Cold Atomic and Molecular Collisions1. Basics2. Feshbach resonances3. Photoassociation

Paul S. Julienne

Quantum Processes and Metrology GroupAtomic Physics Division

NIST

ICAP Summer School 2006University of Innsbruck

And many others, especiallyEite Tiesinga, Carl Williams, Pascal Naidon (NIST),

Thorsten Köhler (Oxford), Bo Gao (Toledo), Roman Ciurylo (Torun)

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)

“Size” of potential V(R)

Rvdw(a0) Evdw(mK) (MHz) 6Li 31 29 614 40K 65 1.0 21 85Rb 83 0.35 7.3 133Cs 101 0.13 2.7

Gribakin and Flambaum, Phys. Rev. A 48, 546 (1993)

Noninteracting atoms

Interacting atoms

Data: Bartenstein et al.,PRL 94, 103201 (2005)

834.1(1.5) G

Gao, Phys. Rev. A 62, 050702 (2000); Figure from E. Tiesinga

Bound states from van der Waals theory

Inelastic collisions

F=2, M=-2

F=3

-1012

For example:

(2,-2)+(2,-2) -->(2,-2)+(2,-1)-->(2,-2)+(2,0)-->(2,-1)+(2,-1)

Probability

because

For s-waves, let

Inelastic collisions

Scattering amplitudes T’ expressed in terms of the elements of the unitary S-matrix S’ = ’ - T’

Final channel:

Scattering channels

Initial channel:

If only a single channel, S= e2i➞ e-2ika as k ➞ 0

If inelastic channels ’≠ exist, unitarity ensures

Thus,

Inelastic collisions continued

Complex scattering length a-ib

How do we get the S-matrix, or bound states?

Coupled channels expansion:

Solve matrix Schrödinger equation

Extract S from solution at large R >> RvdW

Potential matrix V’(R)

conserved

Electronic (Born-Oppenheimer) V(R) does not change

Small spin-dependent potential changes

s-wave Threshold Collisions Summary

Elastic collisions ➞ constant, K ➞ v = 4a2+b2)

Inelastic collisions ➞ 1/v, K ~ constant K = (4h/m)b

Cross section cm2

Rate coefficient K = < v> cm3/s

Complex scattering length a - ib

How fast are (inelastic s-wave) cold collisions?

€

K =h

m4B = (8 ×10−11cm3 /s)

B(a0)

m(amu)

Typical strong event (B ~ x0): 10-10 cm3/s, MOT or BEC

€

dndt

= −2Kn2 = −1τn where

1τ

= 2Kn

In a MOT: ~ 0.1 to 1 sIn a BEC (use K/2): ~ 10 to 100 s

€

1

τ= 2Kn = n

h

m8B = n(1.6 ×10−10cm3 /s)

B(a0)

m(amu)

How fast are inelastic cold collisions (Maxwell-Boltzmann)?

€

1

τ= 2Kn = 2 (nΛT

3 )kBT

hf

€

dndt

= −2Kn2 = −1τn where

1τ

= 2Kn

€

1

QT= ΛT

3 =h

2πμkBT

⎛

⎝ ⎜

⎞

⎠ ⎟

3

2

where

QT = translational partition function

T = thermal de Broglie wavelength

€

K =1QT

kBT

h(2l +1) | S(E) |2

l∑

l=0 only

Probability |S|2 < 1Dynamical factor

PhaseSpacedensity

Upperbound

Dynamics

An Optical Lattice

Laser 1 Laser 2

1 Atom per cell Control

2 Atoms per cell 2-body levels, dynamics

3 Atoms per cell 3-body levels, dynamics

Two atoms in a trap

3D spherically symmetric harmonic trap: ω=2ν

2- :body potential( )V r

Separate the center of mass and relative motion

⎡⎣⎢−

2

2(2m)∇R2 +

1

2(2m)ω 2R2 ⎤

⎦⎥Ψcm

(R ) = EcmΨcm(R )h

2⎡

Centerof mass

−2( )μ

∇r2 +

1

2μ( )ω 2r 2

⎣⎢⎤

⎦⎥Ψ(r) = EΨ(r )V(r)+Relative

Veff(r)

h

Scale size of trap:

d=ωh

V(r)Scale size of

x0

0 v=0v=-1

12

v=-2

3

R6C6-

1

2 ω 2r 2

3

< 5 nm

De > 1012 Hz

Schematic(not to scale)

d

0 10000 20000

Ψ( )R

(Internuclear separation R a0)

1mSee below

( )a

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)

Na F=1,M=+1 Energy levels in a

1 MHz harmonic trap(Bolda et al., 2002)

Points: numerical coupled channelsSolid Line: pseudopotential

E-dependent pseudopotential

a(En) from scattteringcalculation of (En)

Energy-dependentpseudopotential for trap level En:

• 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

Feshbach resonances and Feshbach molecules

Some examples of Feshbach resonances

E. Tiesinga et al., Phys. Rev. A 47, 4114 (1993)

Cesium theory

background

S. Inouye et al Nature 392, 141 (1998)

Sodium BEC

Magnetic field (Gauss)900 910

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

165.2 G

158.4 G

156.4 G

Greiner, M., C. A. Regal, and D. S. Jin, 2003, Nature (London) 426, 537.

Thermal (250 nK)

Making 40K2 molecules

BEC (79 nK)

Tunable scattering resonances used for

Herbig, J., T. Kraemer, M. Mark, T. Weber, C. Chin, H.-C. Nagerl, and R. Grimm, 2003, Science 301, 1510.

133Cs atom cloud

133Cs2 molecule cloud

Making 133Cs2 molecules

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 stateA scattering continuumwith some coupling between them

Resonant Scattering Picture(U. Fano, Phys. Rev. 124, 1866 (1961))

V

Bound state Continuum

Open channel(Background)

Closed channel(Resonance)

shift

width

Energy levels of the Na2 dimer just below the a+a threshold

a+a = 1,1 + 1,1

m=+2 l=0 states

Eaa

A Na example

From Mies, Tiesinga, Julienne, Phys. Rev. A 61, 022721 (2000)

Near 900 G(3 B values)

Near 861 G

4sin2 (E)

(E) = bg(E) + res(E)

915G 920G 930G

Background bg(E)

Threshold Resonant Scattering

V

E=0

Shifted

As E ➞ 0

Basic resonance parameters

abg “background” scattering lengthB0 singularity in a(B) resonance widthmagnetic moment difference

For E --> 0 limit

For finite E and bound states

Effect of interatomic potentialespecially van der Waals -C6/R6

abg relative to

E relative to Evdw

Example 87Rb f=1, m=+1Durr, Voltz, Marte, Rempe, Phys. Rev. Lett. 92, 020406 (2004)

Atom cloudin trap

Atoms Molecules

Stern-Gerlach separation of

From Durr, Voltz, Marte, Rempe, Phys. Rev. Lett. 92, 020406 (2004)

Schematic

Magnetic moment

87Rb + 87Rb

30 THz

10 GHz

87Rb Zeeman substructure

B/h = 1.4 MHz/G

a+a

E(a+a)

87Rb + 87Rb

Bound stateswith m=2

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

http://jilawww.colorado.edu/~jin/pictures.html

Cindy Regal, Marcus Greiner, Deborah JinNIST Boulder

Ultracold 40K atoms, F=9/2,M=-9/2 and F=9/2,M=-7/2

BEC of moleculesNature 426, 537 (2003)

“Fermionic condensate” of paired atomsPhys. Rev. Lett. 92, 040403 (2004)

Bn

E/h (MHz)

QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.

1

0

sin2bg(E,B)

0

20

40

-20

-40160 200 240

B (Gauss)

40K a+b

E-1

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

E/h (MHz)

QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.

1

0

sin2(E,B)

E-1(B)

0

20

40

-20

-40160 200 240

B (Gauss)

40K a+b

B0 BnE-1