  We consider a system of two iso-spectral bosonic modes coupled with a single
two-level systems i.e., a qubit. The dynamics is described by a mode-symmetric
two-modes Jaynes-Cummings. The entanglement, induced between the two bosonic
modes, is analyzed and quantified by negativity. We computed the time evolution
of negativity starting from an initial thermal state of the bosonic sector for
both zero and finite temperature. We also studied the entangling power of the
interaction as a function of mode-qubit detuning and its resilience against
temperature increase. Finally a two-qubit gates based on bosonic virtual
subsystem is discussed.
