Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2.4. Single scattering event
back and forth. In the regime kξ ≪ 1 and ξ ≪ r0, the Thomas-Fermi
formula (2.20) applies.
Excitations of the superfluid ground state in the regime kξ ≪ 1 are longitudinal
sound waves with density fluctuations δn = n − nTF and phase
fluctuations δϕ. The phase is the potential for the local superfluid velocity
vs = (�/m)∇δϕ. The superfluid hydrodynamics is determined by the continuity
equation (2.18a) ∂tn + ∇ · (nvs) = 0 and by the Euler equation for
an ideal compressible fluid, m [∂tvs + (vs · ∇)vs] = −∇(gn + V ), derived
from (2.18b) in the present limit. To linear order in δn and δϕ, these two
equations can be combined to a single wave equation
m � c 2 ∇ 2 − ∂ 2� � �
t δn = ∇ · V (r)∇δn , (2.53)
with the sound velocity c = � µ/m from subsection 2.3.1. The gradientpotential
operator on the right-hand side then causes scattering with an
amplitude proportional to −(k · k ′ )V k ′ −k. Hence, the potential component
V k ′ −k from above, which must appear in all cases to satisfy momentum
conservation, is multiplied with a dipole (or p-wave) characteristic A(θ) =
− cos θ. This scattering cross-section with a node at θ0 = ±π/2 can be
understood, in the frame of reference where the local fluid velocity is zero, as
the dipole radiation pattern of an impurity that oscillates to and fro, quite
similar to the case of classical sound waves scattered by an impenetrable
obstacle [102].
2.4.3. Analytical prediction for arbitrary kξ
Now, the opposite characteristics of the limiting cases discussed above have
to be unified by the Bogoliubov theory that interpolates between both
regimes. Starting point is the equation of motion (2.43) for Bogoliubov
excitations in the mean-field framework. Transforming from time to frequency
domain, equation (2.43) can be written as
γ k ′(ω) = G0(k ′ , ω) 1
L d
2
�
k ′′
� Wk ′ k ′′γ k ′′(ω) + Y k ′ k ′′γ ∗
−k ′′(−ω)� , (2.54)
where G0(k ′ , ω) = [�ω − ɛk ′ + i0]−1 designates the retarded Green function
of the homogeneous condensate, the infinitesimal shift +i0 being due to
is separated into the ini-
causality. The Bogoliubov field γ ′
k = γ (0)
k ′ + γ (s)
k ′
tially imprinted wave γ (0)
k ′ (ω) = γ0(ω) δ ′
k k with γ0(ω) ∝ δ(�ω − ɛk) and the
scattered wave γ (s)
k ′ , which is of order V . The stationary scattering state is
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