Adiabatic Cooling of Planar Motion in a Penning Trap Ion Crystal to Sub-Millikelvin Temperatures

TL;DR

The paper addresses cooling of slow planar $\mathbf{E} \times \mathbf{B}$ modes in two-dimensional Penning-trap ion crystals, where standard laser cooling is ineffective. It introduces an adiabatic ramp of the rotating-wall frequency to dynamically tune nonlinear mode coupling to drumhead modes, enabling sub-millikelvin planar-mode cooling. Using simulations across ideal and thermal ensembles, it shows energy scaling relations $E_f = E_i (\omega_f/\omega_i)$ and $E_f^{E\times B} = E_i^{E\times B} (\beta_f/\beta_i)$, achieving well-resolved drumhead spectra after ramping and post-cooling. The results offer a hardware-light route to high spectral resolution and improved quantum control in large planar ion crystals.

Abstract

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.

Figures (7)

• Figure 1: Comparison of linear and half-cosine rotating wall frequency ramps. A crystal of $N = 50$ ions is initialized in the zero-temperature equilibrium configuration, and the rotation frequency is ramped from 200 to 190 kHz over 1 ms using three protocols: (a) a linear ramp with $\delta$ held constant while $\beta$ is lowered, (b) a linear ramp with fixed $\delta/\beta$, and (c) a half-cosine ramp with fixed $\delta/\beta$. Each panel shows the ion trajectories in the first quadrant of the $xy$-plane during the ramp; color indicates time. Without fixing $\delta/\beta$ [(a)], the crystal elongates and reorganizes. Fixing $\delta/\beta$ [(b)] prevents reconfiguration but excites a rocking mode due to abrupt ramp edges. The half-cosine ramp [(c)] preserves crystal structure and avoids rocking motion. (b,c) Light-gray segments from each ion to the origin indicate global rescaling (collinearity with preserved aspect ratio), and solid black segments connect each ion's initial equilibrium site to its final rescaled equilibrium site from Eq. (\ref{['eq:rescaled_eq']}), tracing the instantaneous-equilibrium path. For the short half-cosine ramp (c) ($T=1~\mathrm{ms}\sim 2\pi/\omega_{\min}$) Euler-force excitation causes brief departures from these paths, but ions return and come to rest at $t=T$, whereas the equal-duration linear ramp (b) leaves residual azimuthal rocking.
• Figure 2: Effect of rotating wall strength on lowest mode frequencies and crystal stability. Ion crystals with $N = 20$, $30$, $40$, and $50$ ions are initialized in the minimum potential energy (zero temperature) configuration at $\omega_r = 2\pi \times 200$ kHz with varying rotating wall strengths in the range $\delta/\beta = 0.1$ to $0.5$. Each system is then ramped to 180 kHz over 1 ms using a half-cosine ramp. (a) The frequency of the lowest $\mathbf{E} \times \mathbf{B}$ mode increases with rotating wall strength, indicating improved adiabatic conditions at larger $\delta/\beta$. (b) The average potential energy offset from $E = 0$ equilibrium, measured over 1 ms following the ramp, decreases with increasing $\delta/\beta$. For $\delta/\beta \gtrsim 0.3$, the system remains near the $E = 0$ equilibrium, indicating minimal heating during the ramp.
• Figure 3: Adiabatic cooling of planar motion compared to theory. An ensemble of 128 different initializations of a 54-ion crystal is generated with all mode amplitudes corresponding to a temperature of 1 mK. A half-cosine ramp is applied to decrease the rotating wall frequency from 200 kHz to 180 kHz over 20 ms. (a) Final energies of the $\mathbf{E} \times \mathbf{B}$ modes are plotted versus mode number. Small gray dots (128 per mode) represent different ion crystal initializations, and large red dots represent the ensemble averages. The average energies are reduced to approximately 0.4 mK, consistent with the adiabatic cooling prediction (red stars). This confirms that the adiabatic energy scaling model accurately describes the cooling process for the $\mathbf{E} \times \mathbf{B}$ modes. (b) Final energies of the axial (drumhead) modes. Small gray dots (128 per mode) represent different ion crystal initializatons, and large green dots represent the ensemble averages.
• Figure 4: Laser cooling of drumhead modes following adiabatic $\mathbf{E} \times \mathbf{B}$ cooling. Laser cooling is applied to the ensemble from Fig. \ref{['fig:adiabaticCooling']} after the adiabatic ramp. Axial cooling beams are applied for 1 ms with fixed $\omega_r = 2\pi \times 180$ kHz. (a) Mode-branch energies during laser cooling. The drumhead (axial) modes cool rapidly toward the Doppler limit, while the $\mathbf{E} \times \mathbf{B}$ modes remain cold and the cyclotron modes absorb recoil heating. (b) Power spectral density of the drumhead modes after 12.5 ms of free evolution following the axial cooling in (a). The ensemble-averaged spectrum (black) shows well-resolved peaks aligned with the theoretical mode frequencies (red lines), indicating high spectral resolution. Thus, after preparation at $1~\mathrm{mK}$ with $\omega_r/2\pi=200~\mathrm{kHz}$ and an adiabatic ramp to $\omega_r/2\pi=180~\mathrm{kHz}$, laser re-cooling of the drumhead modes yields well-resolved drumhead spectra at the target operating point, preserving spectral resolution (cf. Ref. Johnson2024, Fig. 2(e)).
• Figure 5: Laser cooling of thermal ensembles near the planar-to-3D transition. An ensemble of 128 ion crystals with $N = 100$ ions is initialized at 100 mK using the Metropolis-Hastings algorithm. The crystals are initialized at $\omega_r = 2\pi \times 180 \text{ kHz and } 2\pi \times 195 \text{ kHz}$, with $\delta/\beta = 0.5$. Both ensembles are cooled for 20 ms using perpendicular and axial laser beams. (a) Evolution of parallel and perpendicular kinetic energy ($\text{KE}_\parallel$, $\text{KE}_\perp$) and total potential energy (PE) for the 180 kHz ensemble. (Here, $\parallel$ denotes motion along the magnetic field $\mathbf{B}\parallel\hat{z}$ and $\perp$ denotes motion transverse to $\mathbf{B}$; $\text{KE}_\parallel=\sum_i \tfrac{1}{2} m_i \dot z_i^2$ and $\text{KE}_\perp=\sum_i \tfrac{1}{2} m_i (\dot x_i^2+\dot y_i^2)$.) The system remains disordered and does not crystallize. (b) Same for the 195 kHz ensemble. Strong nonlinear coupling between motional branches enables rapid energy exchange and cooling. After $\sim$17.5 ms, the system crystallizes, and mode energies can be computed. Inset: Mode branch averages for the crystallized 195 kHz ensemble: $\mathbf{E} \times \mathbf{B}$ modes are cooled to $\sim$2 mK, drumhead modes to $\sim$1 mK, and cyclotron modes to $\sim$4 mK. The $y$-axis is the average energy in millikelvin (mK) units. The $x$-axis is the last 2.5 ms of the evolution. The solid gray line represents the $\mathbf{E} \times \mathbf{B}$ mode branch average without the inclusion of the COM mode, which is not coupled to the other modes and remains hot.