Download C 6

EuroQUAM satellite meeting, University of Durham, April 18, 2009
Understanding Feshbach molecules
with long range quantum defect theory
Paul S. Julienne
Joint Quantum Institute, NIST and The University of Maryland
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
0812.1486, Feshbach review
0902.1727, Book chapter
0903.0884, MQDT treatment LiK, KRb
3. Long-range potential + quantum defect theory
for atom-atom collisions
Can we get simple, practical models?
E/kB
109 K
106
K
E/h
1000 K
1 THz
1K
1 GHz
1 mK
1 MHz
1 K
1 kHz
1 nK
1 Hz
1 pK
Interior of sun
Surface of sun
Room temperature
Liquid He
Laser cooled atoms
Optical lattice bands
Quantum gases
(Bosons or Fermions)
Ultracold polar molecules are now with us
3. Population
transfer
STIRAP
1. Atom
preparation
100 kHz
100 THz
2. Atom
Association
weakly bound pair
Kohler et al, Rev. Mod. Phys.
4. Polar molecules
78, 1311 (2006)
Dipolar gases, lattices
Chin, et al, arXiv: 0812.1496
Short range
Long range
Separated atoms
A+B
AB
1 eV
10-10 eV (1 K)
10-4 eV
(E) scattering phase
Y
_
a
(E) bound state phase
(Ei)=n at eigenvalue
-C6/R6
“Core”
independent
of E ≈ 0
Analytic
long-range
theory
(B. Gao)
Properties of
separated species
“simple”
Resonance scattering S-matrix theory of molecular collisions
F. H. Mies, J. Chem. Phys. 51, 787, 798 (1969)
where
3
2
where
 h 
1
3
 T  

QT
2

k
T

B 
QT = translational partition function
T = thermal de Broglie wavelength
of pair

Replace
Phase Time Dynamics
Space scale
density
for elastic collisions
Bound states from van der Waals theory
Adapted from Gao, Phys. Rev. A 62, 050702 (2000); Figure from FB review
Spectrum of van der Waals potential
40K87Rb
Blue lines: a = ∞
Singlet
Triplet
Adapted from Fig. 8
Chin, Grimm, Julienne,
Tiesinga, “Feshbach
Resonances in Ultracold
Gases”, submitted to
Rev. Mod. Phys.
arXiv:0813.1496
-10.56 GHz
-3.17 GHz
-0.41 GHz
-3.17 GHz
-3.00 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 mass
abg “background” scattering length
 resonance width
B0 singularity in a(B)
magnetic moment difference
vdW Energy scale
Use vdW solutions for MQDT analysis
Analytic properties of (R,E) across thresholds (E) and between
short and long range (R)
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)
Analytic solutions for -C6/R6 van der Waals potential
B. Gao, Phys. Rev. A 58, 1728, 4222 (1998)
Also 1999, 2000, 2001, 2004, 2005
Solely a function of C6, reduced mass , and scattering length a
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)
From vdW theory, given C6, , a
MQDT theory (1984) gives coupled channels S-matrix and bound states.
Assume a single isolated resonance weakly coupled to the continuum
Yc,bg <<1, Ycc = -Ybg,bg = 0
Bound states
Scattering states
Classification of resonances by strength, arXiv:0812.1496
Resonance strength
See Kohler et al, Rev. Mod. Phys. 78, 1311 (2006)
For magnetically tunable resonances:
Bound state E=0 shifts to
Bound state norm Z as E → 0
Closed channel
dominated
Entrance channel
dominated
“Broad”
“Narrow”
Closed channel
dominated
Entrance channel
dominated
6Li
7Li
ab
aa
Color:
sin2(E)
1
2
E/kB
(mK)
1
0
400
600
B (Gauss)
800
400
600
B (Gauss)
800
0
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 C6
S, T scattering lengths
3 AND ONLY 3 free parameters
arXiv: 0903.0884 Fit 9 s-wave measured resonances in 6Li40K from
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).
To about 2 per cent accuracy (3 G)
40K87Rb
aa resonances
n=-2
n = -3
A(-1)
B(-2)
D(-3)
Ion-atom MQDT elastic and radiative charge transfer
Na + Ca+
Model calculation
only (no real
Potentials)
Ion-atom -C4/R4:
Idziaszek, et al.,
Phys. Rev. A 79,
010702 (2009)
“Universal” van der Waals inelasticity
Chemistry
Long range
Asymptotic
Reflect
A+B
Lost
Cold species
prepared
Scatter off
long-range
potential
Assume
unit probability
of inelastic event
at small R
Transmit
Reflect
“Universal”
van der Waals
model
Applied to RbCs molecular quenching by
Hudson, Gilfoy, Kotochigova, Sage, and De Mille,
Phys. Rev. Lett. 100, 203201 (2008)