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A covariant Wigner-Weyl quantization formalism on the manifold that uses pseudo-differential operators is proposed. The asymptotic product formula that leads to the symbol calculus in the presence of gauge and gravitational fields is presented. The new definition is used to get covariant differential operators from momentum polynomial symbols. A covariant Wigner function is defined and shown to give gauge-invariant results for the Landau problem. An example of the covariant Wigner function on the 2-sphere is also included.

In this thesis, superconducting nanostructures for quantum detection of electromagnetic radiation are studied. In this regard, electrodynamics of topological excitations in 1D superconducting nanowires and 2D superconducting nanostrips is investigated. Topological excitations in superconducting nanowires and nanostrips lead to crucial deviation from the bulk properties. In 1D superconductors, topological excitations are phase slippages of the order parameter in which the magnitude of the order parameter locally drops to zero and the phase jumps by integer multiple of 2\pi. We investigate the effect of high-frequency field on 1D superconducting nanowires and derive the complex conductivity. Our study reveals that the rate of the quantum phase slips (QPSs) is exponentially enhanced under high-frequency irradiation. Based on this finding, we propose an energy-resolving terahertz radiation detector using superconducting nanowires. In superconducting nanostrips, topological fluctuations are the magnetic vortices. The motion of magnetic vortices result in dissipative processes that limit the efficiency of devices using superconducting nanostrips. It will be shown that in a multi-layer structure, the potential barrier for vortices to penetrate inside the structure is elevated. This results in significant reduction in dissipative process. In superconducting nanowire single photon detectors (SNSPDs), vortex motion results in dark counts and reduction of the critical current which results in low efficiency in these detectors. Based on this finding, we show that a multi-layer SNSPD is capable of approaching characteristics of an ideal single photon detector in terms of the dark count and quantum efficiency. It is shown that in a multi-layer SNSPD the photon coupling efficiency is dramatically enhanced due to the increase in the optical path of the incident photon.