This dissertation reports on precision measurements of the atomic structure of calcium ions (⁴⁰Ca⁺), application of trapped ⁴⁰Ca⁺ ions as a probe for a violation of fundamental symmetry and prospects of performing quantum simulations with trapped ion.

We demonstrate a novel technique to perform spectroscopy on the dipole transition of ⁴⁰Ca⁺ that circumvents usual difficulties from dark resonances and Doppler heating. The center of the atomic transition can be detected to a precision of 200 kHz or less with an integration time of 10 minutes. We apply this method to directly measure the influence of micromotion on the fluorescence spectra and confirm the dependence of the modulation index on the radial trap frequency.

We measure the branching fraction of the excited ²P_1/2 state of ⁴⁰Ca⁺ to be 0.93565(7) using a simple experimental scheme readily applicable to many other ion species. Our result for ⁴⁰Ca⁺ distinguishes well among various theoretically calculated values, which is important in guiding further developments of the theoretical work.

We apply the Ramsey spectroscopy technique based on a pair of correlated ions to probe the effect of the violation of local Lorentz invariance (LLI). The energy difference between the two components of the Bell state |Psi_B> = |m_J=5/2,m_J=-5/2>+|m_J=1/2,m_J=-1/2> in the D_5/2 manifold of ⁴⁰Ca⁺ is monitored for 12 hours. We found that the energy component related to the violation of LLI varies less than 17±22 mHz. Assuming a hydrogen-like model of ⁴⁰Ca⁺, the measurement result provides us the bound of the LLI parameter C⁽²⁾₀ to be 1.7±2.2x10^-17.

Based on numerical simulations, we show that the Aubry transition in the Frenkel-Kontorova model with trapped ion can be observed for practical experimental parameters such as the strength and wavelength of the optical lattice, which serves as an external perturbing periodic potential. Moreover, we also show that the normal mode structure of ion chain can change significantly as we vary the strength of the optical lattice.