Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2. The Inhomogeneous Bogoliubov Hamiltonian
Figure 2.3: Bogoliubov dispersion relation
(2.35). Insets: schematic representation
of the amplitudes δΨ⊥
and δΨ� (2.37).
ǫk
µ
10
5
1
δ ˆ Ψ⊥
Ψ0
0
2δ ˆ Ψ�
δ ˆ Ψ⊥
1 kξ 2
transformed to ɛk − �k· v. The static obstacle cannot transfer energy, so
excitations can only be created if v > ɛk/(�k). Thus, there is a critical
velocity vc = mink ɛk/�k = c.
Superfluid flow is one of the key features of Bose-Einstein condensates and
has been subject of many experiments. For example stirring experiments,
where dissipation was observed when the condensate was stirred with velocities
above the critical velocity [56], or persistent flow in a toroidal trap
[100].
In the ideal Bose gas (section 2.1), the excitations are still bare particles
with dispersion relation ɛ 0 k
Ψ0
0
δ ˆ Ψ�
and zero critical velocity. The phase transition to
the Bose-Einstein condensate phase, together with the interactions, results
in superfluidity.
Signature of a Bogoliubov excitation
Let us split the fluctuation operator δ ˆ Ψ(r) = ˆ Ψ(r) − Φ(r) into its components
parallel and perpendicular to the order parameter Φ(r)
δ ˆ Ψ� = 1�
δΨ ˆ + δ ˆ †
Ψ
2
� , δ ˆ Ψ⊥ = 1 �
δΨˆ − δ ˆ †
Ψ
2i
� . (2.37)
With equation (2.27) and the Bogoliubov transformation (2.33), the components
take the form
δ ˆ Ψ� = 1
L d
� ak � ik·r
e ˆγk + e
2 2
k
−ik·r ˆγ † �
k ,
� 1 � ik·r
e ˆγk − e −ik·r ˆγ † �
k ,
δ ˆ Ψ⊥ = 1
L d
2
k
2iak
again with the coefficient ak = � ɛ0 k /ɛk. The signature of a Bogoliubov
, respectively.
excitation is a plane wave with amplitudes ak and a −1
k
32