  We study the quantum quench problem for a class of bosonic conformal
interfaces by computing the Loschmidt echo and the bipartite fidelity. The
quench can be viewed as a sudden change of boundary conditions parameterized by
$\theta$ when connecting two one-dimensional critical systems. They are
classified by $S(\theta)$ matrices associated with the current scattering
processes on the interface. The resulting Loschmidt echo of the quench has long
time algebraic decay $t^{-\alpha}$, whose exponent also appears in the finite
size bipartite fidelity as $L^{-\frac{\alpha}{2}}$. We perform analytic and
numerical calculations of the exponent $\alpha$, and find that it has a
quadratic dependence on the change of $\theta$ if the prior and post quench
boundary conditions are of the same type of $S$, while remaining $\frac{1}{4}$
otherwise. Possible physical realizations of these interfaces include, for
instance, connecting different quantum wires (Luttinger liquids), quench of the
topological phase edge states, etc. and the exponent can be detected in a X-ray
edge singularity type experiment.
