Classical and Quantum Analysis of Phase-Insensitive Systems

Abstract

In Chap. 6 we investigated the quantization of open resonators and of waves on transmission lines. We treated one example of a simple linear system, namely a resonator coupled to a waveguide. Practical electromagnetic systems consist of RLC circuits, resonators, waveguide junctions, fibers, beam splitters, and, of course, amplifiers, to name only a few. Such systems, if linear, are described classically by impedance matrices or scattering matrices (Chap. 2) that are functions of frequency. This formalism is well developed in the classical domain. In this chapter, we review the classical formalism and its generalization to quantum theory. We define Hamiltonians which, via the Heisenberg equations of motion, lead to equations that are in direct correspondence with the classical circuit equations. If the multiports are lossy or exhibit gain, they must contain noise sources in order to conserve commutator brackets from input to output. The commutator brackets determine the minimum amount of noise added to the signal as it passes through the network. Hence one may determine the optimum noise measure achievable in a quantum circuit directly from these relations.