entrance channel, l ν = λ ν + 3 /
2, and k ν = √ µE . Thescattering matrix S is related to K by the following: S = (1 + i K )(1 − i K ) − . (30)Using the convention of Mott and Massey [79], the N-body cross section in d dimensionsis defined as σ fi ( J Π ) = N p (cid:18) πk i (cid:19) d − d ) (cid:88) i (2 J + 1) | S J Π fi − δ fi | , (31)8here Ω( d ) = 2 π d/ / Γ( d/
2) is the total solid angle in d dimensions, and N p is the numberof terms in the permutation symmetry projection operator. In the Rb - Rb - Rb system, N p = 2 !, d = 6, and the total TBR rate is then K = kµ σ = (cid:88) J, Π K J, Π3 = 2! (cid:88) J, Π (cid:88) f,i J + 1) π µk | S J, Π i → f | , (32)where i and f label the three-body continuum (incident) and TBR (outgoing) channels,respectively. σ is the generalized TBR cross section. K J, Π3 is the partial recombinationrate corresponding to J Π symmetry, and k = (2 µE ) / is the wave number in the incidentchannels.Since experiments are performed at fixed temperature instead of fixed energy, the thermalaverage becomes crucial for proper comparison with the experiment. Assuming a Boltzmanndistribution, the thermally averaged recombination rates are given by (cid:104) K (cid:105) ( T ) = (cid:82) K ( E ) E e − E/k B T dE (cid:82) E e − E/k B T dE = 12( k B T ) (cid:90) K ( E ) E e − E/k B T dE. (33)The results presented in III C are given for T = 120 nK .In our calculations, Eq. (9) is solved with 134 SVD sectors and 10 SVD points in eachsector for R < a . In the interval 2 000 a < R <
22 000 a , we use the traditionaladiabatic hyperspherical method with P µν and Q µν calculated by an improved method inRef [54]. The matrix elements of coupling P µν and effective potential W νν can be fitted toan inverse polynomial series at a large distance, and the fitting results of P µν and W ν areused beyond R = 22 000 a . III. RESULTS AND DISCUSSIONA. Hyperspherical potential curves near the interspecies d-wave resonance
The two-body potential υ ( r ij ) used in our calculations is the Lennard-Jones potential,which can be expressed in the form of υ ( r ij ) = C ,ij r ij (cid:20) −
12 ( γ ij r ij ) (cid:21) . (34)In the present study, γ ij is adjusted to give the desired scattering length and number ofbound states. The low-energy behavior of the l th partial wave phase shift for scattering by9 long-range central potential 1 /r s ( s >
2) satisfiestan δ l ( k, ∞ ) ∼ − k l +1 λ l − π s Γ( s − (cid:0) l + − s (cid:1) Γ (cid:0) s (cid:1) Γ (cid:0) l + + s (cid:1) µ b C s k s − , (35)with 2 < s < l + 3 [52, 80, 81]. For the Lennard-Jones potential s = 6, the scattering phaseshift of the d-wave has the following expansion:tan δ ( k, ∞ ) ∼ − k λ − π Γ(5)Γ (2)Γ (3) Γ (11 /
2) 2 µ b C k . (36) a d = λ / is denoted as the “d-wave scattering length” to characterize the findings in termsof the d-wave interactions, which diverges when a d-wave dimer is just about to be bound.Figure 1 shows the s-wave scattering length a s (blue solid line) and d-wave scatteringlength a d (red dashed line) as a function of γ ij . No obvious difference is found between Rb - Rb and Rb - Rb, so we only show the results for one here as an example. Thevertical dotted lines enclose the Rb - Rb parameter range considered in our numericalcalculations. In this range, one two-body s-wave bound state exists before the d-wave boundstate emerges. For homonuclear interaction, we focus on the case in which two Rb atomsare in d-wave resonance (point I in Fig. 1) and the more realistic case in which Rb - Rbinteract via the s-wave scattering length a s = 100 a , which is indicated by arrows II andIII. As shown in Fig. 1, the two Rb atoms have one s-wave bound state and are near thed-wave resonance at point II. At point III, the two Rb atoms are away from the d-waveresonance.In the scattering process, the adiabatic potential curves U ν ( R ) are important in un-derstanding the three-body physics. Figure 2 shows the hyperspherical potential curvesof the Rb - Rb– Rb system with the Rb - Rb d-wave scattering length a d = − a (Fig. 2(a)) and a d = 90 a (Fig. 2(b)). Rb - Rb interact via s-wave scattering length a s = 100 a with the parameter γ ij adjusted to point II in Fig. 1. In each case, the solidpotential curves correspond to the TBR channels and asymptotically approach the dimerbinding energy. The effective potentials for these channels exhibit asymptotic behavior givenby W f ( R ) = l f ( l f + 1)2 µR + E ( f )2 b , (37)where E ( f )2 b is the dimer energy and l f is the relative orbital angular momentum between theatom and dimer. The subscript f distinguishes the recombination channels. The dashed10 IG. 1. (Color online) Two-body s-wave scattering length (blue solid line) and d-wave scatteringlength (red dashed line) as a function of the adjusting parameter γ ij . The Rb - Rb interactionrange is adjusted around the two-body d-wave resonance that is between the two black dottedlines. For the Rb - Rb interaction, d-wave resonance position I and the more realistic cases inwhich Rb - Rb interact via the s-wave scattering length a s = 100 a , points II and III in Fig. 1,are focused on. lines in Fig. 2 denote the three-body breakup channels (or entrance channels), i.e., all threeatoms far from each other as R → ∞ , where the potentials behave as W i ( R ) = λ i ( λ i + 4) + 15 / µR . (38)The values of λ i are nonnegative integers determined by J Π and the identical particle sym-metry [82]. We use the dimensionless quantity of the nonadiabatic coupling strength definedby f vv (cid:48) ( R ) = P vv (cid:48) ( R ) µ [ U v ( R ) − U v (cid:48) ( R )] (39)to characterize the nonadiabatic coupling magnitude, which mainly controls the recombi-nation process. The coupling strength between the highest recombination channel and thelowest entrance channel is shown in the inset of Fig. 2.11 IG. 2. (Color online) Three-body adiabatic potential curves for the Rb - Rb- Rb system with Rb - Rb d-wave scattering length a d = − a in (a) and a d = 90 a in (b). The Rb - Rbs-wave scattering length is a s = 100 a with the parameter γ ij adjusted to point II in Fig. 1. Theinsets show the avoided crossings between the highest recombination channel and first entrancechannel along with their nonadiabatic coupling strength. When the Rb - Rb d-wave scattering length a d is negative, as shown in Fig. 2(a), thenonadiabatic coupling strength between the lowest entrance channel and the first recom-bination channel is localized at a short distance. Here, recombination occurs primarily bytunneling through the potential barrier in the lowest three-body entrance channel to reachthe region of large coupling. Thus, the potential barrier in the lowest three-body entrancechannel plays an important role in the recombination process. Figure 3(a) shows the barrierfor several negative Rb - Rb scattering lengths a d . The height of the barrier decreasesas the Rb - Rb d-wave interaction becomes strong. Diminishing of the potential barrierin the entrance channel is responsible for the recombination enhancement on this scatter-ing side [83]. In addition, when the Rb - Rb interaction becomes sufficiently strong, atwo-body d-wave shape resonance appears [84]. As a result, this resonance will produce aseries of avoided crossings in the three-body potential curves at energies near the positionof the resonance, as shown in Figs. 3(b)-3(d). Ref. [85] also found this phenomenon in athree-fermion system when the two-body scattering volume was negative. The positionsof these avoided crossings might be expected to approach the three-body breakup thresh-old when the Rb - Rb d-wave interaction becomes strong. This can be demonstrated byFigs. 3(b)-3(d). 12
IG. 3. (Color online) (a) Lowest entrance channels with barriers in Rb - Rb- Rb for differ-ent Rb - Rb d-wave scattering lengths; (b), (c) and (d) are potential curves showing a seriesof avoided crossings near the Rb - Rb two-body d-wave resonance. These avoided crossingsapproach the three-body breakup threshold when a d → −∞ . The Rb - Rb s-wave scatteringlength is a s = 100 a with the parameter γ ij adjusted to point II in Fig. 1. For the positive Rb - Rb d-wave scattering length case, the important feature is thebroad avoided crossing between the lowest entrance channel and the highest recombina-tion channel. This can be seen from the inset of Fig. 2(b). Figure 4(a) shows the lowestentrance channel and the first recombination channel of the Rb - Rb - Rb system for dif-ferent Rb - Rb d-wave scattering lengths with the parameter γ ij of the Rb - Rb inter-action adjusted to point II in Fig. 1. The two-body threshold moves towards the three-bodybreakup threshold with increasing Rb - Rb d-wave scattering length. When the top of thebarrier is above the collision energy, recombination will be suppressed by this extra barrier,except possibly at energies matching three-body shape resonances behind this barrier. Theexistence of three-body shape resonance will lead to more or fewer sudden jumps of the13hase shift by π . Meanwhile, the phase shift is well described by the analytical expres-sion δ l ( E ) = δ bg − arctan (cid:16) Γ / E − E R (cid:17) , where E R is the resonance position, Γ is the resonancewidth, and δ bg is a smoothly energy-dependent background phase shift. In Figs. 4(b)-4(d),we plot the corresponding phase shift δ along with the tan δ values as a function of energy.Shape resonances exist when the Rb - Rb scattering length is a d = 114 a and a d = 90 a .By fitting the tan δ points with the analytical expression, we obtain resonance positions of − . × − (Fig. 4(b)) and − . × − (Fig. 4(c)). FIG. 4. (Color online)(a) Lowest entrance and highest recombination channels for three positive Rb - Rb d-wave scattering lengths a d . The parameter of the Rb - Rb interaction is adjustedto point II in Fig. 1. (b), (c) and (d) are the corresponding atom-dimer scattering phase shifts δ and tan δ . . Three-body recombination rates The partial rates K J Π3 and total K for J Π = 0 + , − and 2 + symmetries as a functionof the collision energy with the Rb - Rb d-wave scattering length fixed at a d = − a (Fig. 5(a)) and a d = 90 a (Fig. 5(b)) are shown in Fig. 5. The parameter of the Rb - Rbinteraction is adjusted to point II in Fig. 1. In the zero-energy limit, the recombinationrate obeys the threshold behavior K J Π3 ∼ E λ min , where λ min is the minimum value of λ inEq.(38). For J Π = 0 + , − , + , we have λ min = 0 , , Rb - Rb - Rb system [73, 82].That is, the J Π = 0 + partial rate increases like E from the threshold, while the 1 − and2 + rates behave as E and E , respectively. At high collision energies, K decreases as E − , required by unitary. For the positive d-wave scattering length, Fig. 5(b) shows thatthe Wigner threshold law holds only at small energies E < µK . Note that the Wignerthreshold regime can be characterized as energies smaller than the smallest energy scale,which is typically a molecular binding energy [86]. In the present case, when a d = 90 a ,the newly formed d-wave dimer binding energy is approximately 12 µK . This is in roughagreement with the attained threshold regime of E < µK .Although the J Π = 0 + case is predicted to be the dominant symmetry in the zero colli-sion energy limit, other symmetries may contribute substantially. Therefore, studying theenergy-dependent partial recombination rates corresponding to various symmetries is inter-esting. Figure 5 shows that the contributions from J Π = 1 − and J Π = 2 + partial waves aresignificant when the energy is above 40 µK . C. Nonnegligible role of the intraspecies interaction in the heteronuclear system
In a real ultracold atomic system, the inter- and intraspecies interactions are generallynot controlled independently, and thus, a finite intraspecies scattering length exists. Forthe Rb - Rb - Rb system, near the interspecies d-wave Feshbach resonance, Rb - Rbinteract with each other through a smaller s-wave scattering length a s = 100 a . As shownin Fig 1, Rb - Rb has an s-wave bound state at point II and no s-wave bound state atpoint III with the same Rb - Rb s-wave scattering length.To see how the Rb - Rb interaction influences the TBR, we plot the hypersphericalpotential curves for the first entrance channel and the highest-lying recombination chan-15
IG. 5. (Color online) Partial rates K J Π3 and their total K for J Π = 0 + , − and 2 + symmetriesas a function of the collision energy when the Rb - Rb d-wave scattering length is fixed at a d = − a (Fig. (a)) and a d = 90 a (Fig. (b)) with the parameter of the Rb - Rb interactionadjusted to point II in Fig. 1. The dashed lines represent the threshold laws or unitary limit. nel for the same Rb - Rb d-wave scattering length a d = − a but different Rb - Rb interaction details in Figs. 6(a), 6(c) and 6(e). The interaction details of Rb - Rbgreatly affect the coupling between the lowest entrance channel and the highest recombina-tion channel. The Landau-Zener parameter T ij , which estimates the nonadiabatic transi-tion probabilities [87], can reflect this coupling strength quantitatively and be calculated by T ij = e − δ ij = e − π ∆2 ij αijν , where ∆ ij = U i − U j is evaluated in the transition region and α ij isobtained from P-matrix analysis.When Rb - Rb are in d-wave resonance ( Rb - Rb s-wave scattering length a s =84 a ), point I in Fig. 1, the Landau-Zener parameter approaches 1, as shown in Fig. 6(a),implying that a nonadiabatic transition occurs and the potential well of the recombinationchannel becomes deeper. A d-wave-related trimer state (A peak in Fig. 6(b)) can thusbe supported, which will lead to enhancement of the TBR rate. When we plot the totaland partial J Π = 0 + TBR rates as a function of the Rb - Rb d-wave scattering lengthat fixed Rb - Rb s-wave scattering length a s = 84 a , the total rate exhibits two clearenhancements labeled ”A” and ”B”, as shown in Fig. 6(b). This phenomenon was predictedby Wang et al. [54] in a three-identical-boson system. Peak A corresponds to the d-wave-related trimer state across the collision threshold. According to our analyses of the influenceof the Rb - Rb interaction on the coupling between the first entrance channel and the16
IG. 6. (Color online) The first entrance channel and highest recombination channel hyperpotentialcurves and corresponding Landau-Zener parameter T na for the same Rb - Rb d-wave scatteringlength a d = − a are shown in (a), (c) and (e). The Rb - Rb interaction is at the d-waveresonance (s-wave scattering length a s = 84 a ) in (a). The Rb - Rb s-wave scattering lengths in(c) and (e) are both 100 a , but (c) is close to the d-wave resonance and (e) is far from the d-waveresonance. (b), (d) and (e) show the TBR rate K varying with the Rb - Rb interaction whenthe Rb - Rb interaction is fixed at point II in Fig. 1. K when Rb - Rb is not exactly in d-waveresonance.Figures 6(c) and 6(e) show the hyperspherical potential curves for the same Rb - Rbtwo-body scattering length a d = − a as in Fig. 6(a), while Rb - Rb interact throughs-wave scattering length a s = 100 a with different interaction details. We first focus onthe case in which the parameter of the Rb - Rb interaction is adjusted to point II inFig. 1, where Rb - Rb has one s-wave bound state and is near the d-wave resonance. Inthis case, we find that the position of enhancement A shifts from 1 . r vdW to 1 . r vdW ( r vdW is the van der Waals length between Rb and Rb) compared to the case in which thehomonuclear atoms are in d-wave resonance, as shown in Fig. 6(b). When the parameter γ ij is changed to point III in Fig. 1, where the interaction of Rb - Rb is away from thed-wave resonance, the coupling between the first entrance channel and the highest-lyingrecombination channel becomes weak. Figure 6(e) shows that the Landau-Zener parameteris 0 .
186 in this case, implying that an adiabatic transition occurs. As a result, the potentialwell in the recombination channel may not be deep enough to support the trimer state,leading to the absence of enhancement A in Fig. 6(f).Peak B formed after the d-wave dimer became bound. Our analysis in III A shows thatthis enhancement corresponds to the three-body shape resonance. Its position is also affectedby the interaction details of the two homonuclear atoms. This can be seen from Figs. 6(b),6(d) and 6(f). Our results demonstrate that the intraspecies interaction details play asignificant role in determining the TBR in the heteronuclear system.
IV. CONCLUSIONS
In summary, we have studied the TBR rate for the Rb - Rb - Rb system near the Rb - Rb d-wave resonance. The TBR rates are obtained using quantum calculations inthe frame of the hyperspherical coordinates, which are based on a combination of the SVDmethod, traditional hyperspherical adiabatic method and R-matrix propagation method.Our study reveals two different mechanisms of recombination rate enhancement for positiveand negative Rb - Rb d-wave scattering lengths. When the Rb - Rb d-wave scattering18ength is positive and large, a loosely bound dimer is produced, and enhancement occurs dueto the existence of three-body shape resonance. We have identified two such shape resonanceson the positive Rb - Rb d-wave scattering length side. For the case in which Rb - Rbinteract via a negative d-wave scattering length, the coupling between the lowest entrancechannel and the highest recombination channel is crucial to the formation of the three-bodystate. When the coupling becomes strong, a nonadiabatic transition occurs, which makes thepotential well in the highest recombination channel deeper and thus supports the three-bodystate. The enhancement on the negative interspecies scattering length side corresponds tothe three-body state crossing the three-body threshold.In addition, we have investigated the influence of the finite Rb - Rb interaction on therecombination enhancement. With the same Rb - Rb s-wave scattering length a s = 100 a ,when the interaction is near the d-wave resonance, the coupling between the lowest entrancechannel and the highest recombination channel becomes strong, leading to recombinationrate enhancement. However, if the parameter of the Rb - Rb interaction is adjusted toaway from the d-wave resonance, then the enhancement will disappear. Moreover, our studyalso finds that the intraspecies interaction also affects the Rb - Rb d-wave scattering lengthvalues at which the enhancement appears. Our results have confirmed the main results ofRef. [54] for the homonuclear case and provide numerical evidence that the TBR rates inheteronuclear systems are more complex than those in homonuclear systems.
V. ACKNOWLEDGMENTS
We thank C. H. Greene, Jia Wang, Li You, Meng Khoon Tey and Cui Yue for helpfuldiscussions. Hui-Li Han was supported by the National Natural Science Foundation ofChina under Grants No. 11874391 and No. 11634013 and the National Key Research andDevelopment Program of China under Grant No. 2016YFA0301503. Ting-Yun Shi wassupported by the Strategic Priority Research Program of the Chinese Academy of Sciencesunder Grant No. XDB21030300. [1] E. Nielsen and J. H. Macek, Phys. Rev. Lett. , 1566 (1999).[2] B. D. Esry, C. H. Greene, and J. P. Burke, Phys. Rev. Lett. , 1751 (1999).
3] J. P. D’Incao and B. D. Esry, Phys. Rev. Lett. , 213201 (2005).[4] E. Braaten and H.-W. Hammer, Phys. Rep. , 259 (2006).[5] P. Naidon and S. Endo, Rep. Prog. Phys. , 056001 (2017).[6] E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman,Phys. Rev. Lett. , 337 (1997).[7] J. L. Roberts, N. R. Claussen, S. L. Cornish, and C. E. Wieman, Phys. Rev. Lett. , 728(2000).[8] T. Weber, J. Herbig, M. Mark, H.-C. N¨agerl, and R. Grimm, Phys. Rev. Lett. , 123201(2003).[9] J. Cubizolles, T. Bourdel, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, and C. Salomon,Phys. Rev. Lett. , 240401 (2003).[10] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, C. Chin, J. H. Denschlag, and R. Grimm,Phys. Rev. Lett. , 240402 (2003).[11] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic, andW. Ketterle, Phys. Rev. Lett. , 250401 (2003).[12] V. Efimov, Phys. Lett. , 563 (1970).[13] V. Efimov, Nucl. Phys. A , 157 (1973).[14] J. Wang, J. P. D’Incao, B. D. Esry, and C. H. Greene, Phys. Rev. Lett. , 263001 (2012).[15] Y. Wang, J. Wang, J. P. D’Incao, and C. H. Greene, Phys. Rev. Lett. , 243201 (2012).[16] P. K. Sørensen, D. V. Fedorov, A. S. Jensen, and N. T. Zinner, Phys. Rev. A , 052516(2012).[17] M.-S. Wu, H.-L. Han, C.-B. Li, and T.-Y. Shi, Phys. Rev. A , 062506 (2014).[18] B. Huang, K. M. O’Hara, R. Grimm, J. M. Hutson, and D. S. Petrov, Phys. Rev. A ,043636 (2014).[19] M.-S. Wu, H.-L. Han, and T.-Y. Shi, Phys. Rev. A , 062507 (2016).[20] S. H¨afner, J. Ulmanis, E. D. Kuhnle, Y. Wang, C. H. Greene, and M. Weidem¨uller, Phys.Rev. A , 062708 (2017).[21] H. Han and C. H. Greene, Phys. Rev. A , 023632 (2018).[22] A. N. Wenz, T. Lompe, T. B. Ottenstein, F. Serwane, G. Z¨urn, and S. Jochim, Phys. Rev.A , 040702 (2009).[23] C.-Y. Zhao, H.-L. Han, M.-S. Wu, and T.-Y. Shi, Phys. Rev. A , 052702 (2019).
24] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch,A. Jaakkola, H.-C. N¨agerl, and R. Grimm, Nature , 315 (2006).[25] M. Zaccanti, B. Deissler, C. Derrico, M. Fattori, M. Jona-Lasinio, S. M¨uller, G. Roati, M. In-guscio, and G. Modugno, Nature Physics , 586 (2009).[26] M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C. N¨agerl, F. Ferlaino, R. Grimm, P. S.Julienne, and J. M. Hutson, Phys. Rev. Lett. , 120401 (2011).[27] B. Huang, L. A. Sidorenkov, R. Grimm, and J. M. Hutson, Phys. Rev. Lett. , 190401(2014).[28] T. Lompe, T. B. Ottenstein, F. Serwane, A. N. Wenz, G. Z¨urn, and S. Jochim, Science ,940 (2010).[29] J. R. Williams, E. L. Hazlett, J. H. Huckans, R. W. Stites, Y. Zhang, and K. M. O’Hara,Phys. Rev. Lett. , 130404 (2009).[30] J. H. Huckans, J. R. Williams, E. L. Hazlett, R. W. Stites, and K. M. O’Hara, Phys. Rev.Lett. , 165302 (2009).[31] T. B. Ottenstein, T. Lompe, M. Kohnen, A. N. Wenz, and S. Jochim, Phys. Rev. Lett. ,203202 (2008).[32] S. E. Pollack, D. Dries, and R. G. Hulet, Science , 1683 (2009).[33] N. Gross, Z. Shotan, S. Kokkelmans, and L. Khaykovich, Phys. Rev. Lett. , 163202 (2009).[34] N. Gross, Z. Shotan, S. Kokkelmans, and L. Khaykovich, Phys. Rev. Lett. , 103203 (2010).[35] M. Kunitski, S. Zeller, J. Voigtsberger, A. Kalinin, L. P. H. Schmidt, M. Sch¨offler, A. Czasch,W. Sch¨ollkopf, R. E. Grisenti, T. Jahnke, D. Blume, and R. D¨orner, Science , 551 (2015).[36] G. Barontini, C. Weber, F. Rabatti, J. Catani, G. Thalhammer, M. Inguscio, and F. Minardi,Phys. Rev. Lett. , 043201 (2009).[37] R. S. Bloom, M.-G. Hu, T. D. Cumby, and D. S. Jin, Phys. Rev. Lett. , 105301 (2013).[38] M.-G. Hu, R. S. Bloom, D. S. Jin, and J. M. Goldwin, Phys. Rev. A , 013619 (2014).[39] L. J. Wacker, N. B. Jørgensen, D. Birkmose, N. Winter, M. Mikkelsen, J. Sherson, N. Zinner,and J. J. Arlt, Phys. Rev. Lett. , 163201 (2016).[40] K. Kato, Y. Wang, J. Kobayashi, P. S. Julienne, and S. Inouye, Phys. Rev. Lett. , 163401(2017).[41] R. A. W. Maier, M. Eisele, E. Tiemann, and C. Zimmermann, Phys. Rev. Lett. , 043201(2015).
42] R. Pires, J. Ulmanis, S. H¨afner, M. Repp, A. Arias, E. D. Kuhnle, and M. Weidem¨uller, Phys.Rev. Lett. , 250404 (2014).[43] S.-K. Tung, K. Jim´enez-Garc´ıa, J. Johansen, C. V. Parker, and C. Chin, Phys. Rev. Lett. , 240402 (2014).[44] J. Ulmanis, S. H¨afner, R. Pires, E. D. Kuhnle, Y. Wang, C. H. Greene, and M. Weidem¨uller,Phys. Rev. Lett. , 153201 (2016).[45] J. Ulmanis, S. H¨afner, R. Pires, F. Werner, D. S. Petrov, E. D. Kuhnle, and M. Weidem¨uller,Phys. Rev. A , 022707 (2016).[46] J. Johansen, B. J. DeSalvo, K. Patel, and C. Chin, Nat. Phys. , 731 (2017).[47] X. Xie, M. J. Van de Graaff, R. Chapurin, M. D. Frye, J. M. Hutson, J. P. D’Incao, P. S.Julienne, J. Ye, and E. A. Cornell, Phys. Rev. Lett. , 243401 (2020).[48] J. H. Macek and J. Sternberg, Phys. Rev. Lett. , 023201 (2006).[49] E. Braaten, P. Hagen, H.-W. Hammer, and L. Platter, Phys. Rev. A , 012711 (2012).[50] Y. Nishida, Phys. Rev. A , 012710 (2012).[51] M. A. Efremov, L. Plimak, M. Y. Ivanov, and W. P. Schleich, Phys. Rev. Lett. , 113201(2013).[52] S. Zhu and S. Tan, Phys. Rev. A , 063629 (2013).[53] B. Gao, Phys. Rev. A , 050702 (2000).[54] J. Wang, J. P. D’Incao, Y. Wang, and C. H. Greene, Phys. Rev. A , 062511 (2012).[55] P. Giannakeas and C. H. Greene, Phys. Rev. Lett. , 023401 (2018).[56] B. S. Rem, A. T. Grier, I. Ferrier-Barbut, U. Eismann, T. Langen, N. Navon, L. Khaykovich,F. Werner, D. S. Petrov, F. Chevy, and C. Salomon, Phys. Rev. Lett. , 163202 (2013).[57] R. Chapurin, X. Xie, M. J. Van de Graaff, J. S. Popowski, J. P. D’Incao, P. S. Julienne, J. Ye,and E. A. Cornell, Phys. Rev. Lett. , 233402 (2019).[58] R. J. Fletcher, A. L. Gaunt, N. Navon, R. P. Smith, and Z. Hadzibabic, Phys. Rev. Lett. , 125303 (2013).[59] U. Eismann, L. Khaykovich, S. Laurent, I. Ferrier-Barbut, B. S. Rem, A. T. Grier, M. Dele-haye, F. Chevy, C. Salomon, L.-C. Ha, and C. Chin, Phys. Rev. X , 021025 (2016).[60] A. G. Sykes, J. P. Corson, J. P. D’Incao, A. P. Koller, C. H. Greene, A. M. Rey, K. R. A.Hazzard, and J. L. Bohn, Phys. Rev. A , 021601 (2014).[61] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys. Rev. Lett. , 053201 (2003).
62] J. Zhang, E. G. M. van Kempen, T. Bourdel, L. Khaykovich, J. Cubizolles, F. Chevy, M. Te-ichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon, Phys. Rev. A , 030702(2004).[63] C. H. Schunck, M. W. Zwierlein, C. A. Stan, S. M. F. Raupach, W. Ketterle, A. Simoni,E. Tiesinga, C. J. Williams, and P. S. Julienne, Phys. Rev. A , 045601 (2005).[64] T. Nakasuji, J. Yoshida, and T. Mukaiyama, Phys. Rev. A , 012710 (2013).[65] M. Waseem, J. Yoshida, T. Saito, and T. Mukaiyama, Phys. Rev. A , 020702 (2018).[66] J. Yoshida, T. Saito, M. Waseem, K. Hattori, and T. Mukaiyama, Phys. Rev. Lett. ,133401 (2018).[67] Y. Cui, C. Shen, M. Deng, S. Dong, C. Chen, R. L¨u, B. Gao, M. K. Tey, and L. You, Phys.Rev. Lett. , 203402 (2017).[68] X.-c. Yao, R. Qi, X.-p. Liu, X.-q. Wang, Y.-x. Wang, Y.-p. Wu, H.-z. Chen, P. Zhang, H. Zhai,Y.-a. Chen, and J.-w. Pan, Nat. Phys. , 1745 (2019).[69] L. Fouch´e, A. Boiss´e, G. Berthet, S. Lepoutre, A. Simoni, and T. Bourdel, Phys. Rev. A ,022701 (2019).[70] B. Zhu, H. Stephan, B. Tran, M. Gerken, E. Tiemann, and M. Weidem, arXiv:1912.01264v1.[71] M. Aymar, C. H. Greene, and E. Luc-Koenig, Rev. Mod. Phys. , 1015 (1996).[72] H. Suno, B. D. Esry, C. H. Greene, and J. P. Burke, Phys. Rev. A , 042725 (2002).[73] H. Suno and B. D. Esry, Phys. Rev. A , 062701 (2008).[74] J. Wang, J. P. D’Incao, and C. H. Greene, Phys. Rev. A , 052721 (2011).[75] O. I. Tolstikhin, S. Watanabe, and M. Matsuzawa, J. Phys. B , L389 (1996).[76] K. Baluja, P. Burke, and L. Morgan, Comput. Phys. Commun. , 299 (1982).[77] P. Naidon, S. Endo, and M. Ueda, Phys. Rev. A , 022106 (2014).[78] C. Lin, Physics Reports , 1 (1995).[79] N. P. Mehta, S. T. Rittenhouse, J. P. D’Incao, J. von Stecher, and C. H. Greene, Phys. Rev.Lett. , 153201 (2009).[80] B. R. Levy and J. B. Keller, J. Math. Phys. , 54 (1963).[81] K. Willner and F. A. Gianturco, Phys. Rev. A , 052715 (2006).[82] B. D. Esry, C. H. Greene, and H. Suno, Phys. Rev. A , 010705 (2001).[83] O. I. Kartavtsev and J. H. Macek, Few-Body Systems , 249 (2002).[84] J. P. D’Incao, C. H. Greene, and B. D. Esry, J. Phys. B , 044016 (2009).
85] H. Suno, B. D. Esry, and C. H. Greene, New J. of Phys. , 53 (2003).[86] H. Suno and B. D. Esry, Phys. Rev. A , 062702 (2009).[87] C. W. Clark, Physics Letters A , 295 (1979)., 295 (1979).