Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2.1. Bose-Einstein condensation of the ideal gas
in two dimensions. The considerations so far are valid in the thermodynamic
limit. For experimental applications, the concept has to be adapted to a finite
system size and a finite particle number. This leads to corrections of the
critical temperature [87, 88]. In the thermodynamic limit no Bose-Einstein
condensation is predicted in 1D traps and low-dimensional boxes. Nevertheless,
macroscopic ground-state populations are found in finite systems
[87]. Also particle-particle interactions should enhance the phase coherence.
In experiments, the coherence length is often larger than the largest length
scale and the Bose gas is regarded as quasi-condensate.
Note that the critical temperature of Bose-Einstein condensation is determined
by the particle density. The condensation typically occurs already for
temperatures much higher than the energy gap to the first excited state ɛ1.
Thus, it is fundamentally different from the behavior predicted by the classical
Boltzmann factor e −ɛ/kBT . Bose-Einstein condensation is a statistical
effect, resulting from the indistinguishability and is not caused by attractive
interactions.
2.1.3. Order parameter and spontaneous symmetry
breaking
There is more to Bose-Einstein condensation than just the distribution of
particle numbers. The single-particle state with the macroscopic particle
number defines the condensate function Φ(r) (more precisely, the state associated
with the only macroscopic eigenvalue of the density matrix). This
wave function exists only in the condensed Bose gas and takes the role of
the order parameter of the phase transition to the Bose-Einstein condensate.
The order parameter spontaneously takes a particular phase, breaking the
U(1) symmetry of the non-condensed phase. By virtue of the Goldstone
theorem [72], this spontaneously broken symmetry implies the existence of
Goldstone bosons. Goldstone bosons are excitations related to the broken
symmetry, in this case to a homogeneous phase diffusion with zero frequency
[89]. It will turn out that the Bogoliubov excitations are the Goldstone
bosons of Bose-Einstein condensation.
Experimentally, the phase of the condensate becomes accessible in interference
experiments, where the phase of one condensate with respect to that
of another condensate determines the position of the interference pattern.
21