Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

6. Bloch Oscillations and Time-Dependent Interactions
6.1.3. Time dependent interaction g(t)
Via Feshbach resonances [43–46], ultracold-atom experiments open new possibilities.
The s-wave scattering length can be tuned in a wide range, including
negative values. In particular, a complete suppression of the interaction
is possible. At zero scattering length, very long-living Bloch oscillations
can be observed [40]. However, there are always residual experimental uncertainties,
e.g. an interaction parameter g(t) that fluctuates around zero.
Thus, the question about the effect of such perturbations arises naturally.
We will consider perturbations that are commensurate with the Bloch
frequency. It will turn out that their effect on the dynamics of the Bloch oscillation
depends sensitively on their phase relative to the Bloch oscillation.
Deliberately modulating the interaction can make the Bloch oscillation more
robust against certain perturbations.
6.2. Model
Let us consider particles subjected to a lattice potential. For the description
of Bloch oscillations, the starting point is the Gross-Pitaevskii equation
i� ∂
∂t Ψ = � − �2
2m ∇2 + V (r) � Ψ(r) + g |Ψ(r)| 2 Ψ(r), (6.1)
[subsection 2.2.2, equation (2.16)]. Here, we do not include the chemical
potential because we work at fixed particle number. The setting differs
from the Bogoliubov problem of the extended condensate ground state with
the disorder imprint in part I. Here, the condensate is created in a harmonic
trap and then transferred into an optical lattice, and the trap is switched
off (figure 6.1). Obviously, this is not the ground-state configuration, and
the condensate will have the tendency to spread, both due to repulsive
interaction and due to the usual linear dispersion of matter waves.
The phenomenon of Bloch oscillation takes place in one direction. Thus,
we consider a setup in which the transverse degrees of freedom are frozen.
The transverse harmonic-oscillator ground state is integrated out, leading
to a renormalized interaction parameter in the remaining dimension
g1D = mω⊥
2π� g3D. (6.2)
Here, ω⊥ is the transverse oscillator frequency (or the geometric mean in
anisotropic configurations). The usual three-dimensional interaction pa-
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