Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2.2. Interacting BEC and Gross-Pitaevskii mean-field
Formulation in terms of density and phase
For the physical interpretation it is useful to express the Gross-Pitaevskii
energy functional and the corresponding equation of motion in terms of the
condensate density and its complex phase. The local condensate density
n(r) = |Ψ(r)| 2 is normalized to the total particle number � ddr n(r) = N.
The complex phase of the GP wave-function Ψ(r, t) = � n(r, t) exp{iϕ(r, t)}
is given by ϕ(r, t). The energy functional (2.14) then becomes
�
E[n, ϕ] = d d � 2 �
��∇√ � �
2 2
r
n + n(∇ϕ) + (V (r) − µ)n +
2m
g
2 n2
�
.
(2.17)
The time evolution (2.16) is rephrased and yields the equations of motion
for density and phase
∂ 1 δE
n =
∂t � δϕ
−� ∂ δE
ϕ =
∂t δn
= − �
m ∇ · (n∇ϕ) =: −∇ · (nvs) (2.18a)
�2 ∇
= −
2m
2�n0(r) �2
√ +
n0(r) 2m (∇ϕ)2 + g n0(r) + V (r) − µ .
(2.18b)
The first equation is the continuity equation of an irrotational fluid with
superfluid velocity proportional to the phase gradient vs = �
m (∇ϕ) (the
term superfluid is explained on page 31). The second equation (2.18b)
describes the time evolution of the phase, whose gradient determines the
velocity field. The term containing the derivatives of the density stems
from the quantum mechanical kinetic energy. As it has no classical analog,
it is often called quantum pressure.
The superfluid velocity is proportional to the gradient of the phase of the
condensate wave function and is thus irrotational, i.e. a superfluid cannot
rotate freely. The only possibility to rotate the superfluid is to create vortices
in the superfluid [96, 97].
2.2.3. Ground state
In the following, the dynamics of excitations close to the ground state are
of interest. The ground state Φ(r) = � n0(r), ϕ0(r) = 0 minimizes the
energy functional (2.14) and is a stationary solution of the equations of
motion (2.18). Obviously, the phase has to be homogeneous, i.e. there is no
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