Scaling functions applied to three-body recombination of Cesium-133 atoms
aa r X i v : . [ c ond - m a t . o t h e r] M a y Scaling functions applied to three-body recombination of
Csatoms
L. Platter
1, 2, ∗ and J. R. Shepard † Department of Physics, The Ohio State University, Columbus, OH 43210 Department of Physics and Astronomy,Ohio University, Athens, OH 45701, USA Department of Physics, University of Colorado, Boulder, CO 80309, USA (Dated: November 20, 2018)
Abstract
We demonstrate the implications of Efimov physics in the recently measured recombination rateof
Cs atoms. By employing previously calculated results for the energy dependence of the re-combination rate of He atoms, we obtain three independent scaling functions that are capableof describing the recombination rates over a large energy range for identical bosons with largescattering length. We benchmark these and previously obtained scaling functions by successfullycomparing their predictions with full atom-dimer phase shift calculations with artificial He po-tentials yielding large scattering lengths. Exploiting universality, we finally use these functionsto determine the 3-body recombination rate of
Cs atoms with large positive scattering length,compare our results to experimental data obtained by the Innsbruck group and find excellentagreement.
PACS numbers: 21.45.+v,34.50.-s,03.75.NtKeywords: Renormalization group, limit cycle, cold atoms ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION In atomic physics the term universality refers to phenomena which are a result of a two-body scattering length a much larger than the range R of the underlying potential anddo not depend on any further parameters describing the two-body interaction. The non-relativistic three-body system also exhibits universal properties if a ≫ R , but an additionalthree-body parameter is needed for the theoretical description of observables. Therefore, onethree-body observable can be used (e.g. the minimum of the three-body recombination rate a ∗ ) to predict all other low-energy observables of such systems. A particularly interestingsignature of universality in the three-body system is a tower of infinitely many bound states( Efimov states ) in the limit a = ±∞ with an accumulation point at the scattering thresholdand a geometric spectrum : E ( n ) T = ( e − π/s ) n − n ∗ ¯ h κ ∗ /m, (1)where κ ∗ is the binding wavenumber of the branch of Efimov states labeled by n ∗ . Thethree-body system displays therefore discrete scaling symmetry in the universal limit witha scaling factor factor e π/s . In the case of identical bosons, s ≈ . e π/s ≈ .
7. These results were first derived in the 1970’s by Vitaly Efimov[1, 2] and were rederived in the last decade in the framework of effective field theories (EFT)[3, 4].Recently, experimental evidence for Efimov physics was found by the Innsbruck group[5]. Using a magnetic field to control the scattering length via a Feshbach resonance, theymeasured the recombination rate of cold
Cs atoms and observed a resonant enhancementin the three-body recombination rate at a ≈ − a which occurs because an Efimov stateis close to the 3-atom threshold for that value of a . The three-body recombination ratefor atoms with large scattering length at non-zero temperature has been calculated witha number of different models or based on the universality of atoms with large scatteringlengths [6, 7, 8, 9, 10]. However, a striking way to demonstrate universality is to describeobservables of one system with information which has been extracted from a completelydifferent system. In [11], the authors considered Efimov’s radial laws which parameterizethe three-atom S-matrix in terms of six real universal functions which depend only on adimensionless scaling variable, x = ( ma E/ ¯ h ) / , and phase factors which only containthe three-body parameter. In this work, simplifying assumptions justified over a restricted2ange of x were made to reduce the six universal functions required to parameterize thethree-body recombination rate to just a single function. This function was then extractedfrom microscopic calculations of the recombination rates for He atoms by Suno et al. [12].In a recent paper, Shepard [13] calculated the recombination rates from atom-dimer elasticscattering phase shifts for four different He potentials (the so-called HDFB, TTY, LM2M2and HFDB3FCII potentials) and was able to obtain two universal functions.Here, we relax all but one of the simplifying assumptions made in [11] and extract a set ofthree independent universal functions capable of parameterizing the three-body recombina-tion rate over a wide range of energies. We test the performance of these universal functionsusing “data” generated from phase shift calculations[13] employing artificial short-range Hepotentials. Finally, we use the new universal functions to calculate the scattering length andtemperature dependent recombination rate for
Cs atoms as measured by the Innsbruckgroup[5] and comment on our results.
II. THREE-BODY RECOMBINATION
Three-body recombination is a process in which three atoms collide to form a diatomicmolecule (dimer). If the scattering length is positive and large compared to the range ofthe interaction, we have to differentiate between deep and shallow dimers. Shallow dimershave an approximate binding energy of E shallow ≃ ¯ h / ( ma ) ≪ ¯ h / ( mR ). The bindingenergy of deep dimers cannot be expressed in terms of the effective range parameters and E deep > ∼ ¯ h / ( mR ). If the underlying interaction supports deep bound states, recombinationprocesses can occur for either sign of a . In a cold thermal gas of atoms, recombinationprocesses lead to a change in the number density of atoms n A dd t n A = − L n A , (2)where L denotes the loss rate constant. The recombination coefficient, to which L isproportional, can be decomposed into K ( E ) = K shallow ( E ) + K deep ( E ) , (3)and the recombination rate into the shallow dimer can be further decomposed into contri-butions from the channels in which the the total orbital angular momentum of the three3toms has a definite quantum number J according to K shallow ( E ) = ∞ X J =0 K ( J ) ( E ) . (4)For now, let us consider recombination via the shallow dimer only. If the collision energy E is small compared to the natural energy scale ¯ h / ( mR ), the recombination rate K shallow ( E ) isa universal function of the collision energy E , scattering length a and three-body parameter a ∗ . The universal function depends on the dimensionless scaling variable defined as x = ( ma E/ ¯ h ) / . (5)For J > a ∗ and theimplications of universality are therefore particularly simple, namely K ( J ) = f J ( x )¯ ha /m . (6)However, K (0) depends log-periodically on a ∗ ( this is the signature of Efimov physics!) andis related to the S-matrix for elastic atom-dimer scattering through K (0) ( E ) = kx (1 − | S AD,AD | ) , (7)Efimov’s radial law then gives the dependence on complex universal functions and the three-body parameter a ∗ which defines the scattering length for which the recombination ratehas a minimum as S AD,AD = s ( x ) + s ( x ) e is ln( a/a ∗ ) − s ( x ) e is ln( a/a ∗ ) . (8)The functions s and s are known at threshold s (0) = − e − πs ,s (0) = √ − e − πs e iδ ∞ ,s (0) = e iδ ∞ e − πs , (9)with δ ∞ = 1 . | s (0) | ≃ . i.e. ; ≪
1) for all x and can be ignored.Then the energy dependent recombination rate can be written as K (0) ( E ) = 144 √ π x h − (cid:0) r − r + 2 r r cos[Φ + 2 s log( a/a ∗ )] (cid:1)i ¯ ha m , (10)4here we have set s ij = r ij exp( iφ ij ) and Φ = φ − φ . Under the assumption that s can be neglected the recombination rate depends therefore on the three real-valued function r ( x ), r ( x ) and Φ( x ). It is worth noting that the expression in Eq. (10) is symmetricunder exchange of r and r . However, the threshold conditions in Eq. (9) can be used toto attribute the correct fit solutions to the universal function.As also discussed in Ref. [11], the effects of deep dimers can easily be incorporated throughone additional parameter η ∗ by making the substitutionln a ∗ → ln a ∗ − iη ∗ /s (11)in, e.g. Eq. (10). Employing unitarity the resulting effect on the recombination into shallowdimers can be written as [17] K (0)shallow ( E ) = 144 √ π x (cid:18) − (cid:12)(cid:12) s ( x ) + s ( x ) e iθ ∗ − η ∗ (cid:12)(cid:12) − (1 − e − η ∗ ) | s ( x ) | (cid:19) ¯ ha m . (12)Note that in deriving this expression we assumed again that s ≈
0. In the same mannerone can derive an expression for the recombination rate into deep dimers K deep ( E ) = 144 √ π x (1 − e − η ∗ ) (cid:0) − | s ( x ) | (cid:1) ¯ ha m . (13) III. ALTERNATIVE PARAMETERIZATIONS
Starting with S-matrix element for 3-atom to dimer-atom scattering, it was shown in [11]that under the assumption s = 0 the recombination rate can be written as K (0) ( E ) = C max (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) sin[ s ln( aa ∗ )] (cid:0) h ( x ) + ih ( x ) (cid:1) + cos[ s ln( aa ∗ )] (cid:0) h ( x ) + ih ( x ) (cid:1)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ¯ ha m , (14)where C max ≈ . h i are real-valued functions of x . Additionally,it was assumedthat the imaginary part of the above amplitude can be neglected K (0) ( E ) = C max (cid:12)(cid:12) sin[ s ln( a/a ∗ )](1 + h ( x )) + cos[ s ln( a/a ∗ )] h ( x ) (cid:12)(cid:12) ¯ ha /m . (15)This is well justified by direct calculations of the J = 0 recombination rates for He atomswhich display pronounced minima at approximately E breakup ≃
20 mK [12] and which can5 h i FIG. 1: The scaling functions h (solid line) and h as a function of x. be explained by this assumption. Then the functions h and h can be set to 0 in Eq. (10).The resulting expressions were employed in [13] to extract h and h for x < .
1. Although h and h were determined by fitting to values of K (0) ( E ) calculated using just two of thefour atom-atom potentials considered, they were found to accurately account for the resultsfor all 4 potentials as expected from universality. We have recalculated the h -functions usingthe results for the three-body recombination obtained using the LM2M2 and HFDB3FCIIpotentials and have fitted a polynomial to our results over the energy range 0 < x < . h ( x ) = − . x + 0 . x − . x + 1 . x − . x ,h ( x ) = 0 . x − . x + 0 . x − . x + 0 . x . (16)The functions are displayed in Fig. 1. The effect of deep dimers on the recombinationrate into the shallow dimer can easily be incorporated by making the substitution ln a ∗ → ln a ∗ − iη ∗ /s in Eq. (14) K (0) ( E ) = C max h cosh η ∗ (cid:0) sin[ s ln( a/a ∗ )](1 + h ( x )) + cos[ s ln( a/a ∗ )] h ( x ) (cid:1) + sinh η ∗ (cid:0) cos[ s ln( a/a ∗ )](1 + h ( x )) − sin[ s ln( a/a ∗ )] h ( x ) (cid:1) i ¯ ha m . (17)To take the effects of the recombination rate into deep dimers into account it was assumedin [11] that K deep ( E ) is a function varying slowly with energy and that it can therefore beapproximated with K deep = C − e − η ∗ ) ¯ ha m . (18)6 E[ µ K]10 -29 -28 -27 K ( ) ( E ) [ c m / s ] HFDBTTY
FIG. 2: The exact recombination rates and the corresponding results obtained with scaling (solidlines) and universal functions (dashed lines) of the HFDB (circles) and TTY (triangles) potentials.
IV. EXTRACTION OF THE UNIVERSAL FUNCTIONS
By fitting Eq.(10) to the recombination rates of all four He potentials, we were ableto determine the functions r ( x ), r ( x ) and Φ( x ). Our results are smooth functions for x > . x . For x < .
2, we are not able to find a reliable fit which is indicated by therapid variation of the function Φ in Fig. 3 in this region.To display the qualities of our fit we compare the exact recombination rates obtained withthe TTY and HDFB potentials to the rates calculated with the newly obtained universalfunctions These results for these functions are displayed in Fig. 2. This figure containsalso the recombination rate obtained with the h -functions. While the new set of universalfunctions seem to provide slightly better results for the HFDB potential at larger energies,the h -functions perform equally well for these potentials at lower energies.To test our new parameterizations we have generated three artificial potentials (whichwe call I, II and III) characterized by different three-body parameters a ∗ (with a/a ∗ =1 . , .
188 and 1 . R/a asthe real He potentials used in this work. We have calculated the recombination rates forthese potentials and use these results to benchmark our universal functions. Our results aredisplayed in Fig.4. We find that the new set of functions is capable of describing the recom-7 r r π /20 π /2 φ - φ FIG. 3: The universal functions r , r and Φ as function of x . E[ µ K]10 -28 -27 -26 -25 K ( E ) [ c m / s ] IIIIII
FIG. 4: The exact recombination rates and the corresponding results obtained with the h -functions(solid lines) and the universal functions (dashed lines) for the potentials I (circles), II (squares)and III (triangles) potentials. bination rates of these potentials over a relatively large range of x . Again we benchmarkalso the rates obtained with h and h and find that this set of scaling functions describesthe exact results better at x < r ( x ), r ( x ) and Φ( x ). Thisis surprising at first sight since one certainly expects to obtain a better description of therecombination rate with three instead of two functions. We speculate that the functionalform in Eq. (10) results in stronger constrains on the universal functions than Eq. (14) doeson the h functions. All the potentials, however, contain finite range effects which are notaccounted for in Eq. (8). It is therefore very likely that better fits – using the same ap-proximation – can be obtained from recombination rates calculated in the exact zero-rangelimit. V. RESULTS FOR CESIUM
In the previous section we found that we can obtain a very good overall description of therecombination rate of systems with a large scattering length if we employ the functions h and h for energies smaller than E shallow and the universal function r ( x ), r ( x ) and Φ( x )for energies larger than E shallow . Using these functions at energies close to the minimum inthe recombination guarantees a more appropriate treatment of the effect of deep dimers on9
200 400 600 800 1000 1200a[a ]010002000300040005000 ρ [ a ] FIG. 5: The 3-body recombination length ρ for Cs for a ∗ = 210 a and three different valuesof the parameter η ∗ :0 (solid line), 0.01 (dashed line), and 0.06 (dotted lines) plotted together withthe experimental results of the Innsbruck experiment (triangles) [5]. the recombination rate, which are expected to have the largest effect in this region.The form of the functions f J ( x ) in Eq.(6) and therefore the contribution to the recombi-nation from channels with higher total angular momentum J has been previously analyzedin [11, 13], we thus take these channels into account by using appropriate parameterizationsfor the functions f J ( x ). Cs atoms can recombine into deep and shallow dimers. As men-tioned above, a deep dimer is so strongly bound that it cannot be described within the EFTfor short-range interactions as the binding energy is larger than ¯ h / ( mR ). We account forsuch processes by letting ln a ∗ → ln a ∗ − iη ∗ /s as also discussed above. We then calculatethe temperature dependent recombination rate by calculating α ( T ) = R ∞ dE E e − E/ ( k B T ) K ( E )6 R ∞ dE E e − E/ ( k B T ) . (19)The weight factor E comes from using hyperspherical variables for the Jacobi momenta.In Fig. 5 we display our results for the recombination length ρ = (cid:16) mK √ h (cid:17) / of Csatoms. It can be seen that the results agree very well with the experimental results obtainedby the Innsbruck group at T = 200 nK. 10 I. SUMMARY
In this paper we have used the results from different He atom-atom potentials to extractand to test the predictive power of universal functions. In doing so, we have relaxed allbut one simplifying assumptions which was made in previous work [11, 13]. We have de-termined a third universal scaling function which allows for a description of the three-bodyrecombination rate of systems with large scattering length over a greater range of breakupenergies.We have tested the quality of our parameterizations with artificial finite range potentialswhich are appreciably different from the original Helium potentials but which display uni-versal effects in three-body sector. We have found that our three real universal functions candescribe the recombination of these artificial potentials reasonably well which gives furtherevidence that the assumptions made in [11] were well justified. We also found, however,that the previously calculated scaling functions h and h give an overall better descriptionof the recombination rate for energies E < E shallow . The scaling functions h and h whichcan be represented analytically with a simple polynomial fit given in Eq. (16) are thereforea useful tool to test recombination rate calculations for systems with large scattering length.Finally, we have used both sets of universal functions together to compute the recombi-nation length for Cs atoms for different values of the parameter η ∗ which approximatelyaccounts for the effect of deep dimer states and have compared our results with experimentaldata obtained by the Innsbruck group[5].Although our results show very good agreement with the data, sensitivity to η ∗ is insuf-ficient to permit a precise determination of this parameter. Overall, we consider our resultsto be an excellent example of how few-body systems with large scattering length exhibit uni-versal features. The low-energy properties of He atoms allow us to compute accurately thelow-energy properties of a gas of a completely different element,
Cs, which at first glancehas little in common with He. Nevertheless, we point out that the results cannot be thoughtof as complete treatment of the problem at hand. For example, not only did we make the as-sumption that s does not contribute significantly to the recombination coefficients, we alsoextracted the functions from data sets obtained with finite range potentials. Although theimpact of range corrections is known to be small for realistic Helium atom-atom potentialsas R/a ∼ .
1, it needs to be pointed out that range corrections are expected to be sizable11or large enough energies. To obtain all universal functions s ij relevant to the recombinationrate, a calculation in the limit R → . Furthermore, itis already understood how to include range corrections systematically in the framework ofeffective field theory [14, 15, 16]. Indeed, this approach has already been used to calculaterange corrections to the recombination rate into a shallow dimer [18, 19]. Thus, furthereffort should be devoted to include these effects in the calculation of the energy-dependentrecombination rate. Acknowledgments
We are thankful to Eric Braaten and Daniel Phillips for useful discussions and commentson the manuscript. This work was supported in part by the Department of Energy undergrant DE-FG02-93ER40756, by the National Science Foundation under Grant No. PHY–0354916. [1] V. Efimov, Phys. Lett. , 563 (1970).[2] V. N. Efimov, Sov. J. Nucl. Phys. , 589 (1971).[3] P. F. Bedaque, H.-W. Hammer and U. van Kolck, Phys. Rev. Lett. , 463 (1999).[4] E. Braaten and H.-W. Hammer, Phys. Rept. , 259 (2006).[5] T. Kraemer, M. Mark, P. Waldburger, J.G. Danzl, C. Chin, B. Engeser, A.D. Lange, K. Pilch,A. Jaakkola, H.-C. N¨agerl, and R. Grimm, Nature , 315 (2006).[6] J.P. D’Incao, H. Suno, and B. D. Esry, Phys. Rev. Lett. , 123201 (2004).[7] M.D. Lee, T. Koehler and P.S. Julienne, Phys. Rev. A , 012720 (2007).[8] S. Jonsell, Europhys. Lett. , 8 (2006).[9] M.T. Yamashita, T. Frederico, and L. Tomio, Phys. Lett. A , 468 (2007).[10] P. Massignan and H.T.C. Stoof, Phys. Rev. A 78, 030701 (2008).[11] E. Braaten, D. Kang and L. Platter, Phys. Rev. A , 052714 (2007).[12] H. Suno, B. D. Esry, C . H. Greene, and J. P. Burke, Phys. Rev. A 65, 042725 (2002).[13] J. R. Shepard, Phys. Rev. A , 062713 (2007). This has been done [17] in which appeared after the first submission of this paper
14] P. F. Bedaque, G. Rupak, H. W. Griesshammer and H. W. Hammer, Nucl. Phys. A , 589(2003).[15] H.-W. Hammer and T. Mehen, Phys. Lett. B , 353 (2001).[16] L. Platter and D. R. Phillips, Few Body Syst. , 35 (2006).[17] E. Braaten, H. W. Hammer, D. Kang and L. Platter, Phys. Rev. A , 043605 (2008).[18] H.-W. Hammer, T. A. Lahde and L. Platter, Phys. Rev. A , 032715 (2007).[19] L. Platter, C. Ji and D. R. Phillips, Phys. Rev. A , 022702 (2009)., 022702 (2009).