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