Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2. The Inhomogeneous Bogoliubov Hamiltonian
2.2. Interacting BEC and Gross-Pitaevskii
mean-field
With the concept of Bose-Einstein condensation (section 2.1) in mind, we
now consider interacting Bose gases. For interacting particles, genuine
many-body theory is the starting point. In a mean-field manner, the socalled
Gross-Pitaevskii theory [55, 90] takes advantage of the macroscopically
occupied ground state of the Bose-Einstein condensate. Fluctuations
are neglected and the condensate wave function is computed.
2.2.1. Basic many-body theory
The state of N indistinguishable bosons is described by a N-particle wave
function ΨN(r1, r2, . . . , rN), which is symmetric with respect to the exchange
of two particles. In order to avoid the explicit symmetrization, it is
convenient to use the Fock representation. The Fock space is the direct sum
of correctly symmetrized N-particle Hilbert spaces. It is only specified how
many particles are in each single-particle state. The unphysical information,
which of the particles is which, is not included. Starting from the vacuum
state with no particles, general Fock states are constructed by means of
creation operators that create a particle in a certain single-particle state.
Bosonic (fermionic) commutation relations of the creation and annihilation
operators guarantee that any Fock state is symmetric (antisymmetric). In
real-space, the creators and annihilators are called field operators and are
denoted by ˆ Ψ(r). They obey the bosonic commutator relations
� � � � � �
ˆΨ(r), Ψ ˆ † ′ ′
(r ) = δ(r − r ), ˆΨ(r), Ψ(r ˆ ′
) = 0 = ˆΨ †
(r), Ψ ˆ † ′
(r ) . (2.9)
For fermions, the commutator [·, ·] is replaced with the anticommutator.
Physically, expectation values containing field operators, like the particle
density n(r, t) = � ˆ Ψ † (r, t) ˆ Ψ(r, t) � or the single-particle density matrix
ρ(r, r ′ , t) = � ˆ Ψ † (r, t) ˆ Ψ(r ′ , t) � are the quantities of interest.
The time-evolution of any physical observable is given by the Heisenberg
equation of motion for operators
i� d
dt  = � Â, ˆ H � + i� ∂
Â, (2.10)
∂t
with the Hamiltonian [85]
�
ˆH = d d r ˆ Ψ † � 2 −�
(r)
2m ∇2 �
+ V (r) ˆΨ(r) + g
�
d
2
d r ˆ Ψ † (r) ˆ Ψ † (r) ˆ Ψ(r) ˆ Ψ(r).
(2.11)
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