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

bound state for a general K:

K

E

scattering continuum

0

12t

B

T-matrix and 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!!

Bound state: T(EB)=infty (Bethe-Salpeter eq)

K

E

0

12t

repulsive as>0

attractive -as<0

Repulsive

metastable excited

above band top

nq peaked at q=±pi

Attractive

ground state

below band bottom

nq peaked at q=0

K=0 bound state

assume a 2-body wf: