Realization and application of parity-time-symmetric oscillators in quantum regime

Although people have already artificially formed parity--time ($\mathcal{PT}$) symmetry with gain and loss in a balanced manner, it is still a defect that the gain is restricted to semi--classical but not full quantum. Here we propose and analyze a theoretical scheme to realize full quantum oscillator $\mathcal{PT}$--symmetry. The quantum gain is provided by a dissipation optical cavity with blue detuned laser field. After adiabatically eliminating the cavity modes, we give an effective master equation, which is a more complete quantum description compared with non--Hermitian Hamiltonian, to reveal the quantum behaviors of such a gain oscillator. This kind of $\mathcal{PT}$--symmetry can eliminate the dissipation effect in quantum regime. As examples, we finally apply $\mathcal{PT}$--symmetric oscillators to enhance optomechanically induced transparency and to preserve oscillator non--classical state.