Quantum input-output response analysis is a useful method for modeling the dynamics of complex quantum networks, such as those for communication or quantum control via cascade connections. Non-Markovian and nonlinear effects are expected to be important in networks realized using mesoscopic circuits. Here we first extend the Markovian input-output network formalism to non-Markovian networks, and apply it to various examples. Second, as an natural extension of the input-output formalism of Gardiner and Collet, we develop a new approach based on the so-called quantum Volterra series which can greatly reduce the computational complexity of nonlinear quantum input-output analysis. By this method, we can ignore the internal dynamics of the input-output system and attribute all the information we need to a series of scalar kernal functions. This method can be used to describe the quantum network with both nonlinear components and Bogoliubov components such as quantum amplifers whic h cannot be modelled by the existing methods such as Hudson-Parthasarathy model and quantum transfer function model.
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