Weakly bound atomic trimers in ultracold traps

a r X i v :c o n d -m a t /0206317v 3 15 S e p 2003

Weakly bound atomic trimers in ultracold traps

M.T.Yamashita (a ),T.Frederico (b ),Lauro Tomio (c ),and A.Del?no (d )

(a )

Laborat′o rio do Acelerador Linear,Instituto de F′?sica da USP

05315-970,S?a o Paulo,Brasil

(b )

Dep.de F′?sica,Instituto Tecnol′o gico de Aeron′a utica,Centro T′e cnico Aeroespacial,

12228-900S?a o Jos′e dos Campos,Brasil

(c )

Instituto de F′?sica Te′o rica,Universidade Estadual Paulista,01405-900S?a o Paulo,Brasil (d )

Instituto de F′?sica,Universidade Federal Fluminense,24210-900Niter′o i,RJ,Brasil

(February 1,2008)The experimental three-atom recombination coe?cients of the atomic states 23Na |F =1,m F =?1 ,87Rb |F =1,m F =?1 and 85Rb |F =2,m F =?2 ,together with the corresponding two-body scattering lengths,allow predictions of the trimer bound state energies for such systems in a trap.The recombination parameter is given as a function of the weakly bound trimer energies,which are in the interval 1

The formation of molecules in ultracold atomic traps o?ers new and exciting possibilities to study the dy-namics of condensates [1].It was reported the forma-tion of Rubidium molecules 87Rb 2in a bosonic con-densate,which allowed to measure its binding energy with unprecedented accuracy [2].Ultracold Sodium molecules 23Na 2have also been formed through photoas-sociation [3].However,nothing has been reported till now about formation of molecular trimers in cold traps.The ?rst information one is led to ask is the magnitude of the binding energy of trimers in a cold trap.Two-body scattering lengths of trapped atoms are well known in several cases,as well as their closely related dimer bind-ing energy.In the limit of large scattering lengths,it is necessary to know in addition,one low-energy three-body observable to predict any other one.In this case,the detailed form of the two-body interaction is not im-portant [4,5].The recombination rate of three atoms in the ultracold limit,measured by atomic losses in trapped condensed systems,can supply the necessary informa-tion to estimate the trimer binding energy.For short range interactions,the magnitude of the recombination rate of three atoms is mainly determined by the two-body scattering length,a [6].However,it is important to remark that,still remains a dependence on one typical low-energy three-body scale [4,5].Indeed,it is gratifying to note that all the works on three-body recombination,consistently,present a dependence on a three-body pa-rameter in addition to the scattering length [7–9].The aim of the present work is to report on how one can obtain the trimer binding energy of a trapped atomic system,from the three-body recombination rate and the corresponding two-body scattering length.For this pur-pose,we use a scale independent approach valid in the limit of large positive scattering lengths (or when the interaction range goes to zero),obtained from a renor-malized zero-range three-body theory [4],which relates

the recombination rate,the scattering length and the trimer binding energy.Considering the experimental val-ues of the recombination rates and scattering lengths given in Refs.[10–13],the method is applied to predict the trimer binding energies of 23Na |F =1,m F =?1 ,87

Rb |F =1,m F =?1 ,and 85Rb |F =2,m F =?2 ,where |F,m F is the respective hyper?ne states of the total spin F .We note that the bound-states considered here are in fact high-lying resonances,not true bound states,as they can decay into lower-lying channels.

The validity of our approach is restricted to su?ciently diluted gases,because all the scaling relations are de-rived for three isolated particles.Also,when the scat-tering length is tunned via external ?eld in a trap,the parameters are di?erent from the vacuum values,and consequently our predictions only apply to that particu-lar experimental conditions.For the trapped gases that we are analyzing,the diluteness parameter ρa 3(where ρis the gas density)should not be much larger than one,otherwise one needs to consider higher order correlations between the particles.Indeed,we observe that,in gen-eral,for the analyzed condensed systems,the diluteness parameter is much smaller than one.Even in the case of 85

Rb,where the considered scattering length is obtained via Feshbach resonance techniques [13],the diluteness parameter is about 1/2.

Another relevant remark,pointed out in Ref.[14],is that the recombination into deep bound states can a?ect the theoretical results that are based on calculation of this rate into shallow states alone.This additive contri-bution depends on one more constant,beyond the three-body scale.However,the ?tted contribution of the re-combination into deep bound states,is fortunately much smaller than the contribution of the shallow bound state,as found in the case of 85Rb [14].Such evidence supports our estimatives of trimer energies,when a >0,that are obtained by only considering the contribution of recom-

bination rates into the shallow state.

The values of a are usually de?ned as large in respect

to the e?ective range r0,such that a/r0>>1.The low-energy three-boson system presents,in this limit,the

E?mov e?ect[15],where an in?nite number of weakly bound three-body states appears.The size of such states

are much larger than the e?ective range.The limit a/r0→∞can be realized either by a→∞with r0 kept constant or by r0→0with a constant.In the last case,the limit of a zero-range interaction,corresponds to

the Thomas bound-state collapse[16].In this respect, the E?mov and Thomas limits are equivalent;or,dif-ferent aspects of the same physics[17].The Thomas-E?mov connection is also reviewed in Ref.[5].In the limit a/r0→∞,the details of the interaction for the low-energy three-body system are contained in one typi-cal three-body scale and the two-body scattering length (or the dimer bound-state energy,E2);they are enough to determine all three-body observables[4].Consider-ing,for example,the trimer binding energy(E3)as the three-body scale,any three-body observable(O3)that has dimension of[energy]β,in the limit of r0→0,can be expressed as

O3=Eβ2F2(E2/E3)=Eβ3F3(E2/E3).(1) The dimensional factor in front of the above equation (1)is chosen for convenience as E2or E3.The scaling function in each case is F2or F3.The existence of the scaling limit for zero range interactions was veri?ed in Refs.[18,19].In practice,such limit is approached by the excited state of the atomic trimer obtained in re-alistic calculations,allowing as well the theoretical in-terpretation of those excited states as E?mov states[19]. Here,we observe that the binding energy E3refers to the magnitude of the total energy of the bound-system;the binding energy with respect to the two-body threshold is de?ned as S3≡E3?E2.

The rate of three free bosons to recombine,forming a

dimer and one remaining particle,is given in the limit of zero energy,by the recombination coe?cient[5,6]

K3=ˉh

E2/E3 ,considering that,for large scat-

tering lengths we have1/a=

E2/E3dependence is explicit.Therefore,

α=αmax sin2 ?1.01ln E3+? E3 ,(4)

where? mE3/ˉh2 .Our

next task is the calculation of the scaling function,by

using the renormalized subtracted Faddeev equations[4].

The three-boson recombination coe?cient at zero-

energy is derived from the Fermi’s golden-rule as

K3=

2π

(2πˉh)3|T i→f|

2δ 3

N

dN

3!

K3ρ2.(6)

For each recombination process three atoms are lost,jus-

tifying the factor3in the numerator.The factor3!in the

denominator appears only in case of condensed systems;

it counts for the number of triples in such state[11].

Considering the symmetrized scattering wave-function

for the initial state of three free particles,|Φ0 =

(1/

√

4m 1+ G+0(E)?G0(Eμ) ×

×(T j(E)+T k(E))},(8)

where Eμ≡?μ2/m is the subtraction energy scale with

μa constant in momentum units.It is possible to vary

μwithout changing the physics of the theory as long as the inhomogeneous term of Eq.(8)is modi?ed accord-ing to the renormalization group equations[20].t i is the two-body t?matrix for the subsystem of particles(jk). For E=0and zero-range potential the corresponding matrix elements are given by[4]:

p′|t ?3q24m =13π21

4mE2/3.From Eqs.(8)and(9),and for E=0,the matrix elements of T i are given by:

q i p i|T i(0)| 0 0 =τ(0)δ( q i)+2τ ?3q2i/(4m) h(q i),(10) where the s?wave function h(q)is the solution of

h(q)=?

μ2

3π2kq2(μ2+q2)

?43π ∞0dq′k?q′+i?×

×ln q2+q′2+qq′μ2+q2+q′2+qq′ .(11)

The normalized two-body bound-state wave function, in the zero-range model,to be introduced in Eq.(7),is given by

p|Φb =1ˉh(ˉh/a)2+p2.(12)

By considering the above equations,we obtain the?nal form of the recombination parameter:

α=8(2π)8m2 3 ˉh

3(8π)2 1+16πˉh2k?q+i?ln k2+q2+qk

E2/E3. The calculations were performed in dimensionless units, such that all the momentum variables were rescaled in units ofμ(in other words,μ=1in our calculations). So,the two-atom binding energy is decreased in respect to this scale.In that sense,the Thomas-E?mov states appear for E2/Eμgoing towards zero,which is equiva-lent of having E2?xed andμ→∞.The parameterα, shown in Figure1,is obtained as a function of the most excited trimer state.We have performed numerical cal-culations with at most three E?mov states.The full cir-cles show the results when exists only one bound state. When E2/Eμallows two E?mov states,the results are represented by the solid curve,which is plotted against the energy of the excited state.With full squares we rep-resent the results when E2/Eμallows three E?mov states.The scaling limit is well approached in our calculations. The maximumαoccurs at the threshold(E3=E2)and when(E3/E2)1

deep bound state.Thus,we found instructive to subtract such contribution fromαexp,which is about one unit,as found in Ref.[14].However,the resulting e?ect in the determination of the trimer energy is not so dramatic,as seen in Fig.1.The experimental value ofαfor87Rb|2,2 does not appear in the?gure,as it is well above the maxi-mum.By increasing the value of a from5.8nm to6.8nm we can make the experimental value consistent with our scaling limit approach.We also point out that the trimer can only support E3or E′3,not both simultaneously[19].

TABLE I.For the atomic species A Z|F,m F ,given in the1st column,we present in the6th and7th columns our predicted trimer binding energies,in respect to the threshold,S3≡(E3?E2)and S′3≡(E′3?E2),considering the central values of the experimental dimensionless recombination parametersαexpt(given in the4th column).It is also shown the corresponding two-body scattering lengths a(2nd column),the diluteness parametersρa3(3rd column),and the dimer binding energies E2 (5th column).For87Rb|1,?1 ,the recombination process was obtained in Ref.[11]for noncondensed(?)and condensed(?) trapped atoms.

A Z|F,m F a(nm)ρa3αexpt E2(mK)S3(mK)S′3(mK)

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