Resolved-sideband micromotion sensing in trapped ¹⁷¹Yb⁺ ions using Lindblad/Rate equation

We have investigated excess micromotion for a Yb \(^+\) ion in a Paul trap. In the presence of light modulated by excess micromotion, we calculate the cooling florescence spectrum of the 369.53 nm transition by finding the steady periodic state solution of the Lindblad master equation including the time-dependent Hamiltonian. We have shown that applying a 369.53 nm laser for specific RF parameters does not resolve efficiently micromotion sidebands. To sense well-resolved sidebands micromotion, we solve the Lindblad equation for the four-level optical transitions of the cooling cycle by incorporating the introduced time-dependent Hamiltonian and the branching ratio for spontaneous decay rates at the low intensity limit. However, in the weak-driving regime the rate equation under adiabatic approximation is implemented to detect the excess micromotion effects on the cooling florescence as a function of the repump detuning. This steady state florescence rate of P-state based on the rate equation approach leads to results in agreement with the Lindblad master equation, but with much less computational effort and no numerical issues. We show that by scanning the detuning of the repump laser, the cooling spectrum could be applied as an efficient micromotion sensor at \(Ω _{{c/r}}=0.1 Γ _{{P/B}}\) . Our comprehensive discussion provides a theoretical and practical framework to predict the accurate experimental observations of sideband micromtion sensing in the trapped Yb \(^+\) ion which can be used and also extended.