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