Efficient Three-Dimensional Sub-Doppler Cooling of $^{40}$Ca$^+$ in a Penning Trap

TL;DR

Addresses motional cooling of a single $^{{40}}$Ca$^+$ ion in a Penning trap where Lamb-Dicke confinement is weak. Uses axial dark-resonance cooling with the same beams as Doppler cooling to reach sub-Doppler temperatures, for example reducing $\bar{n}_z$ from 72(23) to 1.5(3) in 800 μs, followed by pulsed sideband cooling to near ground state. Radial modes are cooled by coherently exchanging energy with the axial mode via a parametric quadrupolar drive, enabling 3D sub-Doppler cooling with axial DR beams alone. A semiclassical model based on a Lindblad master equation for the internal states coupled to classical motion reproduces the dynamics and provides guidance on capture range and recoil heating. The approach reduces cooling times and demonstrates a viable path toward scalable quantum information processing with Penning-trap ion arrays, achieving final occupations $\bar{n}_z=0.12(6)$, $\bar{n}_+=15(2)$, $\bar{n}_-=21(4)$.

Abstract

We demonstrate efficient sub-Doppler laser cooling of the three eigenmodes of a $^{40}$Ca$^+$ ion confined in a compact Penning trap operating with a magnetic field of 0.91 T. Using the same set of laser beams as required for the initial Doppler laser cooling operation, we detune the laser frequencies to produce a narrow two-photon dark resonance. The process achieves a 1/e cooling time constant of 108(8) $μ$s, ultimately reducing the mean thermal axial mode occupation from 72(23) to 1.5(3) in 800 $μ$s as measured by resonantly probing an electric quadrupole transition near 729 nm. A parametric drive is applied to the trap electrodes which coherently exchanges the axial mode occupation with that of each radial mode, allowing for three-dimensional sub-Doppler cooling using only the axially-propagating laser beams. This sub-Doppler cooling is achieved for an axial oscillation frequency of $ω_z = 2π~\times~$221 kHz, which places the motion well outside of the Lamb Dicke confinement regime at the Doppler laser cooling limit. Our measured cooling rate and final mode occupation are in good agreement with a semiclassical model which combines a Lindblad master equation solution for ion-photon interactions with classical harmonic oscillator motion of the trapped ion.

Figures (6)

• Figure 1: Cross-section view of the in-vacuum permanent magnet Penning trap assembly with printed circuit boards. The pair of radially magnetized SmCo rings produces a uniform, vertically-oriented magnetic field of 0.91 T at the ion location. The ion is depicted as a white circle centered vertically between the PCBs. The laser propagation directions are shown as blue and red arrows. The scale bar (bottom right) represents 12.7 mm for this drawing.
• Figure 2: The $^{40} \text{Ca}^+$ energy level diagram with relevant laser wavelengths used for this work. The solid and transparent arrows denote laser frequencies used for Doppler and dark resonance cooling, with the distinct sets identifying excitation pathways to each of the $P_{1/2}$ magnetic sublevels. The 729 nm excitation (green dashed arrow) is used to measure axial mode occupation, and a set of 854 nm tones (red dashed arrow) returns population to the $S_{1/2}$ state following 729 nm excitation.
• Figure 3: (Top) Observed ion fluorescence count rate measured for various detunings of the 397B laser from atomic resonance. (Bottom) Zoomed-in view of the two-photon resonance from the frequency scan above. The transparent grey, dashed line shows the detuning of the 397B laser frequency (i.e. optical carrier) during DR cooling. The frequency offsets and relative Rabi rates of the red and blue axial sidebands are plotted on an arbitrary vertical scale for an initial coherent axial occupation of $\bar{n}=72$. Net axial cooling occurs when the photon scatter rate for energy-removing sidebands overwhelms that of the energy-adding sidebands.
• Figure 4: Measured axial mode occupation (points with error bars) of $^{40} \text{Ca}^+$ after application of DR cooling for various durations. The solid blue line is the result of a semiclassical sub-Doppler cooling simulation that takes experimental parameters as inputs. Solid gray lines show the individual cooling (or heating) trajectories for each simulated initial coherent excitation and the blue line is a thermally-weighted mean over all trajectories. We find good agreement between experiment and simulation over nearly two orders of magnitude in axial mode occupation. The simulated DR cooling capture range is $\bar{n}_z < 900$ for these parameters, which include Rabi rates $\Omega_{397A}\sim2\pi\times 2.9$ MHz, $\Omega_{397B}\sim2\pi\times 1.2$ MHz, and $\Omega_{866}\sim 2\pi\times(1.3-2.2)$ MHz.
• Figure 5: Measured axial mode occupation ($\bar{n}_z$) for various parametric exchange pulse durations coupling the axial and magnetron modes. Each experiment begins with Doppler and DR cooling of the axial degree of freedom, followed by application of a parametric axial-magnetron mode coupling excitation for a variable length of time. The axial mode occupation is then extracted from a 729 nm resonant carrier flop. The peak mode occupation measured corresponds to the original magnetron occupation ($\bar{n}_-$) before the parametric exchange. The vertical center of the transparent blue shaded region corresponds to the average occupation of the axial mode after Doppler and DR cooling before any exchanges. Its height corresponds to twice the standard error.