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Scales and Universality in Three-Body Systems

Scales and Universality in Three-Body Systems. Marcelo Takeshi Yamashita yamashita@ift.unesp.br Instituto de Física Teórica – IFT / UNESP. M. R. Hadizadeh IFT. MTY IFT. A. Delfino UFF. T. Frederico ITA. F. F. Bellotti ITA. L. Tomio UFF/IFT. D. S. Ventura IFT.

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Scales and Universality in Three-Body Systems

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  1. ScalesandUniversalityin Three-Body Systems Marcelo Takeshi Yamashita yamashita@ift.unesp.br Instituto de Física Teórica – IFT / UNESP

  2. M. R. Hadizadeh IFT MTY IFT A. Delfino UFF T. Frederico ITA F. F. Bellotti ITA L. Tomio UFF/IFT D. S. Ventura IFT

  3. What's Universality? Independence of the potential Two-bodyscatteringlength >> range ofthepotential • Some consequences Ex: Twoidenticalbosonsinteracting in s-wave Two-body sector K2 50 mK Helium-4 dimer Three-body sector Ex: Threeidenticalbosonsinteracting in s-wave Decrease Decrease Three-body boundstates Three-body boundstates

  4. Infinitethree-body boundstates Efimovstates Appearanceofaneffectivepotential 0 “EvidenceofEfimov quantum states in anultracoldgasofcesiumatoms” ! discoveredby Vitaly Efimov in 1970 T. Kraemer et al. Nature440, 315 (2006)

  5. r0 0 V0 ∞ E2  fixed E3deepest ∞ Thomas collapse in 1935 Efimovstates Describing universal systems 2: scatteringlength (a) / two-bodyenergy Scalingfunction 3: two-bodyenergy + Three-bodyscale • Energy ratiobetweentwoconsecutivestates 515.03 • rmshyperradiusratiobetweentwoconsecutivestates 22.7

  6. momenta energies 0 ε2 Efimovstates (N = 0, 1, 2, ...) Three-bodyboundstateequationwith zero-range interactionwithmomentum cutoff Skorniakovand Ter-Martirosianequation (1956) Λ 1) E2tendsto zero withΛfixed – Efimoveffect 2) Λtendstoinfinitywith E2fixed – Thomas collapse S.K. Adhikari, A. Delfino, T. Frederico, I.D. Goldman and L. Tomio, Phys. Rev. A 37, 3666 (1988) If E2 ≠ 0: whathappenstotheEfimovstatesaftertheydisappear?

  7. E2 virtual E2 bound Im E Im E E3 E3 Re E Re E E2 Increasing |E2| Im E Im E E3 E3 Re E Re E E2 Im E Im E E3 (virtual state) Re E Re E E2 secondRiemannsheet E3 (resonance)

  8. Subtracted T-matrixEquation S.K. Adhikari, T. Frederico and I.D. Goldman, Phys. Rev. Lett. 74, 487 (1995) Three-bodyboundstateequationforzero-range interactionwithsubtraction M.T. Yamashita, T. Frederico, A. Delfino and L. Tomio, Phys. Rev. A 66, 052702 (2002)

  9. Virtual states – extensiontothesecondRiemannsheet M.T. Yamashita, T. Frederico, A. Delfino and L. Tomio, Phys. Rev. A 66, 052702 (2002) Defining wecanwritetheboundstateequation as

  10. Thenwecanwritethecutexplicitly Afterintegrationanddefining

  11. Wehavefinally shouldbeoutsidethecut thus

  12. Efimovstates – Boundand virtual states Lines – Boundstates crosses – ground squares – firstexcited diamonds – secondexcited Symbols – Virtual states circles - referstothefirstexcitedstate triangles – referstothesecondexcitedstate ε2 bound Appearanceofthe virtual state(dotted line) The virtual stateturnsintoanexcitedstate (solidline)

  13. Resonances F. Bringas, M.T. Yamashita and T. Frederico Phys. Rev. A 69, 040702(R) (2004) ε2 unbound iscomplex

  14. Efimovstates - Resonances ε2 virtual

  15. E3 bound E2 virtual E3 bound E2 bound E3 resonance E2 virtual E3 virtual E2 bound FulltrajectoryofEfimovstates T. Frederico, L. Tomio, A. Delfino, M.R. Hadizadehand M.T. Yamashita, FewBodySyst. (2011) online first swave (N=0) Th. Cornelius, W. Glöckle. J. Chem. Phys.85, 1 (1996). s+dwaves (N=0) S. Huber. Phys. Rev. A31, 3981 (1985). x x swave (N=1) B. D. Esry, C. D. Lin, C. H. Greene. Phys. Rev. A 54, 394 (1996). E. A. Kolganova, A. K. Motovilov e S. A. Sofianos. Phys. Rev. A56, R1686 (1997).

  16. Weakly-boundmolecules – Heliumtrimer Ground Firstexcited Symbolsfrom P. Barletta and A. Kievsky Phys. Rev. A 64, 042514 (2001) squares - Groundstate circles - Firstexcitedstate Potentials: HFDB, LM2M2, TTY, SAPT1, SAPT2 M.T. Yamashita, R.S. Marques de Carvalho, L. Tomioand T. Frederico, Phys. Rev. A 68, 012506 (2003)

  17. Range correction for boundstates D. S. Ventura, M.T. Yamashita, L. Tomioand T. Frederico, in preparation From Kokkelmans presentation

  18. Point whereanexcitedthree-bodystatebecomes virtual/bound

  19. The transitionbound-virtual does notdependontheparticlesmassratio M.T. Yamashita, T. Frederico and L. Tomio, Phys. Lett. B 660, 339 (2008); Phys. Rev. Lett. 99, 269201 (2007) 18C 20C (3.5 MeV) Example: n n bound virtual n-18C: 160 (110) keV

  20. boundstate virtual state Root meansquareradii Scalingfunction for theradii A + two-bodyboundstate - two-body virtual state B B g = AorB

  21. BB bound BB bound BB virtual BB virtual Root meansquareradii > > > M.T. Yamashita, L. TomioandT. Frederico, Nucl. Phys. A 735, 40 (2004)

  22. Summary • If at least one two-body subsystem is bound: Efimov state virtual resonance • All two-body subsystems are virtual (borromean case): Efimov state • Range correction for the point where an excited Efimov state disappears Thankyou! http://www.ift.unesp.br/users/yamashita/publist.html

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