Adiabatic hyperspherical study of triatomic helium systems

1. Adiabatic hyperspherical study of triatomic helium systems Hiroya Suno Hiyama Strangeness Nucl. Phys. Lab., RIKEN Nishina Center, RIKEN Formerly: Earth Simulator Center, JAMSTEC April 9, 2012 Collaborator: B.D. Esry (Kansas State Univ.)

2. 4He 4He 4He 4He 3He 4He Triatomic helium systems • Weakly bound systems: binding energy about mK≈100neV. • 4He3 is one of the candidates for seeing “Efimov states”, since 4He2 has a large scattering length ~200x(Bohr radius). • Theoretical treatment is simple since there exists only one dimer state for 4He. • Experimentally, the 4He dimer was observed by Luo et al. and Schöllkopf and Toennies, but Schöllkopf and Toennies could also see the trimer and tetramer.

3. What’s an “Efimov state”? • An INFINITY of three-body bound states appear when the two-body scattering length is large compared to the range of the two-body interaction; a12>>r0. • This occurs even when no bound state exists for the two-body subsystems. • The theory formulated in 1970, but experimentally confirmed only in 2006 in an ultracold gas of 133Cs. • In fact, the evidence of Efimov physics was seen measuring the three-body recombination rates.

4. Three-body recombination • Important loss mechanism for Bose-Einstein condensates. • Collision energies • Ultracold: 1μK≈100peV • Cold: 1mK≈100neV

5. This work • Study the triatomic helium system 4He-4He-4He. • Bound-state spectrum • Cold three-body recombination • Cold atom-dimer elastic scattering • Adopt the current state-of-art interaction potential, including retardation and the three-body term developed by Jeziorska et al. and Cencek et al. • We consider the states with total angular momenta from J=0 to 7. • In addition, we treat the 4He-4He-3He system. • We use the adiabatic hyperspherical representation.

6. Computational method • Smith-Whitten’s hyperspherical coordinates • Three-particle system discribed by six coordinates (one hyperradius, five hyperangles) • Simple to impose the permutation symmetry of identical particles • Adiabatic expansion method • First solve the angular part to obtain the adiabatic potential curves and channel functions • Then solve the hyperradial coupled Eqs. • R-matrix method • Extract the scattering S-matrix from the coupled Eqs.

7. Atomic units • In atomic and molecular physics community, one mostly uses the atomic units for numerical calculations. Atomic unit of length (Bohr radius): Atomic unit of length (electron mass): Atomic unit of energy (Hartree):

8. Smith-Whitten hyperspherical coordinates Hyperradius Hyperangles Euler angles

9. Schrödinger equation • Squared “Grand angular momentum operator” • Interaction potential:

10. Interaction potential where with

11. Interaction potential • Use the helium dimer potential of Jezorska et al. • We can also include retardation effect. • 4He2 • Retarded pot.: E00=-1.564mK, a12=91.81Å. • Unretarded pot. :E00=-1.728mK, a12=87.53Å • No bound state for 4He3He or 3He2. • Use the three-body term of Cencek et al. 4He2

12. Potential energy surface at R=15 a.u. 4He-4He-4He 4He-4He-3He

13. Adiabatic expansion method • Solve the adiabatic equation (R-fixed Schrödinger Eq.): • The total wave function is expanded • Obtain the coupled-radial equation: Nonadiabatic couplings given by

14. Permutation symmetry • Expand the channel function on Wigner D functions • Use a direct product of basis splines for • If all the three particles are identical bosons, we impose the boundaryconditions: • If two particles are identical bosons, we impose the boundaryconditions:

15. Adiabatic hyperspherical potential curves for J=0 • The lowest potential curve corresponds to the atom-dimer channel: • The other higher channels correspond to the three-body continuum states:

16. Adiabatic hyperspherical potential curves for J=0 • Similarly interpreted as those for 4He3. • By symmetry requirement, the atom-dimer channel exists only for the parity-favored cases: Π=(-1)J. • We have calculated the potential curves for JΠ=1-,2+,...

17. Bound state energies • We have found two bound states for 4He3(JΠ=0+), one bound states for 4He23He(JΠ=0+), and none for J>0. • 4He3(JΠ=0+): E0=-130.86mK, E1=-2.5882mK. • Hiyama&Kamimura obtained E0=-131.84mK, E1=-2.6502mK using the PCKLJS potential. • 4He23He(JΠ=0+): E0=-16.237mK. • Retardation (~+3%)is found to be more important than the three-body term (~-0.3%).

18. Wave functions for ground and excited states of 4He3(JΠ=0+)

19. Three-body recombination rates • Three-body recombination rate for 4He+4He+4He→4He2+4He: Hyperradial wave number: Scattering matrix element: • Three-body recombination rate for 4He+4He+3He→4He2+3He:

20. Three-body recombination rates for 4He+4He+4He→4He2+4He E (mK) • Threshold law: at ultracold collision energies,

21. Three-body recombination rates for 4He+4He+3He→4He2+3He • Threshold law: at ultracold collision energies,

22. Collision induced dissociation rates • Collision induced dissociation rate for 4He2+4,3He →4He+4He+4,3He: where

23. Collision induced dissociation rates for 4He2+4He→4He+4He+4He E (mK) • Threshold law: at ultracold collision energies,

24. Collision induced dissociation rates for 4He2+3He→4He+4He+3He

25. Atom-dimer elastic scattering • Elastic cross section for 4He2+4,3He →4He2+4,3He: where

26. Elastic scattering cross sections for 4He2+4He →4He2+4He E-E00 (mK)

27. Elastic scattering cross sections for 4He2+3He →4He2+3He

28. Summary • Studied triatomic helium systems using the adiabatic hyperspherical representation. • Adopted the most realistic helium interaction potential, including two-body retardation corrections and a three-body contribution. • Three-body term plays only a minor role, while the effects of retardation significant. • Subsequently, we have studied the 4He2X systems with X being an alkali-metal atom. • Future work: reproduce these results using the Gaussian Expansion Method (GEM).

29. 4He-4He-7Li and 4He-4He-23Na