  Using the example of the Jaynes-Cummings model, we present a comparison
between time-delayed coherent feedback mediated by reservoirs with continuous
and discrete mode structures and work out their qualitative differences. In
contrast to the discrete-mode case, the continuous-mode case results in the
well-known single-delay dynamics which can, e.g., stabilize Rabi oscillations.
The discrete-mode case, however, shows population trapping, not present in the
continuous-mode model. Given these differences, we discuss the cavity output
spectra and show how these characteristic properties are spectrally
identifiable. This work demonstrates the fundamental difference between the
continuous-mode case, which represents a truly dissipative mechanism, and the
discrete-mode case that is in principle based on a coherent excitation exchange
process.
