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

88 Bogoliubov excitations of Bloch state condensates
¯hω/ER
1.2
1
0.8
0.6
0.4
0.2
0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
¯hq/qB
Figure 7.2: Lowest Bloch band (dashed line) and lowest Bogoliubov band (dash-dotted line)
for s =1and gn =0.5ER compared with the single particle Bloch band (solid line). In the
case of the Bloch bands (solid and dashed line) the groundstate energy has been subtracted .
The solid lines in Fig.7.3 show how the lowest Bogoliubov band changes when the lattice
depth is increased at fixed interaction. At s =1(Fig.7.3a), apart from the formation of the
energy gap close to q = qB, the curve still resembles the dispersion in the uniform case: both
the phononic linear regime at small q and the quadratic regime at larger q are visible. When the
potential is made deeper (s =5, 10; Fig.7.3b,c), the band becomes flatter. As a consequence,
the quadratic regime disappears and the slope of the phononic regime decreases. This reflects
the strong decrease of the velocity of sound as the lattice is made deeper (see discussion below
in section 7.4).
In Fig.7.4, we plot the energy gap between lowest and first excited band as a function of
lattice depth for gn/ER =1. For comparison, we display the corresponding curve for the
energy gap in the Bloch band spectrum of a single particle (see Fig.6.6). In addition, we
plot the gap 2 √ sER between the vibrational levels obtained when approximating the bottom
of a lattice well by a harmonic potential (see Eq.(6.48)). It turns out that the gap in the
Bogoliubov band spectrum approaches the one in the single particle spectrum as s is tuned to
large values. This is the case because in a very deep lattice the first excited band is almost not
affected by interactions and is hence essentially given by the single particle Bloch band while
the lowest Bogoliubov band becomes so flat that the excitation energy at the zone boundary
is negligibly small compared to the energies of the second band. Note that the convergence to
the gap 2 √ sER of the harmonic approximation is very slow, reflecting mainly the role of the
anharmonicity of the potential wells.