Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

4.4. Particle regime
regime, because in the range of validity of the Bogoliubov perturbation theory,
the disorder is weaker than the chemical potential, and the chemical
potential is again much lower than the excitation energy ɛ 0 k .
4.4.3. Transition to really free particles
How are the Bogoliubov excitations in the particle regime related to really
free particles? In the limit of vanishing interaction gn∞ = µ → 0, ξ →
∞, the Gross-Pitaevskii energy functional (2.17) passes over to the usual
Schrödinger energy functional and the Gross-Pitaevskii equation becomes
the Schrödinger equation, which reads
� � 0
�ω − ɛk Ψk =
� d d k ′
(2π) dV k−k ′Ψ k ′ (4.41)
in Fourier space. Also the Bogoliubov excitations seem to pass into free
particles, as the coefficient ak = � ɛ 0 k /ɛk tends to one
ˆγk
(2.32)
= i √ n∞δ ˆϕk + δˆnk/(2 √ n∞) (2.27)
= δ ˆ Ψk. (4.42)
The second equality relies on the fact that the condensate wave function
Φ(r) is extended. It fails for strong disorder V ≫ µ, when the Bose-Einstein
condensate becomes a fragmented Bose glass [74].
Self-energy in the Schrödinger regime
Before working out the difference between the Bogoliubov regime and the
Schrödinger regime, let us consider the problem of the Schrödinger equation
in presence of weak disorder. From equation (4.41), we write the Born
approximation of the self-energy for the disordered Schrödinger problem
ReΣSg = V 2
0 P
� d d k ′
(2π) dσdCd(|k − k ′ |σ)
ɛ0 k − ɛ0 k ′
. (4.43)
For the 1D speckle potential, the correlator Cd is piecewisely linear and the
Cauchy principal value can be computed analytically. Similarly to (4.15)
we find
ReΣSg = 1 V
4
2 � � � �
�
0
ln �
1 + kσ � �
�
�1
− kσ � + kσ ln �
1 − k
�
2σ2 k2σ2 ��
�
�
� , (4.44)
� ɛ 0 k Eσ
see figure 4.13. The self-energy scales like V 2
0 / � ɛ 0 k Eσ, where Eσ = � 2 /(2mσ 2 )
is the energy scale defined by the disorder correlation length.
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