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.)
Triatomic helium systems
4He
4He
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4He
4He
4He
3He
Weakly bound systems: binding energy about mK≈100neV.
4He is one of the candidates for seeing “Efimov states”, since
3
4He has a large scattering length ~200x(Bohr radius).
2
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.
What’s an “Efimov state”?
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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.
Three-body recombination
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Important loss mechanism for Bose-Einstein
condensates.
Collision energies
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Ultracold: 1μK≈100peV
Cold: 1mK≈100neV
This work
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Study the triatomic helium system 4He-4He-4He.
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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.
Computational method
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Smith-Whitten’s hyperspherical coordinates
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Adiabatic expansion method
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Three-particle system discribed by six coordinates (one
hyperradius, five hyperangles)
Simple to impose the permutation symmetry of identical
particles
First solve the angular part to obtain the adiabatic potential
curves and channel functions
Then solve the hyperradial coupled Eqs.
R-matrix method
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Extract the scattering S-matrix from the coupled Eqs.
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):
Smith-Whitten hyperspherical coordinates
Hyperradius
Hyperangles
Euler angles
Schrödinger equation
•Squared “Grand angular momentum operator”
•Interaction potential:
Interaction potential
where
with
Interaction potential
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Use the helium dimer
potential of Jezorska et al.
We can also include
retardation effect.
4He
2
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Retarded pot.: E00=1.564mK, a12=91.81Å.
Unretarded pot. :E00=1.728mK, a12=87.53Å
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No bound state for
4He3He or 3He .
2
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Use the three-body term of
Cencek et al.
4He
2
Potential energy surface at R=15 a.u.
4He-4He-4He
4He-4He-3He
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
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:
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:
Adiabatic hyperspherical potential curves for J=0
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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+,...
Bound state energies
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We have found two bound states for
4He (JΠ=0+), one bound states for 4He 3He
3
2
(JΠ=0+), and none for J>0.
4He (JΠ=0+): E =-130.86mK, E =-2.5882mK.
3
0
1
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Hiyama&Kamimura obtained E0=-131.84mK, E1=2.6502mK using the PCKLJS potential.
 4He23He(JΠ=0+):
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E0=-16.237mK.
Retardation (~+3%)is found to be more
important than the three-body term (~-0.3%).
Wave functions for ground and excited states of 4He3(JΠ=0+)
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:
Three-body recombination rates for
4He+4He+4He→4He +4He
2
E (mK)
•Threshold law: at ultracold collision energies,
Three-body recombination rates for
4He+4He+3He→4He +3He
2
•Threshold law: at ultracold collision energies,
Collision induced dissociation rates
•Collision induced dissociation rate for 4He2+4,3He →4He+4He+4,3He :
where
Collision induced dissociation rates for
4He +4He→4He+4He+4He
2
E (mK)
•Threshold law: at ultracold collision energies,
Collision induced dissociation rates for
4He +3He→4He+4He+3He
2
Atom-dimer elastic scattering
•Elastic cross section for 4He2+4,3He →4He2+4,3He :
where
Elastic scattering cross sections for
4He +4He →4He +4He
2
2
E-E00 (mK)
Elastic scattering cross sections for
4He +3He →4He +3He
2
2
Summary
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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).
4He-4He-7Li
and
4He-4He-23Na