Numerical analysis of motional mode coupling of sympathetically cooled two-ion crystals

A two-ion pair in a linear Paul trap is extensively used in the research of the simplest quantum-logic system; however, there are few quantitative and comprehensive studies on the motional mode coupling of two-ion systems yet. This study proposes a method to investigate the motional mode coupling of sympathetically cooled two-ion crystals by quantifying three-dimensional (3D) secular spectra of trapped ions using molecular dynamics simulations. The 3D resonance peaks of the ⁴⁰Ca⁺-²⁷Al⁺ pair obtained by using this method were in good agreement with the 3D in- and out-of-phase modes predicted by the mode coupling theory for two ions in equilibrium and the frequency matching errors were lower than 2%. The obtained and predicted amplitudes of these modes were also qualitatively similar. It was observed that the strength of the sympathetic interaction of the ⁴⁰Ca⁺-²⁷Al⁺ pair was primarily determined by its axial in-phase coupling. In addition, the frequencies and amplitudes of the ion pair's resonance modes (in all dimensions) were sensitive to the relative masses of the ion pair, and a decrease in the mass mismatch enhanced the sympathetic cooling rates. The sympathetic interactions of the ⁴⁰Ca⁺-²⁷Al⁺ pair were slightly weaker than those of the ²⁴Mg⁺-²⁷Al⁺ pair, but significantly stronger than those of ⁹Be⁺-²⁷Al⁺. However, the Doppler cooling limit temperature of ⁴⁰Ca⁺ is comparable to that of ⁹Be⁺ but lower than approximately half of that of ²⁴Mg⁺. Furthermore, laser cooling systems for ⁴⁰Ca⁺ are more reliable than those for ²⁴Mg⁺ and ⁹Be⁺. Therefore, ⁴⁰Ca⁺ is probably the best laser-cooled ion for sympathetic cooling and quantum-logic operations of ²⁷Al⁺ and has particularly more notable comprehensive advantages in the development of high reliability, compact, and transportable ²⁷Al⁺ optical clocks. This methodology may be extended to multi-ion systems, and it will greatly aid efforts to control the dynamic behaviors of sympathetic cooling as well as the development of low-heating-rate quantum logic clocks.