Triatomic states in ultracold gases

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

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

3. momenta energies 0 ε2 (N = 0, 1, 2, ...) Efimov states 1) E2 tends to zero with Λ fixed – Efimov effect 2) Λ tends to infinity with E2 fixed – Thomas collapse The Efimov effect - Thomas-Efimov equivalence Three-body bound state equation with zero-range interaction with momenta cutoff Skorniakov and Ter-Martirosian equation (1956) Adhikari, Frederico, and Goldman PRL 74, 487 (1995).

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

5. 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 MTY, Frederico, Delfino, and Tomio PRA 66, 052702 (2002) ε2 bound Appearance of the virtual state (dashed line) The virtual state turns into an excited state (solid line) The Efimov states – bound and virtual states

6. Resonances ε2 virtual Bringas, MTY, and Frederico PRA 69, 040702(R) (2004) The Efimov states - resonances

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

8. “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) from http://www.uibk.ac.at/exphys/ultracold/ Excited Efimov state turns into a resonance From the experiment T = 0  a = -898 a0 The Efimov states – triatomic continuum resonances

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

10. The resonance energy can be approximated by We can easily find the solution of ar- for Er Adding the effects of triatomic continuum resonances in the recombination rate L3 for T = 0 where For T = 0 E. Braaten, and H.-W. Hammer, Phys. Rep. 428, 259 (2006) After performing the thermal average of the recombination rate <L3>th we have the recombination length The Efimov states – triatomic continuum resonances

11. Position of the maximum of the recombination length as a function of the temperature. Experimental data from B. Engeser et al., in preparation. 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., Nature440, 315 (2006). The Efimov states – triatomic continuum resonances arxiv:cond-mat/0608542

12. Recombination for positive scattering lengths (two-body bound states) [1] Dimensionless recombination parameter α as a function of the ratio between the binding energies of the diatomic and triatomic molecules. [2] [3] 1 triatomic bound state 2 triatomic bound states 3 triatomic bound states MTY, Frederico, Delfino, and Tomio PRA 68, 033406 (2003) [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). Weakly bound molecules

13. Weakly bound molecules Prediction of trimer binding energies with respect to the threshold, S3=E3-E2 and S’3=E’3-E2, considering the central values of the experimental recombination parameter aexp. It is also shown the respective two-body scattering length and the diluteness parameter ra3. * Non-condensate atoms ** Condensed atoms [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).

14. Scattering length and Recombination coefficient Prediction of trimer energies in atomic traps Summary Complete trajectory of Efimov states for 3 identical bosons 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 !