Self-consistent input-output formulation of quantum feedback

A simple method of analyzing quantum feedback circuits is presented. The classical analysis of feedback circuits can be generalized to apply to quantum systems by mapping the field operators of various outputs to other inputs via the standard input-output formalism. Unfortunately, this has led to unphysical results such as the violation of the Heisenberg uncertainty principle for in-loop fields. This paper shows that this general approach can be redeemed by ensuring a self-consistently Hermitian Hamiltonian. The calculations are based on a noncommutative calculus of operator derivatives. A full description of several examples of quantum linear and nonlinear feedback for optical systems is presented.