Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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1. Interacting <strong>Bose</strong> Gases<br />
1.6. Standing <strong>of</strong> this work<br />
This thesis consists <strong>of</strong> two main parts. Both <strong>of</strong> them try to shed some light<br />
on open questions in the field defined by the corner stones lattice dynamics,<br />
disorder, and interactions.<br />
Part I: The disordered <strong>Bogoliubov</strong> problem<br />
In the first part, the <strong>Bogoliubov</strong> excitations [71] <strong>of</strong> a disordered <strong>Bose</strong>-<br />
<strong>Einstein</strong> condensate are studied. These excitations are essential for the<br />
properties <strong>of</strong> the <strong>Bose</strong> gas. They determine, for example, the critical velocity<br />
<strong>of</strong> superfluidity and thermodynamical properties like the heat capacity.<br />
<strong>Bogoliubov</strong> excitations are the Goldstone modes [72] associated to the U(1)<br />
symmetry breaking <strong>of</strong> the BEC phase transition. They are intimately connected<br />
to that phase transition and have a great importance for the phase<br />
diagram <strong>of</strong> disordered <strong>Bose</strong> gases [73–75]. The question about the impact<br />
<strong>of</strong> disorder on the properties and the phase diagram <strong>of</strong> interacting <strong>Bose</strong><br />
gases can thus be phrased as “How does the disorder potential influence the<br />
elementary excitations <strong>of</strong> the system?”<br />
After the experiment-oriented point <strong>of</strong> view in the previous section, let<br />
us now have a look at fundamental theoretical work. The concepts <strong>of</strong> <strong>Bose</strong>-<br />
<strong>Einstein</strong> statistics were derived in the nineteen-twenties by <strong>Bose</strong> [47] and<br />
<strong>Einstein</strong> [48], including the prediction <strong>of</strong> <strong>Bose</strong>-<strong>Einstein</strong> condensation. A<br />
milestone in the study <strong>of</strong> interacting bosons was <strong>Bogoliubov</strong>’s approach<br />
[71], where the classical treatment <strong>of</strong> the condensate mode leads to the<br />
concept <strong>of</strong> quasiparticles, which interpolate between collective low-energy<br />
excitations and free-particle excitations at high energies (subsection 2.3.1).<br />
More detailed studies <strong>of</strong> the interacting <strong>Bose</strong> gas followed and took into<br />
account the depletion <strong>of</strong> the condensate mode due to interactions [76].<br />
What is known about disordered <strong>Bogoliubov</strong> excitations, what is not?<br />
The works mentioned above aimed mainly on the bulk properties <strong>of</strong> superfluid<br />
helium. Some time later, disordered interacting <strong>Bose</strong> gases came into<br />
focus. Much <strong>of</strong> our present knowledge on disordered BEC traces back to<br />
Huang and Meng [77] and Giorgini, Pitaevskii and Stringari [78]. In both<br />
works, uncorrelated disorder in three dimensions was considered and quantities<br />
like the superfluid fraction and the depletion <strong>of</strong> the zero-momentum<br />
mode due to disorder and interaction were calculated.<br />
Many other different aspects <strong>of</strong> disordered interacting bosons were studied,<br />
but the picture is still far from complete in the details. Many approaches<br />
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