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

64 Stationary states of a BEC in an optical lattice
Exploiting the tight binding formalism, we also derive simple expressions for the compressibility
of the groundstate and the effective coupling constant introduced in chapter 5. Explicit
formulars for the groundstate energy, chemical potential and compressibility in a very deep
lattice are obtained based on a gaussian ansatz for the Wannier function of the lowest band.
Using this ansatz, we also estimate the dependence on lattice depth of the peak density at the
center of a harmonic trap added to the optical lattice.
In [102], we have reported our numerical results for the Bloch band spectra, the group
velocity, the effective mass and their analytical tight binding expressions, as well as the tight
binding expressions for the groundstate compressibility. In this thesis, results for the Bloch
state density profiles and the gap between first and second energy and chemical potential
Bloch band are included. Also, the proof of the relation between the current of a Bloch state
and the energy Bloch bands is added, as well as the discussion of the peak density and the onsite
energy and chemical potential within the gaussian approximation to the Wannier function
of the lowest Bloch band.
6.1 Bloch states and Bloch bands
As discussed in chapter (4), the stationary state of a single particle in a periodic potential
is described by a Bloch function (see Eq.(4.3)). Within GP-theory, the difference between a
single particle and an interacting condensate is accounted for by the nonlinear term in the GPE
(5.4). Now, suppose we are dealing with a solution whose density has period d. Under this
condition, the GPE is invariant under any transformation z → z + ld and we can use the same
arguments as in section (4.1) to show that the respective solution has the form of a Bloch
function
ϕjk(z) =e ikz ˜ϕjk(z) , (6.2)
with the periodic Bloch wave ˜ϕjk(z) = ˜ϕjk(z + ld). Stationary states can be of this kind, but,
due to the presence of the nonlinear term, they do not have to: Other classes of solutions are
associated with density profiles of periodicity 2d, 4d, .... Such solutions have recently been
found in [104, 105]. The groundstate discussed above in chapter 5 is of the form (6.2) with
j =1,k=0. In the following we explore condensates in Bloch states (6.2).
It is convenient to solve the GPE (5.4) for the periodic Bloch waves ˜ϕjk
1
2m (−i¯h∂z +¯hk) 2 + sER sin 2
πz
+ gnd| ˜ϕjk(z)|
d
2
˜ϕjk(z) =µj(k)˜ϕjk(z). (6.3)
From the solution of Eq.(6.3) one gets the functions ˜ϕjk(z) and the corresponding chemical
potentials µj(k). This section is devoted to such solutions.
Density profile
Let us first discuss some solutions for the density |ϕjk(z)| 2 . In Figs. 6.1 and 6.2, we report
results obtained at s =5,gn =0.5ER and s =10,gn =0.5ER respectively for different
bands and quasi-momenta (j =1, 2, 3, ¯hk =0, 0.5qB, 1qB). Density profiles of higher bands
are more modulated and tend to allow for larger particle densities in high potential regions.