Effective range from tetramer dissociation data for cesium atoms

0.74 MB·English

by  M. R. Hadizadeh

Effective range from tetramer dissociation data for cesium atoms
M. R. Hadizadeh1, M. T. Yamashita1, Lauro Tomio1,2, A. Delfino3, and T. Frederico4
1Instituto de F´ısica Teo´rica, Universidade Estadual Paulista, 01140-070, Sa˜o Paulo, SP, Brazil
2Universidade Federal do ABC, 09210-170, Santo Andr´e, SP, Brazil
3Instituto de F´ısica, Universidade Federal Fluminense, 24210-346, Nitero´i, RJ, Brazil
4Instituto Tecnol´ogico de Aerona´utica, DCTA, 12228-900, S˜ao Jos´e dos Campos, SP, Brazil
(Dated: January 11, 2013)
Theshiftsinthefour-bodyrecombinationpeaks,duetoaneffectiverangecorrectiontothezero-
rangemodelclosetotheunitarylimit,areobtainedandusedtoextractthecorrespondingeffective
range of a given atomic system. The approach is applied to an ultracold gas of cesium atoms
3
close to broad Feshbach resonances, where deviations of experimental values from universal model
1
predictionsareassociatedtoeffectiverangecorrections. Theeffectiverangecorrectionisextracted,
0
with a weighted average given by 3.9±0.8R , where R is the van der Waals length scale;
2 vdW vdW
which is consistent with the van der Waals potential tail for the Cs system. The method can
2
n be generally applied to other cold atom experimental setups to determine the contribution of the
a effective range to the tetramer dissociation position.
J

0 PACSnumbers: 67.85.-d,03.65.Ge,05.10.Cc,21.45.-v,
1

] I. INTRODUCTION minima survive in the intermediate regime when the
s
a scattering length a exceeds the characteristic interaction
g range by a relatively small factor [18]. But the positions
The fundamental quantum properties of atomic sys-
- are shifted away from the naive predictions based on the
t tems are under intense investigation in recent cold-atom
n universal theory [14, 15] and finite range potentials have
experiments, in which long-time predicted universal as-
a been used [19, 20] to address such issues. On the other
pects of few-body physics can be probed. In this re-
u
hand, the quantitative reproduction of the experimental
q gard, the increasing number of three-body bound states
findings for the positions of the peaks of the tetramers
. (trimers), emerging when the absolute value of the two-
t resonances has been not yet addressed in detail.
a body scattering length is moved to infinity, known as
m Efimov effect [1], was confirmed in different cold-atom Within our aim in clarifying the shift in the experi-
- laboratories [2–5]. mental data with respect to the zero-range predictions
d
Actually,byextendingtomorethanthreeparticlesthe for the positions of the tetramer resonances at the disso-
n
search for fundamental quantum behavior, in the realm ciation point, we present an approach on how to intro-
o
c of four-body physics it is of interest to test the positions duce the effective range correction in lowest order in the
[ of four-atom resonant recombination peaks [6–9], which zero-rangefour-bodytheory. Weobtainthecorrectionto

appear as the two-body scattering length is tuned close thecorrelationofthepositionsoftwosuccessiveresonant
2
v to an s−wave Feshbach resonance (see e.g. [10]). four-boson recombination peaks. The associated scaling
0 On the theoretical side, the model independence of function is not fully determined by the trimer properties
7 tetramer properties, when few physical scales are fixed, andrequiresafour-bosonscale[11]. Furthermore,weare
2 was recently numerically established by our group [11, showing by our calculations that the effective range cor-
2 12],throughtheanalysisofobservedcorrelationsbetween rectionisnotcompletelyparameterizedbythetrimerand
.
1 tetramerobservables,byconsideringarenormalizedzero- tetramer short-range scales, as one could naively expect.
1 range (RZR) approach. These correlations were realized The contribution of a non-zero effective range affects
2 through appropriate scaling functions, from where one the observed scaling plots, by moving the correspond-
1 can verify that the RZR results are consistent with sev- ing positions in relation to the zero-ranged ones (see e.g.
:
v eral other model calculations [13–16]. Refs. [19, 21] in the case of trimers). Here, it will be
i shown quantitatively how the effective range affects the
X Despite the success of the model independence of the
scaling between negative scattering length ratios, corre-
tetramer scaling functions, the experimental data for ce-
r sponding to the dissociation positions of a trimer and
a siumatoms[6,8,9]areshiftedwithrespecttothecorre-
related tetramers (for two nearby tetramer states, asso-
lation function relating the peaks of successive tetramer
ciated with a given trimer state).
resonances for a zero-range force [11]. Among the pos-
sible effects to explain such a deviation from the results Relevant to the understanding of the effective range
obtained with zero-range two-body interaction, it is nat- contribution to the resonance positions, we should recall
ural to think in a way to introduce the correction due the nice and clear discussion by Chin et al. [10] on the
to a finite effective range in the model, as suggested in role of broad (entrance-channel dominated) and narrow
Ref. [17]. (closed-channeldominated)Feshbachresonances. Asdis-
The experimental data for three-body cold-atom re- cussed, a Feshbach resonance strongly dominated by the
actions show that Efimov resonances and recombination entrance channel allows a description of a two-atom sys-
2
temintermsofasingle-channelshort-rangedmodelwith inverse of the scattering length, up to the same order, is
a van der Waals tail. The broad s−wave Feshbach res-
onance of cesium [6, 8, 9], where tetramer dissociations 1 =±(cid:112)|(cid:15) |− r0|(cid:15) | . (2)
at the four-body continuum threshold were observed, is a 2 2 2
dominated by an entrance-channel. In this case, when
the strength parameter is very large (s >> 1), the Thetwo-bodyamplitude(1)providesacorrectiontothe
res
single-channel description is applicable [10] and conse- STM and FY equations in leading order in r , and as we
0
quently, the effective range should be found around the verified the subtracted forms of these equations are able
values obtained for van der Waals like potentials, tuned to provide finite results dealing with the extra power of
to the scattering length of the resonance. Indeed, as we momentum in the amplitude.
aregoingtodetailinthepresentwork,theestimatedval- Inthefollowing,wepresentsomedetailsontheformal-
ues we have obtained for the effective ranges, from the ism for the subtracted equations we have used, together
positions of tetramer resonances, are somewhat close to with our approach to find the effective range correction
the corresponding effective range of that class of poten- for the position of the tetramer resonances.
tials. Withinourapproachtoextracttheeffectiverange,
we have considered broad Feshbach resonances appear-
ing in ultracold gas of cesium atoms, leading to positive
effective ranges. In contrast, narrow resonances, as dis- B. Subtracted STM equation
cussed by Petrov [22], should lead to negative effective
ranges. In the case of systems with narrow resonances TheFaddeevcomponentsforthree-bosonboundstates
and denser spectrum, we should also mention the recent for the zero-range potential can be rewritten in terms of
work by Sørensen et al. [23]. spectator functions, which are solutions of a subtracted
The present work is organized as follows. In Sect. Skorniakov and Ter-Martirosian (STM) integral equa-
II, the formalism for two, three and four-boson systems tion, given by[24]:
is presented, where we adressed the effective range ex-
pansion of the two-body scattering amplitude, the sub- |K (cid:105)=2τ((cid:15) )G(3) |K (cid:105), (3)
tractedSkorniakovandTer-Martirosian(STM)equation ij,k ij,k ij;ik ik,j
and Faddeev-Yakubovsky (FY) equations. In Sect. III
where(cid:15) isthe(ij)subsystemenergyinthethree-body
theshiftsinthepositionoffour-bodyresonantlosspeaks ij,k
system. Note that we are going to introduce the two-
fromthesolutionoftheSTMandFYequationswiththe
boson amplitude (1) carrying the effective range correc-
effective range correction are given. In Sect. IV, we per-
tion in (3). The subtraction of the free 3-body resolvent
formedananalysisofthepositionsofthefour-bosonres-
regulatesSTMequation,andprojectedoperator,G(3),is:
onance peaks for the cesium data close to wide Feshbach
resonances. The effective range, shown to be compati-
ble with experiments, are compared to the one obtained Gi(j3;)ik =(cid:104)χij|(cid:16)[E−H0]−1−(cid:2)−µ23−H0(cid:3)−1(cid:17)|χik(cid:105),(4)
from van der Waals like potentials for the Cs system.
2
In Sect. V we present our conclusions and perspectives.
where H is the three-body free hamiltonian and −µ2 is
0 3
the subtraction energy scale. The form factor appearing
in the two-body T-matrix of the pair (ij), in the relative
II. FORMALISM FOR TWO, THREE AND momentum p id given by (cid:104)p |χ (cid:105) = 1. The three-
ij ij ij
FOUR-BOSON SYSTEMS bodyregularizationparameterµ keepsundercontrolthe
3
Thomas-collapse, anddetermine thethree-boson observ-
A. Two-body scattering amplitude with effective ables, and canbe parameterized by one s−wavephysical
range correction quantity(see e.g. [36]). The subtracted STM equations
was also used to calculate the position of triatomic con-
The s−wave two-boson scattering amplitude input tinuum resonances for large negative scattering lengths
to the subtracted forms of the trimer Skorniakov and [37], which were observed by the Innsbruck group [2],
Ter-Martirosian equation [24] and the tetramer coupled throughthree-atomrecombinationpeakinacoldcesium
Faddeev-Yakubovskyequations[25]fornegativeenergies gas close to a Feshbach resonance.
√
(cid:15), with the on-shell momentum given by k = i −(cid:15), can
be written as
τ((cid:15))≈ √1/(2π√2) (cid:104)1+ r0 (cid:0)√−(cid:15)±√−(cid:15) (cid:1)(cid:105), (1) C. Subtracted FY equations
− −(cid:15)± −(cid:15) 2 2
2
The tetramer energies are found by solving the
inthelowestorderoftheeffectiveranger . Intheabove, Faddeev-Yakubovsky (FY) equations properly regulated
0
thepoleofthescatteringamplitudeisfixedbythebound inthelimitofthezero-rangeinteraction, whicharewrit-
(+) or virtual state (−) energy, k2 = (cid:15) = −|(cid:15) |. The ten as a set of coupled subtracted integral equations
2
3
[11, 12, 25] given by: extended to include the effective range, as
(cid:34) (cid:32) (cid:33)
aT r
|Kijl,k(cid:105)=2τ((cid:15)ijl,k) Gi(j3;)ik |Kikl,j(cid:105)+ aTN3,N+1 =a−N3A aN−N3,3N,a−N03 , (8)
(cid:18) (cid:19)(cid:35)
+Gi(j4;)ik |Kikj,l(cid:105)+|Hik,jl(cid:105) , (5) where a−N3 is the position of the peak of the three-atom
resonant recombination for a<0, and aT is the scat-
(cid:34) (cid:35) N3,N
tering length for which the excited N-th tetramer dis-
|Hij,kl(cid:105)=τ((cid:15)ij,kl) Gi(j4;)kl 2 |Kkjl,i(cid:105)+|Hkl,ij(cid:105) . (6) sociates and meets the four-body continuum. The solu-
tions of the subtracted STM (3) and FY equations (5,6)
depend only on a, r , and on the three- and four-body
Thesubtractionsinthekernelcomeatthelevelofthefree 0
scaleparameters. Thenforthezeroenergytetramer,the
4-bodypropagatorsinEqs. (5)and(6),whichareusedto
value of the negative scattering length depends only on
regulatetheFYequationsforthecontactpotential. The
r and the short-range three- and four-body momentum
two-bosonscatteringamplitude(1),whichareintroduced 0
scales at the subtraction points. The three-body scale
in the FY equations above, carries the effective range
correction in the calculations that are presented in the can be parameterized by a−N3, while the four-body one
by aT in the determination of aT . In this way
next section. N3,N N3,N+1
The projected 4-body free resolvent, G(4), are sub- (8) can be built. Furthermore, the scaling function A is
tracted at an energy −µ2: calculated with the two-boson scattering amplitude (1)
4 and the solutions of the FY equations, as well as of the
Gi(j4;)ik =(cid:104)χij|(cid:16)[E−H0]−1−(cid:2)−µ24−H0(cid:3)−1(cid:17)|χik(cid:105)(7) STTMheeqduimateinosni,onexlepsasnfduendctuiopntAo fiisrsdtetoerrdmeriniendrb0y. solving
the subtracted STM [24] and FY [25] equations consid-
where H is now the free four-body Hamiltonian. Notice
0 eringtheexpansionoftheatom-atoms−waveamplitude
that when introducing G(3) , eq. (4), in the FY equa-
ij;ik up to order k2 as given by (1) and the corresponding
tions, the free three-body hamiltonian should be substi-
relation between the scattering length and virtual state
tuted by the four-body one. The energy of the two-body
energy(2). ThesolutionsofthesubtractedFYequations
subsystem (ij) appearing as arguments of the two-boson
depend only on a, r , and on the three- and four-body
scattering amplitude in Eq.s (5) and (6), are (cid:15) l and 0
ij,k scaleparameters. Then,forthezeroenergytetramer,the
(cid:15) , associated with a virtual pair in the 3+1 partition
ij,kl value of the negative scattering length depends only on
and in the 2+2 partition, respectively.
r and the short-range three- and four-body momentum
0
Thesubtractionscale−µ2 inG(3) theright-handside scales at the subtraction points. The three-body scale
3 ij;ik
ofEq. (5),fixthetrimerpropertiesconsistentlywiththe canbeparametrizedbya−N3,whensolvingtheSTMequa-
STMequation,butindependentlyonthescaleµ . Look- tion with r included, while the four-body scale is corre-
4 0
ing closer to this term and taking into account the left lated to aT in the determination of aT . Then,
N3,N N3,N+1
hand side, the position of the trimer pole is guarantee to the scaling function (8) is built without reference to the
be same as the bound state energy obtained by solving scale ratio, by eliminating this dependence in the ratio
tFhYe esquubatrtaiocntes,dinST(5M) aenqdua(t6i)o,na.reTrheeguolathriezredtewrmitshianntihne- aTNI3n,2/oau−Nr3cainlctuelramtiosnosfwaTNe3h,1a/vae−N3o.nly obtained results for
dependent subtraction scale −µ2. This method ensures tetramers below the ground state trimer, which should
4
that the three-body subtraction scale fixes the three- be enough for our study, as the values of the scattering
body properties consistently with the subtracted STM lengtharemuchlargerthantheshort-rangelengthscales,
equation, whereas the regularization parameter µ4 fixes corresponding to the three- and four-body subtraction
the four-body observables [25]. points. The scaling function A(cid:0)aT1,1/a−1,r0/a−1(cid:1), which
provides the correction to the position of the four-atom
resonance is shown in the 3D plot of Fig. 1. We have
III. POSITION OF FOUR-BODY RESONANT presentedresultsuptor0/a−1 ∼0.5,for0.40≤aT1,1/a−1 ≤
LOSS PEAKS 0.51, considering the relevant region where recombina-
tion data for cesium have been measured. In this case,
The four-atom recombination resonates when the r0 ∼ aT1,1, and for the validity of the expansion in the
tetramer has zero energy and placed at the four-body effective range one should have r << aT , and the ex-
0 1,1
continuum threshold. In cold atom traps the resonance pansion of the scattering length (2) is questionable. For-
is reached by tuning the negative scattering length close tunately, we found that the coefficients for the effective
to a Feshbach resonance (see e.g. [18]). The theoretical range correction are fairly small, allowing to extend the
description of the scaling function, which correlates the validity of the expansion to the region where data are
valuesofthenegativescatteringlengthswheresuccessive found.
tetramersreachthecontinuum,introducedin[11],isnow WithintheregionoftheplotgiveninFig.1,thesurface
4
The negative value of c decreases the function A for
01
positive effective ranges, which suggests that the region
wherethetwo-bodyamplitude(1)increaseswithrespect
to the zero-range one, for a < 0 and positive effective
0.95 ranges,isimportantforthetetramers. Inthiscase,larger
−1 0.92 variationsofthescatteringlengthtowardnegativevalues,
/a 0.89 compared to the trimer value of a−1, are required to tune
the tetramers to zero energy.
2
T,1, 0.86 Two theoretical results are used for comparison with
a
0.83 the results of the scaling function A, using the plot of
Fig. (2), the first one by Deltuva [16] (0.4435,0.8841),
0.80
0.50 which quotes r0/|a−1| = 0.33 [26] compared to 0.29
extracted from the plot, and another one by von
0.48
a 0.46 0.4 0.5 Stecher [13], (0.45,0.88), with r0/|a−1| =0.38 [27] com-
T pared to 0.36 obtained from the plot. This compari-
1,, 0.44 0.3
1/a1− 0.402.40 0 0.1 0.2r 0 / a −1 stioanlmwoitdhelc[o1m6]palentdellyocdailffGeraeunstsimanodpeoltse,nsteiapla[r1a3b]l,egipvoetseuns-
confidence on the universality and utility of the function
A,toanalyzethedata. Thiscomparisonsuggestsalower
FdeIpGe.n1d.en(ccoeloorfoanTlin/ea)−TihneasTcal/ian−g faunndctrion/aA−.exCpraelcssuilnagtiothnes bound of |r0/a−1|>0.02 in our extraction method.
1,2 1 1,1 1 0 1
areperformedinaleadingorderexpansionofthesolutionsof
the FY equations with the two-body amplitude (1), see text 1.00
for further explanation. Hadizadeh et al. PRL107 (2011) [r0/a_1 = 0]
Gattobigio et al. PRA86 (2012)
Deltuva PRA85 (2012)
0.96 Berninger et al. PRL107 (2011)
Ferlaino et al. FBS51 (2011)
can be parameterized by von Stecher et al. Nature Phys. 5 (2009)
Ferlaino et al. PRL102 (2009)
_ 1
0.92
aT1,2 =A(x,y)= (cid:88) c (x−0.45)myn, (9) /a
mn
a−1 0≤m+n≤2 a aT 1,20.88
where x ≡ aT1,1/a−1, y ≡ r0/|a−1|, and (m,n) ≥ 0. The -r0 /a_1
coefficients are given in the following Table I. 0.84 [[00..01 ,, 00..12]]
[0.2 , 0.3]
[0.3 , 0.4]
[0.4 , 0.5]
TABLE I. Coefficients for the parametrization (9). 0.80
0.42 0.44 0.46 _ 0.48 0.50
aT /a
c c c c c c
00 10 01 11 20 02
1,1 1
0.932 0.724 -0.144 0.347 -0.645 0.001
Although the expansion (9) contains nonlinear terms FIG. 2. (color online) Lower-order effective range correc-
in r , it is built in linear order in the solutions of the tion for the scaling plot aT /a− = A(aT /a−, r /a−) for
0 1,2 1 1,1 1 0 1
subtracted STM and FY equations. The second order thecorrelationbetweenthepositionsofthepeaksforsucces-
termsareinducedbythecurvatureofA(x,y)inxandy. sive tetramer resonances at the continuum threshold, com-
In our case the calculation is performed for the ground paredtoothertheoreticalresultsandexperimentaldata. We
present our results for different r intervals (the upper solid
state trimer and two successive tetramer states. 0
line is from [11]) compared to a few recent calculations as
The coefficients for the effective range correction are
indicated inside the frame [Refs. [13] (triangles), [16] (dia-
expected to be smaller than the ones associated with
monds)and[28](square)]. TheexperimentaldataofRefs.[6],
the variation of aT1,1/a−1. Indeed, the expansion around [9] and [8] are also indicated inside the figure. The lowest
aT1,1/a−1 =0.45showsthatthelinearcoefficientforr0/a−1 band (gray) corresponds to r0 for the van der Waals like
correction gives c /c =−0.15, while c /c =0.78 is potential for cesium atoms with values within the interval
01 00 10 00
5 times larger. The amazing smallness of c01 and c02 0.4 (cid:46)−r0/a−1 (cid:46) 0.5 for a reference value of a−1 (cid:39)−9RvdW
withrespecttotheothercoefficients,reflectsthatafrac- (see Table II).
tion of the effect from the effective range is absorbed by
the variation of the short-range four-body scale, but not In Fig. 2, the 2D plot of our results for the surface
all. Given that, it is quite obvious that the correction A(x,y), corresponding to the scaling function (8), is
coefficients for the effective range contribution, shown in shown. The bands in the plot gives the shift of the scal-
TasasbolceiaIt,esdhowuiltdhbtehseoemxepwanhsaitonsmpaalrlaermtehtaenr athT1,e1/cao−1effi. cients iinntgerfvuanlcstdioenpicdtueedtiontthheefifignuirtee.vFaolureosurofrerf0e/re|an−Nce3|wienatlhsoe
5
plot the theoretical results from [13] and [16], and the set {0.47,0.84}[6] with no quoted errors suggests a large
experimental values from Refs. [6, 8, 9]. The data from value of r0 >5RvdW.
Ref. [7] have large error bars and suggests r ranging
0
from negative to positive values.
TABLE II. Extracted effective ranges from the shift of the
experimental peaks of the four-atom losses for a gas of cold
IV. ANALYSIS OF THE CESIUM DATA cesiumatomsclosetoaFeshbachresonance,usingthescaling
plotshowninFig.2. Theerrorsintheextractedvaluesofr
0
are estimated from the figure as well.
The effective range extracted from the experimental
data is shown in Table II. They were obtained just by Ref. aT /a− aT /a− a− [R ] r [R ]
1,1 1 1,2 1 1 vdW 0 vdW
inspection of Fig. 2, by comparing the experimental val-
[6] 0.47 0.84 -8.7(1) >5
ues with the scaling curve including the effective range
[8] 0.465(34) 0.903(31) -9.54(28) 2.5 ± 1.7
correction. The shift of the data yields values of r0/a−1 [9] 0.47(1) 0.87(1) -8.71 4.8 ± 1.0
larger than the lower bound for the theoretical extrac-
[9] 0.46(2) 0.91(3) -9.64 2 ± 2
tion. We didn’t present results for [7] as it has larger er-
rors. We just notice that the effective ranges from these
data varies from negative to positive values. The ex- It is noticeable that the dissociation point of ce-
traction of r0 from the data for cesium atoms close to sium ground state Efimov trimers for negative scatter-
Feshbach resonances are within errors consistent. The inglengthsarefoundinanarrowband[8],independently
extracted values of r0 vary from 2 to 5 times the van on which Feshbach resonance the system is tuned (see
der Waals length (RvdW), which for the Cs2 system is Table II). Theoretical works [23, 32–35] addressed the
RCs2 = 101.0a [10]. interesting issue of the physical mechanism for the dom-
vdW 0
The effective range of potentials with van der Waals inance of the two-body properties on the position of the
tail −C /r6 at large distances has the approximate for- first Efimov resonance for a < 0, when the trimer meets
6
mula given by [29–31]: the continuum. The results in Table II show a range
of values for r extracted from the tetramer and trimer
0
2Γ(1/4)4 (cid:16)a¯(cid:17)2 (cid:16)a¯ (cid:17)2 resonances, which within errors (excepting one data set)
r = a¯ ϑ and ϑ= + −1 , (10)
0 3 (2π)2 a a are consistent with the effective range values from the
van der Waals type potential. The numerical analysis of
where a¯ = 2π/Γ(1/4)2RvdW (cid:39) 0.955978RvdW is the Wang et al. [32] has shown that two-body interactions,
averagescatteringlengthofvanderWaalspotentials[31], which suppress efficiently the wave function for separa-
with the van der Waals length RvdW = 12(mC6/(cid:126))1/4. tion distances less than r0, have a−1 ∼ −9RvdW, deter-
Considering the van der Waals length one gets that a¯ = minedbytwo-bodyproperties, withthisclassofsystems
96.5a , and closely related to entrance-channel dominated Feshbach
0
resonances. The dominance of the single channel poten-
r0 (cid:39)2.7894RvdWϑ=2.9179a¯ϑ(cid:39)281.7ϑa0. (11) tialwiththevanderWaalstailinthethree-cesiumreac-
tions, with suppression of the short distance wave func-
In the actual cold cesium experiments listed in Table II, tion, putted forward the narrow band where the trimer
thefactorϑvariesfrom1.26(forthesecondtetramerres- dissociation position is found. It also suggests that the
onance) to 1.55 (for the first tetramer resonance), which effective range seen through the shift of the position of
gives r0 within 3.5 − 4.3RvdW. The corresponding in- the four-atom recombination peak with respect to the
terval of 0.4 (cid:46) r0/|a−1| (cid:46) 0.5 for a reference value of zero-range results, should be given by the van der Waals
a−1 (cid:39) −9RvdW, can be identified with the lowest band tail of the potential, as well.
(gray) shown in Fig. 2. It is worthwhile to address the validity of the correc-
The estimated values of r extracted from the shift tion due the effective range, considering the main results
0
of the data with respect to the zero-range calculations, presentedinFig.2andTableII.Inviewoftheexpansion
shown in Fig. 2, are given in Table II. The values for giveninEq.(1),whichatafirstglancewouldbedoubtful,
the effective range are found between 2RvdW and ∼ when r0/|a−1| ∼ aT12/a−1, we found consistence between
5RvdW, while Eq. (11) applied to the set of cesium the present calculations with the results obtained with
data suggests the interval 3.5 − 4.3RvdW represented short-range separable[26] and local[27] interactions. The
by the lowest (gray) band in the figure. The present ex- leadingcoefficientfortheexpansioninr0/|a−1|isonefifth
perimental errors overlaps the cesium data, as one verify ofthecorrespondingcoefficientforthedependenceofthe
in Fig. 2. By performing the error weighted average of scalingfunctioninaT11/a−1 (seeTableI),whichshowsthat
the extracted effective range values from the data given the extracted values of r using the scaling plot comes
0
in Refs. [8] and [9] we obtain 3.9±0.8RvdW. In particu- fromsmallcorrectionstothezero-rangescalingfunction.
lar, the data set {0.47(1),0.87(1)}, provides an effective Therefore, as observed in Fig. 2, the agreement of the
range of about 4.8RvdW, which within 1σ is consistent expansion we have considered can be extended also to
withthevaluesofthevanderWaalspotential. Thedata cases near aT12/a−1 ∼r0/|a−1|.
6
V. CONCLUSION AND OUTLOOK tive ranges were found within the interval 2RvdW (cid:46)
r0 (cid:46) 5RvdW, with a reference value of 3.9±0.8RvdW,
In this work, we present an approach to extract the obtained by the error weighted average of the r0’s ex-
effective range of a given atomic system from the shifts tracted from the data of Refs. [8, 9]. These values are
in the four-body recombination peaks, at the tetramer consistent with the van der Waals potential tail for the
dissociation threshold, with respect to the zero-range re- broad s−wave Feshbach resonances of the Cs2 system,
sults. Wesolvedthetrimerandtetramersubtractedzero- providing r0 within the interval of 3.5 − 4.3RvdW for
range integral equations with two-body amplitudes car- the scattering lengths where the resonances were found.
rying the effective range in lowest order. Besides, the Ourresultsputastrongevidenceontheprevalenceofthe
trimer and tetramer subtracting scales, no further pa- entrance-channeldominanceinthephysicsoftheuniver-
rametersareintroducedinthecalculations. Thecorrela- sal tetramers formed with broad Feshbach resonances.
tion between the negative scattering lengths where suc- The question on how the effective range is sensible
cessivetetramersdissociatesatthefour-bodycontinuum to Feshbach resonance manipulations in the scattering
is extended to include the correction due to the effec- length as suggested by (11) has to be considered. Once
tive range, which is model independent checked against improved experimental data are available, it will be pos-
different short-range potential model calculations. We sible to verify in more detail how Feshbach resonances
found that the effective range correction of the on-shell from different setups affect the r0 values extracted from
two-body amplitude in the calculation of the trimer and thetrimerandtetramerdissociationpoints. Thepresent
tetramer dissociation points, when presented as a cor- analysis, based on the universality of effective range cor-
relation plot of the successive positions of the tetramer rection to the position of tetramer resonances, can be
resonancesisnotcompletelyparameterizedbythetrimer generally applied to any other cold-atom bosonic system
andtetramershort-rangescales, asonecouldnaivelyex- close to Feshbach resonances.
pect. Thisunexpectedpropertyturnedtobeessentialto ACKNOWLEDGMENTS
single out the effective range from the shift in the posi-
tions of the four-atom recombination peaks with respect We thank A. Deltuva and J. von Stecher for send-
to the zero-range theory. ing us some of their effective range results. TF and LT
We applied our proposal to the cesium data mea- also appreciate clarifying discussion with P. Naidon and
sured at broad s−wave Feshbach resonances [6, 8, 9]. S. Endo. Our thanks also to Funda¸c˜ao de Amparo a
In our analysis we considered the shifts of the observed PesquisadoEstadodeS˜aoPauloandConselhoNacional
recombination peaks at the tetramer dissociation posi- de Desenvolvimento Cient´ıfico e Tecnol´ogico for partial
tions with respect to the zero-range results. The effec- support.
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