Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2.5. Exact diagonalization of the Bogoliubov problem
Ψ(r, t = 0). There is no reason why the deviations from the ground state
should affect the number of particles, they merely deform the wave function.
This relation between the phase-related zero-mode and the particle number
does not come as a surprise, as phase and density are conjugate variables.
Frequency pairs
Due to the time-inversion symmetry, solutions of equation (2.68) occur in
pairs. Defining ˆτ = ( 0 1
1 0 ) and using the commutation relations ηˆτ = −ˆτη,
ˆτQ = (Qˆτ) ∗ , we find
ηQ(ˆτa ∗ ν) = ηˆτQ ∗ a ∗ ν = −ˆτ(ηQaν) ∗ = −�ων ˆτa ∗ ν a − ν = ˆτa ∗ ν (2.78a)
Qη(ˆτb ∗ ν) = −Qˆτηb ∗ ν = −ˆτ(Qηbν) ∗ = −�ων ˆτb ∗ ν b − ν = ˆτb ∗ ν . (2.78b)
From these solutions with negative frequencies, another orthogonality relation
follows from equation (2.72) with (b − ν , aµ) = 0
�
d d r [vν(r)uµ(r) − uν(r)vµ(r)] = 0 . (2.79)
Together with (2.75), this identity is used to invert the transformation (2.67)
and express the quasiparticle operators in terms of the field operator
�
ˆβν = d d �
r u ∗ ν(r)δ ˆ Ψ(r) + v ∗ ν(r)δ ˆ Ψ † �
(r) . (2.80)
The identity (2.80) clarifies the meaning of the negative frequencies. The
quasiparticle annihilator ˆ β − ν of the negative-frequency mode reads
ˆβ − ν =
�
d d r
�
vν(r)δ ˆ Ψ(r) + uν(r)δ ˆ Ψ † �
(r) = ˆ β † ν , (2.81)
which coincides with the quasiparticle creator ˆ β † ν. Thus, negative Bogoliubov
frequencies have nothing to do with negative excitation energies, but
simply swap the roles of creators and annihilators. They are not necessary
for the completeness of the Bogoliubov eigenbasis and will be excluded in
the following.
Properties of Bogoliubov eigenstates summed up
• The zero-frequency mode comes from the broken U(1) symmetry and
is not a Bogoliubov excitation in the proper sense.
• Only the excitations with positive frequencies are physically relevant.
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