Quantum manipulation of a single trapped molecular ion

The controlled manipulation of quantum states of single trapped atomic ions forms the basis of some of the most precise measurements preformed to date with proven applications in fundamental physics, time keeping and quantum computing. In this thesis, we extend the toolbox of coherent manipulation of single trapped ions to molecular ions with potential applications including measuring a possible time variation of the proton-to-electron mass ratio, $m_p/m_e$, the implementation of new frequency standards in the mid-infrared regime and the realization of noise-insensitive qubits. We describe in detail the theoretical modeling of molecular energy levels, systematic shifts and transition strengths for the identification of molecular transitions which are useful as a new clock standard and as a molecular qubit. The homonuclear diatomic molecule N$_2^+$ is found to form a noise-insensitive system with clock transitions suitable for precision measurements over a wide range of frequencies. We further describe the experimental implementation of a single-molecule trapped-ion experiment for precision measurements including the design, manufacturing and characterization of a new ion trap and the electronic circuits required for stable operation. We describe several techniques used for laser stabilization and present the techniques developed for cooling the molecular ion from an initial temperature of over 1000 K to the motional ground state of the trap below 10 µK. A new state readout technique is presented which relies on phase-sensitive forces to non-destructively read out and prepare the internal state of the molecule from a large number of possible states. The demonstration of state readout and state preparation of a single ground-state-cooled N$_2^+$ ion signifies the successful implementation of all necessary prerequisites for precision measurements and coherent manipulations of single molecular ions.