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

12.3 Shallow lattice 165
12.3 Shallow lattice
In chapter 5 and 9 we have seen that a condensate loaded in a lattice can be effectively
described as a uniform system of atoms whose motion in the z-direction occurs with effective
mass m∗ and whose interaction, at sufficiently low average density, is described by an effective
coupling constant ˜g. We can then pose the question under what condition this holds also for
the calculation of the quantum depletion. According to the above discussion of the uniform
case, the dispersion of the relevant elementary excitations must be well approximated by
¯hω(px,py,q)= p2 x
2m + p2y 2m + ¯h2 q2 2m∗
p2 x
2m + p2y ¯hq2
+
2m 2m∗
+2˜gn . (12.44)
Note that this dispersion implies different sound velocities in the axial z and transverse x, ydirections
respectively:
cz = ˜gn
, (12.45)
m∗ cx,y = ˜gn
. (12.46)
m
As pointed out in section 12.2, the range of relevant momenta px, py and quasi-momenta q
reaches up to values much larger than mcx,y and much larger than m∗cz respectively. Hence,
a necessary condition for the applicability of the approximation (12.44) is that
m ∗ cz