  We investigate the response of a one-dimensional Bose gas to a slow increase
of its interaction strength. We focus on the rich dynamics of equal-time
single-particle correlations treating the Lieb-Liniger model within a
bosonization approach and the Bose-Hubbard model using the time-dependent
density-matrix renormalization group method. For short distances, correlations
follow a power-law with distance with an exponent given by the adiabatic
approximation. In contrast, for long distances, correlations decay
algebraically with an exponent understood within the sudden quench
approximation. This long distance regime is separated from an intermediate
distance one by a generalized Lieb-Robinson criterion. At long times, in this
intermediate regime, bosonization predicts that single-particle correlations
decay following a stretched exponential. This latter regime is unconventional
as, for one-dimensional interacting systems, the decay of single-particle
correlations is usually algebraic within the Luttinger liquid picture. We
develop here an intuitive understanding for the propagation of correlations, in
terms of a generalized light-cone, applicable to a large variety of systems and
quench forms.
