  A characteristic feature of collective and particle-hole excitations in
neutron-rich nuclei is that many of them couple to unbound neutron in continuum
single-particle orbits. The continuum random phase approximation (cRPA) is a
powerful many-body method that describes such excitations, and it provides a
scheme to evaluate transition strengths from the ground state. In an attempt to
apply cRPA to the radiative neutron capture reaction, we formulate in the
present study an extended scheme of cRPA that describes gamma-transitions from
the excited states under consideration, which decay to low-lying excited states
as well as the ground state. This is achieved by introducing a non-local
one-body operator which causes transitions to a low-lying excited state, and
describing a density-matrix response against this operator. As a demonstration
of this new scheme, we perform numerical calculation for dipole, quadrupole,
and octupole excitations in $^{140}$Sn, and discuss E1 and E2 transitions
decaying to low-lying $2^{+}_{1,2}$ and $3^{-}_{1}$ states. The results point
to cases where the branching ratio to the low-lying states is larger than or
comparable with that to the ground state. We discuss key roles of collectivity
and continuum orbits in both initial and final states.
