Transcript Document

Resonance Scattering in
optical lattices and Molecules
崔晓玲 (IOP, CASTU)
Collaborators: 王玉鹏 (IOP) , Fei Zhou (UBC)
2010.08.02
大连
Outline
• Motivation/Problem: effective scattering in optical lattice
– Confinement induced resonance
– Validity of Hubbard model
– Collision property of Bloch waves (compare with plane waves)
• Basic concept/Method
– Renormalization in crystal momentum space
• Results
– Scattering resonance purely driven by lattice potential
– Criterion for validity of single-band Hubbard model
– Low-energy scattering property of Bloch waves
• E-dependence, effective range
• Induced molecules, detection, symmetry
Motivation I:
biatomic collision under confinements:
induced resonance and molecules
•
3D Free space: s-wave scattering length
see for example: Nature 424, 4 (2003), JILA
as
Feature 1: Feshbach resonance
driven by magnetic field
B0
molecule
Feature 2: Feshbach molecule
only at positive a_s
Eb
B
Motivation I:
biatomic collision under confinements:
induced resonance and molecules
•
•
3D Free space: s-wave scattering length
Confinement Induced Resonance and Molecules
see for example: CIR in quasi-1D
a
z
PRL 81,938 (98); 91,163201(03), M. Olshanii et al
Motivation I:
biatomic collision under confinements:
induced resonance and molecules
•
•
3D Free space: s-wave scattering length
Confinement Induced Resonance and Molecules
see for example: CIR in quasi-1D
Feature 1: resonance
induced by confinement
Feature 2: induced molecule
at all values of a_s
expe: PRL 94, 210401
(05), ETH
Motivation I:
biatomic collision under confinements:
induced resonance and molecules
•
•
3D Free space: s-wave scattering length
Confinement Induced Resonance and Molecules
see for example: CIR in quasi-1D
Q: whether there is CIR or induced molecule in 3D optical lattice?
Motivation II:
Validity of single-band Hubbard model to optical lattice
under tight-binding approximation:
break down in two limits:
 shallow lattice potential
 strong interaction strength
Q: how to identify the criterion quantitatively?
Motivation III:
Scattered Bloch waves near the bottom of lowest band
near E=0,
Ek  k / 2meff , meff 1/ taL
2
2
E
quadratic dispersion defined by band
mass
free space
0
Q: low-energy effective scattering (2 body, near E=0)
free space ?
explicitly, energy-dependence of scattering matrix, effective interaction range,
property of bound state/molecule……
k
Solution to all Qs:
two-body scattering problem in optical lattice for all values of
lattice potential and interaction strength !
however,
•
major difficulty:


state-dependent U
Unseparable: center of mass(R) and relative motion(r)
n1k
n 2 -k
•
n1' k'
U 
n2' -k'
Previous works are mostly based on single-band Hubbard model,
except few exact numerical works (see, G. Orso et al, PRL 95, 060402,
2005; H. P. Buechler, PRL 104, 090402, 2010: both exact but quite timeconsuming with heavy numerics, also lack of physical interpretation
such as individual inter/intra-band contributions, construction of Blochwave molecule…
Our method: momentum-shell renormalization
----from basic concept of low-energy effective scattering
First, based on standard scattering theory,
Lippmann-Schwinger equation :
+
=
T
U
0
E=0
implication of renormalization procedure to obtain low-energy physics!
Our method: momentum-shell renormalization
----from basic concept of low-energy effective scattering
Then, an explicit RG approach:
k'
k
-k -k'

k''


k' k
-k'
-k
-k''
RG eq:
with boundary conditions:
U (  )  U ()  U 2 ()
1
 ...

2

|k |
k
RG approach to optical lattice and results
XL Cui, YP Wang and F Zhou,
Phys. Rev. Lett. 104, 153201 (2010)
•
Simplification of U:
n1k
n1' k'
n 2 -k
n2' -k'
U 
intra-band, specialty of OL
inter-band, to renormalize
short-range contribution
•
two-step renormalization
Step I : renormalize all virtual scattering to higher-band states (inter-band)
Step II : further integrate over lowest-band states (intra-band)
Characteristic parameter: C1 --- interband; C2 --- intraband
Results
1. resonance scattering at E=0:
resonance
scattering of
Rb-K mixture
resonance at
v
C
a /a
L
S
2. Validity of Hubbard model:
To safely neglect inter-band scattering,
Condition I:
C << C
Condition II:
C << a /a
1
1
: deep lattice potential
2
L
S
: weak interaction
Hubbard limit
Under these conditions,
Ueff
on-site U
For previous study in this limit
see P.O. Fedichev et al, PRL
92, 080401 (2004).
In the opposite limit,
(C
1
C or C
2
1
a /a )
L
S
Both intra- and inter-band contribute to low-E effective scattering,
where C1 can NOT be neglected!
set C1=0
cross section
2
2
4πa |χ|
L
, phase shift
a s =-0.5a , v=2.5
L

3. Symmetry between repulsive and attractive bound state:
at large v and |a |
S
a , single-band Hubbard model:
L
simply solvable:
•
K conserved (semi-separated)
•
state-independent U
E
s-band
0
as/aL
Zero-energy resonance scattering
attractive and
repulsive bound state
bound state for a general K:
E
T-matrix and bound state:
scattering continuum
12t


0
K
B
From particle-hole symmetry,
Winkler et al,
Nature 441, 853 (06)
 ( E )   (6t  E )
E
repulsive as>0
scattering continuum
12t


0
attractive -as<0
K
Resonance scattering and bound
states near the bottom of lowest
band for a negative a_s therefore
imply resonance scattering and
bound states near the top of the
band for a positive a_s.
4. E-dependence, effective range :
1
1
m
 
f ( E ), f ( E )  f R  if I
T ( E ) T 4 a
0
L
In Hubbard model regime , when
E  0:
f I  DOS
compare with free space (all E):
Effective interaction range of atoms in optical lattice
is set by lattice constant (finite, >> range in free
space), even for two atoms near the band bottom!
E
This leads to much exotic E-dependence of T-matrix
in optical lattice.
k
Conclusion
 Effective scattering using renormalization approach
 Optical lattice induced resonance scattering (zero-energy)
 Large a_s, shallow v: interband + intraband
 Small a_s, deep v: intraband (dominate)
------- validity criterion for single-band Hubbard model
 Bound state induced above resonance
– Binding energy, momentum distribution (for detection)
– Mapping between attractive (ground state) and repulsive
bound state via particle-hole symmetry
 Exotic E-dependence of T-matrix / effective potential
------- due to finite-range set by lattice constant
Phys. Rev. Lett. 104, 153201 (2010)
Bound state/molecule above resonance (v>vc):
a two-body bound state/molecule :
Real momentum distribution :
Smeared peak at discrete
Q as v increases!!
no interband,
C1=0
Bound state: T(EB)=infty (Bethe-Salpeter eq)
assume a 2-body wf:
E
repulsive as>0
12t
K=0 bound state
Repulsive
Attractive
metastable excited
ground state
above band top
below band bottom
nq peaked at q=±pi
nq peaked at q=0


0
attractive -as<0
K