Gain on ground state of quantum system for truly $\mathcal{PT}$ symmetry

For a truly $\mathcal{PT}$-symmetric quantum system, the conventional non-Hermitian Hamiltonian is $H = \Omega\sigma_x -i\gamma|1\rangle\langle1| + i\gamma|0\rangle\langle0|$, where $\Omega$ and $\gamma$ are real parameters and $\sigma_x$ denotes Pauli X operator. These three terms represent coherent coupling, loss (on state $|1\rangle$), and gain (on state $|0\rangle$), respectively. Although the works in [Phys. Rev. Lett. \textbf{101}, 230404 (2008); Phys. Rev. Lett. \textbf{119}, 190401 (2017); Science \textbf{364}, 878 (2019)] proposed theoretically and/or demonstrate dilation methods for a truly parity-time($\mathcal{PT}$)-symmetric Hamiltonian by embedding into larger Hermitian space, directly realizing the gain term $+i\gamma|0\rangle\langle0|$ has still remained an outstanding challenge for quantum system. While systems omitting this gain term can exhibit a passively $\mathcal{PT}$-symmetric energy spectrum (featuring a parallel imaginary shift) and display related phenomena, they fail to capture the full physical behavior and unique properties inherent to truly $\mathcal{PT}$-symmetric systems. In this manuscript, we propose a method to achieve effective gain on the ground state $|0\rangle$ ($+i\gamma|0\rangle\langle0|$) after averaging all trajectories, by integrating the S{\o}rensen-Reiter effective operator method with the Wiseman-Milburn master equation for continuous measurement and instantaneous feedback control after averaging the evolution over all trajectories. This approach provides a possible pathway to efficiently construct truly $\mathcal{PT}$-symmetric quantum devices, offering a powerful platform for engineering quantum resources vital for quantum information technology applications.