Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2. The Inhomogeneous Bogoliubov Hamiltonian
• All finite-frequency modes fulfill the orthogonality (2.76) with respect
to the ground state. This excludes a real overlap of δΨ with the ground
state Φ.
• All eigenstates pairwisely fulfill the bi-orthogonality relation (2.74).
Bogoliubov ground state
The Bogoliubov quasiparticles satisfy the bosonic commutator relations
[ ˆ βµ, ˆ β † ν] = δµ,ν, [ ˆ β † µ, ˆ β † ν] = [ ˆ βµ, ˆ βν] = 0, (2.82)
which are easily verified using equations (2.75), (2.79) and (2.80). The operators
ˆ β † ν and ˆ βν create and annihilate Bogoliubov excitations. In particular,
the Bogoliubov ground state is the Bogoliubov vacuum |vac〉, defined by the
absence of Bogoliubov excitations
2.5.6. Non-condensed atom density
ˆβν |vac〉 = 0. (2.83)
Bogoliubov quasiparticles can be excited thermally or by external perturbations.
But even in the ground state, not all particles are in the (Gross-
Pitaevskii) condensate state. We are interested in the number of non-
condensed atoms
nnc = 1
Ld �
d d r 〈vac| δ ˆ Ψ † (r)δ ˆ Ψ(r) |vac〉 . (2.84)
It is important to distinguish this density nnc of atoms that are not in the
condensate Φ(r) from the number of particles that are not in the state k = 0.
The latter is sometimes referred to as “condensate depletion” [77, 78], but
it is more a condensate deformation.
The “condensate depletion due to disorder” nR in [77, Eq. (9)] is only due
to the first-order smoothing correction (2.22) of the Gross-Pitaevskii wave
function.
The shift of the non-condensed density due to disorder, which we are
going to compute now, is beyond Huang’s and Meng’s work [77]. In order
to evaluate the non-condensed density, the field operator δ ˆ Ψ(r) is expressed
in terms of the Bogoliubov creators and annihilators, using equation (2.67)
54
nnc = �
ν
1
L d
�
d d r|vν(r)| 2 . (2.85)