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

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