Looking for "avalanche-mechanism" loss at an atom-molecule Efimov resonance
Ming-Guang Hu, Ruth S. Bloom, Deborah S. Jin, Jonathan M. Goldwin
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l Search for “avalanche mechanism” loss at an atom-molecule Efimov resonance
Ming-Guang Hu, Ruth S. Bloom, Deborah S. Jin
JILA, NIST and University of Colorado, Boulder, CO 80309, USA andDepartment of Physics, University of Colorado, Boulder, CO 80309, USA
Jonathan M. Goldwin
Midlands Ultracold Atom Research Centre, School of Physics and Astronomy,University of Birmingham, Edgbaston, Birmingham B15 2TT, UK (Dated: April 16, 2018)The “avalanche mechanism” has been used to relate Efimov trimer states to certain enhanced atomloss features observed in ultracold atom gas experiments. These atom loss features are argued to bea signature of resonant atom-molecule scattering that occurs when an Efimov trimer is degeneratewith the atom-molecule scattering threshold. However, observation of these atom loss featureshas yet to be combined with the direct observation of atom-molecule resonant scattering for anyparticular atomic species. In addition, recent Monte-Carlo simulations were unable to reproduce anarrow loss feature. We experimentally search for enhanced atom loss features near an establishedscattering resonance between K Rb Feshbach molecules and Rb atoms. Our measurementsof both the three-body recombination rate in a gas of K and Rb atoms and the ratio of thenumber loss for the two species do not show any broad loss feature and are therefore inconsistentwith theoretical predictions that use the avalanche mechanism.
I. INTRODUCTION
Evidence for Efimov three-body bound states, whichwere proposed originally in the context of nuclear physics[1], has been observed in a number of ultracold atom gasexperiments [2–14]. In principle, near a magnetic-fieldFeshbach resonance [15] there exists an infinite numberof three-body bound states that follow a discrete scalinglaw. The primary signature of these three-body statesin cold atom gases has been resonantly enhanced three-body loss of trapped atoms. A loss resonance occursat a negative value of the two-body scattering length a ,which is denoted a − , where the energy of the Efimovstate coincides with the scattering threshold energy forthree atoms [16], as shown schematically in Fig. 1 (a).Several experiments have observed multiple Efimov lossfeatures whose locations follow discrete scaling, with each a − larger than that of the last by a factor of e π/s , where s is a universal parameter [17–19].An additional signature of Efimov states can be foundwhen the energy of an Efimov state coincides with thethreshold scattering energy for a Feshbach molecule andan atom. This occurs at a positive value of a de-noted a ∗ and results in resonant collisional loss in atrapped gas mixture of Feshbach molecules and atoms.Atom-molecule loss resonances have been observed for Li [8, 13],
Cs [14], and the mixture of K and Rb[12]. In addition, unanticipated resonances in the lossof trapped atoms at positive a values, without initiallycreating molecules, have been seen for Li [7, 20], K[5], and the mixture of K and Rb [6]. The observedloss features are relatively small, with the increase in theatom loss rate ranging from about a factor of two to afactor of five. The features can be quite narrow, withthe width ranging from a few a to a few hundred a ,where a is the Bohr radius. These resonances are be- lieved to be related to a ∗ and have been attributed toan avalanche mechanism [5, 21, 22], whereby Feshbachmolecules that are produced by non-resonant three-bodyrecombination eject atoms from the trap via resonant,secondary atom-molecule collisions. However, there hasnot yet been an observation of an avalanche feature andan atom-molecule loss resonance in the same system. Inaddition, a recent theoretical simulation suggests that theavalanche mechanism fails to produce a narrow atom lossfeature near the atom-dimer resonance [23].The observation of an atom loss feature does not re-quire the preparation of Feshbach molecules and there-fore can be a simpler method for experimentally locating a ∗ . However, it is important to verify the connectionof observed avalanche peaks with atom-molecule Efimovresonances. In previous work [12], we measured an atom-molecule loss resonance for K Rb Feshbach moleculesand Rb atoms, but did not see any corresponding lossfeature for an atom gas prepared without creating a pop-ulation of trapped Feshbach molecules. The measuredatom-molecule loss rate coefficient β is presented in Fig.1 (b), showing resonant loss around a ∗ = 230(30) a . Be-cause this data could have missed a narrow or a small am-plitude avalanche peak, we present here additional atomloss measurements on the positive a side of the K − RbFeshbach resonance. In particular, in order to search foran Efimov-related avalanche feature, we take many moredata points with a finer spacing in a . In addition, we en-sure uniformity of the initial atom gas conditions as wechange a , since variation of the densities or temperaturecould shift or broaden a resonance feature [21, 23]. Fi-nally, we look for features in both the atom loss as wellas the ratio of the number loss for Rb and K, since theavalanche mechanism should result in additional loss ofRb atoms from the resonant, secondary collisions.The rest of paper is organized as: Sec. II describes FIG. 1. (Color online) Efimov loss processes. (a) Schematicshowing the location of Efimov loss features. The thick blackline corresponds to the threshold energy of three free atoms,the green solid line corresponds to the threshold for a KRbFeshbach molecule plus a free Rb atom, and the brown dashedlines correspond to KRb Efimov bound states. At a − < a ∗ >
0, resonant enhancement of atom-moleculeinelastic collisions is observed and enhanced atom loss due tothe avalanche mechanism has been postulated. In the scat-tering process cartoons, red and blue collision partners repre-sent Rb and K atoms, respectively. (b) Measured atom-molecule loss rate coefficient as a function of a in a mixture ofRb and RbK [12]. Resonant atom-molecule loss was observednear a ∗ = 230(30) a ; the line shows a fit to a theoreticallineshape [12, 24]. how we prepare the ultracold Bose-Fermi mixture andmeasure atom loss, Sec. III presents the experimentalresults, which are compared against predictions based onthe avalanche mechanism near an atom-molecule Efimovresonance, and Sec. IV gives conclusions. II. LOSS MEASUREMENTS
Our measurements start with an ultracold mixtureof bosonic Rb atoms in the | f, m f i = | , i state andfermionic K atoms in the | / , − / i state, where f corresponds to the atomic angular momentum and m f is its projection. An s -wave Feshbach resonanceis used to control the interactions between Rb and K atoms, where a as a function of magnetic field B is given by a = a bg (1- ∆ B − B ), a bg = − a , B =546 .
62 G,∆= − .
04 G [25]. The atom gas is initially prepared at a magnetic field 2 .
07 G below B , which corresponds to a = 88 a . We keep the temperature T of the gas greaterthan 1 . T c as well as greater than 0 . T F , where T c is thetransition temperature for Bose-Einstein condensation of Rb and T F is the Fermi temperature of K.We have investigated atom loss in a single-beam opti-cal dipole trap characterized by trapping frequencies forRb of ω r / π = 600 Hz radially and ω z / π = 6 Hz axially.In our far-detuned optical dipole trap, the trapping fre-quencies for K are larger than those for Rb by a factor of1.4. The optical trap beam propagates along a horizontaldirection, with a beam waist of 20 µ m and a wavelengthof 1090 nm. The atom gas mixture is prepared with aninitial number of Rb atoms, N Rb,i , between 7 . × and9 . × , an initial number of K atoms, N K,i , between2 . × and 3 . × , and an initial temperature be-tween 0.7 µ K and 1.0 µ K.Three-body recombination produces a Feshbachmolecule with a kinetic energy determined by the bindingenergy. In order to make sure that our trapping potentialdoes not confine these KRb molecules or the energeticatoms resulting from scattering with KRb molecules,we want the trap depth to be lower than the bind-ing energy of KRb molecules [26]. As a consequence,we take measurements for a < a in a trap justdeep enough to hold an atom gas with a temperature T max = 1 . µ K. To extend our measurement to largervalues of a where the binding energy of KRb is smaller,for 900 a < a < a we lower our trap depth to bejust deep enough to hold an atom gas with a tempera-ture of T max = 1 . µ K, with ω r / π = 500 Hz radially and ω z / π = 5 Hz axially. For all of the data, the binding en-ergy of the molecules is greater than 1.5 times k B T max ,where k B is the Boltzmann constant.To measure loss, we use a magnetic-field sweep toquickly increase a and then wait for fixed amount of time∆ t as shown in the inset of Fig. 2. The magnetic field isthen returned to the original value where a = 88 a andboth atom species are imaged a few milliseconds afterrelease from the optical trap. The final atom numbersfor Rb and K, which we denote N Rb,f and N K,f , respec-tively, are determined from fits to Gaussian distributions.Combining this with the measured initial atom numbers, N Rb,i and N K,i , yields the loss rate. We take data forvalues of B during the hold time ∆ t that correspond to a from 100 a to 1500 a . The hold time ∆ t is changedfor different ranges of a in order to keep the fractionalnumber loss ( N f − N i ) /N i between 10% and 60%, where N f = N Rb,f + N K,f and N i = N Rb,i + N K,i . The value of∆ t varies from 5 s at small a to 2 . a .Fig. 2 shows a subset of our loss measurement datafor a hold time, ∆ t , of 1 s. This two-point measure-ment approach (measuring the number at time 0 and attime ∆ t ) trades accuracy for precision. Specifically, a fullmeasurement of the loss curve, where the atom numbersare measured at many different times, allows for a moreaccurate determination of the three-body rate coefficientat a particular value of B , however the faster two-point B t B N R b , K t200 250 300 350 40010 -27 -26 ( c m / s ) a (units of a ) FIG. 2. (Color online) An example of data from which weextract the three-body recombination rate. The upper panelshows the measured atom number (circles for Rb and tri-angles for K) after holding at a scattering length a for 1 s.The dashed lines show the measured initial Rb and K atomnumbers. The lower panel shows the extracted three-body re-combination rate coefficient α based on Eq. (1). Inset showsmagnetic-field sweep: the magnetic field B is increased to avalue near the Feshbach resonance in 0 .
25 ms, held at thatvalue for a time ∆ t , and then swept back to the original valuein 0 .
25 ms. measurement minimizes the effect of drifts in experimentparameters and therefore enhances the precision and ourability to detect any small loss peaks as we vary B .In order to combine data taken for different hold times∆ t , we extract an approximate three-body rate coefficient α . For three-body recombination of Rb+ Rb+ K, α is defined by ˙ N ( t ) = − α R d r n K ( r , t ) n ( r , t ) [12],where n Rb ( r , t ) and n K ( r , t ) are number densities of Rband K, respectively. To simplify this differential equa-tion, we can use the fact that K and Rb share almostthe same polarizability in our optical dipole trap and ig-nore the small relative sag between Rb and K clouds.Assuming a Gaussian density profile consistent with aharmonically trapped Maxwell-Boltzmann gas, we canre-write the integral in terms of the total number N andtemperature T as ˙ N ( t ) = − αA ¯ ω N /T [27]. Here,¯ ω = ( ω r2 ω z ) / , A = R (1+ R ) (cid:16) m Rb π √ k B (cid:17) , R is the numberratio N Rb /N K , m Rb is the atom masses of Rb. Al-though the number ratio R can change during a mea-surement, the parameter A is only weakly dependent on R . In the approximation that the temperature and theparameter A are constant during ∆ t , α can be solved foranalytically, α = (cid:20) N − N (cid:21) T A ¯ ω ∆ t . (1)Using the average initial number ratio R = N Rb,i /N K,i =2 . T , we obtain α using Eq. (1) (see Fig. 2, lower panel). As a check, we have comparedthe results from our two-point measurements using Eq.(1) with previous data where α was extracted from manymeasurements of the number of atoms as function of time[12], and we find that they agree to within a factor of 2. III. EXPERIMENTAL RESULTS
We previously identified an Efimov-like resonance be-tween K Rb Feshbach molecules and Rb atoms atthe scattering length a ∗ = 230 a with width of roughly200 a [12]. According to Refs. [5, 21], this atom-molecule resonance can also result in enhanced atom lossat scattering lengths near a ∗ for a gas initially consistingof atoms only. Here, non-resonant three-body recombi-nation of atoms produces energetic K Rb Feshbachmolecules that then collide with atoms multiple times toresult in atom loss. In addition to enhanced total atomloss, our two-species atom gas could provide an addi-tional signature for this avalanche scenario. Namely, thenumber loss ratio ∆ N Rb / ∆ N K should also show a reso-nant increase that coincides with the enhanced atom lossfeature, since the collision channel K Rb+ Rb is en-hanced while K Rb+ K is not [12]. Here, the numberloss ratio is defined by∆ N Rb / ∆ N K = N Rb,f − N Rb,i N K,f − N K,i . (2)Fig. 3 shows our measurement results, where eachpoint on the plots shows the average of four repeatedmeasurements within each dataset having the same valueof ∆ t . The vertical error bars indicate the standard devi-ation of the mean, while the horizontal error bars indicatethe range of a values used in the averaging. The upperpanel in Fig. 3 shows the loss rate coefficient α extractedusing Eq. (1) while the lower panel shows the numberloss ratio ∆ N Rb / ∆ N K . We see no clear evidence for anavalanche peak. Specifically, aside from the deviationfrom a scaling at small values of a , the dominant fea-tures in α appear to be small systematic shifts that occurwhen we combine datasets taken with different values of∆ t , and we can easily rule out the presence of any fea-ture where α is increased by a factor of two or more. Themeasured number loss ratio ∆ N Rb / ∆ N K has an averagevalue of approximately 2, which is the expected value forthree-body recombination with no additional avalanchemechanism loss. The measurement of ∆ N Rb / ∆ N K hasa lower signal-to-noise ratio than α and one can identifysome possible peaks in the data. However, these peakshave no corresponding feature in α . In addition, ourmeasured number loss ratio is qualitatively inconsistentwith predictions from an avalanche mechanism model asshown by dashed and dot-dashed lines in the lower panelof Fig. 3. The amplitude of these potential peaks in ourdata is smaller than that of the model by a factor of 2or more and the width is narrower by a factor of 10 ormore. The avalanche model is described in detail below. -28 -27 -26 -25 -24
100 300 500 700 900 1100 1300 15000246810 N R b / N K ( c m / s ) a (units of a ) FIG. 3. (Color online) Atom loss measurements in a single-beam optical trap. Different datasets are color-coded and havedifferent values of the holding time ∆ t ranging from 5 s to 2.5 ms. The upper panel shows the measured three-body recombinationrate coefficient α (points) versus a . The blue solid line indicates an a dependence, which is expected in the absence of Efimovresonances. The lower panel shows the measured number loss ratio ∆ N Rb / ∆ N K (points) versus a . The black solid linecorresponds to the average value of 1 .
8. The red dashed line and black dot-dashed line come from calculations based on aprobability model from Ref. [21] with η ∗ = 0 .
26 and η ∗ = 0 .
02, respectively.
In Ref. [21], Machtey et al. present an improvedavalanche model based on Ref. [5]. The results of thismodel show qualitative agreement with the two Li ex-periments [7, 20] and the K experiment [5], althoughthe predicted widths for enhanced loss were typicallyseveral times larger than the observed avalanche peakwidths. Monte-Carlo methods have also been used tosimulate the avalanche loss for homonuclear systems [23].These simulations were applied to the Li system and re-sult in an even wider avalanche loss feature, with widthsthat are 10 to 20 times larger than the observed atomloss features [23].For comparison with our data, we have applied themodel of Machtey et al. , which we modify for the het-eronuclear case, to calculate the expected avalanche peakfor our experiment parameters. In the Machtey et al. model, the elastic and inelastic atom-dimer cross sec-tions are used to calculate the probability that a dimercreated by three-body recombination undergoes a spe-cific number of secondary elastic collisions with atomsbefore exiting the trap. A weighted sum of these proba-bilities then yields the expected number of extra atomslost due to the avalanche mechanism. To extend thismodel to a two-species gas, we use the scattering length for a Rb atom and a KRb Feshbach molecule given in Ref.[24]: a AM ( a ) = [ C + C cot( s ln( a/a ∗ ) + iη ∗ )] a withconstants C = 1 . C = 2 .
08, and s = 0 . η ∗ = 0 .
26 and a ∗ = 230 a , are takenfrom the fit to atom-dimer trap loss data in Ref. [12].In terms of a AM , the atom-molecule elastic and inelas-tic cross sections are given by σ el ( a ) = 4 π | a AM ( a ) | and σ inel ( a ) = − π Im( a AM ( a ) /k ), respectively [16]. For theinitial collision, the relative wave number k is calculatedassuming the atom is essentially at rest and the moleculehas a kinetic energy given by [ m Rb / ( m K + 2 m Rb )] E b ,where E b is the binding energy of the molecule. Ineach subsequent collision with an atom at rest, themean energy of the dimer is multiplied by a factor of h ( m K + m Rb ) + m i / ( m K + 2 m Rb ) ≈ . nl . In the calcu-lation, we use the average density of the Rb atoms for n and the geometric mean root-mean-squared width ofthe trapped gas for l . For our data, n = 1 . × cm − and l = 12 µ m. With these parameters, we calculate themean number of Rb atoms lost per three-body recom-bination event, which can be directly compared to ournumber loss ratio data. The red dashed curve in thelower panel of Fig. 3 shows the model result. Becausea much lower Efimov resonance inelasticity parameter, η ∗ , can be extracted from fits to three-body loss data[12], we also show the model result for η ∗ = 0 .
02 (blackdot-dashed curve in Fig. 3). For either value of η ∗ , themodel predicts a wide resonance feature in atom loss andin ∆ N Rb / ∆ N K , which is clearly inconsistent with ourmeasurements.Given that we do not observe a feature consistent withthese predictions for an Efimov avalanche peak, it is use-ful to compare the parameters for our system to thoseof experiments where avalanche peaks have been ob-served. In particular, compared to the Li experimentof Ref. [20], our temperature is very similar (within afactor of 2), the mean size of the trapped gas l is sim-ilar (within 30%), and our atom density is an order ofmagnitude larger, which should be favorable for observ-ing an Efimov avalanche peak. In addition, the Efimovresonance parameters a ∗ and η ∗ used to model the Liloss feature [21] are very similar to those for the K- Rbcase. Our trap aspect ratio is larger than that of Ref. [20]by a factor of 15, but similar to that of Ref. [7], whichalso reported an avalanche peak for Li. Finally, we notethat we have also taken measurements in a crossed-beamoptical dipole trap with aspect ratio of 30, and again no clear avalanche loss feature was observed.
IV. CONCLUSIONS
We have measured the three-body recombination lossrate and the number loss ratio for a K- Rb atom gasmixture at positive scattering length over the range from100 a to 1500 a in a search for a feature connected tothe previously observed atom-dimer Efimov resonance at a ∗ = 230 a . While an avalanche model has been used tointerpret atom loss features seen in other systems as be-ing a consequence of an atom-dimer resonance, our mea-surements do not show a loss feature consistent with thismodel. The fact that there remains no single system inwhich both resonant loss in an atom-dimer gas mixtureand a corresponding loss feature for an atom gas havebeen observed is problematic for validation of this expla-nation of these atom loss peaks. ACKNOWLEDGMENTS
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