ISSN 1054660X, Laser Physics, 2011, Vol. 21, No. 8, pp. 1464–1469.
PHYSICS OF COLD TRAPPED ATOMS
© Pleiades Publishing, Ltd., 2011. Original Text © Astro, Ltd., 2011.
Borromean ThreeBody Heteroatomic Resonances1 L. Tomioa, b*, M. T. Yamashitab, T. Fredericoc, and F. Bringasd a
Instituto de Física, Universidade Federal Fluminense, 24210346, Niterói, RJ, Brazil Instituto de Física Teórica, Universidade Estadual Paulista—UNESP, 01140070, São Paulo, Brazil c Dep de Física, Instituto Tecnológico de Aeronáutica, CTA, 12228900, S. J. dos Campos, Brazil d Cooperative Institute for Marine and Atmospheric Studies, University of Miami, Miami, FL 33149, USA b
*email: [email protected]
Received November 3, 2010; in final form, January 7, 2011; published online July 4, 2011
Abstract—We present an approach to analyze recent experimental evidences of Efimov resonant states in mixtures of ultracold gases, by considering twospecies threebody atomic systems bound in a Borromean configuration, where all the twobody interactions are unbound. For such Borromean threebody systems, it is shown that a continuum threebody swave resonance emerges from an Efimov state as a scattering length or a threebody scale is moved. The energy and width of the resonant state are determined from a scaling function with arguments given by dimensionless energy ratios relating the twobody virtual state subsystem energies with the shallowest threebody bound state. The peculiar behavior of such resonances is that their peaks are expected to move to lower values of the scattering length, with increasing width, as one raises the temperature. For Borromean systems, two resonant peaks are expected in ultralowtemperature regimes, which will disappear at higher energies. It is shown how a Borromean–Efimov excited bound state turns out to a resonant state by tuning the virtual twobody subsystem energies or scattering lengths, with all energies written in units of the next deeper shallowest Efimov state energy. The resonance position and width for the decay into the continuum are obtained as universal scaling functions (limit cycle) of the dimensionless ratios of the two and threebody scales, which are calculated numerically within a zerorange renormalized three body model. DOI: 10.1134/S1054660X11150229 1
The development of new techniques in coldatom laboratories, such as lasercooling mechanisms and the control of twobody interactions by Feshbach res onance techniques [1–3] allow us to explore a pleth ora of interesting nonlinear phenomena driven by the physics of fewbody scattering amplitudes at zero energies. In order to trace the relevance and richness of this physics, among the several reviews, we can mention the [4–6], as well as several recent works exploring the possibilities of controlling the fewbody interaction in nonlinear systems [7–13]. Moreover, the experiments performed nowadays with atoms and molecules at ultralow temperatures, are becoming very relevant, not only to study quantum manybody phys ics, but also fewbody properties which have been con sidered longtime ago in the context of fewnucleon systems. Within such fewbody properties, we can mention the longtime observation of the approxi mately linear correlation between the threenucleon binding energy and the doublet nucleondeuteron scattering length, known as Phillips line ; or the observed correlation between the four and three nucleon binding energies, given by the Tjon line . In particular, it is quite fascinating to verify that, with the actual possibility to alter the twoatom inter 1 The article is published in the original.
action in laboratories (using the wellknown Feshbach resonance techniques) , it is possible to observe experimentally universal correlations between few body observables, as well as to test the extension of their validity. In this respect, one should notice the number of recent experiments in coldatom laborato ries [16–24], which are being dedicated to verify prop erties of the socalled Efimov physics, related to the longtime Efimov prediction of the appearance of an infinite number of threebody bound states when the twobody state energy is exactly at the dissociation threshold . (For the experimental research on this subject, see also the recent reviews done by Ferlaino and Grimm [26, 27] and by Efimov .) The Efimov prediction, done in the nuclear physics context, was first considered as a possible mathemati cal pathology of the threeboson threedimensional equations, in the limit of zero twobody energy, with no possibilities to be verified, considering that the nuclear interaction among the particles is fixed with out a mechanism to change it. Within such picture, the search for Efimov states was extended to existing atomicmolecular systems where twobody binding was very close to zero in comparison to corresponding threebody groundstate  in order to look for some exciting Efimov states, such as the interaction of three Helium atoms systems.
BORROMEAN THREEBODY HETEROATOMIC RESONANCES
The Efimov effect was also shown to be closely related to the collapse of the threebody ground state, when the range of the twobody interaction goes to zero (known as Thomas collapse ), by a scaling transformation . (In , the collapse was studied by considering separable twobody interactions.) In coldatom laboratories, the Efimov states were first identified in scattering using ultracold gas of Cesium atoms at 10 nanokelvin, at the Innsbruck lab oratory . More recently, considering that the asymmetry of masses could favor the observation of such states [32–34], Efimov resonances were identi fied in two heteroatomic channels for KKRb and KRbRb, by using a mixture of 41K and 87Rb . According to the twobody interactions, one can classify generic threebody ααβ systems [35, 36]. Some properties of these states under the change in the interaction parameters are well known. For example, when at least one of the twobody subsystems is bound, an Efimov state emerges from the twobody analytical cut by changing the twobody binding energy [37–39]. Such state, in the second energy sheet, is given by a pole in the real axis and identified as a virtual state. However, Borromean systems are threebody bound states where all the twobody interactions are unbound . In this case, there is no twobody cut, and the Efimov state arrives from a continuum three body resonance . Therefore, we report here and extension of the analysis done in , to Borromean ααβ systems (negative scattering lengths) with two kind of particles. It is shown how the Efimov states emerge from three body swave continuum resonances. As the absolute values of the scattering lengths (given, respectively, by |aαα| and/or |aαβ|) increase, an existing threebody continuum resonance disappears at the scattering threshold with the formation of an Efimov bound state. This process can be replicated by further increasing one or both twobody scattering lengths, with the formation of more Efimov excited states. In the exact Efimov limit, a tower with infinite number of excited states is produced when both twobody scat tering lengths, |aαα| and |aαβ| are infinite. This is the exact Efimov limit, when the corresponding twobody energies are zero. (For the case of identical particles, we define the twobody scattering length by a2 ≡ aαα.) A scaling function defines the region for the exist ence of a threebody resonance, in a parametric space, in terms of scattering lengths measured in units of a 2 –1
threebody length identified with [ m α B 3 /ប ] , where mα is the mass of the α particle and B3 is the threebody boundstate that we use as our threebody scaling parameter. (Our units will be such that ប = 1 and mα = 1, with the mass ratio defined as A ≡ mβ/mα.) This study also completes a previous discussion about the effect of a mass asymmetry on the route of a vir LASER PHYSICS
tual/resonance going to a bound state [37–39, 43]. To show this procedure, we will make use of a zerorange twobody interaction in the threebody problem, reg ularized in a form of a subtracted Faddeev equation. The renormalization is achieved by constructing scal ing functions in terms of observables: the Nth Efimov (N) state energy (resonance or bound E 3 ), the next deeper (N – 1)th Efimov boundstate energy (N – 1) (N – 1) ≡ –E3 ) and the scattering lengths (in the ( B3 present case, negative values corresponding to virtual state energies), achieving the scaling limit  or the limit cycle  for N ∞. It is worthwhile to stress that in our approach the scaling functions are directly related to physical observables, and are calculated numerically. Such scaling functions can also define correlation functions in a generalization of the widely recognized Phillips  and Tjon  plots. Few physical scales are necessary for the quantum description of such large and weakly bound systems. In the limit of zerorange interaction, all the detailed information about the shortrange force, beyond the lowenergy twobody observables, is retained in only one threebody physical information [32, 33]. The existence of a threebody scale implies in the low energy universality found in threebody systems, such that any observable for the case of three identical bosons, in the scaling limit [30, 44], can be described by [46–48] E2 ⎞ η ᏻ ( E, B 3, B 2 ) = ( B 3 ) Ᏺ ⎛⎝ E , ± , B3 B3 ⎠
where ᏻ is a general observable of the threebody sys tem at an energy E, with dimension given by a three body scale energy B3 to the power η. The twobody scales are defined by energies of the twobody virtual states, Eαα and Eαβ, or by the corre spondingly scattering lengths aαα ⯝ –1/ aαβ ⯝ –1/
E αα and
E αβ . Such scales are given in units of a
threebody length scale 1/ B 3 , with B3 defined as the binding energy of the shallowest threebody state. By considering the above, we can write the scaling func tion for the continuum resonance energy E3 as E 3 = B 3 Ᏹ ( a αα B 3, a αβ B 3 ; A ).
In , the corresponding critical boundary, given by the equation
Ᏹ ( a αα B 3, a αβ B 3 ; A ) = 0,
was numerically verified and given in a parametric plane defined by ( E αα /B 3 , E αβ /B 3 ), with all possibilities for bound and virtual twobody states being considered. We should point out that, for the present purpose, the calculations we are reporting
TOMIO et al.
were performed by considering only twobody virtual states (negative aαβ or aαβ). By crossing the critical boundary in the direction of infinity negative scatter ing lengths, an existing continuum threebody reso nance will emerge as a boundstate. In fact, in the par ticular case of identical bosons, a continuum three body state was observed in an ultracold trapped gas of Caesium , through the resonant behavior of three body recombination rate. In an ultracold mixture of 41K and 87Rb gases, the presence of Efimov resonances has been also observed through the threebody collision of KKRb and RbRbK atoms , with two resonantly interacting pairs for positive or negative scattering lengths. Such observation supports the prediction of , resumed in a critical boundary defined by ( ± E αα /B 3 , ± E αβ /B 3 ), for the existence of at least one Efimov bound state. Also by using zerorange interaction, threebody heteronuclear mixtures with resonant interspecies were studied in , where it was dis cussed the critical conditions for formation of Efimov states at the threshold. In the next, we present our formalism, followed by the main results and conclusions.
In order to calculate the scaling function (2), we use subtracted Faddeev equations with zerorange interactions for a threebody system composed by two identical bosons, α, and a third one, β [50, 51]. The following equations, presented for continuum reso nant states, were already written in a similar form in the context of exotic halo nuclei systems [35–39, 52]. After partialwave projection, the swave coupled integral equations of the Faddeev spectator functions χαα and χαβ, can be written as
χ αα ( q ) = 4πτ αα ( q ) p dp dzG 1 ( q, p, z )χ αβ ( p ), –1
χ αβ ( q ) = 2πτ αβ ( q ) p dp dz –1
× [ G 1 ( p, q, z )χ αα ( p ) + G 2 ( q, p, z )χ αβ ( p ) ], –1
τ αα ( q ) = – 2π ∞
+ 2 q 2⎞ dp' , – 4π ⎛ E 3 – A ⎝ ⎠ 2 4A A+2 2 – – p' E q 0 3 4A
2A 2 = – 2π ⎞ ⎝ A + 1⎠
E αβ (4)
A + 2 q2 2A ⎞ E – – 4π ⎛ ⎝ A + 1⎠ 3 2 ( A + 1 ) ∞
d p' , × A + 2 q2 – A + 1 p' 2 – E 0 3 2(A + 1) 2A
1 G 1 ( q, p, z ) = + 1 q 2 2 E3 – A – p – pqz 2A 1 + , 2 2 A + 1 1 + q + p + pqz 2A 1 G 2 ( q, p, z ) = 2 A + 1 + 1 p 2 – pqz E 3 – q – A 2A A 2A 1 + , 2 A + 1 2 pqz A + 1 1 + q + p + 2A A 2A
–1 τ αβ ( q )
where the twobody amplitudes, ταα and ταβ, and the substracted Green functions, G1 and G2, are defined in [38, 39]. The coupled Eqs. (4) are calculated by using a con tour deformation method in the complex momentum plane [53, 54]. The momentum variable as p appears as a function of the deformation angle θ and written as p ≡ |p|e–iθ, with 0 ≤ θ < π/4. The second energy sheet is revealed when the momentum integration path is deformed by a contour along the real axis to the com plex plane with a fixed rotation angle chosen to place the contour path far from the scattering singularities. That exposes the complex resonance energy pole of the scattering matrix. For large enough θ, the solution of the coupled Eqs. (4) in the complex energy plane is found for tan(2θ) > –Im(E3)/Re(E3), where Re(E3) represents the resonance energy and Im(E3) its half width. In the limit when the twobody energies are equal to zero, an infinite number of Efimov states emerges from the solution of the coupled Eqs. (4). At this point, by varying the twobody energies, a given energy E3 will have zero imaginary part, becoming purely real and negative. The Nth Efimov bound state (N) (N) is defined by E 3 ≡ – B 3 with N = 0 indicating the ground state. Its complex energy will be denoted by (N) E 3 when this state becomes a resonance. LASER PHYSICS
BORROMEAN THREEBODY HETEROATOMIC RESONANCES
boundary curve in the parametric space ( – E αα /B 3 ,
– E αβ /B 3 ) (for the case Eαα = Eαβ)  that sepa rates the excited bound state from the continuum. Our plots in Fig. 1 are determined only for N = 1, consid ering that results for higher N’s should coincide with the ones obtained for N = 1. The energy value of the resonance (real part) grows, as the virtual twobody energies increases (by reducing the absolute values of the scattering lengths), up to a maximum and then decreases, while the imaginary part always increases. This makes the resonance to dive deeper in the second energy sheet.
6 4 2 0 −8 −6 −4 −2 0
15 20 25 30 103[|Eαα|/B3(N−1)]
Fig. 1. Resonant behavior of the threebody complex energy as a function of the twobody virtual state energy with Eαα = Eαβ. The real part is given in the upper frame, with the imaginary part in the lower frame. The results are shown for N = 1, with mass ratios A = 1 (solid line), 9 (dashed line), 200 (dotted line).
3. MAIN RESULTS By solving the homogeneous coupled Eqs. (4) in the complex rotated contour for complex energies we obtain the position and width of the continuum three body resonances, which allows us to construct numer ically the scaling function (2). As observed, one can approach quite fast the scaling limit [44, 55], or the limit cycle [56, 57] for N ∞. In two frames, we show in Fig. 1 the real and imaginary parts of the com (N) plex energy E 3 , as functions of the twobody binding energy Eαα, for the case that Eαα = Eαβ. All such energy quantities are dimensionless, given in units of the (N – 1) boundstate energy B 3 . So, in the upper frame of (N)
(N – 1)
Fig. 1, we have Re( E 3 )/ B 3 and, in the lower frame,
(N – 1)
versus Eαα/ B 3
(N) (N – 1) Im( E 3 )/ B 3
(N – 1) Eαα/ B 3 . The Nth resonance energy, given by (N) (N – 1) Re( E 3 )/ B 3 , turns into a bound state when the (N – 1) is decreased. The transition from a ratio Eαα/ B 3
resonant to a bound state follows the same general behavior as verified in the case of three identical bosons . For A = 1, 9, and 200, the values of (N – 1) Eαα/ B 3 at which we have transitions from reso nances to boundstates are, respectively, given by 0.876 × 10–3, 1.67 × 10–3, and 2.15 × 10–3. These values correspond to the ones given the position of the LASER PHYSICS
The present results suggest that, in an ultracold mixture of heteronuclear atoms, the increase of tem perature will move a Borromean continuum resonance towards smaller absolute values of the scattering length (larger twobody virtual energies), while the width increases. This is in agreement with experiment with ultracold Caesium atoms , where the recombina tion peak due to the continuum triatomic resonance moves towards smaller values of |a| as temperature is raised. Therefore such resonance is identified with a state that just arises from an Efimov bound one that dives into the threeboson continuum. The maximum value reached by the real part of the threebody energy, the position of the resonance, presumably creates a curious effect that one could observe in the threebody recombination. As the position of the resonance (N) returns to zero (Re( E 3 ) 0), it may be possible to detect, at a given temperature, the occurrence of two peaks in the threebody recombination, correspond ing to two values of the scattering length. The wider peak should be at smaller values of the absolute value of the scattering length, as one can easily verify by looking at Fig. 1. A more detailed study on these aspects are under investigation, and soon will be reported. The boundaries for the existence of a resonance can be easily verified by considering the calculations done in . The boundary curves will indicate that, at ultralow temperatures, a mixture of heteronuclear gas with two species can have two threebody recombina tion peaks associated with one Efimov state. As aαβ is modified with aαα kept fixed, a wider resonance appears for smaller values of |aαβ|; with a thinner one occurring for larger values. This last case corresponds to an excited bound Efimov state diving into the con tinuum. The two peaks should merge, as the maximum of the resonance is approached, for increasing values of the temperature T. In Fig. 2, for different Eαβ/Eαα ratios, it was plotted (N)
the corresponding maximum values of Re( E 3 ), in (N – 1)
units of B 3 . As shown, the absolute maximum occurs for the symmetric case, when all the twobody
TOMIO et al.
subsystems have the same virtual energy, Eαβ = Eαα, as one can verify by comparing the results with Fig. 1. Therefore, in case of Borromean systems, we found two continuum resonances at different values of the scattering length and same position in energy and dif ferent widths, for heteronuclear and homonuclear sys tems. The absolute maximum for the resonant energy (associated with the system temperature) occurs in both cases. As we observe in Fig. 1 the resonance energy attains a maximum value and then decreases as |a| is moved further towards small values. Therefore, two resonances are located at the same energy for dif ferent values of the negative scattering length. It is expected that the corresponding pair of threebody recombination peaks merges as the temperature is raised, one peak moves towards lower values of the scattering length while the other one to larger values. Using the numerical values of the scattering length for homonuclear systems, one gets that the ratio between the scattering lengths where the real part of the reso nance energy is identical stays between 1.00 and 0.25 as one easily gets by inspection of Fig. 1. As an exper imental challenge for separating the three and four body recombination peaks, we should observe that the second threebody recombination peak is expected to appear in a region where the fourbody recombination peak is also being manifested, with a scattering length ratio of 0.43 [27, 58]. 4. CONCLUSIONS We reported here results of a study on the universal properties of threebody continuum resonances for heteronuclear threeparticle system with two species of atoms, in a Borromean configuration such that all the twobody subsystems are unbound. It is shown how a Borromean–Efimov excited bound state turns out to a resonant state by tuning the virtual twobody subsystem energies or scattering lengths, with all ener gies written in units of the next deeper shallowest Efi mov state energy. The resonance position and width for the decay into the continuum are obtained as uni versal scaling functions (limit cycle) of the dimension less ratios of the two and threebody scales, which are calculated numerically within a zerorange renormal ized threebody model. We discussed how the contin uum resonances for heteronuclear Borromean systems may be observed in mixtures of ultracold gases. Such resonances will be given by their characteristic dis placement with temperature T. As T raises the recom bination peak, that comes from an Efimov state diving into the continuum, should move to lower absolute values of the scattering lengths with an increasing width. Furthermore, two recombination peaks at T = 0 are obtained, with different widths and positioned at different values of the scattering lengths. For a fixed αα scattering length, with the assumption that three body scale does not move, the wider resonance should appear for smaller absolute values of the αβ scattering
1000[Re(E3(N))max/B3(N−1)]1/2 A=1 A=9 A = 200
35 30 25 20 15 10 5 0
8 9 10 (|E|/|E|)1/2 (N)
Fig. 2. Ratio of the maximum values of Re( E 3 (N – 1)th binding energies as a function of ( The curves are labeled as in Fig. 1.
) and the
E αβ / E αα ).
length. As T is raised the two recombination peaks tend to merge. This physical effect may be observed in ultracold trapped heteronuclear systems near a Fesh bach resonance, where the properties of the contin uum resonances offer a rich structure to be explored by tuning the large and negative scattering lengths. ACKNOWLEDGMENTS We thank Fundação de Amparo à Pesquisa do Estado de São Paulo and Conselho National de Desenvolvimento Científico e Tecnológico for partial financial support. REFERENCES 1. E. Timmermans, P. Tommasini, M. S. Hussein, and A. Kerman, Phys. Rep. 315, 199 (1999). 2. S. Inouye, M. R. Andrews, J. Stenger, H.J. Miesner, D. M. StamperKurn, and W. Ketterle, Nature 392, 151 (1998). 3. Ph. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen, and B. J. Verhaar, Phys. Rev. Lett 81, 69 (1998). 4. V. I. Yukalov, Laser Phys. 19, 1 (2009). 5. B. A. Malomed, Soliton Management in Periodic Sys tems (Springer, New York, 2006). 6. F. Kh. Abdullaev, A. Gammal, A. M. Kamchatnov, and L. Tomio, Int. J. Mod. Phys. B 19, 3415 (2005). 7. LiYun Yang, LiBin Fu, and Jie Liu, Laser Phys. 19, 678 (2009). 8. L. Salasnich, F. Ancilotto, N. Manini, and F. Toigo, Laser Phys. 19, 636 (2009). 9. S. K. Adhikari, Laser Phys. Lett. 6, 901 (2009). 10. V. I. Yukalov and D. Sornette, Laser Phys. Lett. 6, 833 (2009). LASER PHYSICS
BORROMEAN THREEBODY HETEROATOMIC RESONANCES
11. V. I. Yukalov and V. S. Bagnato, Laser Phys. Lett. 6, 399 (2009).
32. A. S. Jensen, K. Riisager, D. V. Fedorov, and E. Gar rido, Rev. Mod. Phys. 76, 215 (2004).
12. V. I. Yukalov, Laser Phys. Lett. 7, 467 (2010).
33. E. Braaten and H.W. Hammer, Phys. Rep. 428, 259 (2006).
13. H. L. F. da Luz, F. Kh. Abdullaev, A. Gammal, M. Sal erno, and L. Tomio, Phys. Rev. A 82, 043618 (2010).
34. B. D. Esry, Y. Wang, and J. P. D’Incao, FewBody Syst. 43, 63 (2008).
14. A. C. Phillips, Nucl. Phys. A 107, 209 (1968); Rep. Prog. Phys. 40, 905 (1977).
35. M. T. Yamashita, L. Tomio, and T. Frederico, Nucl. Phys. A 735, 40 (2004).
15. J. A. Tjon, Phys. Lett. B 56, 217 (1975).
36. T. Frederico, M. T. Yamashita, and L. Tomio, Few Body Syst. 45, 215 (2009).
16. T. Krámer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakko la, H.C. Nágerl, and R. Grimm, Nature 440, 315 (2006). 17. T. B. Ottenstein, T. Lompe, M. Kohnen, A. N. Wenz, and S. Jochim, Phys. Rev. Lett. 101, 203202 (2008). 18. S. Knoop, F. Ferlaino, M. Mark, M. Berninger, H. Schoebel, H.C. Nágerl, and R. Grimm, Nature Phys. 5, 227 (2009).
37. M. T. Yamashita, T. Frederico, and L. Tomio, Phys. Rev. Lett. 99, 269201 (2007). 38. M. T. Yamashita, T. Frederico, and L. Tomio, Phys. Lett. B 660, 339 (2008). 39. M. T. Yamashita, T. Frederico, and L. Tomio, Phys. Lett. B 670, 49 (2008). 40. J.M. Richard and S. Fleck, Phys. Rev. Lett. 73, 1464 (1994).
19. M. Zaccanti, B. Deissler, C. D’Errico, M. Fattori, M. JonaLasinio, S. Mller, G. Roati, M. Inguscio, and G. Modugno, Nature Phys. 5, 586 (2009).
41. S. Moszkowski, S. Fleck, A. Krikeb, L. Theussl, J.M. Richard, and K. Varga, Phys. Rev. A 62, 032504 (2000).
20. F. Ferlaino, S. Knoop, M. Berninger, W. Harm, J. P. DIncao, H.C. Náger, and R. Grimm, Phys. Rev. Lett. 102, 140401 (2009).
42. F. Bringas, M. T. Yamashita, and T. Frederico, Phys. Rev. A 69, 040702(R) (2004).
21. J. H. Huckans, J. R. Williams, E. L. Hazlett, R. W. Sti tes, and K. M. OHara, Phys. Rev. Lett. 102, 165302 (2009).
43. I. Mazumdar, A. R. P. Rau, and V. S. Bhasin, Phys. Rev. Lett. 97, 062503 (2006). 44. T. Frederico, L. Tomio, A. Delfino, and A. E. A. Amo rim, Phys. Rev. A 60, R9 (1999).
22. G. Barontini, C. Weber, F. Rabatti, J. Catani, G. Thal hammer, M. Inguscio, and F. Minardi, Phys. Rev. Lett. 103, 043201 (2009).
45. K. G. Wilson, Phys. Rev. D 3, 1818 (1971).
23. N. Gross, Z. Shotan, S. Kokkelmans, and L. Khaykov ich, Phys. Rev. Lett. 103, 163202 (2009).
47. M. T. Yamashita, T. Frederico, A. Delfino, and L. To mio, Phys. Rev. A 66, 052702 (2002).
24. S. E. Pollack, D. Dries, and R. G. Hulet, Science 326, 1683 (2009); S. E. Pollack, D. Dries, M. Junker, Y. P. Chen, T. A. Corcovilos, and R. G. Hulet, Phys. Rev. Lett. 102, 090402 (2009).
48. M. T. Yamashita, T. Frederico, A. Delfino, and L. To mio, Phys. Rev. A 68, 033406 (2003).
25. V. Efimov, Phys. Lett. B 33, 563 (1970).
50. L. Tomio, FewBody Syst. 43, 207 (2008).
26. V. Efimov, Nature Phys. 5, 533 (2009).
51. T. Frederico, A. Delfino, and L. Tomio, FewBody Syst. 31, 235 (2002).
27. F. Ferlaino and R. Grimm, Physics 3, 9 (2010). 28. T. K. Lim, S. K. Duffy, and W. C. Dumert, Phys. Rev. Lett. 38, 341 (1977); H. S. Huber, and T. K. Lim, J. Chem. Phys. 68, 1006 (1978); T. Cornelius and W. Glöckle, J. Chem. Phys. 85, 3906 (1986); B. D. Esry, C. D. Lin, and C. H. Greene, Phys. Rev. A 54, 394 (1996); E. A. Kolganova, A. K. Motovilov, and S. A. Sofianos, Phys. Rev. A 56, R1686 (1997); A. Del fino, T. Frederico, and L. Tomio, J. Chem. Phys. 113, 7874 (2000).
46. A. E. A. Amorim, T. Frederico, and L. Tomio, Phys. Rev. C 56, R2378 (1997).
49. K. Helfrich, H.W. Hammer, and D. S. Petrov, Phys. Rev. A 81, 042715 (2010).
52. T. Frederico, M. T. Yamashita, A. Delfino, and L. To mio, FewBody Syst. 38, 57 (2006). 53. J. Aguilar and J. M. Combes, Commun. Math. Phys. 22, 269 (1971). 54. E. Balslev and J. M. Combes, Commun. Math. Phys. 22, 280 (1971). 55. A. Delfino, T. Frederico, and L. Tomio, FewBody Syst. 28, 259 (2000).
29. L. H. Thomas, Phys. Rev. 47, 903 (1935).
56. S. Albeverio, R. HoeghKrohn, and T. S. Wu, Phys. Lett. A 83, 105 (1981).
30. S. K. Adhikari, A. Delfino, T. Frederico, I. D. Gold man, and L. Tomio, Phys. Rev. A 37, 3666 (1988).
57. R. F. Mohr, R. J. Furnstahl, H.W. Hammer, R. J. Per ry, and K. G. Wilson, Ann. of Phys. 321, 225 (2006).
31. A. Delfino, S. K. Adhikari, L. Tomio, and T. Frederico, Phys. Rev. C 46, 471 (1992).
58. J. von Stecher, J. P. D’Incao, and C. H. Greene, Nature Phys. 5, 417 (2009).