Three-body recombination of two-component cold atomic gases into deep dimers in an optical model
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] M a r Three-body recombination of two-component cold atomic gases into deep dimers in anoptical model
M. Mikkelsen, A. S. Jensen, D. V. Fedorov, and N. T. Zinner
Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark (Dated: August 21, 2018)We consider three-body recombination into deep dimers in a mass-imbalanced two-componentatomic gas. We use an optical model where a phenomenological imaginary potential is added to thelowest adiabatic hyper-spherical potential. The consequent imaginary part of the energy eigenvaluecorresponds to the decay rate or recombination probability of the three-body system. The methodis formulated in details and the relevant qualitative features are discussed as functions of scatteringlengths and masses. We use zero-range model in analyses of recent recombination data. Thedominating scattering length is usually related to the non-equal two-body systems. We accountfor temperature smearing which tends to wipe out the higher-lying Efimov peaks. The range andthe strength of the imaginary potential determine positions and shapes of the Efimov peaks as wellas the absolute value of the recombination rate. The Efimov scaling between recombination peaksis calculated and shown to depend on both scattering lengths. Recombination is predicted to belargest for heavy-heavy-light systems. Universal properties of the optical parameters are indicated.We compare to available experiments and find in general very satisfactory agreement.
I. INTRODUCTION
The Efimov effect is very well established and the en-ergies and radii of the infinitely many three-body boundEfimov states occur in geometric series corresponding towell defined scaling factors depending on the masses ofthe particles. These states have been searched for andfound, but only indirectly as enhanced decay probabil-ities when the two-body interactions are tuned to spe-cific values. Most of these findings are for three identicalatoms where the geometric scaling factor is 22 . II. NOTATION AND BASIC INGREDIENTS
In this section we sketch the details necessary to cal-culate the potential which provides the wave functionsfor recombination computations. The first part describesthe adiabatic expansion method and the dependence onmasses and scattering length parameters for a short-range real potential. The second part discusses the ex-tension to an optical potential by addition of a complexterm.
A. Hyper-spherical expansion
We utilize the formalism developed in [27, 28] totreat the 3-body problem at low energies. The parti-cle coordinates and masses are r i and m i , respectivelyfor i = 1 , ,
3. We describe the three-body systemby hyper-spherical coordinates where the all-importanthyper-radius is defined by ρ = x i + y i = P i We are interested in modelling the recombination fornegative values of all the scattering lengths, that is in the regime where no two-body bound states exist withinthe present zero-range model. We therefore need to in-troduce the final dimer states populated in the final stateof the three-body recombination process. However, theonly information we need is the rate of population, orequivalently the rate at which the three-body system dis-appears. No details of the final states are required, andthe optical model is perfect for this purpose [30]. Thismodel has recently been employed to describe three-bodyrecombination of identical bosons [31].We add an imaginary part to the adiabatic hyper-radial potential in Eq.(5) leading to non-hermiticity andnon-conservation of probability. Thus we can describethe time-dependent probability reduction as an absorp-tion process, since recombination is removal of probabil-ity from the initial three-body system. This is then di-rectly aiming for calculation of the absorption rate. Therecombination process occurs when all three particles si-multaneously are close in space and two can merge intoa dimer, while the third is necessary to conserve energyand momentum. The hyper-radius is an appropriate andconvenient measure of average distance between the threeparticles, see Eq.(1). We therefore modify the potentialin Eq.(5) at small distances, ρ < ρ cut , by2 m ¯ h V ( ρ ) = ν ( ρ ) − ρ if ρ > ρ cutν ( ρ cut ) − ρ cut − V imag · i if ρ ≤ ρ cut , (14)where ρ cut and V imag are constants characterizing theoptical potential, V ( ρ ). This structure regularizes theotherwise diverging, for ρ → 0, real part of the potentialby using a ρ -independent constant for ρ < ρ cut .The solution, f ( ρ ), to Eq.(5) is that of a complexsquare well for ρ ≤ ρ cut , that is f ( ρ ) = A sin( κρ ) , κ = q m ( E − V ( ρ cut )) / ¯ h , (15)where both κ and V are complex quantities. This solu-tion can conveniently be used as initial condition in thenumerical solution for an integration starting in ρ cut .A schematic illustration of the real part of the poten-tial is shown for three identical bosons on fig. 2. Thispotential has a short-distance attractive region, a barrierof height 0 . h / (2 ma ) located at ρ = 1 . a , and adecrease towards zero as 15 / (4 ρ ), where the latter canbe seen from Eq.(5) with ν = 2 obtained from Eq.(10),since sin( νπ/ 2) = 0 in the limit of ρ → ∞ .The small distance attractive behavior outside ρ cut ,but inside the barrier, is proportional to − /ρ . Such apotential produces a number of bound three-body states,which only is limited by the finite value of V for ρ < ρ cut .The energies of these states are related through the scal-ing in Eq.(13). Physically we can think of the recombi-nation process as related to the probability of reachingthe absorptive small distances by tunneling through thebarrier. The recombination probability is then substan-tially enhanced when one of these states has an energyequal to the total three-body energy. ρ / | a | V ( ρ / a ) V ( ρ )Bound statesResonance ρ cut FIG. 2: A schematic illustration of the real part of the opti-cal potential for three identical bosons. Both axes use natu-ral length and energy units of a and ¯ h / (2 ma ). For values ρ > ρ cut it corresponds to the adiabatic potential for identicalbosons. At smaller ρ it has a constant real value correspond-ing to the value of the adiabatic potential in ρ cut . The bluelines correspond to some of the bound states of the − /ρ potential. The dotted red line illustrates a possible resonantstate at positive energy. The scale of this illustration is un-physical, since the value of ρ cut in actual calculations tends tobe much smaller relative to the value of ρ where the potentialhas a maximum. The scattering length is decisive for potential andbound state energies. In particular, for three identicalbosons we define a ( − ) as the value for which a bound stateappears with zero energy. The recombination probabil-ity consequently peaks for the same, very small, energyclose to zero when a = a ( − ) . The parameter ρ cut turnsout to determine the value a ( − ) , whereas the strengthparameter V imag determines the shape and size of thisrecombination peak as a function of a . III. THREE-BODY RECOMBINATION The potential described in the previous section nowhas to be used to calculate the rate of probability dis-appearing in the different three-body channels. We firstdefine formally the different channels and rates, then wedescribe how to compute these rates, first by the tra-ditional method employed for identical particles, and inthe last subsection we discuss a new method intuitivelyrelated to the optical model. A. Rate equations For mass-imbalanced recombination the rate equationsfor the loss-rates are more complicated than for the mass-balanced case. For two components with densities n and n , we can have three-body systems with one distinguish-able and two identical particles. In total we then have4 different possible recombination processes, and we canin principle find the time derivative of either of the twoparticle densities which leads to 6 different recombination coefficients, denoted α , that is˙ n = − α (1)111 n − α (1)112 n n − α (1)221 n n , (16)˙ n = − α (2)222 n − α (2)112 n n − α (2)221 n n . (17)Here the indices 1 and 2 refer to the two different kindsof particles in a two-component gas. The first term inEqs.(16) and (17) corresponds to the recombination co-efficient for identical particles, whereas the last two termscorrespond to the two mass-imbalanced recombinations.We shall in the following formal derivations often imag-ine the example of a Cs-Li gas where Cs is particle 1 andLi particle 2. Then α (1)112 corresponds to the recombi-nation process between two particles of type 1 and oneparticle of type 2, and the rate relates to the change inthe particle density of type 1. Analogously for other sub-and superscripts.The rate equations only describe how many particlesdisappear from a three-body system. The final state isnot specified. We could attempt to further split the mass-imbalanced recombinations into two types depending onthe final structure of which particles form the dimer, thatis1 + 1 + 2 → → (11) + 2 . However, the optical model does not allow distinctionbetween final states, and there is also currently no wayto distinguish the two channels experimentally. Thus, α ( i ) ijk is the full recombination coefficient correspondingto the sum of these different dimer productions.It is obviously much more difficult to find n ( t ) and n ( t ) from Eqs.(16) and (17) than solving one much sim-pler equation corresponding to a gas of identical bosons.However, we assume the coefficients α ( i ) ijk are density andtime-independent, and we are able to calculate these co-efficients directly from the radial equation in the opticalmodel-modified Eq.(5). Also experimental analyses aremore direct by measuring the loss of individual particlesas function of time.We therefore do not attempt to find the full time de-pendence from Eqs.(16) and (17). It is, however, reassur-ing to estimate the time-scale at which the numericallycomputed values of α ( i ) ijk make sense. Let us first assumethat we have a single-species gas corresponding to theequation ˙ n = − α (1)111 n . The solution is n ( t ) n ( t ) = 1 q α (1)111 n ( t )( t − t ) . (18)This square root dependence holds for the single-speciesgas. With typical experimental parameters for the den-sity [19], we find that n is reduced by a factor of 2 overa time period varying from about half a second to a fewnanoseconds as the scattering length changes from -100 a to -20000 a , where a is the Bohr radius. We used herethe values of α (1)111 obtained in the calculations reportedin details in a later section of this paper.Let us then consider Eq.(16) but with n varying slowlyenough to assume it is constant. If now the α (1)112 recom-bination is dominant, we get the solution n ( t ) n ( t ) = 11 + α (1)112 n ( t ) n ( t )( t − t ) . (19)Under these assumptions, we get a half-life ranging froma few milliseconds to a few nanoseconds as the a scat-tering length grows from -100 a to -10000 a . These num-bers correspond to a Cs-Li gas, where the densities areobtained from the experiment [20] and the values of α (1)112 are from calculations reported later in this paper.If the last term is dominating in Eq.(16) we get anal-ogously an exponential time dependence, that is n ( t ) n ( t ) = exp( − α (1)221 n ( t )( t − t )) . (20)Using the densities obtained from the experiment [20],this half-life ranges from about 150 seconds to a few mil-liseconds as the a scattering length increase from about-100 a to -10000 a . The relatively large half-life here re-flects the small values of α (1)221 , which are from calculationsreported later in this paper.We shall return with more discussion on calculated val-ues and relative sizes of the recombination coefficients inEqs.(16) and (17). In general, our results for the ratecoefficients, α ( i ) ijk , combined with our simple approximatesolutions give realistic time-scales compared to experi-mental conditions. When analysing the data in a givenexperiment, a variety of methods are used. Some involvea numerical solution of an equation similar to Eq.(16)[20], while some are more elaborate taking the experi-mental variation of the temperature with time into ac-count [32].In theoretical calculations we generally find the prob-ability loss per unit time for a single three-body system,denoted J ijk , which will be refereed to as the recombina-tion rate. From this the probability loss of single parti-cles involved in the recombination can be found. Let uslook at the particle loss, corresponding to the terms inEq.(16). The number of particles 1 lost per unit time,˙ N , from one three-body system is respectively 3 J , 2 J and 1 J for the (1-1-1), (1-1-2) and (2-2-1) systems. Thetotal number of three-body systems is respectively N , N N and N N for the (1-1-1), (1-1-2) and (2-2-1) sys-tems. Equivalent arguments hold for Eq.(17) and thuswe find the total particle loss per unit time to be˙ N = − J N − J N N − J N N (21)˙ N = − J N − J N N − J N N . (22)The values J ijk depend on the volume in which the three-body systems are confined, and to obtain the α ( i ) ijk coeffi- cients, we need to transform to densities using the iden-tity n i = N i V i . By insertion of this in Eqs.(21,22) andcomparison with Eqs.(16,17) this yields the following re-lation between recombination rates and recombinationcoefficients: α (1)111 = 12 J V , α (2)222 = 12 J V (23) α (1)112 = J V , α (2)112 = 12 J V (24) α (1)221 = 12 J V , α (2)221 = J V . (25)The indices are necessary to distinguish between the fourdifferent three-body systems. The quantity V ijk J ijk , isthen independent of the volume, under the assumptionthat the volume is sufficiently large. The volume usedin theoretical calculations and the volume correspondingto specific experiments are not the same, but V ijk J ijk should be the same and thus it is the quantity for whichcomparisons between experiment and theory is meaning-ful. The difference between V ijk J ijk and the recombi-nation coefficients is the factors related to the numberof three-body systems and particles lost per three-bodyrecombination, all of which have been accounted for inEq.(23)-Eq.(25). For a non-BEC gas there is also a mul-tiplicative symmetry-factor corresponding to the numberof particle permutations P ijk , as described in [33], but fora BEC-gas this symmetry factor disappears [33, 34]. Forthree identical particles P iii = 3!, while for two identicalparticles and one distinguishable particle P iij = 2!.In the following two sections, two methods of calculat-ing J ijk for any three-body system is derived. In order toease the notation the subscripts on J and V are omitted. B. S-matrix method The recombination rate J for a given three-body sys-tem is defined from the missing probability after scat-tering on the optical potential in Eq.(14). Let us assumethe three-body wave-function asymptotically is expressedin Jacobi coordinates as a three-dimensional plane wavenormalized in a box of volume, V . We expand this wave-function in hyper-harmonic free solutions, that is [35] ψ = 1 V e i k x x + i k y y =1 V (2 π ) ( κρ ) X K i K Y ∗K (Ω κ ) Y K (Ω ρ ) J K +2 ( κρ ) , (26)where k x and k y are the Jacobi momenta characterizingthe wave function, κ = ( k x + k y ), and Ω κ denote the fivehyper-angles associated with the directions of k x and k y .The three-body energy is then defined by E = ¯ h κ m . (27)The arguments of the two hyperspherical harmonics, Y K ,are related to the spatial and momentum coordinates,respectively. The collection of angular quantum num-bers are denoted K , where we only need to specify thehyper-momentum quantum number, K . The kinetic en-ergy eigenvalue corresponding to the free wave-functionis K ( K + 4) and at asymptotically large values of ρ related to the eigenvalue of the adiabatic potentials as K = ν ( ρ → ∞ ) − 2. The asymptotic value of ν for thelowest adiabatic potential is 2, corresponding to K = 0.The radial dependence is given in terms of the spher-ical Bessel function, J K +2 , of order K + 2. The hyper-harmonics are normalized to unity Z d Ω Y ∗K ′ Y K = δ K ′ K , (28)where the delta-function express that all quantum num-bers pairwise must be equal. The large-distance asymp-totics, ρ → ∞ , is then obtained from the Bessel function,that is [36] J K +2 ( κρ ) −−−→ ρ →∞ r πκρ e iκρ − iϕ K + e − iκρ + iϕ K , (29)where ϕ K = ( K + 2) π − π . The two terms in Eq.(29)correspond to in- and out-going hyper-sperical waves, re-spectively. We now introduce a short-range optical po-tential and a unit amplitude on the in-coming wave. Theabsolute value of the out-going amplitude is then asymp-totically allowed to differ from unity. From Eqs.(26) and(29) we then obtain the asymptotic from the K = 0 wavefunction, that is1 V (2 π ) ( κρ ) Y ∗ (Ω κ ) Y (Ω ρ ) r πκρ S e iκρ − iϕ + e − iκρ + iϕ , (30)where 0 < | S | < 1, since the optical potential acts asa sink of probability. The hyper-radial current is definedby j ρ = − i ¯ h m (cid:18) ψ ∗ ∂∂ρ ψ − ∂∂ρ ψ ∗ ψ (cid:19) . (31)Using Eqs.(30) and (31) the missing current, ∆ j ρ = j ρ ( S = 1) − j ρ ( S ), is calculated for a given value of S . This amounts to∆ j ρ = 1 V ¯ hm κ (cid:18) πκρ (cid:19) (1 − | S | ) |Y ∗ (Ω κ ) Y (Ω ρ ) | . (32)To get the probability loss per unit time, J , we need tointegrate the missing current over the hyper-surface. Thesurface element must correspond to the physical volumeelement (see [35]). The volume element is thus given by(see Eqs.(2) and (3)): d r x d r y = (cid:18) µ i µ jk (cid:19) / d x d y = m (cid:0) m i + m j + m k m i m j m k (cid:1) / ρ dρd Ω ρ , (33) where we do not have to distinguish between choice ofJacobi coordinates, since the mass factor is symmetricunder exchange of i , j and k . To take into account thedegeneracy of the initial states for a given κ we need toaverage over Ω κ , which we do by integrating over theangles and dividing by R d Ω κ = Ω = π . For a given κ (or energy) , and under the assumption that we are atlarge values of ρ , we then get J = Z m (cid:0) m i + m j + m k m i m j m k (cid:1) / d Ω κ Ω ρ d Ω ρ ∆ j ρ . (34)Exploiting the orthonormality of the hyperspherical har-monics, the missing probability per unit time is then J = m (cid:0) m i + m j + m k m i m j m k (cid:1) / ¯ hκm (cid:18) πκ (cid:19) (1 −| S | ) 1 V . (35)In order to obtain the recombination coefficients we mustmultiply by V . This gives JV = m (cid:0) m i + m j + m k m i m j m k (cid:1) / π ¯ h m (1 − | S | ) E , (36)which corresponds to the general formula for N-body lossgiven in [33]. C. Decay-rate method In this section we present an alternate way to obtainthe recombination rate J for the three-body system. Thisis done by deriving the decay rate of bound states inthe optical potential. First we define the bound statesin a large box with hyper-radius extending from zero to ρ max . The boundary condition is that the wave functionis zero at the edge of the box, that is f ( ρ max ) = 0. Theeigenvalues for the optical potential are complex numbers E = E − i Γ . (37)The imaginary component of the energy describes the de-cay rate of the probability as seen from the time evolutionof the wave-function, defined by | f ( ρ, t ) | = | f ( ρ, t = 0) | exp( − Γ t/ ¯ h ) . The decay rates described by Γ = J ¯ h are three-bodyenergies determined by a box boundary condition. Theoverall energy dependence is then a strong decrease, in-versely proportional to the hyper-radial three-body vol-ume, towards zero as function of box radius ρ max . In or-der to obtain the recombination coefficient Γ¯ h V we thenneed to know V . This volume, V , is defined by equatingtwo ways of calculating the density of three-body statesin the hyper-spherical box extending to ρ max . The first isthe formal expression of integration over given intervalsof coordinates and conjugate momenta, where p x = ¯ h k x and p y = ¯ h k y . The second is direct numerical calcula-tion of the same quantity from the solutions to the hyper-radial equation. The resulting equation is then Z d x d y d p x d p y δ ( E − ( p x + p y ) / (2 m )) = V ′ Ω ¯ h κ dκ/dE = (2 π ¯ h ) dν/dE (38)where V ′ = R d x d y is the volume for the Jacobi co-ordinates, Ω κ dκ/dE is the volume element per unitenergy in momentum space, and dν/dE is the densityof states for a given energy, E , defined by Eq.(27). Thefactor 2 π ¯ h = h is Plancks constant, which is the volumeoccupied by each quantum state.Using Eq.(33) we express V ′ in terms of the physicalvolume, V , that is V = Z d r x d r y = V ′ m (cid:0) m i + m j + m k m i m j m k (cid:1) / . (39)Using Eqs.(38), (27) and (39) we then get V = m (cid:0) m i + m j + m k m i m j m k (cid:1) / E dνdE (cid:18) ¯ h (2 π ) m (cid:19) , (40)where the value of dνdE can be found from solving thehyper-radial equation numerically.We emphasize that the lowest hyper-radial equationonly accounts for both total and partial-wave angularmomentum zero states, while the employed phase-spaceidentity includes all states, independent of angular mo-mentum. The derived relations therefore strongly assumeexcitation energies sufficiently small to exclude contribu-tions from all solutions build on the repulsive higher-lyingadiabatic potentials.The squared volume V scales as V ∝ ρ max , andmeaningful recombination coefficients, independent ofbox size, are therefore only achieved when Γ is propor-tional to ρ − max . This is very demanding for numerical cal-culations, since convergence only is achieved when ρ max is larger than the scattering lengths.The recombination coefficients, expressed in terms ofΓ, are then found from JV = m (cid:0) m i + m j + m k m i m j m k (cid:1) / π ¯ h m E dνdE Γ . (41)Eq.(41) is equal to Eq.(36) then. This leads to the im-mediate conclusion that for an energy E correspondingto an allowed eigenvalue the following relation betweenthe parameters in the two methods should holdΓ = 12 π dEdν (1 − | S | ) . (42) D. Recombination coefficients and finitetemperature One immediate conclusion which is readily obtainedfrom Eq.(23)- Eq.(25)is that α (2)112 = 12 α (1)112 , α (2)221 = 2 α (1)221 , (43)which is reassuring, as for example one particle of type2 disappears for each particle of type 1 in the 1-1-2 re-combination. For identical particles the mass factor in JV (Eq.(36) and Eq.(41)) reduces to m i , where m i isthe physical mass. Numerically, we find that the tun-nelling probability 1 − | S | (for an analytical estimateof this, see [39, 40]) and the decay rate Γ depends onthe physical mass as m i , however, leading to the iden-tity α (1)111 ≈ m m α (2)222 . It is approximate since the Efimovpeaks can have different locations and shapes in the twosystems, leading to non-systematic differences betweenΓ or equivalently S in the two systems. This mass-dependence for identical bosons is in correspondence withearlier work [37].In order to compare our calculations with experimentaldata we need to fold our calculated values of α ( i ) ijk ( E ) witha temperature distribution. The normalised Boltzmanndistribution for 3 particles is given by [31] h α ( i ) ijk i T = 12( k B T ) Z E e − EkbT α ( i ) ijk ( a, E ) dE , (44)where the E factor arises from the three-body phase-space. In order to get good results it is necessary tocalculate α ( i ) ijk in a range of energies around k b T , wherethe integral receives contributions.The highest excitation energies allowed in our low-energy model are given by the energy difference betweenstates build on the first and the neglected second andhigher adiabatic potentials. This energy difference canbe estimated by the difference in potential energies atdistances where the states are located. Since the scatter-ing length is a measure of the sizes of all these Efimov-like states, we use the hyper-radius ρ ≈ a to give alower estimate of the maximum temperature allowed inrealistic calculations. The energy difference between firstand second adibatic potential is for interaction free statesgiven by ¯ h K ( K + 4) / (2 mρ ) where K = 2 [27]. The re-sult of these estimates are temperatures of the order of ≈ µK , which therefore is an upper temperature limitfor realistic calculations.With the temperature distribution implemented byEq.(44) the two methods, that is the S-matrix and the thedecay rate, give essentially the same results for the samechoice of ρ cut and V imag , aside from numerical inaccura-cies. We shall use whichever method is the most conve-nient in the practical calculations. For technical reasonsit is for example generally more convenient to implementthe zero-energy limit by the decay rate method, while itis easier to implement the temperature distribution forthe S-matrix method. IV. PARAMETER DEPENDENCE The recombination coefficients for our two-componentsystem depend on three parameters, that is one massratio and two scattering lengths. We use the notationfrom fig. 1 where the particles 1 and 3 are identical,and 2 is distinctly different. To illustrate the effects weinvestigate first the variation of the adiabatic potentialwhich is the crucial ingredient in all the calculations. Be-fore we investigate the dependence of the recombinationon physical parameters, we investigate the dependenceon the optical model parameters. Then we investigatethe dependence on scattering lengths for masses of sys-tems where experimental results are available. Finallywe compare the different recombinations that can occurin a two-component gas. The factors corresponding toa non-BEC gas are used for all the recombination coeffi-cients.The numerical calculations are technically simple andemploy only homemade standard programs. First thelowest adiabatic potential is found numerically from thecomplex angular eigenvalues, which in tur is found bysolving Eq.(6). The absorption probability is then calcu-lated for different energies from the probability reductionof a plane wave reflected by the optical model poten-tial. This involves solving Eq.(5), at a given energy, forthe modified potential Eq.(14). This is done by numeri-cal integration with initial conditions in ρ cut provided byEq.(15). The result are then fitted to Eq.(30) in order toextract S . The complex eigenvalue of the optical po-tential is calculated by the shooting method, by requiringthat the wave-function is zero at ρ max . The numericalintegration is implemented as in the above S -matrix cal-culations. A. Adiabatic potentials The masses only enter as ratios of masses through the φ ij functions in Eq.(9), that is for our case this leavesonly one parameter, R = m m . We show computed massdependence of adiabatic potentials in fig. 3 as functions ofhyper-radius measured in units of the scattering length, a of the distinguishable particles. The two figures showresults for a vanishing and a relatively large scatteringlength between the identical particles a . We see thatall potentials asymptotically approach / ρ , as in the caseof identical particles.For both values of a we find that increasing R leadsto lower barrier height, and a shift to smaller hyper-radiiof barrier position and ρ -value, where the potential iszero. The dependence is strongest for two heavy and onelight particle, that is for R > 1. Continuing to R < R = 1 curve. Increasing | a | from zero reducesthe peak heights and move the potentials towards largerhyper-radii. ρ | a | ν − / ( ρ / a ) R=1R=2R=4R=8 ρ | a | ν − / ( ρ / a ) R=1R=2R=4R=8 FIG. 3: The lowest adiabatic potentials as function of ρ/a ,calculated from Eqs.(12) and (11). The lower figure is for a = 0 and the upper figure is for a = a . ρ | a | ν − / ( ρ / a ) FIG. 4: The lowest adiabatic potentials as function of ρ/a . The red graph corresponds to Eq.(12). Theother graphs, in descending order, corresponds to a =0 . , . , . , . , . a where a is negative, calculatedfrom Eq.(11). Experimental data are available for a Cs-Li gas wherethe mass ratio is R = 22 . 28. The dependence on thescattering length, a , is shown in fig. 4 for different val-ues of a . The potential is rather small for this valueof R , but still with a distinct barrier where the heightdecreases with increasing values of | a | .To complement we show the dependence on a in fig. 5for different values of a . The barrier decreases withincreasing a /a from a rather large value for vanishing | a | , towards rather small values. For large values of | a | the barrier has become so small and moved so far ρ | a | ν − / ( ρ / a ) FIG. 5: The lowest adiabatic potentials as function of ρ/a .These graphs, in descending order, corresponds to a =0 . , . , . , . , . , . , . , . a where a is neg-ative, calculated from Eq.(11). to the right, that it is no longer visible on the scale chosenfor the figure.These figures show that increasing | a | and | a | hasthe same qualitative effect as increasing the scatteringlength in the case of identical bosons for the opticalmodel, see fig. 2. It decreases the height of the barrierand moves the location of its maximum to the right. B. Optical model The real potential is completely determined for zero-range two-body interactions in terms of scatteringlengths and masses. The box-like imaginary potentialhas hyper-radius and strength as the two phenomenolog-ical parameters. This radius can only assume discretevalues determined by the requirement that the scatter-ing length is reproduced for the occurrence of a specificrecombination peak. The corresponding strength corre-latedly varies shape and size of the chosen peak. On fig. 6the results of varying V imag , while ρ cut is a constant valueof 1 . a are shown. −28 −26 −24 −22 −20 −18 −16 | a | [ a ] α ( ) h c m s i V imag = 16 ¯ h ma V imag = 160 ¯ h ma V imag = 1 . ¯ h ma FIG. 6: The recombination coefficient for Cs-Cs-Cs as func-tion of the scattering length a for different choices of V imag ,at ρ cut = 1 . a We see that increasing the size of V imag makes the peaks more pronounced while also making the absolutevalue for the rest of the graph somewhat smaller. De-creasing the size of V imag has the opposite effect. Thisis the exact same qualitative effect which was seen whenvarying the absorption parameter n ∗ describing deeplybound states in [24].From one set of parameters another equally goodchoice is found by scaling the square well hyper-radiusup or down by the Efimov factor and at the same timescaling the strength in opposite direction by the square ofthe same Efimov factor. Using these parameters repro-duces exactly the same shape of the Efimov resonanceand the same absolute value of the recombination. Thismeans that V imag ρ cut for a given peak has a constantvalue where ρ cut can discretely vary by the Efimov fac-tor. The largest allowed hyper-radius must be at a hyper-radius where the real potential precisely has the inverseradial square behavior. The lowest value is only limitedby the requirement that it must be finite, since otherwisethe already removed zero-range divergency reappears. Inpractical calculations we generally pick values in the or-der of a to be certain that we do not approach any ofthese limiting cases. C. Recombination For identical particles we know that the recombinationcoefficient is proportional to the fourth power of the scat-tering length. For a two-component system this relationmust be extended to account for two different scatteringlengths as well as for a mass-ratio dependence. We shallillustrate with two mass ratios R = 22 . R = 2 . Li- Cs [20, 21] and K- Rb gases. We used theseisotopes to have well-defined mass ratios in the calcula-tions. The characteristic small mass variations betweenisotopes would only marginally change the results, pro-vided the boson-fermion characters remain the same orbecome irrelevant as when only one identical atom par-ticipates in the process. Here we shall only investigatethe pure mass dependence.The dominating recombination coefficient is related to α (1)112 = 2 α (2)112 , where label 2 corresponds to the light par-ticle. This means that we here consider the heavy-heavy-light three-body systems with mass ratios, R = 22 . R = 2 . 23. We calculate all the recombination coefficientsin the limit of zero three-body energy.The periodic structure of enhanced recombination oc-curs each time the scattering length is multiplied by theEfimov scaling factor, s = exp( π/ | ν ( ρ = ∞ ) | ), found forinfinite scattering lengths of the contributing systems.This scaling parameter depends first of all on the massratio. The Efimov effect requires that | a | = ∞ , while | a | can assume any finite or infinite values. The resultsfor the two limiting cases, | a | = 0 and ∞ , are given intable I as a function of mass ratio, R . For large R the twocases are almost identical, but there is a big difference for0small mass-ratios and this trend continues for R < 1. Ifall three scattering lengths are infinitely large the scalingfor R = 1 is s = 22 . R approaches a con-stant of s = 15 . 7. For a more detailed discussion of thetwo different cases see [26]. TABLE I: Efimov scaling factor s for different values of R R = m m a = a = ∞ a = ∞ , a = 0 1986 153.8 23.3 9.76 6.64 5.25 The influence of a can conveniently be studiedthrough the recombination of the two chosen systems.For R = 22 . a = 0 and 4.7989 for | a | = ∞ and the peakpositions should therefore remain. In contrast the scalingbetween peaks for R = 2 . 23 should move between the ex-treme limits of 121.1 for a = 0 and 20.28 for | a | = ∞ as a function of a . −26 −24 −22 −20 −18 −16 | a | [ a ] α r e c , h c m s i a = 0 a = − a a = − a a = − a a = − a FIG. 7: The recombination coefficient, α (1)112 , as a function of a , for different values of a , where R = 22 . 2. The param-eters of the optical potential are V imag = 68 ¯ h ma , with ρ cut in the interval, 0.24-0.32 a as a varies from 0 to -20000 a ,adjusted to maintain the position of the first Efimov peak at a ≈ − a . Let us first focus on the recombination coefficient, α (1)112 ,for a large mass ratio of 22 . 28. We calculate α (1)112 asa function of a , for different constant values of a .The strength of the optical potential is V imag = 68 ¯ h ma ,whereas ρ cut is adjusted slightly to reproduce the peakposition | a ( − )1 | ≈ − a , corresponding to the experi-mental peak position of Cs-Cs-Li [20, 21], for the differentvalues of a .The main results of these calculations are summed upin fig. 7, where we show the coefficient as a function of a for different values of a . The most striking fea-ture of fig. 7 is that the different values of a lead toa different overall dependence of α (1)112 as function of a .For small values of | a | there is a a dependence cor-responding to the a scaling for identical bosons, but forbigger values of | a | this relation is no longer valid. As | a | grows bigger, the recombination is enhanced, which is most visible at smaller values of | a | . This is consis-tent with larger values of each of the scattering lengths, | a | and | a | , lowering the potential barrier, making iteasier to reach small distances and thereby enhancing therecombination.We also know that a has a stronger influence on thepotential than a , which corresponds well with a be-ing the most important parameter for the recombinationshown in fig. 7. Specifically, we see that α (1)112 changesmore as a runs from 0 to -20000 a , than it does when a varies from 0 to -20000 a .All in all, for realistic values of the scattering lengths,we can view a as moderating the a dependence andsimultaneously enhancing the recombination. More vio-lent changes of a cannot be excluded when the interac-tions are controlled by the Feshbach resonance technique.However, this is not the case in any of the physical sys-tems discussed here. −28 −26 −24 −22 −20 −18 −16 | a | [ a ] α ( ) h c m s i a = 0 a = − a a = − a a = − a a = − a a = − a FIG. 8: The recombination coefficient, α (1)112 as a function of a calculated for different values of a , where R = 2 . 23. Thestrength of the potential is as in fig. 7 while ρ cut is adjustedto give one peak at about a ≈ − a . We now move to the mass ratio R = 2 . 23 where theEfimov scaling has a strong dependence on a as seenfrom the extreme limits given in table I. The relativepositions of the recombination peaks must then vary withthe finite value of a . Again we calculate α (1)112 as afunction of a for different values of a . The results ofsuch a series of calculations are shown in fig. 8, where ρ cut is adjusted to produce the same somewhat arbitrarypeak position at about | a ( − )1 | ≈ − a for all a . Thenext Efimov peak then has to move in the interval of a from 20 . · a ( − ) ≈ − a and 121 . · a ( − ) ≈ − a . TABLE II: The ratio, a ( − )2 /a ( − )1 , between the a scatteringlengths for the first two peaks as function of a when R =2 . 23. The last column give the related ρ cut values. a [ a ] 0 -1000 -5000 -10000 -20000 -50000 a ( − )2 a ( − )1 ρ cut [ a ] 1.21 0.97 1.14 1.18 1.2 1.22 The most noticeable thing in fig. 8 is that the loca-1tion of the second Efimov resonance changes with thefinite value of a . This variation of the ratio between a values of second and first peak is shown in table IIfor different a values. The rather modest variation of ρ cut is also shown in this table. We notice the correctcontinuous variation between the two extreme limits ofthe Efimov scaling with a , although the value of ρ cut for a = 0 reflects the singularity moving between Eqs.(10)and (12).We emphasize that small changes of ρ cut is fine-tuningto maintain the first peak at the same position. Themuch larger shift of the second peak is entirely due tothe variation of a , and essentially completely indepen-dent of the parameters of the imaginary potential. Asubstantial change in the second peak position requiresa value of a ≈ − a . However, the change is fairlygradual and there is no ”magic value” where a sud-denly begins to contribute. The ideal Efimov scaling forthree resonant interactions of 20.28 is essentially reachednumerically when a = − a .The second prominent feature in fig. 8 is that a finitevalue of a modifies the overall dependence of α (1)112 as afunction of a . This is the same qualitative feature asseen in fig. 7 for the large mass ratio R = 22 . 2. Finite val-ues of a leads to a higher absolute value of α (1)112 within ahuge a interval, a ∈ [0 , − a ]. This means thatincreasing the size of | a | enhances the recombinationcoefficient, like we found for the R = 22 . D. Different recombination processes We have so far only considered recombination fromthree-body systems with two heavy and one light parti-cle such as Rb-Rb-K and Cs-Cs-Li. However, the sametwo-component gas can also decay by the other 3 combi-nations, two light and one heavy particle (Rb-K-K, Cs-Li-Li), and 3 identical heavy (Rb-Rb-Rb, Cs-Cs-Cs) orlight particles (K-K-K, Li-Li-Li). In order to estimate theimportance of the terms in Eqs.(16) and (17), it is nec-essary to calculate all these recombination coefficients.To make a realistic comparison we first consider the Cs-Li gas, where the Efimov resonances are fixed by experi-mental data for Cs-Cs-Li [20], Cs-Cs-Cs [18] and Li-Li-Li[17], and the parameters of Li-Li-Cs are assumed to bethe same as for Cs-Cs-Li. We assume that a = a = 0.This would underestimate the recombination coefficientsas shown in the previous subsection, but it allows on theother hand a clean comparison where the fourth powerdependence applies, α (1) ijk ∝ a . Actual correlated finitevalues of these scattering lengths obtained through theFeshbach technique could then be important and quan-titatively alter the comparison.On figure 9 all the recombination coefficients for theCs-Li gas have been plotted. The optical model param-eters are chosen to reproduce the position of the lowestmeasured recombination peaks [17, 18, 20]. The exper-imental data for Li is actually for Li [17], but our aim is here only to test the mass dependency. In the Cs- Li experiment, Li-Li-Li and Li-Li-Cs recombinations areexpected to be suppressed due to Fermi statistics, whichis not taken into account in our model. So the resultsare not directly comparable to the experiment. In ad-dition this is not a direct comparison between the mass-balanced and mass-imbalanced case, since the recombina-tion coefficient is a function of different scattering lengthsin the different cases.Furthermore, for a = 0 it was not possible to lo-cate an Efimov resonance within the range of a ∈ [0 , − a ] for neither the Li-Li-Cs nor the K-K-Rbmass ratio, even though a wide range of ρ cut were tested.This is because the Efimov scaling factor now is so bigthat it is hard to locate an interval with even one Efi-mov resonance. In these comparisons we shall focus onrecombination coefficients for zero energy. −30 −28 −26 −24 −22 −20 −18 | a ij | [ a ] α ( ) i j k h c m s i α (1)112 ( a ) α (1)221 ( a ) α (1)111 ( a ) −30 −28 −26 −24 −22 −20 −18 −16 | a ij | [ a ] α ( ) i j k h c m s i α (2)112 ( a ) α (2)221 ( a ) α (2)222 ( a ) FIG. 9: The top shows the recombination coefficients result-ing in Cs loss, and the bottom shows the recombination co-efficients resulting in Li loss, both as a function of a . Theoptical parameters in the calculations are ρ cut = 0 . a and V imag = 240 ¯ h ma is used for Cs-Cs-Li and Li-Li-Cs. For theCs-Cs-Cs recombination ρ cut = 1 . a and V imag = 16 ¯ h ma areused and for Li-Li-Li ρ cut = 0 . a and V imag = 68 ¯ h ma areused. The absolute sizes on fig. 9 show that the Cs-Cs-Lirecombination is much more likely than the Li-Li-Cs re-combination for the same value of a . The Cs-Cs-Csrecombination only depends on a , which is used as the x -coordinate for this process in fig. 9. This scatteringlength is expected to be of less importance compared to a in the mixed recombination coefficients, and therefore2assumed to be zero in those estimates.The comparison is then not straightforward but stilluseful, since a finite value of a would increase α (1)112 be-yond the curve in fig. 9. Thus we deduce that α (1)112 ( a =0 , a ) ≫ α (1)111 ( a = a ), and we therefore believe thatthe Cs-Cs-Li recombination is much more likely to oc-cur than the Cs-Cs-Cs recombination in realistic systems.We also see in fig. 9 that α (2)222 ( a = a ) ≫ α (2)112 ( a =0 , a ). This does not allow any conjecture about relativesizes in a realistic system because the intra-species scat-tering lengths usually are much smaller than the neces-sary large (for the Efimov effect) inter-species scatteringlength. The recombination coefficients between identicalparticles are then expected to be relatively small. −28 −26 −24 −22 | a | [ a ] α ( ) i j k h c m s i α (1)112 α (1)221 FIG. 10: Recombination coefficients for R = 2 . 23 as a func-tion of a . The optical parameters are ρ cut = 2 . a and V imag = 20 ¯ h ma for both systems. To complement we now investigate the much smallermass ratio R = 2 . 23 (corresponding to a Rb-K gas). Theresults are shown on fig. 10 for different recombinationprocesses. The optical model parameters are chosen togive a peak for a ≈ − a in the α (1)112 coefficient.As for the larger mass ratio we again conclude that theheavy-heavy-light recombination process is more likelythan the ligt-light-heavy recombination process.Figs. 9 and 10 also lead to another conclusion. Abigger mass-ratio seems to give a bigger recombinationcoefficient for two heavy particles and one light. Thisis in accordance with a corresponding decrease of size ofthe adiabatic potentials, which intuitively suggests higherprobability for recombination at small hyper-radii.A bigger mass-ratio also seems to give a somewhatsmaller recombination coefficient for two light particlesand one heavy particle, although not as pronounced.This means that the bigger the mass-ratio, the bigger thedifference between the heavy-heavy-light recombinationand the light-light-heavy recombination. For both smalland big mass-ratios, the α ( i )221 terms can be neglected inEqs.(16) and (17) based solely on these mass-related ar-guments.We emphasize these conclusions are only strictly validfor small | a | and | a | . Finite values imply more compli- cated relations between the different recombination coef-ficients which only can be determined by taking the Fes-hbach resonances of a specific system into account. So intheory, it is possible to have a specific system where thelight-light scattering length | a | is much bigger than theheavy-heavy scattering length | a | , which in turn leadsto dominance of the light-light-heavy over the heavy-heavy-light process. V. COMPARISON WITH EXPERIMENT In this section we will confront our theoretical resultswith recent experiments that have been carried out for a Cs- Li gas [20, 21], as well as with a recent experimentfor a Cs gas, in which a second Efimov resonance hasbeen observed [19]. The variable parameter in experi-ments is the magnetic field B , which via the mechanismof Feshbach resonances can be used to change the scatter-ing lengths, a . The phenomenological relation between a and B is a ( B ) = a bg (cid:18) B − B (cid:19) (45)where ∆ , a bg and B are determined experimentally foreach individual system. Eq.(45) applies for most systemsand will be used in the cases investigated in this section.We first focus on the equal mass process after which wecontinue with the dominating heavy-heavy-light recom-bination process. A. The Cs- Cs- Cs recombination Recently a second Efimov resonance was observed in a Cs gas [19], and we can conveniently test the modelagainst this experimental confirmation of the original Efi-mov scenario. The first Efimov resonance for a Csgas was observed earlier [18] to have a peak for a ( − ) ≈− a . The experimental data from these measure-ments also allow us to adjust both ρ cut to give the peakposition and subsequently tune the strength, V imag , tothe shape of this first peak. With the scaling mass, m , of Cs, ρ cut = 1 . a and V imag = 16 ¯ h ma , the first Efimovresonance is then rather well reproduced.We show the calculated results on fig. 11, where thesecond Efimov peak is obtained without any adjustmentsbeyond the first peak. The experimental results are notcompletely commensurable, since the first experimentwas done at a temperature of about 15 nK, and thesecond experiment at 9.6 nK. We can circumvent thisby using the fact that temperatures of this value are in-significant at small values of the scattering length. Thedata for the first peak from the first experiment [18] cantherefore be assumed to arise for the same temperatureof 9.6 nK as the second peak in the second experiment3 −26 −24 −22 −20 −18 | a | [ a ] α ( ) h c m s i Experimental data at T=9.6 nKT= 1 nKT=9.6 nK FIG. 11: The recombination coefficient for Cs-Cs-Cs, α (1)111 , atdifferent temperatures plotted with experimental data from[18, 19]. The optical model parameters are ρ cut = 1 . a and V imag = 16 ¯ h ma . [19]. Both temperatures are much smaller than the up-per temperature limit estimated to be 1 µK for a realisticcalculation.The absolute value around the first peak and forbig a is generally remarkably close to the experimentalvalue. However, the calculated recombination coefficientis much too small at small a , where the temperature hasno influence. The theoretical results follow the overall a rule of recombination dependence and these small a deviations must therefore arise from other processes con-tributing to the experimental values.The calculated temperature dependence for largerscattering lengths is in very good agreement with themeasurements. This is remarkable, since a reduction intemperature to 1 nK results in a fairly dramatic changeof the recombination curve at large values of a . The con-sequence is that a moderate temperature of a few nK al-ready smears out the second Efimov peak, and prohibitsobservation. We then conclude that using the correcttemperature gives a shape that is in pretty good agree-ment with the experimental results. However, the secondpeak is dislocated compared to the ideal Efimov scalingfactor of 22.7, which predicts this peak to be at around a = − a . In the experiment, it is found at ap-proximately − a , while the calculation gives thepeak at − a .Overall, the results of this comparison with experi-ments for identical bosons are encouraging. The tem-perature effects are well accounted for and the shape andlocation of the second Efimov peak is also in broad agree-ment with data, for phenomenological parameters fittedto the first peak. Our model seems to work for the well-known mass-balanced case, and comparison to data formass-imbalanced systems should then be considered. B. The Cs- Cs- Li recombination The crucial parameter is the scattering length, a = a L iCs , which has to be very large to provide the Efimoveffect. However, in addition also a = a C sCs is impor-tant for quantitative predictions. The overall a scalingis modified for a finite value of a , which is determinedby the magnetic field through the Feshbach resonance ofthe system as described in Eq.(45). It is then interest-ing to look for effects in the recently obtained two setsof experimental data [20, 21]. They are in broad agree-ment, although the details are a little different. In [21]3 Efimov resonances are reported, where the third oneis very hard to distinguish from the background due tofinite temperature effects. In [20] the recombination coef-ficient is given as a function of a , which allows an easycomparison with our calculations.The experimental conditions in [20] provide the pa-rameters in Eq.(46) for the interspecies (Cs-Li) Feshbach-resonance of the prepared spin-states, that is a bg = − . a , ∆ = 61 . G , B = 842 . G , (46)which by insertion in Eq.(45) gives the variation of a .The a scattering length is estimated to run between − a and − a [20], so we have chosen a constantintermediate value of a = a CsCs = − a . −24 −22 −20 −18 | a | [ a ] α ( ) h c m s i Experimental data at T=450 nKSystematic uncertaintyT=1 nK, a = − a T=1 nK, a = 0T=450 nK, a = − a T=450 nK, a = − FIG. 12: The recombination coefficient for Cs-Cs-Li, α (1)112 as afunction of the interspecies scattering length, for a = 0 and a = − a , plotted with the experimental data from [20].The optical model parameters are ρ cut = 0 . a , V imag =100 ¯ h ma for a = − a and ρ cut = 0 . a , V imag =240 ¯ h ma for a = 0. One small temperature of 1 nK and theexperimental temperature of 450 nK are shown. We compare theoretical and experimental results infig. 12 where both temperature and a dependence areshown. We first note that there is a pretty good corre-spondence between the experimental data at 450 nK andthe theoretical prediction at 450 nK with the suggestedfinite value of a . The shape is nicely reproduced andthe absolute value is within the systematic uncertainty.We emphasize that the calculations for vanishing a = 0does not describe the data quantitatively at 450 nK. Theabsolute value and the shape is only correct when the4enhancement of α (1)112 at small values of a , due to finitevalues of a , is taken into account. We also draw atten-tion to the fact that for a substantial finite value of | a | , α (1)112 is no longer independent of energy at small valuesof | a | . The temperature of 450 nK is not far from 1 µK ,that is upper temperature limit in a realistic calculation.But the data is described very well.It seems like the calculated values of α ( i ) ijk describe therelative behaviour of different recombinations well. Therecombination coefficient for Cs-Cs-Li is larger than forCs-Cs-Cs for both the experimental and the calculatedvalues. The calculated values of both are within experi-mental uncertainty. The relative location of the Efimovresonances depends on the finite value of a . For theCs-Cs-Li system the limiting values are 4.8766 or 4.7989,however, which means that we do not expect any note-worthy difference. The 3 Efimov resonances in [21] arewell suited for testing the theoretical prediction for theEfimov resonances.We use the value, B = 942 . 75, of the resonant mag-netic field. The two reported different values of B arewithin the experimental uncertainties of both experi-ments. In table III, we compare predicted and exper-imental values for the Efimov peaks when the opticalmodel parameters are adjusted to reproduce either first( ρ cut = 0 . a ) or second peak ( ρ cut = 0 . a ). TABLE III: The values of ∆ B = B − B determined to givethe first or second Efimov peak for Cs-Cs-Li, while the othertwo peaks ( second or first as well as third) are predicted withthat set of corresponding optical model parameters.1st peak 2nd peak 3rd peakExperimental value of ∆ B [G] 5.61(16) 1.07(2) 0.22(4)∆ B [G] fitted to 2nd peak 5.8 1.07 0.21∆ B [G] fitted to 1st peak 5.65 1.03 0.2 Then in both cases the third peak is in agreement withcalculations within the experimental uncertainty. How-ever, this does not say much, since a small change inmagnetic field results in a huge change in the scatter-ing length, see Eq.(46) and Eq.(45). Due to this, it isalso hard to determine an accurate ρ cut value which re-produces the very weak third peak. The two predictedpeaks in the two fits are at the edge allowed by the ex-perimental uncertainty, and as such they are still veryclose to the observed values. The finite temperature ef-fects are also likely to move the locations slightly, whichmight explain smaller deviation.We can now calculate the inter-species scatteringlengths corresponding to the experimental peak valuesby using Eq.(46) with B = 942 . 75, which yields | a ( − )1 | =331 . a , | a ( − )2 | = 1621 a and | a ( − )3 | = 7777 . a . Thisgives a Efimov scaling of 4.8911 between the first twopeaks and 4.7980 between the last two peaks. These val-ues are in agreement with the Efimov scaling found bygoing to the universal limits of infinite scattering lengths, 4.8766 or 4.7989, depending on whether there is two orthree contributing resonant interactions. C. Universal properties of optical parameters For identical bosons ref.[31] proposes a possible rela-tion between the optical model parameters and the vander Waals length of the system. This is motivated by therecent findings in equal mass systems [18] of a univer-sal relation between the three-body parameter and thetwo-body van der Waals length [41–47]. We can expandthis idea to the mass-imbalanced system. The three vander Waals lengths which we will compare with is for Cs-Cs, Li-Li and Cs-Li respectively 202 a , 65 a and 44 . a [48, 49]. The crucial parameter is ρ cut which by definitionis a hyper-radius. From the definition in Eq.(1) we cananalogously define a hyper-radial van der Waals lengthas ρ vdW = 2 R R µ m r vdW, + 1 + R / R µ m r vdW, , (47)where the distance between particles is replaced by thecorresponding two-body van der Waals lengths, and therelated two-body reduced masses, µ and µ , are intro-duced and R = m m .For identical particles with m equal to the mass of theparticles, this hyper-radius reduces to the van der Waalslength, ρ vdW = r vdW, , which in ref.[31] was comparedto ρ cut by simple division. For three identical particles wefind respectively ρ cut /ρ vdW = 0 . ρ cut /ρ vdW =0 . a , between the identicalheavy particles is relatively small. Furthermore, when a is finite and begins to contribute the recombinationpeak moves. To keep the peak location independent of a we therefore varied ρ cut slightly with a . This ad-dition to ρ cut has to come from the first term of Eq.(47)which should vary from zero to a given finite contributionwhen | a | = ∞ . Thus we can parametrize by ρ vdW = 2 R R µ m | a | ( f a + | a | R f ( R )) r vdW, + 1 + R / R µ m r vdW, , (48)where we suggest to use f = 10 . For equal masses and a → ∞ this reduces to Eq.47 for equal masses. For un-equal masses the R factor ensures that for a → ∞ , theterm still doesn’t become dominant, in correspondencewith the moderate change in ρ cut , even as big values of | a | were used numerically. We use f ( R ) = 1, but per-haps a more complicated function is required. For the5experimental value, a = − a ( ρ cut = 0 . a ), wethen get ρ cut /ρ vdW = 0 . V imag , can also be compared to thenatural unit for a van der Waals interaction, V vdW =¯ h /mρ rdW . which amounts to V imag /V vdW = 6 . · , . · , . · for Cs-Cs-Cs, Li-Li-Li, and Cs-Cs-Li, respectively. The value of m ¯ h V imag ρ cut is constantfor a given location and shape in a specific system. Thenumerical values from our calculation are for this quan-tity 31.3600, 11.4308, 6.6564 for the systems Cs-Cs-Cs,Li-Li-Li and Cs-Cs-Li.The upshot of these comparisons is that a possible ap-proximate value of ρ cut for a system can be obtainedby ρ cut ≈ . ρ vdW . In addition the value of V imag to within a factor of two seems obtainable by V imag ≈ · V vdW . Finally the value m ¯ h V imag ρ cut in differentsystems are within the same order of magnitude. VI. DISCUSSION AND OUTLOOK We have developed a method for calculating the three-body recombination coefficient of mass-imbalanced three-body systems at negative values of the scattering lengthsand for small energies. In order to do this we employedthe hyper-spherical method, zero-range potentials andthe Faddev decomposition to calculate the lowest-orderadiabatic potential.In addition we formulated and explored the opticalmodel to calculate recombination processes into deepdimers. This method introduces two phenomenologicaloptical parameters, strength and range, which respec-tively determine location and shape of the Efimov res-onances. We then developed two methods for findingthe recombination coefficients of both mass-balanced andmass-imbalanced systems. By means of the traditionalS-matrix method and by a new method, where the de-cay rate of bound states in a box due to the presence ofthe optical potential is calculated. In general the resultsobtained with the two methods are essentially indistin-guishable, although differing in numerical inaccuracies.As they tend to complement each other, we choose themost convenient method in actual calculations.The model was tested against the experimental dataon recombination coefficients in Cs-Cs-Cs and Cs-Cs-Lisystems as functions of the Feshbach tuned scatteringlengths. In both cases, after fitting the range and the strength of the imaginary potential to the first peak, themodel was able to describe quantitatively the whole curveincluding the temperature effects.The two-parameter fits of strength and hyper-radiusof the imaginary part of the optical potential are veryefficient for both position and shape of the recombina-tion peaks. It is also remarkable that the absolute valuesof the experimental recombination coefficients are repro-duced within the experimental uncertainty.Using the developed methods we have reached a num-ber of conclusions. The main conclusions are that re-combination is dominated by the heavy-heavy-light pro-cess and mainly determined by the heavy-light scatteringlength. But the heavy-heavy scattering length is impor-tant for obtaining the correct value of the recombinationcoefficients, as it enhances the recombination probabilityand modifies the behaviour as a function of the heavy-light scattering length. For a large mass ratio the Efimovscaling is determined entirely by the heavy-light scatter-ing length, but for a smaller mass ratio finite values of theheavy-heavy scattering length becomes important. TheEfimov scaling then moves continuously between valuesfrom two and three resonating subsystems.The focus of this paper has been the case where allscattering lengths are negative, but it could be just as rel-evant to investigate two negative, one positive or two pos-itive, one negative scattering length cases. Since a finitevalue of the heavy-heavy scattering length can substan-tially alter the absolute value of a given two-componentrecombination, it may be interesting to test the three-body recombination for the same system at different Fes-hbach resonances experimentally, in order to further con-firm this prediction.For small mass-ratios we find a big difference in theEfimov scaling between peaks as the heavy-heavy scat-tering length is increased. It may be fruitful to investi-gate the intermediate area between two and three reso-nant subsystems in more theoretical detail. In additiona Feshbach resonance in a two-component gas with smallmass-ratio, in which the heavy-heavy scattering length isbig in the same area as the inter-species scattering lengthwould allow investigation of this effect.The authors thank PK Sørensen for invaluable helpon the numerical details. The authors are grateful forenlightening discussions with A. G. Volosniev, N. Winter,N. B. Jørgensen, L. J. Wacker, J. F. 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