Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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2.3. <strong>Bogoliubov</strong> <strong>Excitations</strong><br />
At high energies, the coefficient ak tends to one and the components<br />
<strong>of</strong> the wave parallel and perpendicular to the order parameter are equal.<br />
Then, the <strong>Bogoliubov</strong> wave resembles the plane wave <strong>of</strong> a free particle δΨ ∝<br />
exp � i(k0 · r − ɛk0 t/�)� (inset in figure 2.3).<br />
At low energies, the spectrum (2.35) changes from quadratic to linear, but<br />
it is still gapless: at low momenta, excitations with arbitrarily low energy<br />
exist. The coefficient ak tends to zero, which means, that the <strong>Bogoliubov</strong><br />
excitation γk has hardly any signature in the density, but all the more in<br />
the phase [see inset in figure 2.3]. In the limit k → 0, the <strong>Bogoliubov</strong><br />
excitations pass over to a homogeneous shift <strong>of</strong> the condensate phase. This<br />
connects the <strong>Bogoliubov</strong> excitations to the broken U(1) symmetry <strong>of</strong> <strong>Bose</strong>-<br />
<strong>Einstein</strong> condensate. The <strong>Bogoliubov</strong> excitations are the Goldstone modes<br />
[72] associated to this broken symmetry.<br />
2.3.2. The inhomogeneous <strong>Bogoliubov</strong> problem<br />
Let us come back to the Hamiltonian (2.29) and include the external potential<br />
V (r) and its imprint <strong>of</strong> the external potential in the condensate density<br />
n0(r). This inhomogeneity breaks translation invariance, which allowed the<br />
diagonalization by the <strong>Bogoliubov</strong> transformation (2.33) in the free problem,<br />
with two consequences: Firstly, there will be scattering among the Bogoli-<br />
ubov modes, and secondly, the anomalous terms ˆγ †<br />
kˆγ† −k ′ and their Hermitian<br />
conjugates do not vanish, which leads to anomalous coupling.<br />
As the Hamiltonian (2.29) is quadratic in the fluctuations without any<br />
term mixing δˆn and δ ˆϕ, the general structure <strong>of</strong> the disorder part F (V ) =<br />
F − F (0) is<br />
ˆF (V ) [δˆn, δ ˆϕ] = 1<br />
2L d<br />
2<br />
�<br />
�<br />
n∞δ ˆϕ †<br />
k ′S ′<br />
k kδ ˆϕk + δˆn†<br />
k,k ′<br />
k ′R k ′ kδˆnk<br />
4n∞<br />
�<br />
. (2.38)<br />
The density and phase coupling matrices S and R vanish in absence <strong>of</strong><br />
disorder. Comparing with (2.29), we find that both the phase couplingelement<br />
and the density coupling element are related to the imprint <strong>of</strong> the<br />
external potential in the condensate wave-function � n0(r) = Φ(r).<br />
Non-perturbative couplings<br />
The phase coupling is proportional to the condensate density n0(r) = Φ(r) 2<br />
and can be written as<br />
� �2 m k′ ·k Φ ′<br />
k −qΦq−k. (2.39)<br />
S ′<br />
k k = 1<br />
L d<br />
2<br />
q<br />
33