Analysis And Feedback Network Synthesis For Triplet-Type Linear Quantum Stochastic Systems

This paper is mainly concerned with the new quantum feedback network synthesis algorithm fortriplet-type quantum stochastic systems that are usually encountered in linear quantum optics and inphenomenological models of linear quantum circuits. The synthesis theory sought in this case canbe naturally viewed as a quantum analogue of linear electrical network synthesis theory and as suchhas potential for applications beyond the realization of coherent quantum feedback controllers. theprimary results obtained in this paper include the following three parts.In the first part, the general linear quantum stochastic systems and the triplet-type quantumstochastic systems are discussed from a control system theoretic perspective. First, we presentthe mathematical modelings of general linear quantum stochastic systems and triplet-type quantumstochastic systems. Second, another mathematical modeling of triplet-type quantum stochastic sys-tems is given, which plays an important role in the following conclusion. Then the necessary andsufficient condition for the equivalent of general linear quantum stochastic systems and the triplet-type quantum stochastic systems is put forward.In the second part, a new quantum feedback network synthesis algorithm for triplet-type quan-tum stochastic systems are developed. First, we recall and improve the quantum feedback networktheories and the concatenation and series products of triplet-type quantum stochastic systems. Thenwe show how an arbitrary triplet-type quantum stochastic system can be synthesized approximately ina systematic way from one degree of freedom generalized open quantum harmonic oscillators whichare easy to implement in practice. Especially, a new algorithm is developed to determine the cou-pling matrix of these one degree of freedom oscillators, which plays a significant role in constructingthe Hamiltonian of the triplet-type quantum stochastic system that will be synthesized. Moreover,an explicit synthesis example is provided to illustrate the realization of any two degrees of freedomtriplet-type quantum stochastic system.In the final part, the physical realization of any given triplet-type quantum stochastic systemis described based on the feedback network synthesis algorithm developed above. First, we detailthe construction of arbitrary one degree of freedom generalized open quantum harmonic oscillators,in the context of quantum optics, using various linear and nonlinear quantum optical components.Then using the feedback network synthesis algorithm developed in this paper we construct a quantumfeedback network to physically realize the triplet-type quantum stochastic system.