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