  We provide the first solution of a time-dependent metric operator for the
non-Hermitian Jaynes-Cummings Hamiltonian. We use this solution to calculate
the entanglement between two identical isolated such Hamiltonians. The presence
of a non-Hermitian interaction term leads to a spontaneously broken
$\mathcal{PT}$-symmetric regime which manifests itself in the exotic
time-evolution of entanglement. When the symmetry is broken, oscillatory modes
transition into decay. As such that there is a drastic difference in behaviour
between the broken and unbroken regimes.
