  The purpose of this paper is to present a comprehensive study of a coherent
feedback network where the main component consists of two distant double
quantum dot (DQD) qubits which are directly coupled to a cavity. This main
component has recently been physically realized (van Woerkom, {\it et al.},
Microwave photon-mediated interactions between semiconductor qubits, Physical
Review X, 8(4):041018, 2018). The feedback loop is closed by cascading this
main component with a beamsplitter. The dynamics of this coherent feedback
network is studied from three perspectives. First, an analytic form of the
output single-photon state of the network driven by a single-photon state is
derived; in particular, it is observed that coherent feedback elongates
considerably the interaction between the input single photon and the network.
Second, excitation probabilities of DQD qubits are computed when the network is
driven by a single-photon input state. Moreover, if the input is vacuum but one
of the two DQD qubits is initialized in its excited state, the explicit
expression of the state of the network is derived, in particular, it is shown
that the output field and the two DQD qubits can form an entangled state if the
transition frequencies of two DQD qubits are equal. Finally, the exact form of
the pulse shape is obtained by which the single-photon input can fully excite
one of these two DQD qubits at any controllable time, which may be useful in
the construction of $2$-qubit quantum gates.
