Semi-Classical Methods and N-Body Recombination Seth Rittenhouse ITAMP, Harvard-Smithsonian Center for Astrophysics, Cam

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

4. Jose P. D’Incao Nirav Mehta Chris H. Greene Javier von Stecher

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 Zaccanti et. al., Nature Phys. 5, 586 (2009). Ultra cold Li7 gas: Rice group (soon to be published)

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. 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 For example, The energy landscape 3 Bodies 2-D ~ 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 l non-integer valued or complex.

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

13. Relevant examples of potential curves Three bosons with negative scattering length: Transition region Here be dragons! Repulsive universal long-range tail Attractive inner region

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

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

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

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

21. WKB to the rescue 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:

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

24. 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. Applying the WKB Recombination formula 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 = 1/22.72=0.00194, i.e. if one bound state is found at E0= -1 in some system of units, then the next level will be found at E1= -0.00194, and E2= -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. Lowest adiabatic hyperradial channel a<0 Short range stuff Universal region Transition region Universal region for identical bosons

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

45. Observing the Efimov effect: three-body recombination a < 0 K.E. • 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 6Li 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.