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 potential0812.1486, Feshbach review 0902.1727, Book chapter0903.0884, MQDT treatment LiK, KRb

3. Long-range potential + quantum defect theory for atom-atom collisionsCan we get simple, practical models?

Surface of sunRoom temperature

Liquid He

Laser cooled atoms

(Bosons or Fermions)

Interior of sun

Optical lattice bandsQuantum gases

1 pK

1 nK

1 K

1 mK

1 K

1000 K

106 K

109 K

E/kB

E/h

1 MHz

1 GHz

1 THz

1 kHz

1 Hz

Ultracold polar molecules are now with us

1. Atom preparation

3. Populationtransfer

STIRAP

2. AtomAssociation

weakly bound pair

100 kHz

100 THz

4. Polar moleculesDipolar gases, lattices

Kohler et al, Rev. Mod. Phys. 78, 1311 (2006)

Chin, et al, arXiv: 0812.1496

Long range

-C6/R6

Analyticlong-range

theory(B. Gao)

a_

10-4 eV

Separated atoms

Properties ofseparated species

“simple”

10-10 eV (1 K)

A+B

Y

1 eV

AB

“Core”independent

of E ≈ 0

Short range

(E) scattering phase

(E) bound state phase(Ei)=nat eigenvalue

Resonance scattering S-matrix theory of molecular collisionsF. H. Mies, J. Chem. Phys. 51, 787, 798 (1969)

where

€

1

QT= ΛT

3 =h

2πμkBT

⎛

⎝ ⎜

⎞

⎠ ⎟

3

2

where

QT = translational partition function

T = thermal de Broglie wavelengthof pair

Replace

for elastic collisionsPhaseSpacedensity

Timescale

Dynamics

Adapted from Gao, Phys. Rev. A 62, 050702 (2000); Figure from FB review

Bound states from van der Waals theory

Spectrum of van der Waals potential

Adapted from Fig. 8Chin, Grimm, Julienne,Tiesinga, “FeshbachResonances in UltracoldGases”, submitted to Rev. Mod. Phys.arXiv:0813.1496

Singlet

Triplet

Blue lines: a = ∞

40K87Rb

-0.41 GHz-3.17 GHz-10.56 GHz

-3.00 GHz-3.17 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

C6 van der Waals coefficient reduced massabg “background” scattering length

resonance widthB0 singularity in a(B)magnetic moment difference

vdW Energy scale

Analytic properties of (R,E) across thresholds (E) and betweenshort and long range (R)

Analytic solutions for -C6/R6 van der Waals potentialB. Gao, Phys. Rev. A 58, 1728, 4222 (1998)Also 1999, 2000, 2001, 2004, 2005Solely a function of C6, 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, 012708 (2000)P. S. Julienne and B. Gao, in Atomic Physics 20, ed. by C. Roos,

H. Haffner, and R. Blatt (2006) (physics/0609013)

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 C6, , a

Assume a single isolated resonance weakly coupled to the continuumYc,bg <<1, Ycc = -Ybg,bg = 0

Bound states

Scattering states

Classification of resonances by strength, arXiv:0812.1496

For magnetically tunable resonances:

Bound state norm Z as E → 0

Bound state E=0 shifts to

Resonance strength

See Kohler et al, Rev. Mod. Phys. 78, 1311 (2006)

Closed channeldominated

Entrance channeldominated

“Broad”

“Narrow”

400 600 800

B (Gauss)

6Li ab

0

1

2

E/kB(mK)

400 600 800

B (Gauss)

0

1

7Li aa

Closed channeldominated

Entrance channeldominated

Color:sin2(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 resonancesPoints: coupled channels Lines: box model

Closed-channelcharacter

Energy

Can we get simple models for bound and scattering states?

Use vdW solutions for MQDT treatment

Ingredients:Atomic hyperfine/Zeeman propertiesAtomic-molecule basis set frame transformationVan der Waals coefficient C6

S, T scattering lengths

arXiv: 0903.0884 Fit 9 s-wave measured resonances in 6Li40K 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, 053201 (2008).

3 AND ONLY 3 free parameters

40K87Rb aa resonances

n=-2

n = -3

A(-1)

D(-3)

B(-2)

Ion-atom MQDT elastic and radiative charge transferNa + Ca+

Ion-atom -C4/R4:

Idziaszek, et al., Phys. Rev. A 79, 010702 (2009)

Model calculationonly (no realPotentials)

A+B

Long range Asymptotic

Cold speciesprepared

Chemistry

Scatter offlong-rangepotential

Assumeunit probability

of inelastic eventat small R

“Universal” van der Waals inelasticity

LostReflect

Transmit

Reflect

“Universal” van der Waals

model

Applied to RbCs molecular quenching byHudson, Gilfoy, Kotochigova, Sage, and De Mille, Phys. Rev. Lett. 100, 203201 (2008)