In this dissertation, we study the ultra-cold atomic many-body quantum phenomena in a quantum cavity. In this study of atom-light interaction, the ground state energy spectra of two-level atoms in a three-dimensional photonic crystal resonant cavity at absolute zero are calculated. We theoretically predict the reversible phenomenon of the quantum phase transition from Mott insulator to superuid and
also extended it to two atomic species systems. In this study of atom-atom interaction, we find out the analytical solution for steady-state of a Bose-Einstein condensate through pure mathematical analysis. Current experiments can conduct this measurement. These theories, if further developed, will be very useful for the successful development of a quantum computer chip.

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