Understanding Feshbach molecules with long range quantum defect theory Paul S. Julienne Joint Quantum Institute, NIST and The University of Maryland EuroQUAM satellite meeting, University of Durham, April 18, 2009 Collaborators (theory) Tom Hanna, Eite Tiesinga (NIST) Thanks also to Bo Gao (U. of Toledo) and Cheng Chin (U. Chicago) J. K. Freericks (Georgetown U.), M. Maśka (U. Silesia), R. Lemański (Wroclaw)

Outline 1.Sone general considerations 2. The significance of the long-range potential , Feshbach review , Book chapter , MQDT treatment LiK, KRb 3. Long-range potential + quantum defect theory for atom-atom collisions Can we get simple, practical models?

Surface of sun Room temperature Liquid He Laser cooled atoms (Bosons or Fermions) Interior of sun Optical lattice bands Quantum gases 1 pK 1 nK 1  K 1 mK 1 K 1000 K 10 6 K 10 9 K E/k B E/h 1 MHz 1 GHz 1 THz 1 kHz 1 Hz

Ultracold polar molecules are now with us 1. Atom preparation 3. Population transfer STIRAP 2. Atom Association weakly bound pair 100 kHz 100 THz 4. Polar molecules Dipolar gases, lattices Kohler et al, Rev. Mod. Phys. 78, 1311 (2006) Chin, et al, arXiv:

Long range -C 6 /R 6 Analytic long-range theory (B. Gao) a _ eV Separated atoms Properties of separated species “simple” eV (1  K) A+B Y 1 eV AB “Core” independent of E ≈ 0 Short range  (E) scattering phase (E) bound state phase (E i )=n  at eigenvalue

Resonance scattering S-matrix theory of molecular collisions F. H. Mies, J. Chem. Phys. 51, 787, 798 (1969) where Q T = translational partition function  T = thermal de Broglie wavelength of pair Replace for elastic collisions Phase Space density Time scale Dynamics

Adapted from Gao, Phys. Rev. A 62, (2000); Figure from FB review Bound states from van der Waals theory

Spectrum of van der Waals potential Adapted from Fig. 8 Chin, Grimm, Julienne, Tiesinga, “Feshbach Resonances in Ultracold Gases”, submitted to Rev. Mod. Phys. arXiv: Singlet Triplet Blue lines: a = ∞ 40 K 87 Rb

-0.41 GHz GHz GHz

-3.00 GHz GHz

Goal: Simple, reliable model for classification and calculation * Now: Full quantum dynamics with CC calculations All degrees of freedom with real potentials Exact, but not simple * vdW-MQDT: Reduction to a simpler representation Parameterized by C 6 van der Waals coefficient  reduced mass a bg “background” scattering length  resonance width B 0 singularity in a(B)  magnetic moment difference vdW Energy scale

Analytic properties of  (R,E) across thresholds (E) and between short and long range (R) Analytic solutions for -C 6 /R 6 van der Waals potential B. Gao, Phys. Rev. A 58, 1728, 4222 (1998) Also 1999, 2000, 2001, 2004, 2005 Solely a function of C 6, reduced mass , and scattering length a Generalized Multichannel Quantum Defect Theory (MQDT): F. H. Mies, J. Chem. Phys. 80, 2514 (1984) F. H. Mies and P. S. Julienne, J. Chem. Phys. 80, 2526 (1984) Ultracold: Eindhoven (Verhaar group), JILA (Greene, Bohn) P. S. Julienne and F. H. Mies, J. Opt. Soc. Am. B 6, 2257 (1989) F. H. Mies and M. Raoult, Phys. Rev. A 62, (2000) P. S. Julienne and B. Gao, in Atomic Physics 20, ed. by C. Roos, H. Haffner, and R. Blatt (2006) (physics/ ) Use vdW solutions for MQDT analysis

For coupled channels case Given the reference the single-channel functions: for scattering (E>0)  (E), C(E), tan (E) and bound states (E<0) (E) MQDT theory (1984) gives coupled channels S-matrix and bound states. From vdW theory, given C 6, , a Assume a single isolated resonance weakly coupled to the continuum Y c,bg <<1, Y cc = -Y bg,bg = 0 Bound states Scattering states

Van der Waals MQDT bound state equation Use Gives for binding energy when Solution as k b --> 0 with

Classification of resonances by strength, arXiv: For magnetically tunable resonances: Bound state norm Z as E → 0 Bound state E=0 shifts toResonance strength See Kohler et al, Rev. Mod. Phys. 78, 1311 (2006)

Closed channel dominated Entrance channel dominated “Broad” “Narrow”

B (Gauss) 6 Li ab E/k B (mK) B (Gauss) Li aa Closed channel dominated Entrance channel dominated Color: sin 2  (E)

Two-channel “box” model Corresponds to vdW MQDT when “box” width is chosen to be Bound state equation for level with binding energy with

Bound state E and Z for selected resonances Points: coupled channels Lines: box model Closed-channel character Energy

Can we get simple models for bound and scattering states? Use vdW solutions for MQDT treatment Ingredients: Atomic hyperfine/Zeeman properties Atomic-molecule basis set frame transformation Van der Waals coefficient C 6 S, T scattering lengths arXiv: Fit 9 s-wave measured resonances in 6 Li 40 K from To about 2 per cent accuracy (3 G) E. Wille, F. M. Spiegelhalder, G. Kerner, D. Naik, A. Trenkwalder, G. Hendl, F. Schreck, R. Grimm, T. G. Tiecke, J. T. M. Walraven, et al., Phys. Rev. Lett. 100, (2008). 3 AND ONLY 3 free parameters

40 K 87 Rb aa resonances

n=-2

n = -3 A(-1) D(-3) B(-2)

Ion-atom MQDT elastic and radiative charge transfer Na + Ca + Ion-atom -C 4 /R 4 : Idziaszek, et al., Phys. Rev. A 79, (2009) Model calculation only (no real Potentials)

A+B Long range Asymptotic Cold species prepared Chemistry Scatter off long-range potential Assume unit probability of inelastic event at small R “Universal” van der Waals inelasticity Lost Reflect Transmit Reflect

“Universal” van der Waals model Applied to RbCs molecular quenching by Hudson, Gilfoy, Kotochigova, Sage, and De Mille, Phys. Rev. Lett. 100, (2008)