Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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„ Φq ′<br />
¯Φ q ′<br />
«<br />
V<br />
„ Φq ′ −q<br />
¯Φ q ′ −q<br />
«<br />
„<br />
ˆγk<br />
ˆγ †<br />
« „<br />
ˆγk+q<br />
−k<br />
ˆγ †<br />
«<br />
−(k+q)<br />
2.3. <strong>Bogoliubov</strong> <strong>Excitations</strong><br />
Figure 2.4: Universal <strong>Bogoliubov</strong><br />
scattering vertex (2.41). The<br />
pseudo-spinor � ˆγk, ˆγ † �T −k scatters<br />
with the condensate wavefunction<br />
Φ and its inverse pr<strong>of</strong>ile<br />
¯ Φ.<br />
The matrix W k ′ k contains the usual scattering between different momentum<br />
states, induced by the inhomogeneous potential. The anomalous scattering<br />
element Y k ′ k allows the creation or annihilation <strong>of</strong> excitation pairs. The<br />
reason for the anomalous coupling Y , between ˆγk and the conjugate oper-<br />
ators ˆγ †<br />
−k ′ lies in the expansion <strong>of</strong> the nonlinear term and the <strong>Bogoliubov</strong><br />
approximation.<br />
The anomalous coupling can be handled in a linear manner by defining a<br />
pseudo-spinor ˆ Γk = � ˆγk, ˆγ † �T −k , which leads to the equation <strong>of</strong> motion<br />
i� ∂<br />
∂t ˆ �<br />
1 0<br />
Γ ′<br />
k =<br />
0 −1<br />
�<br />
�<br />
V ′ ˆ<br />
k kΓk . (2.44)<br />
� �<br />
ɛ ′ ˆ<br />
k Γk ′ + 1<br />
L d<br />
2<br />
The product <strong>of</strong> the matrix diag(1, −1) and V from equation (2.41) makes<br />
the time evolution non-Hermitian, which is discussed in more detail in section<br />
2.5. As depicted in figure 2.4, the pseudo-spinor ˆ Γk is scattered by the<br />
matrix-valued potential V, with the momentum transfer provided by the<br />
condensate wave function Φ and the inverse pr<strong>of</strong>ile ¯ Φ (c.f. equations (2.39),<br />
(2.40) and (2.41)).<br />
In order to keep track <strong>of</strong> the momentum balance, the momentum <strong>of</strong> the<br />
creator in the second component <strong>of</strong> the pseudo-spinor ˆ Γk has been chosen<br />
reversed. This concept <strong>of</strong> a pseudo-spinor containing ˆγk and ˆγ †<br />
k<br />
−k<br />
has been<br />
adopted from the Nambu formalism, employed in the BCS theory <strong>of</strong> superconductivity<br />
[16, Section 18.5] and will be particularly useful later in<br />
section 3.3.<br />
2.3.3. Disorder expansion <strong>of</strong> the <strong>Bogoliubov</strong> Hamiltonian<br />
The coupling elements depend non-perturbatively on the condensate wavefunction,<br />
which is the solution <strong>of</strong> the stationary Gross-Pitaevskii equation<br />
(2.16). In practice this solution is either obtained numerically [cf. subsection<br />
2.4.4, section 4.3] or by the systematic perturbative expansion <strong>of</strong><br />
subsection 2.2.4. For the analytical treatment, we choose the perturbative<br />
expansion (2.21) <strong>of</strong> Φ and ¯ Φ, and expand the scattering elements in orders<br />
<strong>of</strong> the inhomogeneous potential.<br />
35