Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2.3. Bogoliubov Excitations
and phase fluctuations to quasiparticle creation and annihilation operators
ˆγ †
k and ˆγk
�
ˆγk
ˆγ †
� �
ak a
=
−k
−1
k
−ak a −1
� � √
i n∞δ ˆϕk
δˆnk
k
2 √ �
. (2.32)
n∞
A transformation of this kind, with the free parameter ak, guarantees that
the quasiparticles obey bosonic commutation relations analogous to (2.1).
Inserting the inverse of the above transformation
� i √ n∞δ ˆϕk
δˆnk
2 √ n∞
�
= 1
2
� a −1
k
ak
−a−1
k
ak
� � ˆγk
ˆγ †
−k
into the Hamiltonian (2.31), we find off-diagonal terms ˆγ †
k ˆγ†
−k
�
(2.33)
and ˆγ−kˆγk.
These can be eliminated by choosing the free parameter as ak = � ɛ0 k /ɛk
with ɛk = � ɛ0 k (2µ + ɛ0 k ). Then the Hamiltonian takes its diagonal form
ˆF (0) = �
k
ɛkˆγ †
k ˆγk. (2.34)
The famous Bogoliubov dispersion relation [71]
�
ɛk = ɛ0 k (2µ + ɛ0 k ) (2.35)
replaces the kinetic energy of free particles ɛ0 k (figure 2.3). In the high-energy
regime, the chemical potential µ is negligible compared with ɛ0 k and the free
dispersion relation, shifted by the chemical potential, is recovered
Superfluidity
ɛk = ɛ 0 k + µ + O(µ/ɛ 0 k). (2.36)
In the low-energy regime, the interaction g n = µ dominates over the kinetic
energy. A single excitation involves many individual particles, comparable
to classical sound waves. The dispersion relation is linear ɛk = ck, with
sound velocity c = � µ/m. According to Landau’s argument [54, chapter
10.1], this linear dispersion at low energies implies superfluidity:
In general, an obstacle moving through a fluid with velocity v can dissipate
energy by creating elementary excitations. By a Galilei transformation
to the reference frame of the obstacle, the energy of the excitation ɛk is
31