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

and spectroscopy were described in [79]. In [72, 79, 80] effects related to the non-adiabatic
loading of the sample into the lattice were explored. The motion of the lattice was used
to do dispersion management of matter wave packets [81]. It was found to have a lensing
effect on the condensate [82] and was applied recently to generate bright solitons [83, 84].
Finite temperature effects were studied in [85, 86]. In particular, a change of the critical
temperature of Bose-Einstein condensation was observed reflecting the two-dimensional nature
of the cloud in each well of a deep potential [85], while in [86] the temperature-dependent
transport properties of the system were demonstrated. The phase coherence of a condensate
loaded into a two-dimensional lattice was investigated in [87]. Recently, a two-dimensional
lattice was used successfully to prepare a one-dimensional Bose gas [88].
Remarkable progress has been made also in the study of regimes where the GP-description
breaks down: A first advance in this direction was the observation of number squeezing in a
superfluid ultracold atomic gas in a one-dimensional lattice [89]. Further on, the superfluidinsulator
transition of cold atoms in a three-dimensional lattice was observed [90, 91]. The
transition to the insulating phase has been used to demonstrate the collapse and revival of the
matter wave field of a Bose-Einstein condensate [92]. Atoms in the insulating phase promise
to be a precious resource for quantum computing: Their spin-dependent coherent transport
between lattice sites and the controlled creation of entanglement have already been achieved
[93, 94, 95].
This thesis deals with repulsively interacting three-dimensional dilute-gas Bose-Einstein
condensates in optical lattices at zero temperature. We concentrate on the range of lattice
depths where GP-theory is valid, i.e. the gas is almost completely condensed and exhibits full
coherence. We deal with one-dimensional lattices in the first place. The generalization of many
of the results to cubic two-dimensional lattices is straightforward and will be commented on. As
a general strategy, we first exclude harmonic trapping from our considerations. Since typically,
the particles are distributed over many lattice sites in the presence of harmonic trapping, its
effects can be included in a second step, as will be shown in detail. In so far as we neglect
harmonic trapping effects and concentrate on one-dimensional optical lattices, we study the
properties of condensates as described by the Gross-Pitaevskii equation
i¯h ∂Ψ(z,t)
∂t
= − ¯h2 ∂
2m
2
∂z2 + sER sin 2
πz
+ g |Ψ(z,t)|
d
2
Ψ(z,t) , (3.14)
where the order parameter Ψ fulfills the normalization condition
dr |Ψ(z,t)| 2 = Ntot , (3.15)
with Ntot the total number of particles. Because we assume the system to be confined in a
box of length L along x, y and we exclude dynamics involving these transverse directions, the
order parameter Ψ depends only on z. The dependence of Ψ on x, y comes into play only once
we allow for the effects of harmonic trapping.
The linear response of the system and the depletion of the condensate are treated within
Bogoliubov theory (see [1]). In the latter case, also elementary excitations in the transverse
directions have to be taken into account.
Particular attention is paid to the analogies and differences with respect to the single
particle case and with respect to the case of a uniform or harmonically trapped condensate.
29