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Cold Atomic and Molecular Collisions 1. Basics 2. Feshbach resonances

ICAP Summer School 2006 University of Innsbruck. Cold Atomic and Molecular Collisions 1. Basics 2. Feshbach resonances 3. Photoassociation Paul S. Julienne Quantum Processes and Metrology Group Atomic Physics Division NIST. And many others, especially

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Cold Atomic and Molecular Collisions 1. Basics 2. Feshbach resonances

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  1. ICAP Summer School 2006 University of Innsbruck Cold Atomic and Molecular Collisions 1. Basics 2. Feshbach resonances 3. Photoassociation Paul S. Julienne Quantum Processes and Metrology Group Atomic Physics Division NIST And many others, especially Eite Tiesinga, Carl Williams, Pascal Naidon (NIST), Thorsten Köhler (Oxford), Bo Gao (Toledo), Roman Ciurylo (Torun)

  2. Some resources Jones et al, Rev. Mod. Phys. 78, 483 (2006) van der Waals properties + PA review Julienne and Mies, J. Opt. Soc. Am. B 6, 2257 (1989) WKB/quantum connections and threshold laws Burnett et al, Nature 416, 225 (2002) “box”interpretation of A + simple review of cold collisions Kohler, Goral, Julienne, cond-mat/0601429 (in press, Rev. Mod. Phys.) Production of cold molecules via magnetically tunable Feshbach resonances In progress, Chin, Grimm, Tiesinga, Julienne, Rev. Mod. Phys. Feshbach resonances in ultracold gases (look at preprint server by end of 2006)

  3. Gribakin and Flambaum, Phys. Rev. A 48, 546 (1993) “Size” of potential V(R)

  4. Interacting atoms Noninteracting atoms

  5. 834.1(1.5) G Data: Bartenstein et al.,PRL 94, 103201 (2005)

  6. Bound states from van der Waals theory Gao, Phys. Rev. A 62, 050702 (2000); Figure from E. Tiesinga

  7. For example: (2,-2)+(2,-2) -->(2,-2)+(2,-1) -->(2,-2)+(2,0) F=3 -->(2,-1)+(2,-1) Probability because F=2, M=-2 -1 0 1 2 For s-waves, let Inelastic collisions

  8. Scattering channels Initial channel: Final channel: Inelastic collisions Scattering amplitudes T’expressed in terms of the elements of the unitary S-matrix S’ = ’ - T’

  9. If inelastic channels ’≠ exist, unitarity ensures Thus, Inelastic collisions continued If only a single channel, S= e2i ➞ e-2ika as k ➞ 0 Complex scattering length a-ib

  10. Coupled channels expansion: Solve matrix Schrödinger equation conserved Electronic (Born-Oppenheimer) V(R) does not change Small spin-dependent potential changes How do we get the S-matrix, or bound states? Extract S from solution at large R >> RvdW Potential matrix V’(R)

  11. Elastic collisions ➞ constant, K ➞ v  = 4a2+b2) • Inelastic collisions • ➞ 1/v, K ~ constant K = (4h/m)b s-wave Threshold Collisions Summary Cross section cm2 Rate coefficient K = <  v> cm3/s Complex scattering lengtha - ib

  12. How fast are (inelastic s-wave) cold collisions? Typical strong event (B ~ x0): 10-10 cm3/s, MOT or BEC In a MOT: t ~ 0.1 to 1 s In a BEC (use K/2): t ~ 10 to 100 ms

  13. where QT = translational partition function Probability |S|2 < 1 Dynamical factor T = thermal de Broglie wavelength How fast are inelastic cold collisions (Maxwell-Boltzmann)? l=0 only Phase Space density Upper bound Dynamics

  14. An Optical Lattice Laser 2 Laser 1

  15. 1 Atom per cell Control 2 Atoms per cell 2-body levels, dynamics 3 Atoms per cell 3-body levels, dynamics

  16. Scale size of V (r) x0

  17. 1 2 2 r ( ) w m 2 3 2 1 0 v=0 d v=-1 3 C 6 - R 6 v=-2 Schematic (not to scale) 12 D > 10 Hz e < 5 nm

  18. Na F=1,M=+1 Energy levels in a 1 MHz harmonic trap (Bolda et al., 2002) Energy-dependent pseudopotential for trap level En: a(En) from scatttering calculation of (En) Points: numerical coupled channels Solid Line: pseudopotential E-dependent pseudopotential D. Blume and Chris H. Greene, Phys. Rev. A 65, 043613 (2002) E. L. Bolda, E. Tiesinga, and P. S. Julienne, Phys. Rev. A 66, 013403 (2002)

  19. Feshbach resonances and Feshbach molecules • What is a scattering resonance? • Basic properties of threshold resonance scattering and bound states • Halo molecules • Simple parameterization by 5 key parameters: • abg (background scattering length), C6 (van der Waals coefficient), m (mass) •  (width),  (magnetic moment difference) • Basic molecular physics of alkali atom resonances • Illustration of typical resonances: 6Li, 85Rb, 87Rb, 40K, Cs • Resonance dynamics—making and dissociating molecules • Resonances in traps

  20. Some examples of Feshbach resonances background Sodium BEC Cesium theory 910 900 Magnetic field (Gauss) E. Tiesinga et al., Phys. Rev. A 47, 4114 (1993) S. Inouye et al Nature 392, 141 (1998)

  21. 165.2 G 158.4 G 156.4 G An example for E➞ 0 Cornish, Claussen, Roberts, Cornell, Wieman, Phys. Rev. Lett. 85, 1795 (2000) 85Rb BEC (below 15 nK) 162.3 G 157.2 G

  22. Making 40K2 molecules Greiner, M., C. A. Regal, and D. S. Jin, 2003, Nature (London) 426, 537. BEC (79 nK) Thermal (250 nK) Making 133Cs2 molecules 133Cs atom cloud 133Cs2 molecule cloud Herbig, J., T. Kraemer, M. Mark, T. Weber, C. Chin, H.-C. Nagerl, and R. Grimm, 2003, Science 301, 1510. Tunable scattering resonances used for

  23. Long history of resonance scattering O. K. Rice, J. Chem. Phys. 1, 375 (1933) U. Fano, Nuovo Cimento 12, 154 (1935) J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (1952) H. Feshbach, Ann. Phys. (NY) 5, 357 (1958); 19, 287 (1962) U. Fano, Phys. Rev. 124, 1866 (1961) Separation of system into: An (approximate) bound state A scattering continuum with some coupling between them

  24. Bound state Continuum V Closed channel (Resonance) Open channel (Background) width shift Resonant Scattering Picture (U. Fano, Phys. Rev. 124, 1866 (1961))

  25. a+a = 1,1 + 1,1 m=+2 l=0 states Eaa Energy levels of the Na2 dimer just below the a+a threshold

  26. Near 900 G (3 B values) Background bg(E) 4sin2(E) 915G 920G 930G (E) = bg(E) + res(E) Near 861 G A Na example From Mies, Tiesinga, Julienne, Phys. Rev. A 61, 022721 (2000)

  27. V E=0 As E ➞ 0 Shifted Threshold Resonant Scattering

  28. abg relative to E relative to Evdw Basic resonance parameters For E --> 0 limit abg “background” scattering length B0 singularity in a(B)  resonance width  magnetic moment difference For finite E and bound states Effect of interatomic potential especially van der Waals -C6/R6

  29. Stern-Gerlach separation of Molecules Atoms Atom cloud in trap Example 87Rb f=1, m=+1 Durr, Voltz, Marte, Rempe, Phys. Rev. Lett. 92, 020406 (2004)

  30. Schematic Magnetic moment From Durr, Voltz, Marte, Rempe, Phys. Rev. Lett. 92, 020406 (2004)

  31. 87Rb + 87Rb 30 THz 10 GHz

  32. 87Rb Zeeman substructure B/h = 1.4 MHz/G

  33. 87Rb + 87Rb a+a E(a+a) Bound states with m=2

  35. “Fermionic condensate” of paired atoms Phys. Rev. Lett. 92, 040403 (2004) BEC of molecules Nature 426, 537 (2003) Cindy Regal, Marcus Greiner, Deborah Jin NIST Boulder Ultracold 40K atoms, F=9/2,M=-9/2 and F=9/2,M=-7/2 http://jilawww.colorado.edu/~jin/pictures.html

  36. 40 Bn E-1 1 sin2bg(E,B) 20 E/h (MHz) 0 0 -20 40K a+b -40 200 240 160 B (Gauss)

  37. 40 Bn B0 E-1 1 sin2(E,B) 20 E/h (MHz) 0 0 E-1(B) -20 40K a+b -40 200 240 160 B (Gauss)

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