Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2. The Inhomogeneous Bogoliubov Hamiltonian
and expand the problem in the quantum fluctuations δ ˆ Ψ(r). 1 Later, in
subsection 2.5.6, we will see that the small parameter of this expansion is
the gas parameter � na3 s [95], i.e. the range of the interaction as compared
1 − with the average particle spacing n 3.
The Gross-Pitaevskii mean-field approximation consists in neglecting the
quantum fluctuations δ ˆ Ψ(r), i.e. the field operators in (2.11) are replaced by
a complex field Ψ(r). Equivalently, the Gross-Pitaevskii approximation is
obtained from the Hamiltonian Ê = ˆ H −µ ˆ N by a Hartree-Fock ansatz
�
of the
−N/2 many-particle wave function as a pure product ΨN(ri) = N i Ψ(ri).
Then, the operator (2.11) reduces to the Gross-Pitaevskii energy functional
E[Ψ, Ψ ∗ �
] = d d � 2 �
r
2m |∇Ψ(r)|2 + � V (r) − µ � |Ψ(r)| 2 + g
2 |Ψ(r)|4
�
.
(2.14)
The equation of motion can be derived from the variation of the action
� d d r dtL with the Lagrangian
�
L =
d d r i�
2
�
Ψ ∗∂Ψ
�
− Ψ∂Ψ∗ − E[Ψ, Ψ
∂t ∂t
∗ ], (2.15)
see e.g. [54, Chapter 7]. The so-called Gross-Pitaevskii equation describes
the time evolution in terms of a functional derivative with respect to the
conjugate field
i� ∂ δE
Ψ =
∂t δΨ∗ = � − �2
2m ∇2 + V (r) − µ � Ψ(r) + g |Ψ(r)| 2 Ψ(r). (2.16)
Alternatively to the Lagrangian prescription, which might appear a bit ad
hoc at this place, the Gross-Pitaevskii equation is obtained straightforwardly
from the Heisenberg equation of motion (2.12) of the many-particle problem
by inserting equation (2.13) and neglecting the fluctuations. Formally, the
Gross-Pitaevskii equation is very similar to the Schrödinger equation of a
single particle. Kinetic and potential energy appear in the same manner,
the only modification is the interaction term g|Ψ| 2 . The Gross-Pitaevskii
equation is also called nonlinear Schrödinger equation and appears in many
different fields of physics [9, 10].
1 The non-vanishing expectation value of the field operator ˆ Ψ(r) can be rigorously defined in a coherent
state: The macroscopic population of the ground state, together with the grand canonical ensemble
with variable particle number, allows the construction of coherent states from superpositions of states
with different particle numbers in the condensate state [94].
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