1. Semi-Classical Methods and N-Body Recombination Seth Rittenhouse ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138 Efimov States in Molecules and Nuclei, Oct. 21 st 2009

2. Hard Problems with Simple Solutions Seth Rittenhouse ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138 Efimov States in Molecules and Nuclei, Oct. 21 st 2009

3. WKB is Smarter than You Think Seth Rittenhouse ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138 Efimov States in Molecules and Nuclei, Oct. 21 st 2009

4. Jose P. D’IncaoNirav MehtaJavier von Stecher Chris H. Greene

5. Review of Recombination Experiments 2006: First solid evidence of an Efimov State was seen in Innsbruck

6. Since then, several other groups have seen Efimov states Ottenstein et.al., PRL. 101, 203202 (2008) Huckans et. al., PRL 102, 165302 (2009)

7. Since then, several other groups have seen Efimov states Ultra cold Li 7 gas: Rice group (soon to be published) Zaccanti et. al., Nature Phys. 5, 586 (2009).

8. More recently: Four body effects have been observed! Ferlaino et. al., PRL 102, 140401 (2009) Rice group

9. Hyperspherical Coordinates: the first step for easy few body scattering. General idea: treat the hyperradius adiabatically (think Born-Oppenheimer). Provides us with a convenient view of the energy landscape ~ R

10. For example, The energy landscape 3 Bodies 2-D Hyperspherical Coordinates: the first step for easy few body scattering. General idea: treat the hyperradius adiabatically (think Born-Oppenheimer). Provides us with a convenient view of the energy landscape ~ R

11. When the hyperradius is much different from all other length scales, the adiabatic potentials become universal, e.g. which is the non-interacting behavior at fixed hyperradius. The potentials for other length scale disparities look very similar, but with  non-integer valued or complex.

12. Relevant examples of potential curves Three bosons with negative scattering length:

13. Repulsive universal long-range tail Attractive inner region Transition region Here be dragons! Relevant examples of potential curves

14. Four bosons with negative scattering length: Relevant examples of potential curves

15. Four bosons with negative scattering length: Repulsive four-body potentials Efimov trimer threshold Attractive inner wells Broad avoided crossing Relevant examples of potential curves

16. Sometimes things can get ugly, so be careful! Not-so relevant examples of potential curves: a cautionary tale

17. Let’s get quantitative Once hyperradial potentials have been found, it might be nice to have scattering crossections and rate constants. Three-body: Esry et. al., PRL 83, 1751 (1999); Fedichev et. al., PRL 77, 2921 (1996); Nielsen and Macek, PRL, 83 1566 (1999); Bedaque et. al., PRL 85, 908 (2000); Braaten and Hammer, PRL 87 160407 (2001) and Phys. Rep. 428,259 (2006); Suno et. al., PRL 90, 053202 (2003).

18. Through some hyperspherical magic this can be generalized to the N-body cross section and rate Mehta, et. al., PRL 103, 153201 (2009) This is messy, but there already is some good physics buried in here.

19. At very low incident energies, only a single incident channel survives. Using the unitary nature of the S-matrix, this simplifies things quite a bit

20. If know about scattering in the initial channel, then we know everything about the N-body losses!!! This only depends on the incident channel! Still a fairly nasty multi-channel problem, how can we solve this?

21. Specify a little bit more, consider N-bosons with a negative two body scattering with at least one weakly bound N-1 body state. The lowest N-body channel will have a very generic form: WKB to the rescue

22. Approximate the incident channel S-matrix element using WKB phase shift with an imaginary component. = WKB phase inside the well = WKB tunneling = Imaginary phase (parameterizes losses)

23. Putting this all together gives the recombination rate constant

25. Some things to note: This only holds when the coupling to deep channels is with the scattering length. If coupling exists at large R, we must go back to the S-matrix, or find another cleaver way to describe losses. This assumes the S matrix element is completely controlled by the behavior of the incoming channel. If outgoing channel is important, as in recombination to weakly bound dimers, a more sophisticated approximation of the S-matrix is needed.

26. Re-examine three bosons Assume that all of the tunneling occurs in the universal large R region, and that all phase accumulation occurs in the universal inner region.

27. Re-examine three bosons Assume that all of the tunneling occurs in the universal large R region, and that all phase accumulation occurs in the universal inner region. This gives a recombination rate constant of In agreement with known results

28. A little discussion of four-boson potentials [Von Stecher et. al., Nature Phys. 5, pg 417] Look at potentials in this region. Negative scattering length with at least one bound Efimov state.

29. Just after first Efimov state becomes bound Two four body bound states are attached to each Efimov threshold.. (Hammer and Platter, Euro. Phys. J. A 32, 113; von Stecher, D’Incao and Greene Nature Phys. 5, 417).

30. Slightly larger scattering length

31. Attractive region becomes deep enough to admit a four-body state

32. Second Efimov state becomes bound. Two four-body states can be supported for each Efimov state.

33. Applying the WKB Recombination formula

34. 4-body resonances Second Efimov state becomes bound. (Cusp?)

35. Can 4-body effects actually be seen? Surprisingly, yes. Measurable four-body recombination occurs to deeply bound dimer states: (No weakly bound trimers)

36. More recently: Four body recombination to Efimov Trimers has been measured.

37. N>4 Without potentials we can’t say too much, but recent work has shown where we could expect resonances. Can 5 or more body physics be seen,

38. Can 5 or more body physics be seen? Without strong resonances, back of the envelope approximation says, probably not. N = 4 N = 5 N = 6

39. Summary N-body recombination becomes intuitive when put into the adiabatic hyperspherical formalism Getting the potentials is hard, but even without them, scaling behavior can be extracted. Low energy recombination can be described by the scattering behavior in a single channel. WKB does surprisingly well in describing the single channel S-matrix Four body recombination can actually be measured in some regimes.

40. In 1970 a freshly-minted Russian PhD in theoretical nuclear physics, Vitaly Efimov, considered the following natural question: What is the nature of the bound state energy level spectrum for a 3 particle system, when each of its 2-particle subsystems have no bound states but are infinitesimally close to binding? Efimov’s prediction: There will be an INFINITE number of 3-body bound states!! This exponential factor = 1/22.7 2 =0.00194, i.e. if one bound state is found at E 0 = -1 in some system of units, then the next level will be found at E 1 = -0.00194, and E 2 = -3.8 x 10 -6, etc….

41. The Efimov effect (restated) [Nucl. Phys. A. (1973)]

42. Qualitative and quantitative understanding of Efimov’s result At a qualitative level, it can be understood in hindsight, because two particles that are already attracting each other and are infinitesimally close to binding, just need a whiff of additional attraction from a third particle in order to push them over that threshold to become a bound three-body system. Quantitatively, Efimov (and later others) showed that a simple wavefunction can be written down at each hyperradius.

43. Transition regionUniversal region Short range stuff Lowest adiabatic hyperradial channel a<0 for identical bosons

44. K.E. a < 0 Observing the Efimov effect: three-body recombination

45. K.E. a < 0 Observing the Efimov effect: three-body recombination Three-body recombination can be measured through trap losses. Shape resonance occurs when an Efimov state appears at 0 energy. Spacing of shape resonances is geometric in the scattering length.

46. Only one resonance, need two to show Efimov scaling Second resonance at Need low temperatures:

47. Other possible Efimov states He trimer

48. Other possible Efimov states Recently, three hyperfine states of 6 Li Ottenstein et.al., PRL. 101, 203202 (2008)Huckans et. al., arXiv:0810.3288 (2008)

49. Real two-body interaction are multi-channel in nature. Simplest thing: Zero-range model

50. How does this translate to three bodies? Start by looking at a simplified model: no coupling.

51. Make excited bound state resonant with second threshold Coupled Uncoupled Parameters for an excited threshold resonance

52. Full calculation looks a bit ugly. First 300 potentials [PRA, 78 020701 (2008)]

53. Simplified picture: Cartoon of two important curves. Efimov Diabat Three free particles Actually an avoided crossing Efimov states Super-critical 1/R 2 potential leads to geometrically spaced states. Coupling leads to quasi-stability: Three-body Fano-Feshbach Resonances With no long-range coupling, widths scale geometrically

54. K.E. Three particles come together at low energy with respect to the first threshold.Excite the system with RF photons.If photon energy is degenerate with Efimov state energy, expect strong coupling to lower channels. Photon and binding energies are released as kinetic energy The Experiment

55. Cartoon three body loss spectrum. 1 st state2 nd statemany states

56. Four Bosons and Efimov’s legacy Figure from von Stecher et. al., eprint axiv/0810.3876

57. A little review of von Stecher’s work on four-boson potentials eprint axiv/0810.3876 Look at potentials in this region. Negative scattering length with at least one bound Efimov state.

58. Just after first Efimov state becomes bound

59. Slightly larger scattering length

60. Attractive region becomes deep enough to admit a four-body state

61. Second Efimov state becomes bound. Two four-body states can be supported for each Efimov state.

62. Simplest way to see four-body physics is through four-body recombination. N-body recombination rate coefficient, in terms of the T matrix, is given by: For four bosons in the low energy regime this reduces to

63. The behavior T matrix element is dominated by the lowest four- body channel.

64. If a four-body state is present, a shape resonance occurs.

65. Using a simple WKB wavefunction gives the four-body recombination rate coefficient up to an overall factor. a 7 scaling (predicted by asymptotic scaling potential) 4-body resonances Second Efimov state becomes bound

66. Four-body behavior scales with the three-body Efimov parameter. We can expect Log periodic behavior! Position of four-body resonances is universal: Observation of four-body resonances can give another handle on identifying Efimov states

67. Summary 3-bodies and Efimov Physics: PRA 78, 020701 (2008) – Zero-range multichannel interactions predict an Efimov potential at an excited three-body threshold. – Coupling to lower channels gives bound states coupled to the three-body continuum: 3-body Fano-Feshbach resonances! – Quasi-stable Efimov states may, possibly, be accessed via RF spectroscopy allowing for the observation of multiple resonances. 4-bosons – 4-body recombination shows universal resonance behavior. – Postitions of 4-body resonances give a further handle on idetifying an Efimov state.

68. Four-Fermions

69. Jacobi and “Democratic” Hyperspherical Coordinates “H” - type 1 2 3 4 Body-fixed “democratic” coodinates (Aquilantii/Cavalli and Kuppermann): Parameterize moments Inertia with R,  1 and  2 : 2 1 Rotate Jacobi vectors Into body-fixed frame: 3 Parameterize body-fixed Vectors with three-more angles:

70. Variational basis for four particles: (Assume L=0) Note: There is no (shallow) three-body bound state for (up-up-down) fermions. Dimer+Dimer: Dimer+Three-body continuum: Four-Body continuum:

71. After just a few thousand cpu hours: Potentials! With potentials, we can start looking at scattering

72. Dimer-dimer scattering length a dd (0)= 0.6 a Petrov, PRL (2004) With effective range: von Stecher, PRA (2008)

73. Energy dependence means any finite collision energy leads to deviation from the zero energy results

74. What about dimer relaxation? or

75. Unfortunately, there are an infinite number of final states!

76. Fermi’s golden rule leads to a simple expression for the rate: is the WKB tunneling probability is the WKB wave number is the density of final states near R is probability that three particles are close together at hyerradius R.

77. By performing the integral over different hyperradial regions, we can isolate different types of process. Integration over only very small hyperradii isolates relaxation channels where all four particles are involved.

78. Four-body processes Three-body processes influenced by presence of fourth particle Three-body only processes

79. Petrov (2004) Small R contribution Intermediate scaling behavior [arXiv:0806.3062]