Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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