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 />
Schrödinger equation, which describes, for example, wave packets <strong>of</strong> water<br />
waves [9] and self-focusing laser pulses in nonlinear optical media [10].<br />
After Anderson’s discovery <strong>of</strong> localization [1], disordered systems <strong>of</strong> electrons<br />
and bosons have been studied for decades. A major difficulty comes<br />
from the interplay <strong>of</strong> disorder with interactions. At first, electronic systems<br />
were <strong>of</strong> interest [11], but with the experimental research on superfluid<br />
Helium, also bosons came into focus. Repulsive interaction among bosons<br />
prevents the condensation into the localized single-particle ground state and<br />
keeps the gas extended. Early works on the so-called dirty boson problem<br />
used renormalization techniques in one dimension [12, 13]. Others studied<br />
bosons on disordered lattices, the disordered <strong>Bose</strong>-Hubbard model [2, 14, 15],<br />
where a random on-site potential models disorder.<br />
Because <strong>of</strong> to the complexity <strong>of</strong> the problem, there are still many open<br />
questions. There is a vast parameter space to cover: lattice vs. continuous<br />
systems, uncorrelated vs. correlated disorder, and the dimension <strong>of</strong> the<br />
system. The theoretical interest is kept alive by experimental progress in<br />
both lattice systems and continuous systems <strong>of</strong> ultracold atoms, where interaction,<br />
artificial disorder and the effective dimension can be controlled<br />
practically at will.<br />
In the following, we consider the basic ingredients lattice, disorder, interaction<br />
and cold-atom experiments in some more detail.<br />
1.1. Lattices<br />
Physics in lattice potentials is very important for our understanding <strong>of</strong><br />
solids. A typical question is, for example, why certain materials are electrical<br />
conductors, while others are insulators.<br />
Many solids, like metals, ice or graphite, have a crystal structure, i.e. the<br />
atoms or molecules <strong>of</strong> the material are arranged on a lattice with perfect<br />
periodicity. The electrons experience this lattice as a periodic potential.<br />
Even weak lattice potentials have dramatic effects when the de Broglie wave<br />
length <strong>of</strong> the particle comes close to the lattice period. In momentum space,<br />
this point marks the edge <strong>of</strong> the Brillouin zone, where a band gap occurs<br />
(section 6.1). If the lattice potential is strong, the system is efficiently<br />
described with a tight-binding ansatz, i.e. a single-band description.<br />
2