Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

The order parameter’s temporal evolution in the external potential V (r,t) obeys the Gross-
Pitaevskii equation (GPE)
i¯h ∂Ψ(r,t)
∂t
27
= − ¯h2
2m ∇2 + V (r,t)+g |Ψ(r,t)| 2
Ψ(r,t) . (3.10)
Two-body interaction between atoms is accounted for by the nonlinear term which is governed
by the coupling constant
g = 4π¯h2 a
, (3.11)
m
where a is the s-wave scattering length. Throughout this thesis, we will focus on repulsively
interacting atoms (a >0). The criterion for the diluteness of the gas reads
na 3 ≪ 1 , (3.12)
with n the density.
With the external potential V (r,t) given by an optical lattice, the GP-equation differs
from the Schrödinger equation of a particle in a crystal structure by the nonlinear mean field
term, opening up the possibility to explore analogies and differences with respect to solid state
physics.
A dilute-gas condensate in a lattice at very low temperatures is well described by GP-theory
only if the potential is not too deep. An increase of the lattice depth is in fact accompanied
by a drop of the condensate fraction and a loss of coherence. This is due to the enhanced role
played by correlations between the particles. The gas can even loose its superfluid properties:
At a critical value of the lattice depth at zero temperature the gas undergoes a quantum phase
transition to an insulating phase. Complete insulation is achieved provided the number of
particles is a multiple of the number of sites.
The different physical regimes experienced by a cold atomic gas in an optical lattice can
be described using a Bose-Hubbard Hamiltonian (for a review see [65]). This Hamiltonian is
obtained by expanding the atomic field operators of the many-body Hamiltonian in the single
particle Wannier basis. Terms due to higher bands are omitted. From the lowest band, only
on-site and nearest-neighbour contributions are retained. In this framework, the state of the
system is expressed in the Fock basis {|N1,...,Nl,...〉} where l labels the Wannier functions,
or equivalently, the lattice sites and the numbers Nl give the number of atoms at site l. With
ˆ† b l as the creation operator for an atom at site l and ˆnl the associated number operator, the
Bose-Hubbard Hamiltonian reads
ˆH = −δ
ˆnl(ˆnl − 1) . (3.13)
l,l ′ =l±1
ˆ† b l ˆ U
bl ′ +
2
The parameters U and δ govern the on-site interaction and the tunneling of particles to neighbouring
sites respectively. They are associated with two competing tendencies of the system:
On one side, the atoms try to reduce their interaction energy by localizing at different lattice
sites thereby reducing occupation number fluctuations. On the other side, they tend to spread
over many sites in order to minimize the kinetic energy. The physical characteristics of the zero
temperature groundstate depends on the ratio U/δ between tunneling and on-site interaction:
For U/δ ≪ 1, the particles are delocalized over all sites. In this case, all particles occupy the
l