Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2. The Inhomogeneous Bogoliubov
Hamiltonian
In this chapter, the framework for describing dilute Bose gases in weak external
potentials is derived. Before starting with the actual problem of the
interacting gas in a given external potential, we shortly review the mechanism
of Bose-Einstein condensation at the example of the ideal Bose gas
(section 2.1). Then, in section 2.2, we formulate the interacting manyparticle
problem and perform the mean-field approximation, which allows
the Gross-Pitaevskii ground state to be computed.
The essential of this chapter is the saddlepoint expansion around the
disorder-deformed condensate state (section 2.3). This yields the Hamiltonian
and the equations of motion for Bogoliubov excitations in presence of
the external potential. For illustration and as a numerical test, scattering
of a Bogoliubov excitation at a single impurity is discussed analytically and
compared to a numerical integration (section 2.4).
The Bogoliubov excitations disclose information beyond the mean-field
ground-state, in particular the fraction of non-condensed atoms that are
present even in the ground state. In section 2.5, important properties of
Bogoliubov eigenstates, in particular their orthogonality relations, are discussed.
The orthogonality to the zero-frequency mode will be of particular
importance when choosing the basis for the disordered problem in chapter 3.
In order to be self-contained, this chapter reports some basic topics that
can be found in books and review articles. For more details, the reader is
referred to the reviews by Dalfovo, Giorgini, Pitaevskii and Stringari [85]
and by Leggett [86], and the books by Pethick and Smith [54] and Pitaevskii
and Stringari [55].
2.1. Bose-Einstein condensation of the ideal gas
In the following, the basic ideas of the phenomenon of Bose-Einstein condensation
are presented, using the example of the ideal Bose gas. For the
ideal gas, the partition function can be calculated analytically, which leads
to the derivation of Bose-Einstein statistics.
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