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arXiv:2505.04459 · Physical Review A

Numerical modeling for trapped-ion thermometry using dark resonances

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

Published Jul 29, 2025• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/117l-94nf

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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.
We experimentally demonstrate a method to determine the temperature of trapped ions which is suitable for monitoring fast thermalization processes. We show that observing and analyzing the lineshape of dark resonances in the fluorescence spectrum provides a temperature measurement which is accurate over a large dynamic range, applied to single ions and small ion crystals. Laser induced fluorescence is detected over a time of only , allowing for rapid determination of the ion temperature. In the measurement range of 10 −1 – mK we reach better than accuracy. Tuning the cooling laser to selected resonance features allows us to control the ion temperatures between mK and more than mK. Experimental work is supported by a solution of the eight-level optical Bloch equations when including the ions' classical motion. This technique paves the way for many experiments, including heat transport in ion strings, heat engines, non-equilibrium thermodynamics or thermometry of large ion crystals.
This thesis describes work on numerical modelling of the 43Ca+ ion in a Paul trap using the optical Bloch equations. This is a challenging system to study, with many states involved in the internal dynamics. A major outcome is the development of a cooling scheme for the 146.09 gauss atomic clock transitions which makes use of a dark resonance. It is much more effective than methods that avoid coherent effects. The scheme is realised in experiment. Complicated fluorescence data is modelled very well, and predictions for the ion's motional temperature show good agreement with measured values. Data and fits for an ion that has been Doppler cooled below the Doppler limit are presented. I describe GLOBES, a set of routines that simulates an arbitrary ion in the presence of an arbitrary system of laser beams using the optical Bloch equations. Techniques used to efficiently calculate the steady state, analyse fluorescence scans and solve time-dependent problems for small and large times are discussed. A new routine SILVER IMPER that leapfrogs over the initial dynamics to model the approach to the steady state is introduced. Doppler cooling in 40Ca+ is analysed and two extensions made to the basic theory. The ‘excursion method’ of calculation takes account of the non-linear variation with velocity of the scattering rate. The ‘dynamic method’ allows for the fact that the ion may not be in equilibrium with the incident radiation during its oscillations, a necessity as the timescale of the external motion is of order the characteristic timescale of the internal motion for standard secular frequencies. This 'dynamic effect' is a general property of trapped ion systems and is also observed in a two-state system. A two-variable fluorescence scan taken from a four-laser, five-level system in 40Ca+ is analysed. Techniques to fit large data sets and automatically resolve resonant features are demonstrated. A general treatment of resonant behaviour in three, four and five level pump/probe systems is used to describe the data. This is verified by a second, tailor-made set of scans.
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.
Doppler cooling with lasers is essential to ions' trapping and also a preliminary step towards achievement of ultracold ions. Due to lack of effective tools, experimentally monitoring the ions' temperature and the laser-ion coupling is difficult in Doppler cooling. Here we analytically explore the Doppler cooling process of trapped ions, exemplified by 40^{40}40Ca+^{+}+, by solving the friction coefficient in the Doppler cooling with respect to a thermal bath, particularly, to a bath with large heating rate. We show four regions for cooling and heating induced by the three-level electromagnetically induced transparency and propose a practical method for measuring the Rabi frequency by the Doppler cooling window. In addition, the final temperature of the laser-cooled ions can be obtained analytically in the case of a weak thermal bath, whereas for the strong thermal bath this requires numerical treatment due to involvement of the Doppler shift and large Lamb-Dicke parameter. Our analytic results would help for understanding many experimental observations, such as configuration phase transition, phonon laser and thermodynamics regarding hot trapped ions.
Coulomb crystals of cold trapped ions are a leading platform for the realization of quantum processors and quantum simulations and, in quantum metrology, for the construction of optical atomic clocks and for fundamental tests of the standard model. For these applications, it is not only essential to cool the ion crystal in all its degrees of freedom down to the quantum ground state but also to be able to determine its temperature with a high accuracy. However, when a large ground-state cooled crystal is interrogated for thermometry, complex many-body interactions take place, making it challenging to accurately estimate the temperature with established techniques. In this work, we present a new thermometry method tailored for ion crystals. The method is applicable to all normal modes of motion and does not suffer from a computational bottleneck when applied to large ion crystals. We test the temperature estimate with two experiments, namely with a one-dimensional linear chain of four ions and a two-dimensional crystal of 19 ions and verify the results, where possible, using other methods. The results show that the new method is an accurate and efficient tool for thermometry of ion crystals.
Coulomb crystals of cold trapped ions are a leading platform for the realization of quantum processors and quantum simulations and, in quantum metrology, for the construction of optical atomic clocks and for fundamental tests of the standard model.For these applications, it is not only essential to cool the ion crystal in all its degrees of freedom down to the quantum ground state but also to be able to determine its temperature with a high accuracy.However, when a large ground-state cooled crystal is interrogated for thermometry, complex many-body interactions take place, making it challenging to accurately estimate the temperature with established techniques.In this work, we present a new thermometry method tailored for ion crystals.The method is applicable to all normal modes of motion and does not suffer from a computational bottleneck when applied to large ion crystals.We test the temperature estimate with two experiments, namely with a one-dimensional linear chain of four ions and a two-dimensional crystal of 19 ions and verify the results, where possible, using other methods.The results show that the new method is an accurate and efficient tool for thermometry of ion crystals.