  We study a harmonically confined atomic gas which is subjected to an
additional external potential such as an optical lattice. Using a linear
response formulation, we determine the response of the gas to a small,
time-dependent displacement of the harmonic trap and derive a simple exact
relation showing that the centre-of-mass position of the atomic cloud is
directly related to the global optical conductivity of the system. We
demonstrate the usefulness of this approach by calculating the optical
conductivity of bosonic atoms in an optical lattice. In the Mott insulating
phase, there is clear evidence of an optical Mott gap, providing a
proof-of-principle demonstration that the global optical conductivity gives
high-quality information about the exci- tations of strongly-correlated quantum
gases.
