Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2.2. Interacting BEC and Gross-Pitaevskii mean-field
The first part contains the kinetic energy and the potential energy due to the
external potential V (r). The interaction potential Vint = gδ(r − r ′ ) in the
second part represents two-body collisions. As the atoms are neutral, this
interaction is short range and the actual physical interaction has been replaced
by a point interaction. In three dimensions, the parameter g depends
on the s-wave scattering length as as g = 4π� 2 as/m. This approximation
is good in the dilute-gas limit, where the average particle distance n −1/3
is much larger than the scattering length as. The external potential V (r)
is typically given by the harmonic trapping potential, possibly superposed
with scattering impurities or a disorder potential.
In the Fock representation it is specified how many atoms are in each
single-particle state. From the start, the Fock representation is capable of
handling variable particle numbers. Thus, it is straightforward to relax the
constraint of a fixed particle number by Legendre-transforming to the grand
canonical Hamiltonian Ê = ˆ H −µ ˆ N. Here, the chemical potential µ controls
the average particle number. In the grand canonical picture, the equation
of motion of the field operator reads
i� d
dt ˆ Ψ(r) = � − �2
2m ∇2 + V (r) − µ � ˆ Ψ(r) + g ˆ Ψ(r) † ˆ Ψ(r) ˆ Ψ(r), (2.12)
according to equation (2.10) with ˆ H replaced by Ê = ˆ H − µ ˆ N.
2.2.2. Gross-Pitaevskii energy functional and equation of
motion
The description in terms of the many-particle Hamilton operator (2.11)
holds very generally, but suitable approximations are desirable for practical
use.
At sufficiently low temperatures, also the interacting Bose gas is expected
to Bose-Einstein condense. In three dimensions, this can be proven rigorously
[91, 92]. In one and in two dimensions, at least a quasi-condensate [93]
should exist, where the phase coherence is not truly long-range, but should
extend over the experimentally relevant length scales.
The macroscopically populated single-particle state is called the condensate
state Φ(r). It is defined as the eigenstate associated to the only macroscopic
eigenvalue of the single particle density matrix. The condensate wave
function takes a particular phase and spontaneously breaks the U(1) symmetry
of the non-condensed system, such that Ψ(r) = � ˆ Ψ(r) � is non-zero.
One can separate the field operator into its mean value and its fluctuations
ˆΨ(r) = Ψ(r) + δ ˆ Ψ(r) (2.13)
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