Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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5.3. Theoretical outlook<br />
potential. The imprint and the detection <strong>of</strong> the <strong>Bogoliubov</strong> waves follows<br />
Vogels et al. [61], using Bragg spectroscopy techniques. The crucial point is<br />
the detection <strong>of</strong> the scattered <strong>Bogoliubov</strong> wave, because it will be one or two<br />
orders <strong>of</strong> magnitudes smaller than the imprinted plane wave. In addition,<br />
the detection has to take place with angular sensitivity.<br />
Despite these challenges, the continuous transition from a p-wave scattering<br />
amplitude in the sound-wave regime to an s-wave scattering amplitude<br />
in the particle regime promises to be an interesting object <strong>of</strong> experimental<br />
study.<br />
Measurement <strong>of</strong> the speed <strong>of</strong> sound in disordered BEC<br />
The predicted renormalization <strong>of</strong> the speed <strong>of</strong> sound due to disorder, as discussed<br />
in chapter 4, can in principle be measured experimentally. However,<br />
in the perturbative regime <strong>of</strong> the predictions, the corrections scale quadratically<br />
with the disorder strength, and consequently they are very small. In<br />
a direct observation in real space, as in the experiment [57], the difference<br />
in the propagation speed should be hardly separable from the noise in the<br />
data. A more sophisticated ansatz is measuring the static structure factor<br />
by Bragg spectroscopy and to extract the dispersion relation, similar to the<br />
experiments [59, 62], but with a speckle disorder potential superposed to<br />
the trapping potential. In contrast to the first ansatz, it is possible to excite<br />
certain k values selectively, which allows also probing the transition from<br />
sound-like to particle-like excitations.<br />
5.3. Theoretical outlook<br />
With the inhomogeneous <strong>Bogoliubov</strong> Hamiltonian (section 2.3) and the<br />
block matrix notation (subsection 3.3.2) at hand, many other questions can<br />
be approached.<br />
Localization<br />
It should be particularly interesting to characterize the localization <strong>of</strong> <strong>Bogoliubov</strong><br />
excitations across the transition from sound waves to particles. In<br />
three dimensions, phonon and particle regime have opposite characteristics<br />
concerning their localization [29]: Delocalized low-energy phonon states are<br />
separated by a mobility edge ω ∗ from localized high-energy states. Inversely,<br />
electrons are localized at the lower band edge, separated from extended<br />
states at the mobility edge E ∗ . The upper band edge does not apply in the<br />
case <strong>of</strong> bosons in a continuous system. Phonon regime and particle regime<br />
111