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Born-Oppenheimer approximation for open quantum systems within the quantum trajectory approach

Based on the quantum trajectory approach, we extend the Born-Oppenheimer (BO) approximation from closed quantum system to open quantum system, where the open quantum system is described by a master equation in Lindblad form. The BO approximation is defined and the validity condition is derived. We find that the dissipation in fast variables benefits the BO approximation that is different from the dissipation in slow variables. A detailed comparison between this extension and our previous approximation (that is based on the effective Hamiltonian approach, see X. L. Huang and X. X. Yi, Phys. Rev. A 80, 032108 (2009)) is presented. Several new features and advantages are analyzed, which show that the two approximations are complementary to each other. Two examples are taken to illustrate our method.