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

96 Bogoliubov excitations of Bloch state condensates
Bogoliubov Bloch amplitudes of the lowest band
Using the normalization conditon for the Bogoliubov amplitudes (7.24), Eq.(7.35) yields
Uq = εq +¯hωq
2 ,
¯hωqεq
(7.41)
Vq = εq − ¯hωq
2 ,
¯hωqεq
(7.42)
where εq =2δsin2 (qd/2) is the lowest energy Bloch band (6.29) from which the groundstate
energy has been subtracted.
According to (7.22,7.23) the coefficients of the Fourier expansion of the Bogoliubov Bloch
waves ũ, ˜v read
bql = Uqaql , (7.43)
cql = Vqaql , (7.44)
where aql are the coefficients of the Fourier expansion of the condensate Bloch wave ˜ϕq. InFig.
7.8 we compare the square moduli |bql| 2 , |cql| 2 obtained from the approximation (7.43,7.44)
with those obtained from the exact Fourier expansion (7.17,7.18) of the numerical solutions of
the Bogoliubov equations (7.14,7.15). As previously, we evaluate the expressions (7.41,7.42)
for Uq, Vq using the numerical results for κ and m ∗ . We find that the agreement between the
full numerical and the tight binding results is very good.
The large-s limit of the ratio of Uq/Vq for non-zero interaction gn/ER reads
Uq
Vq
≈− 1+ √
2δκ |sin (qd/2)| . (7.45)
The second term is small and decreases rapidly as a function of s implying that in the tight
binding regime Uq and Vq are of the same order of magnitude in the whole Brillouin zone.
This shows that all excitations of the lowest Bogoliubov band acquire quasi-particle character.
For large gn/ER (small κ) this behavior is enhanced. Note that the second term decreases
with increasing s like the band height of the lowest Bogoliubov band in the large-s limit (see
Eq.(7.39) and Fig. 7.7).