Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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6.1. Introduction<br />
Figure 6.3: Sketch <strong>of</strong> the intensity <strong>of</strong> two counterpropagating<br />
laser beams, which creates the lattice<br />
potential for the ultracold atoms. Taken from<br />
[37].<br />
wave packet oscillates in space. This phenomenon is known as Bloch oscillation<br />
and was predicted quite some time ago by Bloch and Zener [131, 133].<br />
Notably, the edge <strong>of</strong> the Brillouin zone π/d coincides with the momentum,<br />
where the de Broglie wave length equals the lattice period. Simply<br />
speaking, Bragg reflection <strong>of</strong> the particle leads to the oscillating motion. In<br />
subsection 6.2.2, we will consider Bloch oscillations in a more rigorous way<br />
than in this introductory paragraph.<br />
6.1.2. Experimental realization<br />
Bloch oscillations rely on the coherent reflection <strong>of</strong> waves and are very sensitive<br />
to any kind <strong>of</strong> dephasing, as interaction effects or impurities in the<br />
lattice. In solid-state systems, the lattice spacing d is given by atomic distances,<br />
which is so short that the electrons suffer from scattering events long<br />
before they reach the edge <strong>of</strong> the Brillouin zone π/d. For the experimental<br />
observation <strong>of</strong> Bloch oscillations, it was necessary to increase artificially the<br />
lattice spacing, which was achieved in semiconductor superlattices [134, 135].<br />
Later, Bloch oscillations were observed in cold atomic gases in optical<br />
lattice potentials [7, 8]. In cold-atom experiments, the atoms are trapped<br />
in the optical dipole potential (subsection 3.1.3, or [39]) <strong>of</strong> a laser standing<br />
wave, figure 6.3. The strength <strong>of</strong> the lattice can be adjusted via the laser<br />
intensity and detuning, and the lattice spacing can be selected via the laser<br />
wave length and the relative angle <strong>of</strong> the beams [136].<br />
There is also an optical analog <strong>of</strong> Bloch oscillations. In the experiments<br />
[137, 138], an array <strong>of</strong> weakly coupled wave guides was prepared with an<br />
index <strong>of</strong> refraction that increased linearly across the array. Then, a light<br />
beam propagating along the wave-guide array oscillates along the transverse<br />
direction. The longitudinal direction <strong>of</strong> the array takes the role <strong>of</strong> the time<br />
axis c t.<br />
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