Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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6. Bloch Oscillations and Time-Dependent Interactions<br />
Figure 6.2: Sketch <strong>of</strong> the dispersion relation<br />
in a lattice. The dashed line shows<br />
the free dispersion ɛ 0 k = �2 k 2 /(2m),<br />
folded back into the first Brillouin<br />
zone. The lattice with period d couples<br />
the degenerate states k = +π/d<br />
and k = −π/d at the edges <strong>of</strong> the Brillouin<br />
zone, which leads to band gaps<br />
(avoided crossings). In a deep lattice<br />
potential, the band gap becomes large<br />
and excitations from the lowest band<br />
to higher bands can be neglected at low<br />
temperatures. − k<br />
π<br />
ǫ<br />
d<br />
n = 3<br />
n = 2<br />
n = 1<br />
on the particles. Even if the lattice is weak, in the sense that each site is<br />
only a weak scatterer, the coherent superposition <strong>of</strong> waves reflected at the<br />
periodic lattice can lead to total reflection if the scattered waves interfere<br />
constructively.<br />
6.1.1. Bloch oscillation <strong>of</strong> a single particle<br />
The lattice is symmetric under discrete translations by integer multiples <strong>of</strong><br />
the lattice vectors. As a consequence, the Bloch theorem [131] states that<br />
every energy eigenstate ψ(r) can be written as the product <strong>of</strong> a plane wave<br />
eik·r and a lattice-periodic function u (n)<br />
k (r). The energy states are organized<br />
in energy bands labeled with the index n (figure 6.2).<br />
If the lattice potential is deep, the gap separating the lowest band from<br />
higher bands becomes large, such that tunneling to higher bands (Landau-<br />
Zener tunneling [132]) can be neglected and a description restricted to the<br />
lowest band is sufficient. The dispersion relation ɛ(k) is proportional to<br />
− cos(kd) (subsection 6.2.1).<br />
The basic concept <strong>of</strong> Bloch oscillations can be understood within a semiclassical<br />
reasoning. Let us consider a wave packet with a narrow momentum<br />
distribution in a lattice. If the wave packet is accelerated by a<br />
uniform force −F (like gravity or an electric field for charged particles),<br />
then its momentum �k = −F t will increase linearly. In a lattice, however,<br />
momentum and velocity do not coincide. Indeed, the velocity is<br />
given by the group velocity, i.e. the first derivative <strong>of</strong> the dispersion relation<br />
vg = ∂kɛ(k) ∝ − sin(kd) = sin(F t). Instead <strong>of</strong> accelerating uniformly, the<br />
116<br />
π<br />
d