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

86 Bogoliubov excitations of Bloch state condensates
where ũj ′ q(z), ˜vj ′ q(z) are periodic with period d and the index σ in Eqs.(7.6,7.7) is replaced by
the band index j ′ and the quasi-momentum q of the excitation. Because the Eqs.(7.6,7.7) are
linear, the set of Bogoliubov Bloch amplitudes (7.11,7.12) represents all possible solutions, in
contrast to the case of the nonlinear equation (5.4), where states of Bloch form constitute only
the class of solutions which are associated with a density profile of period d. The set of solutions
(7.11,7.12) is associated with a band spectrum ¯hωj ′(q) of the energies of elementary excitations
(“Bogoliubov band spectrum”). In order to conform with periodic boundary conditions, the
quasi-momentum q must belong to the spectrum
q = 2π
ν, ν=0, ±1, ±2,... . (7.13)
L
The solutions (7.11,7.12) and the corresponding Bogoliubov band spectrum ¯hωj ′(q) depend
on the particular stationary condensate ϕjk. In the following, we will restrict ourselves to
discussing small perturbations of the groundstate ϕ. The Bogoliubov equations (7.6,7.7) for
this case read
− ¯h2 ∂
2m
2
∂z2 + sER sin 2
πz
+2dgn| ˜ϕ(z)|
d
2
− µ ujq(z)+gnd˜ϕ 2 (z)vjq(z) =¯hωj(q)ujq(z) (7.14)
− ¯h2 ∂
2m
2
∂z2 + sER sin 2
πz
+2dgn| ˜ϕ(z)|
d
2
− µ vjq(z)+gnd˜ϕ ∗2 (z)ujq(z) =−¯hωj(q)vjq(z) (7.15)
where we have replaced the index σ by the band index j and the quasi-momentum q of the
excitation. Evidently, the Bogoliubov band spectrum takes only positive values in this case.
The numerical solution of Eqs.(7.14,7.15) is conveniently obtained by expanding the periodic
functions ũjq, ˜vjq, ˜ϕ in a Fourier series.
The Bogoliubov equations (7.14,7.15) have been solved for example by [119, 120]. Numeric
solution for the bands ¯hωj(q) of (7.6,7.7) with values of the condensate quasi-momentum k = 0
has been reported in [118]. The damping of Bogoliubov excitations in optical lattices at finite
temperatures has been studied by [121].
Bogoliubov band spectrum
Similarities and differences with respect to the well-known Bogoliubov spectrum in the uniform
case (s =0) are immediate. As in the uniform case, interactions make the compressibility
finite, giving rise to a phononic regime for long wavelength excitations (q → 0) inthelowest
band. In high bands the spectrum of excitations instead resembles the Bloch dispersion (see
Eq.(6.4)), as it resembles the free particle dispersion in the uniform case. The differences
are that in the presence of the optical lattice the lattice period d and the Bragg momentum
qB =¯hπ/d emerge as an additional physical length and momentum scale respectively and
that the Bogoliubov spectrum develops a band structure. As a consequence, the dispersion
is periodic as a function of quasi-momentum and different bands are separated by an energy
gap. In particular, the phononic regime present in the lowest band at q =0is repeated at
every even multiple of the Bragg momentum qB.
In Fig.7.1 we compare the Bologoliubov bands at s =1for gn =0and gn =0.5ER. In
the interacting case, one notices the appearance of the phononic regime in the lowest band,