We study the formation and collision-aided decay of an ultra-cold atomic
Bose-Einstein condensate in the first excited band of a double-well 2D-optical
lattice with weak harmonic confinement in the perpendicular $z$ direction. This
lattice geometry is based on an experiment by Wirth et al. The double well is
asymmetric, with the local ground state in the shallow well nearly degenerate
with the first excited state of the adjacent deep well. We compare the band
structure obtained from a tight-binding (TB) model with that obtained
numerically using a plane wave basis. We find the TB model to be in
quantitative agreement for the lowest two bands, qualitative for next two
bands, and inadequate for even higher bands. The band widths of the excited
bands are much larger than the harmonic oscillator energy spacing in the $z$
direction. We then study the thermodynamics of a non-interacting Bose gas in
the first excited band. We estimate the condensate fraction and critical
temperature, $T_c$, as functions of lattice parameters. For typical atom
numbers, the critical energy $k_BT_c$, with $k_B$ the Boltzmann constant, is
larger than the excited band widths and harmonic oscillator energy. Using
conservation of total energy and atom number, we show that the temperature
increases after the lattice transformation. Finally, we estimate the time scale
for a two-body collision-aided decay of the condensate as a function of lattice
parameters. The decay involves two processes, the dominant one in which both
colliding atoms decay to the ground band, and the second involving excitation
of one atom to a higher band. For this estimate, we have used TB wave functions
for the lowest four bands, and numerical estimates for higher bands. The decay
rate rapidly increases with lattice depth, but stays smaller than the tunneling
rate between the $s$ and $p$ orbitals in adjacent wells.
