  A trijunction made of three topological semiconducting wires, each supporting
a Majorana bound state at its two extremities, appears as one of the simplest
geometry in order to perform braiding of Majorana fermions. By embedding the
trijunction into a microwave cavity allows to study the intricate dynamics of
the low-energy Majorana bound states (MBSs) coupled to the cavity electric
field under a braiding operation. Extending a previous work (Phys. Rev. Lett.
2019, 122, 236803), the full time evolution of the density matrix of the
low-energy states, including various relaxation channels, is computed both in
the adiabatic regime, as well as within the Floquet formalism in the case of
periodic driving. It turns out that in the stationary state the observables of
the system depend on both the parity of the ground state and on the non-Abelian
Berry phase acquired during braiding. The average photon number and the second
order photon coherence function $g^{(2)}(0)$ are explicitly evaluated and
reveal the accumulated non-Abelian Berry phase during the braiding process.
