Depicted: A big universal molecule with 4 atoms, attached to a 3-body Efimov state. Thanks, to NSF and the AFOSR- MURI!

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1 N-Body Recombination and the Efimov effect Chris Greene, with Jose D Incao, Nirav Mehta, Seth Rittenhouse (at ICAP) and Javier von Stecher, JILA and Physics, U. of Colorado-Boulder March 2010 overview Depicted: Thanks, to NSF and the AFOSR- MURI! A big universal molecule with 4 atoms, attached to a 3-body Efimov state

2 Thanks at the outset to my collaborators/coauthors on the ultra-cold N-body project: Jose D Incao, Senior Research Associate at JILA Javier von Stecher, a JILA Senior Research Associate Seth Rittenhouse, postdoc at Harvard/ITAMP Nirav Mehta Faculty,Trinity Univ., San Antonio Yujun Wang, JILA postdoc Recent PRL on 3- dipole Efimov physics + long time collaborators, Brett Esry and Doerte Blume

3 Ugo Fano s vision around 1981, following a promising line of research dating back to earlier work by: Delves, Smirnov, Macek, Lin, Schatz, Kuppermann, A + BC + D Small hyperradius reaction volume All reactive processes can be viewed as a single coordinate, the hyperradius R, evolves from large to small and then to large values again ACD + B

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5 Elements of Fano s Vision 1. The vital role of eigenmodes of the most relevant operators in any given problem, usually including all or part of the Hamiltonian 2. The transformative importance of a picture, to help see pathways and mechanisms, as in the Born- Oppenheimer potential curves for reactive processes in chemistry Ugo Fano, sketched by Zdenek Herman 3. Qualitative insight can often be extracted powerfully from semiclassical pictures, as in WKB, Landau-Zener-Stueckelberg, etc. J=0 adiabatic hyperspherical potential curves for He+He+He (with Suno, Esry, & Burke, 2002 PRA)

6 Strategy of the adiabatic hyperspherical representation: convert the partial differential Schroedinger equation into an infinite set of coupled ordinary differential equations: To solve: First solve the fixed-r Schroedinger equation, for eigenvalues U n (R): Next expand the desired solution into the complete set of eigenfunctions And the original T.I.S.Eqn. is transformed into the following set which can be truncated on physical grounds, with the eigenvalues interpretable as adiabatic potential curves, in the Born-Oppenheimer sense.

7 Warning: In problems with realistic interactions, the adiabatic potential curves often look like the following for Li+Li+Li, calculated by Esry and Burke, 1999-ish: Then a modified strategy works well to handle the numerous close avoided crossings, formulated by Tolstikhin, Watanabe, and Matsuzawa, J Phys B 29, L389 (1996), called the Slow Variable Discretization (SVD)

8 Various strategies for getting the potential curves: 1. Expand the adiabatic eigenfunction into hyperspherical harmonics and diagonalize. A difficulty convergence is slow, but some large systems can be tackled at least approximately. Example: Ps - adiabatic hyperspherical potential curves (Botero&CHG 1986 PRL) Eigenvectors in the primitive hyperspherical harmonic basis Eigenvectors in the prediagonalized hyperspherical harmonic basis

9 Botero & CHG 1986 PRL, Ps- hyperspherical potential curves for 1 P o symmetry

10 An extreme example of this is to use a single basis function, only the lowest hyperspherical harmonic allowed by symmetry (the K-harmonic method) 1998

11 Rittenhouse, Cavagnero, von Stecher, CHG Energy levels in this simple treatment agree quite well with a Hartree- Fock treatment of the Fermi contact potential

12 Various strategies for getting the potential curves: 2. Augment the hyperspherical harmonic basis by a few functions that represent the asymptotic channels relevant for the problem being studied: Sadeghpour & CHG, 1990 PRL 1 P o hyperspherical potential curves for H - doubly excited states Many sharply diabatic crossings approximately conserved quantum numbers (molecular interpretation of the 2-electron system)

13 Various strategies for getting the potential curves: 3. Expand the adiabatic eigenfunction into a local basis in the hyperangles, such as B-splines, finite elements, DVR, etc. For example, for the 3-body problem, this leads to a two dimensional eigenproblem to be solved at each hyperradius, typically 100x100 = 10 4 total basis functions See e.g. Zhou, Lin, & Shertzer, J Phys B J. Wang & CHG, unpublished Esry, Lin, CHG 1996 PRA This method is very accurate for 3 bodies with arbitrary interactions, but difficult to extend to N>3.

14 D Incao, Esry, CHG

15 Experiments in BOSONS 1. Zaccanti, Inguscio, Modugno et al. Nature Phys. 5, 586 (2009) 2. Barontini, Thalhammer, Inguscio, Minardi, et al. PRL 103, (2009) 3. Gross, Shotan, Kokkelmans, Khaykovich, PRL103, (2009) 4. Pollack, Dries, Hulet, Science 326, 1683 (2009) 5. Knoop,Ferlaino,Nagerl, Grimm, et al. Nature Phys. 5, 227 ( 09) DISTINGUISHABLE FERMIONS 6. Ottenstein, Lompe, Kohnen, Wenz, Jochim, PRL 101, (2008) 7. Huckans, Williams, O Hara et al, PRL 102, (2009) 8. Williams, Huckans, O Hara et al. PRL 103, ( 09)

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17 Vitaly Efimov(R) gives Randy Hulet(L) a congratulatory high-five at the Rome conference on Efimov physics, October 2009 : experiment catches up with theory!

18 Nielsen and Macek, 1999 PRL; Esry, Greene, and Burke, 1999 PRL-3-body recombination at large a from an adiabatic hyperspherical perspective Floerchinger, Schmidt, Moroz, Wetterich, 2009 functional renormalization approach Other groups subsequently rederived the Efimov physics in the universality regime of large two-body scattering lengths, especially relevant for 3-body recombination, using other methods: Braaten and Hammer, Effective field theory approach Shepard, 2007 Fadeev treatment in momentum space, effective theory Lee, Köhler, Julienne, body Green s function approach based on a transition matrix; basic formulation was developed in nuclear physics by Sandhas, Alt, and Grassman. Gogolin, Mora, Egger, 2008 Analytic solution of a model

19 Various strategies for getting the potential curves: 4. Diffusion Quantum Monte Carlo, a promising method for N=4 to 20 particles, for at least the lowest adiabatic hyperspherical potential curve: Blume & CHG, J Chem Phys. 2000

20 Note: good agreement found for He cluster excited state energies in this recent preprint, arxiv: : This is a way to augment Quantum Monte Carlo calculations to determine a class of excited state properties, albeit, within the adiabatic hyperspherical approximation.

21 Strategy 5. Green s function method to find the hyperspherical potentials analytically e.g PRA, Rittenhouse, Mehta, CHG

22 To understand the Efimov effect, look at the effective potential energy curve at unitarity, as a function of the hyperradius: Potential energy Fall to the center Hyperradius, R The Efimov potential curve varies like: mR 2 Mathematical Detail. Once you have this effective dipole-type attractive potential curve, the rest is TRIVIAL! Here, trivial means that the solutions are simply Bessel functions (of imaginary order, and imaginary argument). E n 1 E n e 2 / s, where s is a universal constant.

23 Various strategies for getting the potential curves: 6. Correlated Gaussian Hyperspherical Method von Stecher & CHG, PRA 2009 A+B+A+B AB+A+B AB+AB

24 2-component fermions: key observable that has to come out correctly is that dimer-dimer scattering length A dd Phys. Rev. Lett 93, (2004) Predicted A dd =0.6 a +/- 0.01a Previous (ancient) theory had been based on the perturbative result A dd =2 a Others confirming this 0.6a result include, e.g. Radzihovsky and Gurarie Subsequent work: von Stecher and Greene, 2007 PRL: Our new results, from subtracting the noninteracting energy of two bosonic molecules from the ground state energy on the BEC side of the crossover, have now pinned down the second digit: A dd =0.608 a +/ a (distributed Gaussian diagonalization) Add=0.604 a (4-body hyperspherical calculation) (several other groups have similar results as well by now)

25 Some of the intermediate steps in this calculation integrals at fixed hyperradius such as: See also J. von Stecher PhD thesis, Scaling with number of particles: N=3 all matrix elements can be evaluated analytically at fixed R N=4 matrix elements at fixed R require 1D numerical integration N-3 dimensional integrals would in general require numerical evaluation at fixed R, for each matrix element in the basis set.

26 Applications Theory (Mehta et al., PRL 103, (2009)) of N-body recombination processes, e.g. A + A + A + A A 3 + A The 4-boson system: 4-body recombination and its surprising importance von Stecher, D Incao, & CHG, Nature Phys D Incao, von Stecher, & CHG, PRL 2009

27 Previous important studies of the 4-boson system in the universality regime in 3D: We have conjectured, that there are always two four-body resonances between any two three-body states. (i.e. below each Efimov state) + Also, no 4-body param. also found general correlations between N- body bound levels and (N-1)-body bound levels conclude that a 4-body parameter is in fact needed, but they only studied low (non-universal states), which is presumably why they reach a different conclusion from that of Platter and Hammer and also different from ours.

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29 Our findings Our results are consistent with Hammer & Platter s insightful conjecture, and also with Hanna & Blume s calculations

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32 Revisiting the 2006 Grimm group experiment that was the first to see 3-body Efimov states p.417 von Stecher, D Incao, CHG 2009

33 But before we could actually calculate the rate of 4-body recombination in an ultracold gas, we had to develop some scattering theory: And here it is, THE FORMULA for N-body recombination, i.e. for the process: A+A+A+.+..A A N-1 +A or A N-2 +A+A + etc.

34 Scattering amplitude, purely hyperradial potential, phaseshifts Some formal work to generalize scattering theory to d- dimensions

35 And using WKB ideas, we can derive a semianalytic expression giving the structure of the N- body recombination rate at zero energy: a 3N-5 4-body recomb. rate versus a

36 Unitarity Limits for N-body recombination, e.g. A+A+A+A+ A 3 +A+ N-body reduced mass

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38 Measurement of four-body state 1 Measurement of four-body state 2 Estimated size of this 4-body molecule is around nm

39 Scott Pollack, Dan Dries, and Randy Hulet, Science, Dec.2009

40 Scott Pollack, Dan Dries, and Randy Hulet, Science, Dec.2009 measured universality ratios

41 MORE THAN 4 BOSONS: von Stecher s J. Phys. B article in 2010: combined study using correlated Gaussians, and diffusion Monte Carlo arxiv: increasing attraction

42 0.46(1) 0.65(2) 0.73(1) are latest revised/improved values from J. von Stecher, arxiv: von Stecher - arxiv: and 2010 JPB

43 Some physics with 3 dipoles Wang, D Incao, CHG, References: Wang, D Incao, CHG, arxiv: (fermionic dipoles)

44 Spherical Adiabatic treatment for Two-Dipoles in Q2D Adiabatic separation: Radial Equations : : adiabatic potentials : non-adiabatic couplings (inelastic transitions) Angular Equations : (fix r, solve for the angles) Important length scales: (short-range) (dipolar) (van der Waals length) (dipole length) (oscilator length) (confinement)

45 Dipolar gases We are interested in the strongly dipolar limit: : Q2D Two-body dipolar physics becomes universal : Three-body dipolar physics becomes universal (Efimov states)

46 Three-Dipoles in 3D The A=6 problem:wang, D Incao, & CHG Strategy: Hyperspherical!!! Angular momentum is not conserved! Ouch!

47 Three-Dipoles in 3D Structure of the effective three-dipole potentials

48 Three-Dipoles in 3D Three-body Efimov States!?

49 Dipolar Efimov Effect Repulsion for Long-Lived, universal states!!!

50 Three-Dipoles in 3D Dipolar Efimov States : Repulsion for Long-Lived, universal states!!!

51 Dipolar Efimov Effect Predicted Efimov pattern versus dipole length Long-Lived, universal states!!!

52 Conclusions: Much progress has been made using a broad range of different methods to determine adiabatic hyperspherical potential curves and couplings Some are applicable to A>4 problems, others less so given current computational limitations Efimov physics shows a signature for A=3,4,5,. bosonic systems, and also for some fermionic systems Even for a system with angular momentum nonconservation, i.e. the A=6 problem with 3 oriented dipoles, there is an Efimov effect predicted by recent work