We study the dynamics of a one-dimensional Bose gas at unit filling
in both shallow and deep optical lattices and obtain the dynamic
structure factor \bm{S(k,\omega)}𝐒(𝐤,𝛚)
by monitoring the linear response to a weak probe pulse. We introduce a
new procedure, based on the time-dependent variational Monte Carlo
method (tVMC), which allows to evolve the system in real time, using as
a variational model a Jastrow-Feenberg wave function that includes pair
correlations. Comparison with exact diagonalization results of
\bm{S(k,\omega)}𝐒(𝐤,𝛚)
obtained on a lattice in the Bose-Hubbard limit shows good agreement of
the dispersion relation for sufficiently deep optical lattices, while
for shallow lattices we observe the influence of higher Bloch bands. We
also investigate non-linear response to strong pulses. From the power
spectrum of the density fluctuations we obtain the excitation spectrum,
albeit broadened, by higher harmonic generation after a strong pulse
with a single low wave number. As a remarkable feature of our
simulations we furthermore demonstrate that the full excitation spectrum
can be retrieved from the power spectrum of the density fluctuations due
to the stochastic noise inherent in any Monte Carlo method, without
applying an actual perturbation.
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