Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2. The Inhomogeneous Bogoliubov Hamiltonian
r ′ ). With this, we expand the grand canonical Hamiltonian Ê = ˆ H − µ ˆ N
(2.11) in orders of δ ˆ Ψ and δ ˆ Ψ † . The linear part vanishes, because Φ(r) is
the ground-state of the Gross-Pitaevskii equation. The relevant part is then
the quadratic part ˆ F of the Hamiltonian Ê
Ê = E0 + ˆ F [ ˆ δn, δ ˆϕ]. (2.28)
Third-order and forth-order terms in the fluctuations are neglected. Here,
E0 = E[n0(r), ϕ0(r)] is the Bogoliubov ground-state energy. The quadratic
Hamiltonian is found as
�
ˆF = d d � �� 2 �
r ∇
2m
δˆn
2Φ(r)
� 2
+
� ∇ 2 Φ(r) �
4Φ 3 (r) δˆn2 + Φ 2 (r)(∇δ ˆϕ) 2
�
+ g
2 δˆn2
�
.
(2.29)
The potential V (r) does not appear directly in the equations of motion
for the excitations. Instead, it enters via the condensate function Φ(r),
according to the nonlinear equation (2.19).
With (2.29), the problem is reduced to a Hamiltonian that is quadratic
in the excitations. To this order, there are no mixed terms of δˆn and δ ˆϕ,
but the Heisenberg equations of motion (2.10) for δ ˆϕ and δˆn
∂δ ˆϕ
∂t
1 � �
= δ ˆϕ, Fˆ ,
i�
∂δˆn
∂t
are coupled, because � δˆn(r), δ ˆϕ(r ′ ) � = i δ(r − r ′ ).
2.3.1. The free Bogoliubov problem
1 � �
= δˆn, Fˆ i�
(2.30)
Before including the external potential, we consider the excitations of the
homogeneous system V (r) = 0, n0(r) = µ/g. In this case, the Bogoliubov
Hamiltonian (2.29) reduces to
ˆF (0) �
=
= �
k
d d � 2 �
r
2m
��
∇ δˆn
2 √ �2 n∞
+ (∇ √ n∞δ ˆϕ) 2
�
�
ɛ 0 kn∞δ ˆϕkδ ˆϕ−k + (2µ + ɛ 0 k) δˆnkδˆn−k
4n∞
+ 2g n∞
� δˆn
2 √ n∞
� 2�
�
, (2.31)
with ɛ 0 k = �2 k 2 /(2m). The Fourier representation is already diagonal in k,
but the equations of motion (2.30) are still coupled. This can be resolved by
the Bogoliubov transformation [71], a transformation that couples density
30