BEC in an Optical Potential

Optical Potential

IcgEe

e eLgg

3

0

22

2

3

Second order perturbation theory:

Ic

2

30

2

2

3'

Scattering rate:

EH int

Many-Body Hamiltonian

rrrrr

rrGrr

ˆˆˆˆ2

ˆ1cos2

ˆ22

dg

md

Negligible Interactions

)()(),( ruRrurue nknknkikr

nk

G

iGrGeVrV )(

0,2VVV GG

Bloch Theorem: periodic potential

1',',

2

' 2/2

2 llllRll ssG

klEH

xklGilnk ecr )(

m

GER 2

2 22 REs /

'

''l

llll EccH

Limiting Cases

E

kk ,1

2/2/2/,1 GGG

Weak Lattice: Bandgap

Wave functions:

Near boundary:

Tight Binding: Harmonic potentialrr EE

E

m

G

8

22

Wannier functions )(1

)( ,

..0

redkv

Rrf kniRk

ZB

n Eigenstates

Simulations

5

0

Rabi Oscillationsk,ω1 -k,ω2

m

k

k

E

kv R

2

/4

2

OR: Lattice fixed, and quasimomentum of BEC is non zero

Adiabatic Versus Sudden Release

Adiabatic:

2

0,,E

kt

Hnk

Bloch Oscillations & Landau Zener Tunneling

1.2 ms

Weak Interactions - GPE

Can solve and find:2

2

2*

11

km

nn

1

If interactions are weak:

nsg~1

dzgdgd

d

gn

2/

2/

04~

GPE

Bloch Wave with no linear counterpart:

c

vcV x 2

Im22

*

ixix eaaek 2/

2

1

Bloch waves are not only solutions!

Landau Instability

In lack of lattice:

Cqk 4/0 2

Dynamical Instability

If v=0

Dark: Dynamical Instability

Light: Landau Instability

Triangles: V=0

Circles :