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Simulating mixed-state evolution in open quantum systems is crucialfor various chemical physics, quantum optics, and computer scienceapplications. These simulations typically follow the Lindblad master equationdynamics. An alternative approach known as quantum state diffusion unravelingis based on the trajectories of pure states generated by random wave functions,which evolve according to a nonlinear It\^o-Schr\"odinger equation (ISE). Thisstudy introduces weak first- and second-order solvers for the ISE based ondirectly applying the It\^o-Taylor expansion with exact derivatives in theinteraction picture. We tested the method on free and driven Morse oscillatorscoupled to a thermal environment and found that both orders allowed practicalestimation with a few dozen iterations. The variance was relatively smallcompared to the linear unraveling and did not grow with time. The second-ordersolver delivers much higher accuracy and stability with bigger time steps thanthe first-order scheme, with a small additional workload. However, thesecond-order algorithm has quadratic complexity with the number of Lindbladoperators as opposed to the linear complexity of the first-order algorithm.

Weak second-order quantum state diffusion unraveling of the Lindblad master equation