Solution of the master equation for the PT-symmetry processes

Following the work of Bender et al. [1], systems with parity and time symmetry (that is, they are invariant under a mirror reflection and complex conjugation) have been extensively studied. The interesting aspect of PT-symmetric theory is to replace the traditional condition that the Hamiltonian of a quantum system must be Hermitian with the weaker condition that it combines parity and time symmetry. This allows us to study a wide class of non-Hermitian Hamiltonians that would previously have been ignored. Indeed, depending on external parameters the PT symmetry of the systems may be spontaneously broken and the existence of this threshold point is essential to those unique characteristics of PT systems, such as the nonreciprocal transmission of light [2] and simultaneous lasing and absorption [3]. Most of the works to date are based on closed systems and ignore quantum noises. However, gain and loss process are open quantum process which will add quantum fluctuations to the system. In addition, due to the quantum fluctuations, some interesting features different from those in classical systems have been found. Therefore to fully understand the PT symmetry process, it is necessary to have a complete quantum-mechanical description of the system with the evolution of quantum fluctuations. In this work, we start from the quantum master equation and develop a formalism suitable for describing PT-symmetry processes in open quantum systems and we propose a new term called effective density matrix. From this new term we can get the evolution of the input state and predict the result of any measurement performed on the system. Our results reveal the full quantum nature of the PT-symmetry process.