Optimizing Doppler laser cooling protocols for quantum sensing with 3D ion crystals in a Penning trap

Abstract

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, $N$, in the crystal increases. Here we develop a powerful numerical framework to simulate laser cooling of up to $10^5$ 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 $\boldsymbol{E}\times\boldsymbol{B}$ 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.

Figures (11)

• Figure 1: The setup to cool the crystal relies on two axial and one perpendicular cooling beam. The axial beams have a low intensity which we approximate as uniform across the crystal in our simulations. They are detuned by $\Delta_{\parallel_1}=\Delta_{\parallel_2} = -\gamma_0/2$ from resonance with the $2s\;^2S_{1/2}\rightarrow 2p\;^2P_{3/2}$ transition. The perpendicular beam is offset from the trap center by a distance $d$ and has detuning $\Delta_{\perp}$. Unlike in the case of a 2D crystal, here, the perpendicular beam waist in both directions, $w_y$ and $w_z$, affect the cooling rate. Figure is reproduced from zaris2024.
• Figure 2: (a) Two examples of 3D crystal normal modes are illustrated. The same equilibria configuration for a $N=1000$ ion crystal is shown in each panel. The shell structure characteristic of ultracold mesoscopic crystals is evident. The relative phase of each ion in the mode is represented by its color and the relative amplitude of the ion's motion is represented by its size. The left-hand plot shows an $\boldsymbol{E}\times\boldsymbol{B}$ mode in which the ions above $z=0$ have a nearly uniform phase and ions below $z=0$ have a different, but also nearly uniform phase (mode 2). For this plot, the phase is calculated using $\phi_p = arg\{u_n^x\}$, where $u_n^x$ is the $x$-component of the positional part of eigenmode $n$. The right-hand panel illustrates the axial center of mass mode (mode 1884), in which all ions move in phase at frequency $\omega_z$. Here, the phase is $\phi_p = arg\{u_n^z\}$. (b) The normal mode frequencies for the crystal shown in (a) are plotted in blue. The mode branches occupy distinct frequency bands and are separated by frequency gaps, as is seen in 2D crystals. In order of increasing frequency, they are the $\boldsymbol{E}\times\boldsymbol{B}$, axial, and cyclotron branches. In more prolate 3D crystals, the gap between branches can vanish. The axial component of each mode, $f_n^z$, is shown in red. It is computed using Eq. \ref{['eq8']}. The $\boldsymbol{E}\times\boldsymbol{B}$ modes gain a large axial component, even in 3D crystals with moderate aspect ratios. The axial modes have a small, but noticeable, planar component in this case. Note that the axial center-of-mass mode, for which $f^z=1$, is no longer the highest frequency mode, as is the case in 2D crystals.
• Figure 3: (a) The shape of $N=1000$ ion crystals are plotted for different values of $\omega_r$. The equilibrium configurations plotted correspond to the following trap parameters: $B_z =4.4588$ T, $\omega_z = 2\pi\times1.59$ MHz, $\delta=0.0104$. The transition from a 2D crystal to a 3D crystal occurs near $\omega_r = 2\pi\times 178.15$ kHz. The equilibrium crystal corresponding to this frequency is nearly 2D, except the most central ions have popped out of the plane and have axial coordinates of up to $\sim 1$$\mu$m. As $\omega_r$ is increased further, the axial extent of the crystal increases and the shell structure becomes evident. (b) The $\boldsymbol{E}\times\boldsymbol{B}$ (mode numbers 1-1000) and axial (mode numbers 1001-2000) normal mode frequencies are plotted for the ion crystals in (a). Most 2D crystals have $\boldsymbol{E}\times\boldsymbol{B}$, axial, and cyclotron modes which occupy distinct frequency bands (orange curve). However, near the 2D-3D transition, the highest frequency $\boldsymbol{E}\times\boldsymbol{B}$ and lowest frequency axial modes approach one another (red curve). The frequency gap reappears immediately after the transition, but again vanishes as $\omega_r$ continues to increase. Therefore, resonant mode coupling in large $\beta$ 3D crystals may result in enhanced Doppler laser cooling. (c) The axial component of the $\boldsymbol{E}\times\boldsymbol{B}$ and axial modes is shown for each of the crystals. As $\omega_r$ increases, $\boldsymbol{E}\times\boldsymbol{B}$ modes gain larger axial components and axial modes gain a larger planar component. The $\boldsymbol{E}\times\boldsymbol{B}$ modes of 3D crystals may, therefore, be cooled by the axial beams in the NIST setup. (d) The potential energy fraction of the $\boldsymbol{E}\times\boldsymbol{B}$ and axial modes. For crystals with larger $\omega_r$, the $\boldsymbol{E}\times\boldsymbol{B}$ modes have a larger kinetic energy component. Since Doppler laser cooling directly reduces the kinetic energy, this may result in enhanced cooling of large $\beta$ crystals.
• Figure 4: Theory estimates for the maximum $\boldsymbol{E}\times\boldsymbol{B}$ mode frequency and the minimum axial frequency versus plasma rotation frequency are shown in orange and blue curves, respectively. These plots are obtained using Eq. (\ref{['eq10']}) and Eq. (\ref{['eq13']}). The theoretical treatment predicts that $\omega_{\parallel min}>\omega_{E max}$ for $\omega_r \lesssim$ 500 kHz. For larger values of $\omega_r$, $\omega_{\parallel min}$ becomes smaller than $\omega_{E max}$. This behavior is observed in the mode frequencies of the crystals from Fig. \ref{['fig3']}, which we plot here using orange and blue dots. Note that the difference $\omega_{\parallel min}-\omega_{E max}$ nearly vanishes in the simulation data from $\omega_r=178.15$ kHz. This occurs near the 2D to 3D transition. The theory plots do not account for this behavior, but this effect is well-understood (see text).
• Figure 5: The potential energy cooling is illustrated for crystals with different values of $\omega_r$ and $\delta$. The kinetic and potential energies are initialized at 10 mK and then laser cooling is simulated for 16 ms. Solid lines (small $\delta$): The potential energy of 2D crystals is generally difficult to cool, as shown by the yellow curve. By tuning the trap parameters such that the crystal is near the 2D-3D transition, the potential energy can be cooled efficiently (red curve). After the transition, an oblate 3D crystal is produced and, again, the potential energy cools relatively slowly (green curve). By further increasing $\beta$, a combination of the increased axial component of the $\boldsymbol{E}\times\boldsymbol{B}$ modes and a reduced gap between mode branches leads to enhanced cooling (blue and purple curves). Dashed lines (large $\delta$): Increasing the strength of the rotating wall potential, $\delta$, can reduce the potential energy of 3D crystals. This is seen most dramatically in the case of the $\omega_r=2\pi\times220$ kHz crystal.