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Numerical Studies of Universality in Few-Boson Systems

Numerical Studies of Universality in Few-Boson Systems. Javier von Stecher. Department of Physics and JILA University of Colorado. In collaboration with Chris H. Greene Jose P. D’Incao. Acknowledgements: Doerte Blume Seth Rittenhouse Nirav Mehta NSF.

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Numerical Studies of Universality in Few-Boson Systems

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  1. Numerical Studies of Universality in Few-Boson Systems Javier von Stecher Department of Physics and JILA University of Colorado In collaboration with Chris H. Greene Jose P. D’Incao Acknowledgements: Doerte Blume Seth Rittenhouse Nirav Mehta NSF Probable configurations of a weakly-bound four-boson state.

  2. How does Efimov physics and universality extends beyond three bodies?

  3. Universality in Few-Boson Systems Outline: • Universality in four-boson systems • Numerical formulation (CG and CGH) • Four-bosons predictions • Extension to larger systems • Model Hamiltonian • Cluster states

  4. Universality in Few-Boson Systems Outline: • Universality in four-boson systems • Numerical formulation (CG and CGH) • Four-bosons predictions • Extension to larger systems • Model Hamiltonian • Cluster states

  5. Four-body Formulation Solving the Schrödinger equation model interactions: Hamiltonian: r d0 V(r) - a or combinations of Gaussians a>0 a<0 V0 scattering length dimer energy

  6. Four-body Formulation Correlated Gaussian basis set expansion: • Analytical form of matrix elements (accurate and fast). • Efficient Optimization: Stochastical variational method • Good for bound states. • Difficult to describe scattering properties • remove CM and set L=0 • basis functions: Symmetrization: spin statistic 3 r34 4 r13 r14 r24 r23 1 2 r12

  7. Four-body Formulation R J. von Stecher, C. H. Greene PRA (2009) Hyperspherical representation: R: overall size (collective motion) Divide and conquer: typical potential curves …solved at fixed R. Coupled 1D equations Use correlated Gaussian to expand channel functions Find efficient way to evaluate matrix elements • Accurate description of four-body systems!!

  8. Four-body Formulation Hyperspherical Picture …for scattering properties. Fragmentation thresholds 1+1+1+1 R U(R) 2+1+1 Interaction region 2+2 3+1 3+1 All particles close together: Bound and quasi-bound states

  9. Four-body Formulation Trap system predictions: J. von Stecher, D. Blume, C. H. Greene PRL 2007, PRA 2007, 2008 S.Y. Chang, and G. F. Bertsch PRA (R) (2007) I. Stetcu, B. Barrett, U. van Kolck, J. Vary. PRA (2007) Y. Alhassid, G. F. Bertsch, and L. Fang PRL (2008) ….. J. von Stecher, C. H. Greene PRA (2009) Hyperspherical Picture D.S. Petrov, C. Salomon, G.V. Shlyapnikov PRL (2004) …for studying universality Four fermion system at unitarity • Universality arguments predict that the hyperspherical potential curves are of the type (Tan, Werner & Castin, …): Connection to trapped system: trap frequency Predictions: s0=2.0092 (CGH) s0=2.0096 (CG) • Appropriate framework to analyze universality See also: J. P. D’Incao et al PRA (2009)

  10. Universality in Few-Boson Systems Outline: • Universality in four-boson systems • Numerical formulation (CG and CGH) • Four-bosons predictions • Extension to larger systems • Model Hamiltonian • Cluster states

  11. Four-Bosons J. von Stecher, J. P. D’Incao and C. H. Greene, Nat. Phys. (2009) Universality in four bosons a= Efimov scaling Efimov scaling , … 3 body 4 body , … Two four-body states , … same scaling relations!!! In agreement with Hammer & Platter EJPA (2007) (Hammer presentation)

  12. Four-Bosons Hyperspherical Picture (a=) 3 bodies 4 bodies R R Four-body potential curves follow the same scaling of the Efimov states !!! Four-body physics is universal!!

  13. Four-Bosons Hyperspherical Picture (a=) 3 bodies 4 bodies • Different Efimov families • Different interaction potentials R R Four-body potential curves follow the same scaling of the Efimov states !!! Three-body parameter: Four-body physics is universal!!

  14. Four-Bosons Hyperspherical Picture (a=) 3 bodies 4 bodies R R Pair correlation of lowest four-body states Four-body potential curves follow the same scaling of the Efimov states !!! Three-body parameter: Four-body physics is universal!!

  15. Four-Bosons J. von Stecher, J. P. D’Incao and C. H. Greene, Nat. Phys. (2009) Four-body states a= Hammer & Platter prediction: Agreement with Hammer & Platter (2007)

  16. Four-Bosons J. von Stecher, J. P. D’Incao and C. H. Greene, Nat. Phys. (2009) Spectrum: Extended Efimov plot • Calculations at more than 500 scattering lengths!!! dimer-two atoms threshold (red lines) atom-trimer thresholds (green lines) dimer-dimer threshold (red lines) four-body states (black lines) 1/a

  17. Four-Bosons Universality away from unitarity

  18. Four-Bosons Universality away from unitarity

  19. Four-Bosons Universality away from unitarity

  20. Four-Bosons Universality away from unitarity a<0

  21. Four-Bosons Universality away from unitarity a<0 a>0 • Important for collisional properties • Experimental observations This afternoon: Universal four-body resonances in ultracold atomic and molecular gases J. P. D’Incao (Innsbruck, LENS)

  22. Universality in Few-Boson Systems Outline: • Universality in four-boson systems • Numerical formulation (CG and CGH) • Four-bosons predictions • Extension to larger systems • Model Hamiltonian • Cluster states

  23. N-Bosons • How to extend the analysis of universality to larger systems?

  24. N-Bosons Extension to larger systems Helium and spin-polarized tritium: similar behavior • How many parameters are needed to describe N-boson systems? Hanna & Blume, PRA (2006): • Linear correlations between cluster energies (like Tjon lines) Blume et al., PRL(2002) Other studies: Lewerenz, JCP (1997). Blume & Greene, JCP(2000). Hanna & Blume, PRA (2006). … Hammer & Platter, EPJA (2007).

  25. Universality in Few-Boson Systems Outline: • Universality in four-boson systems • Numerical formulation (CG and CGH) • Four-bosons predictions • Extension to larger systems • Model Hamiltonian • Cluster states

  26. N-Bosons J. von Stecher arXiv:0909.4056 (2009) Model Hamiltonian • Goal: construct a minimal Hamiltonian that would lead to a “universal ground state”. Hyperspherical potential at unitatity 3 bodies R dilute and weakly bound ~ universal controlled by short-range physics short-range physics Universal region

  27. N-Bosons J. von Stecher arXiv:0909.4056 (2009) Model Hamiltonian • Goal: construct a minimal Hamiltonian that would lead to a “universal ground state”. Hyperspherical potential at unitatity 3 bodies RC dilute and weakly bound ~ universal !! Lowest state is universal !!! short-range physics Universal region

  28. N-Bosons J. von Stecher arXiv:0909.4056 (2009) Model Hamiltonian Many-body Hamiltonian: controls three-body physics: 3-body parameter controls two-body physics: scattering length

  29. N-Bosons J. von Stecher arXiv:0909.4056 (2009) Model Hamiltonian • Two parameters (a and Rc) to independently control two and three-body physics. Many-body Hamiltonian: 2 bodies 3 bodies Bethe Peierls boundary condition RC U(R) (rij) Hard core three-body potential rij

  30. N-Bosons J. von Stecher arXiv:0909.4056 (2009) Numerical methods • Correlated Gaussian basis set expansion up to N=6. • Gaussian potential for two and three-body interactions • Diffusion Monte Carlo simulations up to N=13. • Hard Wall 3-body potential and an attractive square 2-body potential • Trial wave function: HR correlations 3-body correlations 2-body correlations

  31. Universality in Few-Boson Systems Outline: • Universality in four-boson systems • Numerical formulation (CG and CGH) • Four-bosons predictions • Extension to larger systems • Model Hamiltonian • Cluster states

  32. N-Bosons J. von Stecher arXiv:0909.4056 (2009) Weakly-bound Clusters Universal description: (just from dimensional analysis) Universal function Numerical predictions: Comparison with other predictions Comparison between CG and DMC • At least one N-body cluster state for each Efimov trimer!!

  33. N-Bosons J. von Stecher arXiv:0909.4056 (2009) Weakly-bound Clusters Properties: • “Super” Borromean states • Linear Correlations (Tjon lines): • N-body resonant enhancement of losses: Positions of resonances This afternoon: Position of five-body resonance: Semi-Classical Methods and N-Body Recombination S. Rittenhouse

  34. N-Bosons J. von Stecher arXiv:0909.4056 (2009) Dilute Clusters Pair correlation function Five-body cluster state: N=6 N=3 Sphere radius: ~ 4r0 ~Rc Trayectories extracted from Quantum Monte Carlo simulations rm~ 7Rc ~ 30 r0 ~ 2Rc

  35. N-Bosons J. von Stecher arXiv:0909.4056 (2009) Hyperspherical Picture …sketching potentials N bodies Helium clusters Monte Carlo + Hyperspherical 1+1+1 R 3+1 4+1 5+1 D. Blume & C. H. Greene, JCP 2001 • Potential curves scale with size and energy of cluster states • All clusters follow Efimov scaling relations

  36. Outlook: • Extract width of four-body states • Different species or/and spin statistic? • Universality? • Four-body states? • Model Hamiltonian • with Quantum Monte Carlo • Cluster states • Large N limit

  37. Energy per particle at unitarity

  38. Four-Bosons J. von Stecher, J. P. D’Incao and C. H. Greene, Nat. Phys. (2009) Four-body states a= Hammer & Platter prediction:

  39. Four-Bosons J. von Stecher, J. P. D’Incao and C. H. Greene, Nat. Phys. (2009) Four-body states a= Agreement with Hammer & Platter (2007) atom- trimer correlation Pair correlations trimer correlation • Weakly bound atom trimer state: • Large and positive atom-trimer scattering length

  40. a<0 a>0 Blue: atom-trimer Red: dimer-dimer Green dimer-2-atoms

  41. Four-Bosons J. von Stecher, J. P. D’Incao and C. H. Greene, Nat. Phys. (2009) Deviations from universality • Different potentials • Different Efimov states Non universal Universal less than 5% deviations Testing universality: • Scattering length ratios for a<0 depends weakly on nonuniversal effects!! Non Universal • Deviations: • 2nd order four-body corrections or numerical limitations? Universal

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