resonance scattering in optical lattices and molecules
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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
see for example: Nature 424, 4 (2003), JILA
molecule
BB0
as
Eb
Feature 1: Feshbach resonance driven by magnetic field
Feature 2: Feshbach molecule only at positive a_s
Motivation I:
biatomic collision under confinements: induced resonance and molecules
• 3D Free space: s-wave scattering length
Motivation I:
z
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
PRL 81,938 (98); 91,163201(03), M. Olshanii et al
a
biatomic collision under confinements: induced resonance and molecules
• 3D Free space: s-wave scattering length• Confinement Induced Resonance and Molecules
Motivation I:
Feature 1: resonance induced by confinement
Feature 2: induced molecule at all values of a_s
see for example: CIR in quasi-1D
expe: PRL 94, 210401 (05), ETH
biatomic collision under confinements: induced resonance and molecules
• 3D Free space: s-wave scattering length• Confinement Induced Resonance and Molecules
Motivation I:
Q: whether there is CIR or induced molecule in 3D optical lattice?
see for example: CIR in quasi-1D
Validity of single-band Hubbard model to optical lattice
Motivation II:
under tight-binding approximation:
Q: how to identify the criterion quantitatively?
break down in two limits: shallow lattice potential strong interaction strength
Scattered Bloch waves near the bottom of lowest band
Motivation III:
0 k
E
near E=0,
2 / 2 ,k effE k m 21/eff Lm ta
quadratic dispersion defined by band mass free space
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……
Solution to all Qs:
two-body scattering problem in optical lattice for all values of lattice potential and interaction strength !
• major difficulty:
however,
U
1n k
2n -k
1n ' k'
2n ' -k'
state-dependent U Unseparable: center of mass(R) and relative motion(r)
• 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 time-consuming with heavy numerics, also lack of physical interpretation such as individual inter/intra-band contributions, construction of Bloch-wave molecule…
----from basic concept of low-energy effective scattering
First, based on standard scattering theory,
Lippmann-Schwinger equation :
= +T
0U
E=0
Our method: momentum-shell renormalization
implication of renormalization procedure to obtain low-energy physics!
k
-kk'
-k'
k''
-k''
k
-k
k'-k' RG eq:
with boundary conditions:
----from basic concept of low-energy effective scattering
Our method: momentum-shell renormalization
Then, an explicit RG approach:
2
| |
1( ) ( ) ( ) ...
2k k
U U U
RG approach to optical lattice and results
• Simplification of U:
XL Cui, YP Wang and F Zhou, Phys. Rev. Lett. 104, 153201 (2010)
U
1n k
2n -k
1n ' k'
2n ' -k'
inter-band, to renormalize short-range contribution
intra-band, specialty of OL
• 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
1. resonance scattering at E=0:
C L Sv a /a
Results
resonance scattering of Rb-K mixture
resonance at
For previous study in this limit see P.O. Fedichev et al, PRL 92, 080401 (2004).
effU on-site U
2. Validity of Hubbard model:
To safely neglect inter-band scattering,
Condition I: : deep lattice potential1 2<< CC
Under these conditions, Hubbard limit
1 L S<< a /aCCondition II: : weak interaction
Lsa =-0.5a , v=2.5
cross section , phase shift2 2
L4πa |χ|
set C1=0
In the opposite limit,
Both intra- and inter-band contribute to low-E effective scattering, where C1 can NOT be neglected!
1 2 1 L S( C or a /a )C C
as/aL
E
s-band
0
3. Symmetry between repulsive and attractive bound state:
S Lat large v and |a | a , single-band Hubbard model:
simply solvable:
• K conserved (semi-separated)
• state-independent U
Zero-energy resonance scattering
attractive and repulsive bound state
repulsive as>0
attractive -as<0
Winkler et al, Nature 441, 853 (06)
From particle-hole symmetry, ( ) (6 )E t E
K
E
scattering continuum
0
12t
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 :
0
1 1( ), ( )
( ) 4L
R I
mf E f E
Ti
E T af f
E 0 :
compare with free space (all E):
DOSIf
In Hubbard model regime , when
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!
This leads to much exotic E-dependence of T-matrix in optical lattice.
k
E
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
Conclusion
Phys. Rev. Lett. 104, 153201 (2010)
Bound state/molecule above resonance (v>vc):
a two-body bound state/molecule :
Real momentum distribution :no interband, C1=0
Smeared peak at discrete Q as v increases!!