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Microscopic origin and universality classes of the three-body parameter. Pascal Naidon. The University of Tokyo. Masahito Ueda. Shimpei Endo. 3 particles (bosons or distinguishable) with resonant two-body interactions. single-channel two-body interaction no three-body interaction.
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Microscopic origin and universality classes of the three-body parameter PascalNaidon The University of Tokyo Masahito Ueda Shimpei Endo
3 particles (bosons or distinguishable) with resonant two-body interactions • single-channel two-body interaction • no three-body interaction
The Efimov 3-body parameter Parameters describing particles at low energy The Efimov effect (1970) y x (2-body parameter) Scattering length a z Hyperradius Zero-range condition with Three-body parameter - R dimer Efimov attraction trimer 3-body parameter
The Efimov 3-body parameter Parameters describing particles at low energy The Efimov effect (1970) y x (2-body parameter) Scattering length a z Hyperradius Zero-range condition with 22.72 Three-body parameter - trimer R dimer Efimov attraction 3-body parameter
Universality for atoms Microscopic determination? no universality of the scattering length 133Cs 4He 85Rb triplet scatt. length r 7Li 23Na 39K 87Rb short-range details van der Waals 6Li Two-body potential universality of the 3-body parameter HyperradiusR short-range details Efimov Effective three-body potential
Three-body with van der Waals interactions Phys. Rev. Lett. 108 263001 (2012) J. Wang, J. D’Incao, B. Esry, C. Greene Lennard-Jones potentials supporting n = 1, 2, 3, ...10 s-wave bound states Efimov HyperradiusR Three-body repulsion at
Interpretation: two-body correlation Two-body correlation Asymptoticbehaviour Strong depletion Interatomic separation r Resonance
Interpretation: two-body correlation Kinetic energy cost due to deformation Excluded configurations squeezed equilateral Excluded configurations induced deformation deformation Efimov elongated
Confirmation 1: pair correlation model FModel = FEfimovx j(r12) j(r23) j(r31) (product of pair correlations) (hyperangular wave function) 3-body potential Pair model (for Lennard-Jones two-body interactions) Exact Energy E [] Efimov attraction Hyperradius R []
Confirmation 2: separable model Parameterised to reproduce exactly the two-body correlation at zero energy. • Reproduces the low-energy 2-body physics • Scattering length • Effective range • Last bound state • …. -
Confirmation 2: separable model Parameterised to reproduce exactly the two-body correlation at zero energy. n Pair model Separable model Exact Energy Integrated probability HyperradiusR HyperradiusR
Confirmation 2: separable model Parameterised to reproduce exactly the two-body correlation at zero energy. Other potentials at most 10% deviation -0.366 -6.55 Separable model 0.204 -11.0 -4.47 0.472 Exact calculations -12.6 0.173 -16.3 0.128 -6.23 0.350 S. Moszkowski, S. Fleck, A. Krikeb, L. Theuÿl, J.-M.Richard, and K. Varga, Phys. Rev. A 62 , 032504 (2000).
Summary two-body correlation universal effective range three-body deformation three-body repulsion universal three-body parameter effective range
Two-body correlation universality classes Faster than Power-law tails Power-law tails Probability density Van der Waals 4.0 3.0 1.0 2.0 0.0 Interparticle distance () Universal correlation Step function correlation limit
Separable model 3-body parameter in units of the two-body effective range (= size of two-body correlation) Nuclear physics Atomic physics Binding wave number at unitarity ? Number of two-body bound states
Summary The 3-body parameter is (mostly) determined by the low-energy 2-body correlation. Reason: 2-body correlation induces a deformation of the 3-body system. • Consequences: the 3-body parameter • is on the order of the effective range. • has different universal values for distinct classesof interaction P. Naidon, S. Endo, M. Ueda, PRA 90, 022106 (2014) P. Naidon, S. Endo, M. Ueda, PRL 112, 105301 (2014)
