Using the quantum fast Fourier transform in linear optics the input mode annihilation operators ͕â 0 , â 1 ,. .. ,â s−1 ͖ are transformed into output mode annihilation operators ͕b 0 , b 1 ,. .. ,b s−1 ͖. We show how to implement experimentally such transformations based on the Cooley-Tukey algorithm, by the use of beam splitters and phase shifters in a linear optical system. Optical systems implementing 1,2, and 3 qubits discrete Fourier transform (DFT) are described, and a general method for implementing the n-qubit DFT is analyzed. These transformations are used on various input radiation states by which phase estimation and order finding can be computed.
Theoretical and experimental studies of Berry and Pancharatnam phases are reviewed. Basic elements of differential geometry are presented for understanding the topological nature of these phases. The basic theory analyzed by Berry in relation to magnetic monopoles is presented. The theory is generalized to nonadiabatic processes and to noncyclic Pancharatnam phases. Different systems are discussed including polarization optics, n-level atomic system, neutron interferometry and molecular topological phases.
The microsphere optical nanoscopy (MONS) technique recently demonstrated the capability to break the optical diffraction limit with a microsphere size of 2-9 µm fused silica. We report that larger polystyrene microspheres of 30, 50 and 100 µm diameters can overcome the diffraction limit in optical imaging. The sub-diffraction features of a Blu-ray Disc and gold nano-patterned quartz were experimentally observed in air by coupling the microspheres with a standard optical microscope in the reflected light illumination mode. About six to eight times magnification was achieved using the MONS. The mechanism of the MONS was theoretically explained by considering the transformation of near-field evanescent waves into far-field propagating waves. The super-resolution imaging was demonstrated by experiments and theoretical simulations.
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