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arXiv:2402.01594 · Frontiers in Quantum Science and Technology

Dark resonance spectra of trapped ions under the influence of micromotion

Nicolás A. Nuñez Barreto, Muriel Bonetto, Marcelo Alejandro Luda, Cecilia Cormick, Christian Tomás Schmiegelow

Published Mar 27, 2024• 1 citation
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Abstract

We study the influence of micromotion on the spectrum of trapped ions with a lambda-type level scheme, leading to dark resonances due to coherent population trapping. We work with calcium ions trapped in a ring-shaped Paul trap, in which one can compensate excess micromotion for only one ion of the crystal. We observe that micromotion affects the shapes of the dark resonances and causes the appearance of “echoes” separated by intervals given by the drive frequency. We present a theoretical model that provides good fits to the measurements and can be used to estimate the amplitude of the micromotion modulation of the atomic motion. We estimate an effective temperature of the ions from the spectra and observe clear micromotion heating as well as impaired cooling for sufficiently large excess micromotion.

DOI
10.3389/frqst.2024.1381117

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We study the influence of micromotion on the spectrum of trapped ions with a lambda-type level scheme, leading to dark resonances due to coherent population trapping. We work with calcium ions trapped in a ring-shaped Paul trap, in which one can compensate excess micromotion for only one ion of the crystal. We observe that micromotion affects the shapes of the dark resonances and causes the appearance of “echoes” separated by intervals given by the drive frequency. We present a theoretical model that provides good fits to the measurements and can be used to estimate the amplitude of the micromotion modulation of the atomic motion. We estimate an effective temperature of the ions from the spectra and observe clear micromotion heating as well as impaired cooling for sufficiently large excess micromotion.
Single, rf-trapped ions find various applications ranging from metrology to quantum computation. High-resolution interrogation of an extremely weak transition under best observation conditions requires an ion almost at rest. To avoid line-broadening effects such as the second order Doppler effect or rf heating in the absence of laser cooling, excess micromotion has to be eliminated as far as possible. We have tested a novel method to probe the motional state of a confined three-level ion. Actually, dark resonances observed with two counterpropagating laser beams are very sensitive to the trapped ion's velocity. Their contrast increases with reduced micromotion and allows in this way to control the oscillation amplitude of the stored ion. The influence of different parameters such as laser linewidth and power has been investigated experimentally and numerically.
The simulation of vibrational energy transport and quantum thermodynamics with trapped ions requires good methods for the estimation of temperatures. One valuable tool for this purpose is based on the fit of dark resonances in the fluorescence spectrum. However, this fit demands numerical simulations of the coupled electronic-motional dynamics which usually involve a trade-off between accuracy and speed. Here, we discuss several techniques with simplified dynamical equations for the simulation of the spectrum of a trapped ion that undergoes thermal motion, identifying the advantages and limitations of each method. We start with a three-level model to provide a better insight into the approximations involved, and then move on to tackle the experimentally relevant case of an eight-level calcium ion. We observe that mimicking the effect of thermal motion by means of additional dephasing is computationally very convenient, but can lead to significant errors in the estimation of the temperature. Nevertheless, this can be counteracted by a proper calibration, supporting the use of dark resonances as a practical thermometer.
The accurate evaluation of the second-order Doppler frequency shift (SODFS) of trapped ions in a linear Paul trap has been studied with experiments and molecular dynamics (MD) simulations. The motion of trapped ions in the trap has three contributions, and we focus on the ion excess micromotion, which is rarely discussed when evaluating the SODFS of ions. Based on the hypothesis that the ion density is uniformly distributed in the radial direction, we propose a model to accurately evaluate the total SODFS for an ion ensemble with temperature less than 100 mK, which consists of three contributions of ion motion. The effectiveness of the model has been verified by MD simulation, and the experimental results of the excess micromotion of trapped ions are consistent with the theory. We believe that our model offers advantages in accurately evaluating the SODFS for the ion trap, especially those of laser-cooled ion microwave clocks based on large ion clouds.
Micromotion of ions in Paul traps has several adverse effects, including alterations of atomic transition line shapes, significant second-order Doppler shifts in high-accuracy studies, and limited confinement time in the absence of cooling. The ac electric field that causes the micromotion may also induce significant Stark shifts in atomic transitions. We describe three methods of detecting micromotion. The first relies on the change of the average ion position as the trap potentials are changed. The second monitors the amplitude of the sidebands of a narrow atomic transition, caused by the first-order Doppler shift due to the micromotion. The last technique detects the Doppler shift induced modulation of the fluorescence rate of a broad atomic transition. We discuss the detection sensitivity of each method to Doppler and Stark shifts, and show experimental results using the last technique.
Laser cooling of a single trapped ion in a Paul trap is discussed theoretically in the Lamb-Dicke limit, with full consideration of the time dependence of the trapping potential. Resulting mean kinetic energies are defined as time averages over one period of the micromotion and are compared with final temperatures expected from the laser-cooling treatment with harmonic traps. For laser-atom detunings close to the micromotion frequency the results differ significantly from those expected for a harmonic trap potential. A physical interpretation is given and simple formulas are derived for the strong confinement case.
As relative systematic frequency uncertainties in trapped-ion spectroscopy are approaching the low 10−18 range, motional frequency shifts account for a considerable fraction of the uncertainty budget. Micromotion, a driven motion fundamentally connected to the principle of the Paul trap, is a particular concern in these systems. In this article, we experimentally investigate at this level three common methods for minimizing and determining the micromotion amplitude. We develop a generalized model for a quantitative application of the photon-correlation technique, which is applicable in the commonly encountered regime where the transition linewidth is comparable to the rf drive frequency. We show that a fractional frequency uncertainty due to the 2nd-order Doppler shift below |Δν/ν|=1×10−20 can be achieved. The quantitative evaluation is verified in an interleaved measurement with the conceptually simpler resolved sideband method. If not performed deep within the Lamb-Dicke regime, a temperature-dependent offset at the level of 10−19 is observed in resolved sideband measurements due to sampling of intrinsic micromotion. By direct comparison with photon-correlation measurements, we show that the simple to implement parametric heating method is sensitive to micromotion at the level of |Δν/ν|=1×10−20 as well.