1. Triatomic states in ultracold gases Marcelo Takeshi Yamashita Universidade Estadual Paulista - Brazil  Lauro Tomio – IFT / Unesp  Tobias Frederico – ITA  Francis Bringas - ITA  Antonio Delfino - UFF  Collaborators Work partially supported by

2. Guidelines Summary The Efimov states Bound states Virtual states Resonances Triatomic continuum resonances Three-body recombination for virtual and bound two- body states in ultracold traps

3. The Efimov effect - Thomas-Efimov equivalence Three-body bound state equation with zero-range interaction with momenta cutoff momentaenergies ε2ε2 0 (N = 0, 1, 2,...) Efimov states 1) E 2 tends to zero with Λ fixed – Efimov effect 2) Λ tends to infinity with E 2 fixed – Thomas collapse Adhikari, Frederico, and Goldman PRL 74, 487 (1995). Skorniakov and Ter-Martirosian equation (1956)

4. The Efimov states – bound, virtual and resonances Three-body bound state equation with zero-range interaction with subtraction Three-body resonances Three-body energy is complex x y Contour deformation method Three-body virtual states

5. The Efimov states – bound and virtual states Lines – Bound states crosses – ground squares – first excited diamonds – second excited Symbols – Virtual states circles - refers to the first excited state triangles – refers to the second excited state Appearance of the virtual state (dashed line) The virtual state turns into an excited state (solid line) ε 2 bound MTY, Frederico, Delfino, and Tomio PRA 66, 052702 (2002)

6. The Efimov states - resonances ε 2 virtual Resonances Bringas, MTY, and Frederico PRA 69, 040702(R) (2004)

7. The Efimov states – trajectory of Efimov states Complete trajectory of Efimov states E3 bound E2 virtual E3 resonance E2 virtual E3 bound E2 bound E3 virtual E2 bound

8. The Efimov states – triatomic continuum resonances from http://www.uibk.ac.at/exphys/ultracold/ “Evidence of Efimov quantum states in an ultracold gas of cesium atoms” ! T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-C. Nägerl & R. Grimm, Nature 440, 315 (2006) Excited Efimov state turns into a resonance From the experiment T = 0  a = -898 a 0

9. Real part Imaginary part x 0.1 Triatomic continuum resonances in an ultracold gas of cesium atoms From calculations Analytic approximations The Efimov states – triatomic continuum resonances

10. Adding the effects of triatomic continuum resonances in the recombination rate L 3 for T = 0 where The resonance energy can be approximated by We can easily find the solution of a r- for E r After performing the thermal average of the recombination rate th we have the recombination length For T = 0 E. Braaten, and H.-W. Hammer, Phys. Rep. 428, 259 (2006)

11. Recombination length in a cesium trapped gas as a function of the scattering length and temperature. Solid curves from up to bottom 10, 100, 200, 300, 400 and 500 nK. Symbols are the experimental results for 10 nK (full circles), 200 nK (full triangles) and 250 nK (open diamonds) from T. Kraemer et al., Nature 440, 315 (2006). Position of the maximum of the recombination length as a function of the temperature. Experimental data from B. Engeser et al., in preparation. The Efimov states – triatomic continuum resonances arxiv:cond-mat/0608542

12. Weakly bound molecules Recombination for positive scattering lengths (two-body bound states) 1 triatomic bound state 2 triatomic bound states 3 triatomic bound states [1] [2] [3] [1] E. A. Burt et al. Phys. Rev. Lett. 79, 337 (1997). [2] D. M. Stamper-Kurn et al. Phys. Rev. Lett. 80, 2027 (1998). [3] N. R. Claussen, E. A. Donley, S. T. Thompson e C. E. Wieman. Phys. Rev. Lett. 87, 160407 (2001); J. L. Roberts, N. R. Claussen, S. L. Cornish e C. E. Wieman. ibid. 85, 728 (2000). Dimensionless recombination parameter α as a function of the ratio between the binding energies of the diatomic and triatomic molecules. MTY, Frederico, Delfino, and Tomio PRA 68, 033406 (2003)

13. Weakly bound molecules [1] E. A. Burt et al. Phys. Rev. Lett. 79, 337 (1997). [2] D. M. Stamper-Kurn et al. Phys. Rev. Lett. 80, 2027 (1998). [3] N. R. Claussen, E. A. Donley, S. T. Thompson e C. E. Wieman. Phys. Rev. Lett. 87, 160407 (2001); J. L. Roberts, N. R. Claussen, S. L. Cornish e C. E. Wieman. ibid. 85, 728 (2000). [4] J. Söding et al. Appl. Phys. B69, 257 (1999). * Non-condensate atoms** Condensed atoms Prediction of trimer binding energies with respect to the threshold, S 3 =E 3 -E 2 and S’ 3 =E’ 3 -E 2, considering the central values of the experimental recombination parameter  exp. It is also shown the respective two-body scattering length and the diluteness parameter  a 3.

14. Summary Complete trajectory of Efimov states for 3 identical bosons Prediction of trimer energies in atomic traps Scattering length and Recombination coefficient Inclusion of the triatomic continuum resonance effect in the recombination length Recombination length at finite temperatures Good description of the position of resonance as a function of the temperature Thank you !