Quantum dynamical phase transition in a system with many-body interactions

We introduce a microscopic Hamiltonian model of a two level system with many-body interactions with an environment whose excitation dynamics is fully solved within the Keldysh formalism. If a particle starts in one of the states of the isolated system, the return probability oscillates with the Rabi frequency ${\omega }_0$. For weak interactions with the environment $1\mathrm{/}{\tau }_{\mathrm{S}\mathrm{E}}<2{\omega }_0,$ we find a slower oscillation whose amplitude decays with a decoherence rate $1/\tau_{\phi}=1/(2\tau_{$\mathrm{S}\mathrm{E}$% })$\mathrm{.}However,beyondafinitecriticalinteractionwiththeenvironment,$1/\tau_{$\mathrm{S}\mathrm{E}$}>2\omega_{0}$,thedecoherenceratebecomes$1/\tau_{\phi}\propto(\omega_{0}^{2})\tau_{$\mathrm{S}\mathrm{E}$}$. The oscillation period diverges showing a \emph{quantum dynamical phase transition}to a Quantum Zeno phase.