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Scientific Reports

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

Nadia Salami, M. Aslaninejad, Shahram Solaymani, Jamshid Sabbaghzadeh

Published Oct 20, 2026• 0 citations
Publisher

Abstract

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 \(\Omega _\mathrm{{c/r}}=0.1 \Gamma _\mathrm{{P/B}}\) . 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.

DOI
10.1038/s41598-026-76231-3

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Resolved-sideband micromotion sensing in trapped 171^{171}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 (\Omega _\mathrm{{c/r}}=0.1 \Gamma _\mathrm{{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.
Ions displaced from the potential minimum in an RF Paul trap exhibit excess micromotion. A host of well-established techniques are routinely used to sense (and null) this excessive motion in applications ranging from quantum computing to atomic clocks. The rich atomic structure of Yb+, a heavy ion, includes low-lying 2D3/2 states that must be repumped to permit Doppler cooling, typically using a 935 nm laser coupled to the 3D[3/2]1/2 states. In this article, we demonstrate the use of this transition to make resolved-sideband measurements of 3D micromotion in 172Yb+ and 171Yb+ ions. Relative to other sensing techniques, our approach has very low technical overhead and is distinctively compatible with surface-electrode ion traps.
An experimental and theoretical study of laser sideband cooling of a trapped ion in a radiofrequency trap is presented. The influence of the micromotion in the time-dependent potential of a Paul trap can lead to a counterintuitive situation, where sideband cooling is possible for positive values of the laser detuning, i.e., for a laser frequency that is higher than the resonance frequency of the ion at rest. The cooling rate and the capture range are calculated in a semiclassical model. Experimental results of laser sideband cooling at positive detuning are demonstrated with a single trapped In+{\mathrm{In}}^{+}In+ ion stored in a miniature Paul trap.
An experimental study of sideband laser cooling of trapped ions in miniature radio-frequency traps is presented. In+{\mathrm{In}}^{+}In+ ions are laser cooled by exciting the narrow intercombination line ${5s}^{2}{}^{1}{S}_{0}\ensuremath{\rightarrow}5s5p{}^{3}{P}_{1}$, which shows resolved sidebands due to the vibrational motion of the ions in the trap. The influence of the micromotion at the frequency of the oscillating trapping field is investigated. Heating processes through coupling with the trapping field occur for certain regions of negative laser detuning. A method of bichromatic sideband cooling is introduced that combines efficient cooling from high temperatures with the possibility of reaching low mean vibrational quantum numbers 〈n〉<1〈n〉<1〈n〉<1 by using narrow optical transitions. This technique allows recording of high-resolution excitation spectra of the cooling transition in the Lamb-Dicke regime and thus determination of 〈n〉〈n〉〈n〉. Single ions as well as two-ion Coulomb crystals have been cooled to temperatures below $100 \ensuremath{\mu}\mathrm{K}$, and population of the ground state of motion to more than 50% over long time scales is demonstrated. These values are close to the lower limit for the detection of vibrational excitation with the present method.
We demonstrate minimization of ion micromotion in a linear Paul trap with the use of a high finesse cavity. The excess ion micromotion projected along the optical cavity axis or along the laser propagation direction manifests itself as sideband peaks around the carrier in the ion-cavity emission spectrum. By minimizing the sideband height in the emission spectrum, we are able to reduce the micromotion amplitude along two directions to approximately the spread of the ground state wave function. This method is useful for cavity QED experiments as it describes the possibility of efficient 3-D micromotion compensation despite optical access limitations imposed by the cavity mirrors. We also show that, in principle, sub-nanometer micromotion compensation is achievable with our current system.
We demonstrate minimization of ion micromotion in a linear Paul trap with the use of a high finesse cavity. The excess ion micromotion projected along the optical cavity axis or along the laser propagation direction manifests itself as sideband peaks around the carrier in the ion-cavity emission spectrum. By minimizing the sideband height in the emission spectrum, we are able to reduce the micromotion amplitude to approximately the spread of the ground state wave function. This method is useful for cavity QED experiments as it allows for efficient 3-D micromotion compensation despite optical access limitations imposed by the cavity mirrors. We also show that sub-nanometer micromotion compensation is possible with our current system.
We investigate sideband spectroscopy of a trapped ion using a probe laser phase modulated at the trap drive frequency. The enhanced sensitivity of our technique provides us 1000-fold improvement in our control of excess micromotion. It allows us to monitor the evolution of stray fields on a timescale of a few minutes and detect differential phases of $5\phantom{\rule{0.16em}{0ex}}\textmu{}\mathrm{rad}$ between applied ac potentials. We also demonstrate the ability to suppress Doppler shifts from excess motion to well below the limit imposed by the intrinsic motion of the ion in its motional ground state. The technique we introduce can be readily implemented in any ion-trap system that utilizes sideband spectroscopy for micromotion compensation and can be seamlessly integrated into experiments in a fully automated way.