An Engineer’s Brief Introduction to Microwave Quantum Optics

Abstract

Classical microwave circuit theory is incomplete in that it is incapable of fully modeling some phenomena at the quantum level. Various theoretical frameworks can be employed to incorporate single-photon statistical effects in the treatment of microwave networks. Such methods include quantum input-output network (QION) theory and SLH theory. A synthesis of the quantum and classical circuit treatments requires a description of second quantization within classical microwave theory. In order to make these topics understandable to an electrical engineer, we demonstrate some underpinnings of quantum optics in terms of microwave engineering. For example, we relate traveling-wave phasors for transmission-lines, such as voltage and current, to bosonic field operators. The second quantization and necessary components for a quantum treatment of microwave circuit theory are summarized in a table that maps microwave scattering parameters to bosonic operators in a transmission-line. To illustrate the need for second quantization, we use the results of second quantization along with first principles of quantum input-output network theory to determine a state-space representation and a transfer function of a single port quantum network. The same results could be obtained from SLH theory, and this serves as a case study for applying microwave theory to open quantum systems. The results of this work could be used to treat NISQ and other superconducting hardware. Additionally, these results could be incorporated into an applied curriculum, assisting in the engineering education of the future quantum workforce.