  We study inhomogeneous quantum quenches in the attractive regime of the
sine-Gordon model. In our protocol, the system is prepared in an inhomogeneous
initial state in finite volume by coupling the topological charge density
operator to a Gaussian external field. After switching off the external field,
the subsequent time evolution is governed by the homogeneous sine-Gordon
Hamiltonian. Varying either the interaction strength of the sine-Gordon model
or the amplitude of the external source field, an interesting transition is
observed in the expectation value of the soliton density. This affects both the
initial profile of the density and its time evolution and can be summarised as
a steep transition between behaviours reminiscent of the Klein-Gordon, and the
free massive Dirac fermion theory with initial external fields of high enough
magnitude. The transition in the initial state is also displayed by the
classical sine-Gordon theory and hence can be understood by semi-classical
considerations in terms of the presence of small amplitude field configurations
and the appearance of soliton excitations, which are naturally associated with
bosonic and fermionic excitations on the quantum level, respectively. Features
of the quantum dynamics are also consistent with this correspondence and
comparing them to the classical evolution of the density profile reveals that
quantum effects become markedly pronounced during the time evolution. These
results suggest a crossover between the dominance of bosonic and fermionic
degrees of freedom whose precise identification in terms of the fundamental
particle excitations can be rather non-trivial. Nevertheless, their interplay
is expected to influence the sine-Gordon dynamics in arbitrary inhomogeneous
settings.
