Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

Chapter 7
Bogoliubov excitations of Bloch state
condensates
Small perturbations of a stationary Bloch state condensate can be analyzed in terms of its
Bogoliubov excitations. We show that these excitations have Bloch symmetry and hence are
labeled by their band index j and their quasi-momentum ¯hq (see section 7.2). Accordingly,
the Bogoliubov dispersion takes the form of a band spectrum (“Bogoliubov band spectrum”)
which depends on the stationary condensate Bloch state whose excitations are considered.
The physical meaning underlying the Bogoliubov band spectrum is very different from the
one of the energy Bloch bands discussed in the previous chapter: The Bloch bands refer to
states which involve a motion of the whole condensate through the lattice. In contrast, the
Bogoliubov bands describe small perturbations which involve only a small portion of atoms.
The non-perturbed condensate acts as a carrier or, in other words, as a medium through which
the perturbed portion is moving. This physical picture explains why interaction effects on the
Bogoliubov spectrum are more significant than on the Bloch energies.
We calculate the Bogoliubov bands of the groundstate condensate (see section 7.2). The
lowest band exhibits a phononic regime at small quasi-momenta while higher Bogoliubov bands
are found to be little affected by interactions. An analysis of the Bogoliubov amplitudes of the
lowest band shows that the v-amplitude becomes comparable to the u-amplitude in the whole
Brillouin zone as the lattice is made deeper. Hence, all excitations of the lowest band acquire
quasi-particle character, even in the range of quasi-momenta where the dispersion is not linear
and excitations are not phonons.
We develop a formalism which is suitable to describe the lowest Bogoliubov band in the
tight binding regime (see section 7.3). Analytic formulas for the lowest Bogoliubov band
and the respective Bogoliubov amplitudes are found. They involve the tunneling parameter δ
describing the lowest energy Bloch band in the tight binding regime (see chapter 6.2 above)
and the compressibility κ of the groundstate. When the lattice is made very deep, the band
takes the form of the modulus of a sin-function. The band height is given by 2 √ δκ and
decreases much more slowly as a function of the lattice depth as the height 2δ of the lowest
energy Bloch band (see chapter 6.2 above). The relative difference between Bogoliubov u
and v-amplitude at the boundary of the Brillouin zone goes to zero like √ 2δκ, indicating the
quasi-particle character of excitations.
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