Quantum input-output theory plays a very important role for analyzing the
dynamics of quantum systems, especially large-scale quantum networks. As an
extension of the input-output formalism of Gardiner and Collet, we develop a
new approach based on the quantum version of the Volterra series which can be
used to analyze nonlinear quantum input-output dynamics. By this approach, we
can ignore the internal dynamics of the quantum input-output system and
represent the system dynamics by a series of kernel functions. This approach
has the great advantage of modelling weak-nonlinear quantum networks. In our
approach, the number of parameters, represented by the kernel functions, used
to describe the input-output response of a weak-nonlinear quantum network,
increases linearly with the scale of the quantum network, not exponentially as
usual. Additionally, our approach can be used to formulate the quantum network
with both nonlinear and nonconservative components, e.g., quantum amplifiers,
which cannot be modelled by the existing methods, such as the
Hudson-Parthasarathy model and the quantum transfer function model. We apply
our general method to several examples, including Kerr cavities, optomechanical
transducers, and a particular coherent feedback system with a nonlinear
component and a quantum amplifier in the feedback loop. This approach provides
a powerful way to the modelling and control of nonlinear quantum networks.Comment: 12 pages, 7 figure