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