Mott Insulator-Superfluid transition a introduction

							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