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

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

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

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

5. Solving the Schrödinger equation Hamiltonian: Four-body Formulation model interactions:rV(r) d0d0d0d0 a<0 a>0 V0V0V0V0a dimer energy scattering length - or combinations of Gaussians

6.  Analytical form of matrix elements (accurate and fast).  Efficient Optimization: Stochastical variational method  Good for bound states.  Difficult to describe scattering properties Correlated Gaussian basis set expansion: r 12 r 13 r 24 r 34 r 14 r 23 12 3 4 remove CM and set L=0 Symmetrization: spin statistic basis functions: Four-body Formulation

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

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

9. s 0 =2.0096 (CG) s 0 =2.0092 (CGH)  Universality arguments predict that the hyperspherical potential curves are of the type (Tan, Werner & Castin, …): J. von Stecher, C. H. Greene PRA (2009) 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) ….. Trap system predictions: Four fermion system at unitarity  Appropriate framework to analyze universality D.S. Petrov, C. Salomon, G.V. Shlyapnikov PRL (2004) Connection to trapped system: trap frequency Predictions: See also: J. P. D’Incao et al PRA (2009) Hyperspherical Picture …for studying universality Four-body Formulation

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

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

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

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

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

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

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

17. Universality away from unitarity

20. a<0

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

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

23.  How to extend the analysis of universality to larger systems?

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

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

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

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

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

29. Model Hamiltonian Many-body Hamiltonian: 3 bodies 2 bodies r ij  (r ij ) Bethe Peierls boundary condition  Two parameters (a and R c ) to independently control two and three-body physics. U(R U(R) RCRC J. von Stecher arXiv:0909.4056 (2009) Hard core three-body potential

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

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

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

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

34. Dilute Clusters Pair correlation function Trayectories extracted from Quantum Monte Carlo simulations Sphere radius: ~ 4r 0 ~R c Five-body cluster state: r m ~ 7R c ~ 30 r 0 ~ 2R c N=3 N=6 J. von Stecher arXiv:0909.4056 (2009)

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

36. 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 Outlook:

38. Energy per particle at unitarity

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

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

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

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