Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2. The Inhomogeneous Bogoliubov Hamiltonian
superfluid flow. The density profile fulfills the stationary Gross-Pitaevskii
equation
− �2 ∇
2m
2Φ(r) Φ(r) + g |Φ(r)|2 = µ − V (r) . (2.19)
In a flat potential V (r) = 0, the density n(r) = n∞ = µ/g is constant.
Interaction and kinetic energy define the characteristic length scale of the
BEC, the healing length ξ = �/ √ 2mgn∞. The condensate changes its density
on this length scale. In presence of an external potential V (r), the
solution of the nonlinear equation (2.19) is non-trivial. The limits of very
strong interaction and no interaction are understood as follows:
• In the Thomas-Fermi (TF) regime, the kinetic energy (quantum pressure)
is negligible compared with the interaction energy. The density
profile is determined by the balance of the external potential and the
interaction:
nTF(r) = (µ − V (r))/g for µ > V (r), otherwise zero. (2.20)
Often, the Thomas-Fermi approximation is a reasonable approximation.
However, special care has to be taken at the edges of the trap,
where the condensate density seems to vanish abruptly.
• In the opposite case of a non-interacting system g = 0, the Gross-
Pitaevskii equation reduces to the linear Schrödinger equation. In a
harmonic trap, the ground state is given by the Gaussian wave function
of the harmonic-oscillator ground state. In the homogeneous system,
the ground state is the k = 0 mode with homogeneous density.
However, this state is very sensitive to weak perturbations, like a weak
disorder potential, because there is no interaction that counteracts the
localization of the wave function. Thus, the unstable non-interacting
gas is not a convenient starting point for perturbation theory of the
ground state.
2.2.4. The smoothed potential
In the case of an extended condensate that is modulated by a weak potential,
one can perform a weak disorder expansion in the small parameter
V/µ [99]
� n0(r) = Φ(r) = Φ0 + Φ (1) (r) + Φ (2) (r) + . . . . (2.21)
With this expansion, the stationary Gross-Pitaevskii equation (2.19) is
solved order by order. There are two mechanisms: (i) scattering of atoms
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