One port impedance quantization for a class of annihilation operator linear quantum systems

Theoretical developments in the field of superconducting microwave circuits involving continuous measurement feedback control as well as coherent feedback control are essential for advances in the domain of quantum technology. The smallest structures that can be fabricated by lithographic technologies request the inevitable development of the quantum control theory in order for quantum systems to function reliably in an engineering sense. Within these perspectives, this paper develops the positive real properties for a class of quantum systems involving annihilation operators only and then relates them to the corresponding bounded real properties and consequently to physical realizability conditions developed earlier by the authors. Based on these positive real properties of the annihilation operators quantum systems, it is shown that it is possible to use the Brune algorithm in order to find electrical circuits that can physically implement the type of quantum systems involving annihilation operators only. This would involve Josephson junctions, microwave cavities, and voltage sources. This theory, illustrated for one-port circuits, is essential for the implementation of microwave circuits especially those related to quantum filters found in the field of quantum computing.