Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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1.4. Cold-atoms—Universal model systems<br />
Even without disorder, the <strong>Bose</strong>-Hubbard model <strong>of</strong>fers rich physics and its<br />
phase diagram is still a subject <strong>of</strong> very active research [34].<br />
In part II <strong>of</strong> this work, an interacting bosonic system will be studied in<br />
the mean-field version <strong>of</strong> the <strong>Bose</strong>-Hubbard model, which is known as the<br />
discrete Gross-Pitaevskii equation.<br />
1.4. Cold-atoms—Universal model systems<br />
In experiments with ultracold atoms, lattice potentials, disorder and interactions<br />
are brought together. Ultracold atoms are a very exciting topic for<br />
their own sake, but they are also very useful to model problems from different<br />
fields <strong>of</strong> physics [35–37]. In solid-state physics it is difficult to access<br />
and to manipulate system parameters directly. Let us, for example, consider<br />
electrons in a metal. There is no way <strong>of</strong> changing the lattice spacing, the<br />
lattice strength and the Coulomb interaction. In addition, the experimental<br />
access is limited.<br />
Thanks to the experimental progress in the past years, cold atoms in<br />
magnetic and optical traps have become very well controllable. They can<br />
be used to model solid-state systems, with the system parameters selectively<br />
tunable and a more flexible access for measurements.<br />
Tailoring potentials<br />
By virtue <strong>of</strong> the Zeeman effect, atomic energy levels are shifted by magnetic<br />
fields, depending on the magnetic quantum number. The Zeeman shift is<br />
proportional to the magnitude <strong>of</strong> the magnetic field. This allows trapping<br />
a spin-polarized gas in suitable magnetic configurations [38].<br />
The trapped atoms can then be manipulated by optical means. If the field<br />
<strong>of</strong> a laser couples to an internal transition <strong>of</strong> the atoms, it induces a light<br />
shift <strong>of</strong> the energy levels [39]. This results in a potential proportional to the<br />
intensity <strong>of</strong> the laser field, and allows the potential to be controlled on length<br />
scales <strong>of</strong> the laser wave length. Optical lattice potentials can be realized<br />
with two counter-propagating laser beams. This allows producing optical<br />
crystals that are much cleaner than real crystals. The phenomenon <strong>of</strong> Bloch<br />
oscillations in tilted lattices, for example, is so sensitive to dephasing that<br />
it cannot be observed in real solids. In ultracold atoms in optical lattices,<br />
however, Bloch oscillations can be observed [40, 41]. In highly anisotropic<br />
traps, the dynamics in certain directions can be completely frozen. This<br />
allows effectively one-dimensional and two-dimensional experiments to be<br />
realized [42].<br />
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