  We propose a new method of optical lattice modulation spectroscopy for
studying the spectral function of ultracold bosons in an optical lattice. We
show that different features of the single particle spectral function in
different quantum phases can be obtained by measuring the change in momentum
distribution after the modulation. In the Mott phase, this gives information
about the momentun dependent gap to particle-hole excitations as well as their
spectral weight. In the superfluid phase, one can obtain the spectrum of the
gapless Bogoliubov quasiparticles as well as the gapped amplitude fluctuations.
The distinct evolution of the response with modulation frequency in the two
phases can be used to identify these phases and the quantum phase transition
separating them.
