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. 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 boundary conditions:
If two particles are identical bosons, we impose the boundary conditions:

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= mK, E1= mK.
Hiyama&Kamimura obtained E0= mK, E1= mK using the PCKLJS potential.
4He23He(JΠ=0+): E0= mK.
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