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arXiv:2610.29718

Semiclassical molecular dynamics simulations of electromagnetically induced transparency cooling of 2D and 3D ion crystals in a Penning trap

John Zaris, Wes Johnson, Krishna Balaji Sogathur, Athreya Shankar, John J. Bollinger, Allison L. Carter, Murray J. Holland, Scott E. Parker

quant-phphysics.atom-phSubmitted Oct 21, 2026• Announced Oct 22, 2026• 0 citations
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Abstract

Motional modes of trapped-ion crystals in Penning traps can now be cooled to near their ground states via electromagnetically induced transparency (EIT) cooling. Numerical simulations are valuable tools for interpreting EIT cooling results and optimizing experimental protocols, but existing molecular dynamics codes are limited to the simulation of Doppler cooling. In this work, we fill this gap by implementing a semiclassical model of EIT cooling in a novel MD code. By simultaneously evolving the ions' internal quantum states and classical motion, we are able to calculate the instantaneous EIT laser forces acting on the ions. We first validate our code by comparing the single-ion cooling rate to fully quantum simulation results. Next, we analyze cooling of the drumhead modes in planar ion crystals, quantifying the effects of planar ion motion and a finite laser beam width. Finally, we demonstrate efficient EIT cooling of the axial modes in 3D ellipsoidal crystals containing up to 500 ions. Such crystals are a promising platform for future quantum sensing experiments with enhanced sensitivity. We find that the mode spectra of 3D crystals can be engineered to enable sympathetic cooling of the low-frequency $\boldsymbol{E}\times\boldsymbol{B}$ modes. This numerical framework provides a powerful tool for modeling sub-Doppler cooling techniques and for optimizing future quantum sensing and quantum information experiments.

Comments
25 pages, 15 figures

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Motional modes of trapped-ion crystals in Penning traps can now be cooled to near their ground states via electromagnetically induced transparency (EIT) cooling. Numerical simulations are valuable tools for interpreting EIT cooling results and optimizing experimental protocols, but existing molecular dynamics codes are limited to the simulation of Doppler cooling. In this work, we fill this gap by implementing a semiclassical model of EIT cooling in a novel MD code. By simultaneously evolving the ions' internal quantum states and classical motion, we are able to calculate the instantaneous EIT laser forces acting on the ions. We first validate our code by comparing the single-ion cooling rate to fully quantum simulation results. Next, we analyze cooling of the drumhead modes in planar ion crystals, quantifying the effects of planar ion motion and a finite laser beam width. Finally, we demonstrate efficient EIT cooling of the axial modes in 3D ellipsoidal crystals containing up to 500 ions. Such crystals are a promising platform for future quantum sensing experiments with enhanced sensitivity. We find that the mode spectra of 3D crystals can be engineered to enable sympathetic cooling of the low-frequency E×B\boldsymbol{E}\times\boldsymbol{B}E×B modes. This numerical framework provides a powerful tool for modeling sub-Doppler cooling techniques and for optimizing future quantum sensing and quantum information experiments.
Penning traps, with their ability to control planar crystals of tens to hundreds of ions, are versatile quantum simulators. Thermal occupations of the motional drumhead modes, transverse to the plane of the ion crystal, degrade the quality of quantum simulations. Laser cooling using electromagnetically induced transparency (EIT cooling) is attractive as an efficient way to quickly initialize the drumhead modes to near ground-state occupations. We numerically investigate the efficiency of EIT cooling of planar ion crystals in a Penning trap, accounting for complications arising from the nature of the trap and from the simultaneous cooling of multiple ions. We show that, in spite of challenges, the large bandwidth of drumhead modes (hundreds of kilohertz) can be rapidly cooled to near ground-state occupations within a few hundred microseconds. Our predictions for the center-of-mass mode include a cooling time constant of tens of microseconds and an enhancement of the cooling rate with increasing number of ions. Successful experimental demonstrations of EIT cooling in the NIST Penning trap [Jordan et al., [Phys. Rev. Lett. 122, 053603 (2019)] validate our predictions.
Large, 3D trapped ion crystals offer improved sensitivity in quantum sensing protocols, and are expected to be implemented as platforms in near-future experiments. However, numerical techniques used to study the laser cooling of such crystals are inefficient as the number of ions, NNN, in the crystal increases. Here we develop a powerful numerical framework to simulate laser cooling of up to 10510^5105 ions stored in a Penning trap. We apply this framework to characterize and optimize the cooling of ellipsoidal 3D crystals. We document new pathways to enhanced cooling based on the addition of an axial component to the potential energy-dominated E×B\boldsymbol{E}\times\boldsymbol{B}E×B modes. Furthermore, we observe greatly enhanced cooling of the perpendicular kinetic energy to below 1 mK in prolate ion crystals, enabling a simplified cooling beam setup for such crystals. We propose specific values of trap and laser beam parameters which lead to optimal cooling in a variety of examples. This work illustrates the feasibility of preparing large 3D crystals for high-sensitivity quantum science protocols, motivating their use in future experiments.
We simulate the dynamics, including laser cooling, of three-dimensional (3-D) ion crystals confined in a Penning trap using a newly developed molecular dynamics-like code. The numerical integration of the ions’ equations of motion is accelerated using the fast multipole method to calculate the Coulomb interaction between ions, which allows us to efficiently study large ion crystals with thousands of ions. In particular, we show that the simulation time scales linearly with ion number, rather than with the square of the ion number. By treating the ions’ absorption of photons as a Poisson process, we simulate individual photon scattering events to study laser cooling of 3-D ellipsoidal ion crystals. Initial simulations suggest that these crystals can be efficiently cooled to ultracold temperatures, aided by the mixing of the easily cooled axial motional modes with the low frequency planar modes. In our simulations of a spherical crystal of 1000 ions, the planar kinetic energy is cooled to several millikelvin in a few milliseconds while the axial kinetic energy and total potential energy are cooled even further. This suggests that 3-D ion crystals could be well suited as platforms for future quantum science experiments.
We experimentally study electromagnetically induced transparency cooling of the drumhead modes of planar two-dimensional arrays with up to N≈190 Be^{+} ions stored in a Penning trap. Substantial sub-Doppler cooling is observed for all N drumhead modes. Quantitative measurements for the center-of-mass mode show near ground-state cooling with motional quantum numbers of n[over ¯]=0.3±0.2 obtained within 200 μs. The measured cooling rate is faster than that predicted by single particle theory, consistent with a quantum many-body calculation. For the lower frequency drumhead modes, quantitative temperature measurements are limited by frequency instabilities, but near ground-state cooling of the full bandwidth is strongly suggested. This advance will greatly improve the performance of large trapped ion crystals in quantum information and metrology applications.
We theoretically propose and experimentally demonstrate a parallel electromagnetically induced transparency (parallel EIT) cooling technique for ion crystals in the Paul trap. It breaks the stringent cooling requirements of the standard EIT cooling, thus allowing, in principle, to simultaneously cool the motional mode spectrum with an arbitrary range. Experimentally a large cooling range over 5 MHz is proved by using a single trapped 40Ca+{}^{40}{\mathrm{Ca}}^{+}40Ca+ ion. We also observe simultaneous near-ground-state cooling for all motional modes of a 4-ion chain with the best average phonon number of about 0.2. For cooling a large number of modes, we only need to modulate the probe beam of the standard EIT cooling with several rf frequencies. This cooling scheme has no selectivity for the types of ions (e.g., 171Yb+{}^{171}{\mathrm{Yb}}^{+}171Yb+, 40Ca+{}^{40}{\mathrm{Ca}}^{+}40Ca+), therefore it would be a powerful tool for initialization of all kinds of operational trapped-ion quantum computers or simulators.
Two-dimensional planar ion crystals in a Penning trap are a platform for quantum information science experiments. However, the low-frequency planar modes of these crystals are not efficiently cooled by laser cooling, which can limit the utility of the drumhead modes for quantum information processing. Recently, it has been shown that nonlinear mode coupling can enhance the cooling of the low-frequency planar modes. Here, we demonstrate in numerical simulations that this coupling can be dynamically tuned by adiabatically changing the rotation frequency of the ion crystal during experiments. Furthermore, we show that this technique can, in addition, produce lower temperatures for the low-frequency planar modes via an adiabatic cooling process. This result allows cooling of the planar modes to sub-millikelvin temperatures, resulting in improved spectral resolution of the drumhead modes at experimentally relevant rotation frequencies, which is crucial for quantum information processing applications.