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