Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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2. The <strong>Inhomogeneous</strong> <strong>Bogoliubov</strong> Hamiltonian<br />
this process, a particular <strong>Bogoliubov</strong> mode is excited very strongly 2 , which<br />
allows treating the <strong>Bogoliubov</strong> excitations in the mean-field manner <strong>of</strong> subsection<br />
2.3.4. Equivalently, the time-dependent Gross-Pitaevskii equation<br />
(2.16) can be integrated numerically.<br />
During its propagation, the wave is scattered at the impurity and the scattered<br />
wave is detected at scattering angle θ. For the experimental detection,<br />
again a Bragg-spectroscopy technique should be used in order to separate<br />
the excitation from the condensate background making it observable in a<br />
time-<strong>of</strong>-flight image. This technique has already been used experimentally<br />
for the excitation and detection <strong>of</strong> <strong>Bogoliubov</strong> excitations by Vogels et al.<br />
[61], following a proposal by Brunello et al. [60]. The only thing to change<br />
is to detect the scattered wave at an angle θ with respect to the imprinted<br />
wave.<br />
2.4.2. Limiting cases <strong>of</strong> the elastic scattering amplitude<br />
The <strong>Bogoliubov</strong> dispersion relation (2.35) interpolates between the soundwave<br />
regime at low energies and the particle regime at high energies. Before<br />
considering the scattering amplitude in the entire energy range, it is instructive<br />
to consider the limiting cases from a separate point <strong>of</strong> view.<br />
Scattering <strong>of</strong> particles<br />
In the single-particle regime <strong>of</strong> the <strong>Bogoliubov</strong> spectrum, excitations are<br />
plane matter waves with dispersion relation ɛ 0 k = �2 k 2 /2m. From the momentum<br />
representation <strong>of</strong> the Schrödinger equation it follows that the am-<br />
plitude <strong>of</strong> a single-scattering process k ↦→ k ′ = k + q is proportional to the<br />
d �<br />
− Fourier component Vq = L 2 d −iq·r d rV (r)e <strong>of</strong> the scattering potential. If<br />
the potential V (r) varies on a characteristic length r0, the scattering may<br />
be anisotropic if the wave can resolve this structure, qr0 ≥ 1 [22, 23]. In<br />
the opposite case <strong>of</strong> a small obstacle such that qr0 ≪ 1, also known as the<br />
s-wave scattering regime, the scattering amplitude is simply proportional to<br />
V0 and can therefore only be isotropic.<br />
Scattering <strong>of</strong> sound waves<br />
In the other limit, the characteristics <strong>of</strong> sound waves are very different.<br />
Similar to classical sound waves in air, many particles oscillate collectively<br />
2 This technique was employed in an experiment [61], where the number <strong>of</strong> 40 000 excited atoms corresponds<br />
to about 20 000 <strong>Bogoliubov</strong> quasiparticles at kξ = p/( √ 2mc) ≈ 0.27.<br />
40