A robust method to reach the motional quantum regime of (anti-)protons in cryogenic multi-Penning traps

Abstract

Sympathetic laser cooling is a key concept in precision spectroscopy and quantum state control of charged particles. Significant challenges arise in the metrologically relevant case where the effective interaction between the particles is weak and the particle to be cooled exhibits significant initial motional energy. Here we specifically address the most generally applicable case where the laser-cooled ion and the particle of interest are confined to two spatially separate potential wells with equal motional frequency for resonant enhancement of the cooling dynamics. We analyze the latter through numerical simulations and find that anharmonicities of the potential wells can prevent maintaining the resonance condition throughout the cooling process and thus inhibit a significant reduction in motional energy. We propose a cooling scheme that sweeps the trapping frequency of the potential wells. We show that this scheme enables efficient cooling from cryogenic temperatures all the way to the quantum regime of motion. As a specific application scenario, we analyze the sympathetic cooling of (anti-)protons into the quantum regime of motion for quantum-logic-spectroscopy-based tests of CPT invariance at the quantum limit in Penning traps. Nevertheless, our results and cooling strategies are generally applicable to other laser-inaccessible ion species.

Figures (11)

• Figure 1: Multi-Penning trap.a) Cross-sectional view of an exemplary multi-Penning trap design for the coupling experiments, with trap sections shown in different colors. A $^9$Be$^+$ ion is depicted in blue and an (anti-)proton in red. The inner trap surface is circularly symmetric around the trap axis. The macroscopic trap electrodes have inner diameters of 9mm and are made of gold-plated, oxygen-free high thermal conductivity copper. An axial resonator, consisting of a superconducting coil and a cryogenic amplifier, is connected to a precision trap electrode for (anti-)proton detection and 4K cooling. Purple arrows illustrate the Raman beams used for cooling and temperature measurement of the $^9$Be$^+$ ion. b) Detailed cross-sectional view of the Coupling trap, consisting of nine microfabricated electrodes. The positions of the (anti-)proton and $^9$Be$^+$ ion are shown, along with the particle separation $s_0$ and offset $\Delta s_0$ relative to the central electrode. Further details are provided in the text.
• Figure 2: Double-well potential.a) Proton coupled with a $^9$Be$^+$ ion. b) Antiproton coupled with a $^9$Be$^+$ ion. The double-well potentials were generated using potential minima separation $s_0=$ 0.7mm and axial trap frequency $f=\omega/2\pi=$ 500kHz for both potential minima. The antiproton potential well is inverted due to its negative charge. The purple shaded region indicates the positions where the (anti-)proton can be trapped, while the green shaded region represents positions where it experiences a harmonic potential. The energy ranges of the trapping and harmonic regions are described in more detail in Fig. \ref{['fig:harmoniccoupling']}.
• Figure 3: Simulation sequence.(a) Harmonic coupling. (b) Frequency sweep. The simulation sequences are shown using a proton as an example; the same studies are performed for the antiproton. Both subfigures are divided into three steps: i) initialization of the (anti-)proton with energy $E_{\rm{p/\bar{p}}, init}$ and preparation of the $^9$Be$^+$ ion in its motional ground state (GS); ii) energy exchange between the particles; iii) analysis of the final (anti-)proton energy $E_{\rm{p/\bar{p}},fin}$. The plot in subfigure (a.ii) shows an example of the energy evolution of the (anti-)proton and $^9$Be$^+$ ion during the coupling process in the time-independent double-well potential.
• Figure 4: Energy ranges of the double-well potential. Harmonic energy ranges for the (a) proton and (b) antiproton, and trapping energy ranges for the (c) proton and (d) antiproton, as functions of trap frequency. Shaded regions in (a) and (b) denote initial energies from which the antiproton can be cooled below $1mK\,\times k_\mathrm{B}$ via harmonic coupling; their upper boundaries are given by linear fits (shown in the legend), with coefficients in $mK\per kHz$. In (c) and (d), the data are fitted quadratically (shown in the legend), with coefficients in $mK\per kHz\squared$. Potentials with parameters $\omega$ and $s_0$ above the red-shaded region trap more than 90% of $4K$ (anti-)protons.
• Figure 5: Stability of the energy transfer.a) Half width at half maximum ($\gamma$) of the energy transfer resonance curve as a function of the trap frequency. The plot was calculated using Eq. \ref{['eqn:transfer_time']} and the relation $2\gamma=1/\tau_{ex}$. In the top right corner, an example resonance curve for $f=$ 500kHz and $s_0=$ 0.6mm (marked with a blue triangle) is fit to a Lorentzian function. The data are evaluated using the percentage of energy transferred from a proton oscillating at frequency $f$ to a $^9$Be$^+$ ion oscillating at frequency $f+\Delta f$, where $\Delta f$ is the deviation from the resonance frequency. b) Required power supply stability to achieve an 80% energy transfer. For each $s_0$, the maximum initial energy of the (anti-)proton is taken from Fig. \ref{['fig:harmoniccoupling']}. The calculation of the voltage stability is described in the appendix. Dashed lines indicate a power supply stability of $V_{pp}=$ 0.5µV_pp. Data points above the $1\sigma$ stability line are expected to achieve the desired transfer at least 68.3% of the time, while those above the $2\sigma$ stability line are expected to do so at least 95.4% of the time.