  We present a theoretical study of the effects of the next-nearest-neighbor
(NNN) hopping ($t_2$) on the properties of non-interacting bosons in optical
lattices in the presence of an Aubry-Andr\'{e} quasi-disorder. First we
investigate, employing exact diagonalization, the effects of $t_2$ on the
localization properties of a single boson. The localization is monitored using
an entanglement measure as well as with inverse participation ratio. We find
that the sign of $t_2$ has a significant influence on the localization effects.
We also provide analytical results in support of the trends found in the
localization behavior. Further, we extend these results including the effects
of a harmonic potential which obtains in experiments. Next, we study the
effects of $t_2$ on Bose-Einstein condensation. We find that, a positive $t_2$
strongly enhances the low temperature thermal depletion of the condensate while
a negative $t_2$ reduces it. It is also found that, for a fixed temperature,
increasing the quasi-disorder strength reduces the condensate fraction in the
extended regime while enhancing it in the localized regime. We also investigate
the effects of boundary conditions and that of the phase of the AA potential on
the condensate. These are found to have significant effects on the condensate
fraction in the localization transition region.

Quantum Gases