# Mott Insulator-Superfluid transition a introduction by hmb46803

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```									Mott Insulator-Superfluid transition:
a introduction

Tzu-Chieh Wei
University of British Columbia

Collaborators:
Brian DeMarco (UIUC)
Smitha Vishveshwara (UIUC)
Courtney Lannert (Wellesley)
Dec. 22, 2009
Outline
Motivations

Optical lattice and Bose-Hubbard model

Mott state and Superfluid states

Mott-SF Transition

Probing wedding-cake structure

Conclusion
Observation of Mott-SF transition
[Greiner et al. ‘02]

V0 ↑: w ↓, U ↑
Bose-Hubbard model

<i,j> nearest neighbor on lattice, created by standing
laser fields

Lattice spacing = λ/2
Bose-Hubbard model

Hopping from wavefunction overlap

Interaction U
Mott States: w=0
No hopping: w=0

Ground state has fixed number per site

Ground state has n=k
and energy gap

k-1   k   k+1           n
Phase diagram near w=0
Mott state is incompressible and has a gap

Ground state has n=k
and energy gap
except at µ/U= integer
k-1   k   k+1     n
Incompressible

 Expected
phase Diagram
near w=0
Mean-field treatment

Expect emergence of SF state Ginzburg-Landau
free energy for order parameter ψ

Transition occurs at r=0   phase boundary

Approach: decouple hopping term

Self-consistency:                 Z: coordination number
Mean-field Hamiltonian

Idea: Use mean-field ground state to evaluate HB and
use Ψ as variational parameter

Per site:

Z: coordination number
Next, evaluate

using perturbations
Mean-field Hamiltonian per site

denote as
Suppose near Mott n0
Treat the first term as perturbation

Energy:
Mean-field Hamiltonian per site

State:                     denote as
Mean-field energy

Perturbed state:

<b> and <b+>:

r
Thus,
Mott-SF transition

r
r=0
SF is compressible
Perturbed state:

Average density:

Mott: incompressible, as

SF: compressible, as
Observation of Mott-SF transition
[Greiner et al. ‘02]

V0 ↑: w ↓, U ↑
Optical Lattice Configuration

e.g. 87Rb atoms
N ~ 106
T ~ 1 nK

magnetic confining trap                       lasers

a = 425 nm
For V0 = 30 ER,
U
e.g. ER ~ 150 nK                                           V0
w ~ 0.5 nK , U ~ 100 nK

t
Wedding-cake structure

Overall trapping potential

r increases
Effective chemical potential
varies with space

Layers of Mott states
e.g.
Probing wedding-cake structure –
microwave spectroscopy
Utilize transition between two states a & b
via oscillating (microscopic) B field
U bb                U aa
H = ℏωab +       nb ( nb − 1) +      na ( na − 1) + U ab na nb − ∑ µ ( i ) ⋅ B
2                   2                               i

− ∑ µ ( i ) ⋅ B = ℏΩ  eiω ( t ) t + e − iω ( t ) t  ∑ i ( ai bi† + bi ai† )
                              
i

Energy spectrum            U ab = U aa ,
U bb = β U aa
Spectroscopic method for probing
wedding-cake structure
[DeMarco, Lannert, Vishveshwara & Wei ‘05]

Spectrum for transitions and            Rapid Adiabatic Passage simulations
sweep frequencies
Experiments: Bloch’s group
[Bloch group ‘06]
Experiments: Ketterle’s group
[Ketterle group ‘05]
Prof. Chin’s experiment

see his talk
Conclusion
Introduce Bose-Hubbard model and construct a
mean-field phase boundary

Propose a spectroscopic scheme to probe spatial
dependence of Mott state

A tour de force probe: see Prof. Cheng Chin’s talk

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