Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2. The Inhomogeneous Bogoliubov Hamiltonian
for the Bogoliubov states. This condition on the eigenstates takes the place
of the orthogonality in the Schrödinger problem. For Bogoliubov modes
with different frequencies ων �= ωµ, the integral in equation (2.74) has to
vanish. For degenerate modes, the choice of the eigenbasis is not unique,
but can be chosen such that the integral vanishes for different members of
the same eigenspace as well. Together with a proper normalization (needed
for µ = ν), equation (2.74) takes the form
�
d d r [u ∗ ν(r)uµ(r) − v ∗ ν(r)vµ(r)] = δµ,ν
(2.75)
for all modes except for the zero-frequency mode. At this point, we forestall
that the normalization constant is +1. Below, we will show that this
corresponds to the bosonic character of the ˆ βν.
The zero-mode from subsection 2.5.3 cannot be normalized. However,
condition (2.74) still holds, thus all Bogoliubov modes with finite frequency
fulfill
�
d d r Φ ∗ (r) [uν(r) − vν(r)] = 0 , (2.76)
the orthogonality condition with respect to the ground state.
Mean-field total particle number
In the mean-field interpretation of Bogoliubov excitations (subsection 2.3.4),
the Hilbert space representing the excitations has doubled from one complex
function δΨ, to a pseudo-spinor containing u and v. When expanding a
given initial state in the Bogoliubov mean-field framework (subsection 2.3.4)
into Bogoliubov eigenmodes, one has to determine the coefficients αν in
Ψ = Φ + �
ν [ανuν(r) − α∗ νv∗ ν(r)]. If the ground state has no superfluid flow
(i.e. no vortices, no 1D flow like e.g. in [110]), then the ground-state wave
function and the Bogoliubov modes uν(r) and vν(r) can be chosen real. The
deviation from the ground state then reads
δΨ(r, t) = uν(r)e −iωνt − vν(r)e iωνt
= � uν(r) − vν(r) � cos(ωνt) − i � uν(r) + vν(r) � sin(ωνt). (2.77)
The real part shifts the local condensate density by δn(r) = 2Φ(r)ReδΨ(r).
The orthogonality condition (2.76) with respect to the ground state has a
simple interpretation: it simply states that the total particle number is not
changed by the excitations. This is reasonable, because the Gross-Pitaevskii
ground state is computed with the particle number given by the initial state
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