Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2.4. Single scattering event
theory for a weak potential. The trapping potential is neglected, instead,
the impurity potential is placed into a finite homogeneous system with periodic
boundary conditions. In order to investigate the interesting envelope
function (2.59) in the scattering amplitude (2.61), the potential factor V k ′ −k
should allow scattering in all directions. That means, the impurity should
be pointlike in the sense that its extension r0 should be much smaller than
the wave length kr0 ≪ 1. First, the Gross-Pitaevskii ground state is calculated
by imaginary-time propagation, using the forth-order Runge-Kutta
method [104]. Then a plane-wave Bogoliubov excitation of the unperturbed
system is superposed. During the real-time evolution, the wave propagates
and is scattered at the impurity. After some time t, the stationary scattering
state should be reached in a square with edge length L ≈ c t around the
impurity. This part of the system is Fourier transformed and compared to
the condensate ground state without the excitation. In the density plot of
the scattered wave |δΨ k ′| 2 , figure 2.7(a), one can already observe the node
of vanishing scattering amplitude (2.60) very clearly.
The data shown in figure 2.7(a) still shows the Gross-Pitaevskii wave
function. One can get more insight in terms of Bogoliubov amplitudes γ k ′ =
uk ′δΨk ′+vk ′δΨ∗
−k ′, with uk = (ɛk+ɛ0 k )/(2�ɛkɛ0 k ) and vk = (ɛk−ɛ0 k )/(2�ɛkɛ0 k ).
The corresponding plot of the imaginary part of the Bogoliubov-transformed
amplitude in figure 2.7(b) clearly shows the amplitude sign change across the
scattering node. Note that figure 2.7(b) is much clearer than figure 2.7(a),
because there Bogoliubov excitations with opposite ±k ′ interfere in the wave
function densities |δΨk ′| 2 . One may wonder why the superposition of nodes
stemming from opposite wave vectors still gives a density dip as clear as in
figure 2.7(a). In fact, in the single-particle case kξ ≫ 1, vk/uk tends to zero
such that only the node of one component is observed, whereas for sound
waves kξ ≪ 1 both components contribute equally, but now with symmetric
nodes at ± π
2
that superpose exactly. This node robustness should facilitate
the experimental observation.
We extract the scattering amplitude from the data on the elastic scattering
circle in figure 2.7(b) and plot it together with the analytical prediction
(2.61) as function of the scattering angle θ at various wave vectors kξ, as
shown in figure 2.8. The agreement is very good, with residual numerical
scatter due to transients and boundary effects.
Equation (2.61) and figure 2.8 show that the overall magnitude of scattering
peaks at kξ ≈ 1, at the crossing from waves to particles. Physically,
this results from two competing scalings in (2.56):
• The Bogoliubov scattering amplitude W (k, θ) ∝ ɛ 0 k /ɛk is proportional
to k for kξ ≪ 1 and saturates to a constant for kξ ≫ 1.
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