Theory of Microwave to Optical Photon Upconversion Using Erbium Doped Crystals

The subject of this thesis is coherent conversion from microwave photons to optical photons. This has applications for quantum computing, and would allow interfacing between superconducting qubits and optical fibres. Specifically, we will focus on photon conversion via erbium ions in a doped crystal. We model a device which has an erbium doped crystal inside overlapping microwave and optical cavities. Input microwave photons are combined with photons from an optical pump laser via interactions with the erbium ions. This will produce optical output photons carrying the same quantum information as the input microwave photons. We develop a description of the interaction between the atoms and the light fields, accounting for both loss effects in the single atoms, and inhomogeneous broadening of the ensemble. Our theory is compared with experimental data, and shows good agreement. We explore various phenomena which arise in this system, including the effects of temperature and microwave power. It is shown that the description of this device from earlier work is not valid in the regime where the conversion efficiency is greatest. Hence, this work is necessary to be able to predict the maximum conversion efficiency. We develop a linearised model which is accurate in the regime where the microwave and optical fields are small, such as the regime used for quantum information. This model is used to maximise the photon number conversion efficiency of the device. We predict conversion efficiencies above 20%, far higher than has been achieved experimentally with using rare earth ions. Modifying our device should further increase the conversion efficiency to above 80%