Sympathetic cooling schemes for separately trapped ions coupled via image currents

The ability to prepare (ultra-)cold particles in a controlled environment through laser cooling has been a major driving force in the enhanced precision of recent experiments in the field of atomic, molecular and optical physics [1]. However, many particle species of interest offer no suitable transitions for direct laser cooling, so that for these particles sympathetic cooling techniques have to be employed [2–7]. Established sympathetic cooling schemes rely on Coulomb interactions between the target particle and the laser-cooled ions. In order to obtain a sufficiently high coupling rate, the two species must either be co-trapped in the same potential well [7, 8] or trapped in separate potential wells [9, 10] that enable particles distances of 10 − 100 μm. However, trap designs that enable these length scales and the Coulomb interaction between the particle species impose large systematic shifts on potential precision measurements in Penning traps [11].

One coupling technique which is not accompanied by these drawbacks is image-current coupling. Here, the charged particle species are coupled through image currents induced in adjacent trap electrodes, allowing a macroscopic separation of the target species and cooling species. Recently, our group has published the first experimental results on sympathetically cooling a single proton in a Penning trap by coupling it to a cloud of laser-cooled ⁹Be⁺ ions located in a separate trap [12]. In this experiment, the coupling between the ions was mediated by a common superconducting RLC resonator, which is usually used for particle detection [13]. A temperature reduction of the axial mode from 17.0 (2.4) K to 2.6 (2.5) K was achieved. Other recent experimental efforts have demonstrated the coupling between two highly charged ions with a comparable experimental setup [3] or the coupling of two laser-cooled ions stored in a Paul trap by a floating room-temperature wire [14].

The recent developments [12] will be one of the cornerstones for the next generation of precision measurements at low energies. For example, our collaboration (BASE) probes the fundamental charge, parity and time (CPT) reversal symmetry by performing ultra-high-precision Penning-trap measurements on protons and antiprotons [15]. In particular, their charge-to-mass ratio is compared at a relative precision of 69 × 10⁻¹² [16] and their magnetic moments are measured with fractional resolutions of 3 × 10⁻¹⁰ [17] and 1.5 × 10⁻⁹ [18], respectively; all being the most precise measurements of these quantities to date. However, especially the magnetic moment measurements are limited by the nonzero particle temperature of ≳1 K. Other experimental programs that are subject to comparable limitations include the spectroscopy of highly-charged ions [8, 19], the cooling of antiprotons and positrons for the synthesis of cold antihydrogen [5, 20, 21], mass measurements of heavy elements [6, 22], or high-precision metrology with atomic clocks [7, 23].

In this work, we study image-current coupling mechanisms by means of combined analytical calculations and numerical simulations. Since the particles and the RLC resonator constitute a complex stochastic system of three coupled oscillators, we use the numerical simulations to study the dynamical behavior and optimize cooling times and temperature limits. The relevant physics is implemented into the simulation code from first principles and experimentally established models only. After demonstrating agreement between our simulation results and existing experimental data, we assess the feasibility and performance of three different cooling schemes. We find that with two of the techniques, coupling via a common electrode and off-resonant coupling via a common RLC circuit, temperatures of ≈10 mK can be reached with cooling time constants on the order of 5–20 s. Compared to the established method of resistive cooling with an RLC resonator at 4–20 K [24], this constitutes a temperature reduction by a factor of $\approx 100$ and a cooling time reduction by a factor of $\approx 10$. The third technique, on-resonance coupling with a common RLC resonator, can provide cooling time constants below 1 s but is limited to particle temperatures in the hundred mK-regime with the current experimental setup.

Although applicable to many systems, we find the image-current cooling technique to be of immediate relevance to ultra-high precision measurements in Penning traps and therefore motivate the requirements on the cooling performance with a future measurement of the proton magnetic moment.

2.1. Penning-trap basics

In a Penning trap, charged particles are trapped radially by a constant homogeneous magnetic field, in cylindrical coordinates (ρ, φ, z) oriented in the axial direction,

$$\begin{equation}\vec{B}={B}_{0}{\vec{e}}_{z}\text{,}\end{equation} \tag{ 1 }$$

and axially by a harmonic electrostatic potential,

$$\begin{equation}\phi (\rho ,z)={C}_{2}{U}_{0}({z}^{2}-{\rho }^{2}/2)\text{,}\end{equation} \tag{ 2 }$$

usually created by a stack of five cylindrical electrodes in a compensated, orthogonal design [25]. C₂ is a geometrical parameter defined by the trap size and U₀ the electrical potential difference between the endcap and the central ring electrode [26]. A simplified schematic is shown in figure 1(a). If higher-order contributions to the magnetic field and axial potential are neglected, the axial mode in z-direction is independent of the other coordinates and has frequency

$$\begin{equation}{\nu }_{\mathit{\text{z}}}=\frac{1}{2\pi }\sqrt{\frac{2q{C}_{2}{U}_{0}}{m}},\end{equation} \tag{ 3 }$$

where q and m are the charge and the mass of the particle, respectively. The radial motional modes split up into a slow and fast mode, referred to as magnetron (ν₋) and modified cyclotron (ν₊) modes with frequencies

$$\begin{equation}{\nu }_{-}=\frac{1}{2\pi }\frac{q{B}_{0}}{2m}-\sqrt{{\left(\frac{1}{2\pi }\frac{q{B}_{0}}{2m}\right)}^{2}-\frac{{\nu }_{z}^{2}}{2}}\end{equation} \tag{ 4 }$$

$$\begin{equation}{\nu }_{+}=\frac{1}{2\pi }\frac{q{B}_{0}}{2m}+\sqrt{{\left(\frac{1}{2\pi }\frac{q{B}_{0}}{2m}\right)}^{2}-\frac{{\nu }_{z}^{2}}{2}}.\end{equation} \tag{ 5 }$$

At B₀ ≈ 2 T and U₀ ≈ 1–10 V, typical values for the frequencies of the proton in the trap system used in our experiments are ν_z ≈ 600 kHz, ν₋ ≈ 7 kHz and ν₊ ≈ 29 MHz.

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The motion of the particle induces image currents in the trap electrodes that permit a non-destructive measurement of the particle's motional frequencies. Fundamental properties of the trapped particle such as its charge-to-mass ratio and magnetic moment can be derived from these frequency measurements [27]. The axial mode is most commonly detected [28], for which the magnitude of the induced image current is

$$\begin{equation}{I}_{\text{ind}}=\frac{q}{D}\dot{z}.\end{equation} \tag{ 6 }$$

Here D is a specific length related to the trap size and the chosen pickup geometry [28, 29] and can be considered an 'effective electrode distance'. $\dot{z}$ is the particle's axial velocity. Detectors for these currents consist of ultra low-noise amplifiers connected to superconducting NbTi coils, in our experiment with inductances L of several mH, which are cooled down to 4.2 K by a liquid helium bath [13, 30]. Due to parasitic effects, the coil needs to be modelled by an additional capacitance C_res and a resistance R, forming a parallel RLC circuit with eigenfrequency ν₀ as depicted in figure 1(a). The quality factor Q = R/(2πν₀ L) of such an axial resonator is typically on the order of 20 000, including limitations caused by the input resistance of the cryogenic amplifiers and parasitic resistive losses [13, 30]. The thermal Johnson–Nyquist noise of the parallel resistance, $\langle {U}_{\text{noise}}^{2}\rangle =4{k}_{\text{B}}{T}_{\text{res}}R{\Delta}f$ for a frequency interval Δf, charges up the whole RLC circuit and a resonance signal at the circuit's eigenfrequency is created.

By connecting this resonator to a trap electrode, a trapped charged particle is exposed to the noisy oscillations of the RLC circuit and the axial energy dynamically samples a Boltzmann distribution with temperature T_res. The coupling strength between ion and resonator depends among other parameters on the ion's frequency detuning relative to the resonator. On resonance, where the coupling is strongest, the particle effectively shorts the resonator noise, creating a characteristic dip in the fast Fourier transform (FFT) spectrum [28]. An example of such a spectrum is shown in figure 3 of section 5.

In the context of our studies, we will refer to energy as the total oscillator energy, which is, although given in units of K ⋅ k_B, defined deterministically for every point in time. For example, for the axial mode of a particle, ${E}_{\mathit{\text{z}}}=\left(q{C}_{2}{U}_{0}{z}^{2}+\frac{1}{2}m{\dot{z}}^{2}\right)$. In contrast, we will refer to temperature as the average energy of the oscillator divided by the Boltzmann constant, i.e. T_z = ⟨E_z ⟩/k_B.

2.2. Magnetic moment measurements and the need for advanced cooling techniques

Penning-trap based magnetic moment measurements of the proton and antiproton determine the free cyclotron frequency ${\nu }_{\text{c}}=\frac{1}{2\pi }\frac{q}{m}B$ and the spin precession frequency (Larmor frequency) ${\nu }_{\text{L}}=\frac{1}{2\pi }\frac{g}{2}\frac{q}{m}B$ simultaneously. The ratio of the two quantities is independent of the magnetic field and charge-to-mass ratio of the particle and yields the dimensionless g-factor:

$$\begin{equation}g=2\frac{{\nu }_{\text{L}}}{{\nu }_{\text{c}}}.\end{equation} \tag{ 7 }$$

ν_c is determined using the Brown–Gabrielse invariance theorem ${\nu }_{\text{c}}=\sqrt{{\nu }_{+}^{2}+{\nu }_{\mathit{\text{z}}}^{2}+{\nu }_{-}^{2}}$ [26] and measuring ν₊, ν_z , ν₋ via image current techniques and sideband coupling [31] or phase-sensitive detection [32]. However, a direct measurement of the Larmor frequency is not possible as the spin orientation cannot be detected by image currents in the trap electrodes. Instead, spin flips are detected by utilizing the continuous Stern–Gerlach effect [15]. Here, a strong magnetic inhomogeneity is superimposed to the homogeneous field, B_z (ρ, z) = B₀ + B₂(z² − ρ²/2), where B₂ characterizes the strength of the magnetic inhomogeneity. This results in the axial frequency becoming a function of both the energy of the radial modes as well as the spin state of the trapped particle. Spin flips can be induced by applying an external rf drive with frequency close to ν_L. Measuring the spin-flip probability as a function of the drive frequency yields a resonance line from which ν_L can be extracted [33]. However, due to frequency shifts caused by the nonzero mode temperatures, the strong magnetic bottle poses limitations to the measurement precision of ν_c and ν_L. This limitation can be overcome by employing multi-trap techniques [34–36]: the frequency measurements for the determination of ν_c and ν_L are conducted in the precision trap (PT) with a homogeneous magnetic field, while the spin state is determined in the analysis trap (AT) featuring the magnetic bottle. In the AT, despite using a strong B₂ = 300 000 T m⁻², a spin flip of a single proton corresponds to an axial frequency shift of only $\approx 170$ mHz compared to ν_z ≈ 600 kHz for our experimental parameters. As a result, the axial frequency stability is a key parameter for high-fidelity identification of the spin-state [35, 37]. We observe that a major contribution of axial frequency fluctuations can be attributed to modified cyclotron quantum transitions [35]. The transition rate ζ₊ is proportional to the mode energy:

$$\begin{equation}{\zeta }_{+}\propto {n}_{+}\propto {E}_{+}.\end{equation} \tag{ 8 }$$

Here n₊ denotes the quantum number of the cyclotron mode with energy E₊ = h ν₊(n₊ + 1/2) = k_B T₊. The effect of noise-driven cyclotron transitions on a sequence of axial frequency measurements or simulations is illustrated in figure 1(b) for the two cases of low and high cyclotron temperatures. At low T₊, the spin flip is easily resolved from residual fluctuations. In contrast, at high T₊ the superimposed fluctuations prevent a high-fidelity identification of a spin flip. Since the measurement of ν₊ increases the modified cyclotron energy, the preparation of the cyclotron mode with a subthermal energy must be conducted in each double-trap measurement cycle.

Our latest magnetic moment measurements employed the technique of resistive feedback cooling with a cyclotron resonator [38, 39]. When coupled to the resonator, the cyclotron mode thermalizes with the resonator and samples a Boltzmann distribution, in the case of the proton measurement with an average temperature of 3 K [24]. However, at the same time the acceptance threshold T_thresh for unambiguous spin-state determination was set to T₊ ⩽ T_thresh = 600 mK, so that a time-consuming stochastic selection of the cyclotron energy was necessary. In fact, 2/3 of the duration of one full experimental cycle of about 90 min were needed for the preparation of the particle in a state of low cyclotron energy and the subsequent spin-state detection [17]. As a consequence, the measurement of the proton g-factor is limited by the statistical uncertainty, and the development of cooling schemes that achieve colder particle temperatures in a shorter time is crucial for overcoming the current limit in precision. Ultimately, more powerful cooling methods would not only enable an enhanced spin state detection fidelity, but also allow to deterministically prepare the proton with a cold cyclotron mode in much shorter times, which is one of the cornerstones for achieving $< 1{0}^{-10}$ fractional precision in the next generation of (anti-)proton magnetic moment measurements.

In order to realize this boost in precision experimentally, we develop technologies and methods to sympathetically cool the proton with a cloud of laser-cooled ⁹Be⁺ stored in a separate trap a few cm away, for which we have recently performed the proof-of-principle measurement [12]. The coupling is achieved through image currents induced in a common electrode or a common resonator. Since image-current coupling requires matching an oscillation frequency of the two species involved and the cyclotron and magnetron frequencies are, in contrast to the axial frequency, not easily adjustable in our experiment, we currently apply the sympathetic cooling technique to the axial mode. The subsequent coupling between the cyclotron mode and axial mode is usually performed via a quadrupolar radiofrequency drive (sideband coupling) [31]. However, with this technique the mode temperatures are related by ${T}_{+}=\frac{{\nu }_{+}}{{\nu }_{\mathit{\text{z}}}}{T}_{\mathit{\text{z}}}$, in our case T₊ ≈ 60 T_z for the proton. In order to achieve a significantly improved cyclotron mode preparation at the previous threshold of T₊ ⩽ 600 mK, cooling techniques that reach axial proton temperatures of T_z ≲ 10 mK are required. For example, T_z = 10 mK axial temperature would enhance the success rate of the cyclotron energy selection from $\approx 18$% to $\approx 63$%. Furthermore, the temperature distribution after the $\leqslant 600$ mK-selection is shifted towards lower temperatures, leading to an improved spin-state detection fidelity and a reduction in time for determining the spin state. Since the main objective is to reduce the experimental cycle time and the cyclotron mode needs to be prepared for each measurement cycle, the cooling scheme itself should require less than one minute. Apart from the cooling performance, the technique must be applicable to both protons and antiprotons and should not induce relative systematic frequency shifts larger than 1 × 10⁻¹¹.

One technique which satisfies the challenging requirements described in sections 1 and 2.2 is image-current coupling, where the proton is sympathetically cooled by a cloud of laser-cooled ⁹Be⁺ stored in a separate trap and the particles are coupled through image currents. A trap stack consisting of five traps was developed for our experiment, of which four are used to implement the coupling schemes and one is used for spin state analysis [12, 40, 41]. From an instrumentation point of view, our apparatus allows two different types of image-current couplings, namely coupling via a common endcap electrode or coupling via a common RLC resonator. The experimental configurations are depicted schematically in figure 2 and described in more detail in references [2, 12, 42].

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When performing the common-endcap coupling (figure 2(a)), the proton and beryllium ions are stored in the coupling trap (CT) and beryllium trap (BT), respectively. These traps are designed such that they share a common endcap with which the coupling scheme in the proposals of references [2, 42] can be realized. The image current induced by the particles charges up the capacitance of the common-endcap electrode. The resulting voltage generates a force on the particles, so that a system of two coupled oscillators is created. By laser cooling the beryllium ions with the 313 nm ²S_1/2 →²P_3/2 transition (natural linewidth: $\approx 2\pi \times 20\enspace \text{MHz}$), they act as a heat sink and the proton will be sympathetically cooled as well. The resonator, which is still required for detecting the particles and matching their axial frequency, can be decoupled by adding or subtracting an additional parallel capacitance using a cryogenic switch [41]. This switch changes the resonator's eigenfrequency by a preset amount which we adjust to $> 100$ kHz. Using such a switch has the advantage that the particle frequencies can be matched on resonance and no adjustment of the trapping potentials is necessary when starting the coupling procedure. Furthermore, at $> 100$ kHz resonator detuning no significant source of heating is present, which is the advantage of this specific approach. Alternatively, we can trap the particles in the PT and the source trap (ST), which are connected by a common RLC resonator, as depicted in figure 2(b). Since the resonator is an oscillator itself, this configuration represents a system of three coupled oscillators. The advantage of the RLC resonator is that it amplifies the image current signal by the Q-factor of the resonator, resulting in a fast energy exchange between the particles. In turn, however, the cooling effect competes with the stochastic heating by the noise, ultimately limiting the minimal proton temperature. This resonator-coupling scheme was employed in the first demonstration of sympathetic cooling through image currents recently conducted by our group [12]. The particle-to-detector coupling rate, which governs both the heating and cooling rate, can be reduced by detuning the particles from the resonator eigenfrequency. The reduction in heating rate allows for colder particle temperatures and this specific scheme will be numerically investigated in section 6.3.

Next, we describe the numerical implementation of the equations governing our simulations. The proton and ⁹Be⁺ ions induce image currents [28]

$$\begin{equation}{I}_{\text{res/com,p}}=\frac{q}{{D}_{\text{res/com,p}}}{\dot{z}}_{\text{p}}\qquad \text{and}\qquad {I}_{\text{res/com,Be}}=\frac{q}{{D}_{\text{res/com,Be}}}{\dot{z}}_{\text{Be}}\end{equation} \tag{ 9 }$$

in the resonator or the common endcap electrode, denoted with subscript 'res' and 'com', respectively. In all cases D_res/com,p/Be is a geometrical parameter defined by the trap geometry and can be considered an 'effective electrode distance'. The common endcap is modeled as an ideal capacitor C_com charged by the image currents. The potential U_com due to the accumulated charge leads to a force on the particles, ${F}_{\text{com,p/Be}}=\frac{q}{{D}_{\text{com,p/Be}}}{U}_{\text{com}}$. Similarly, the particles are driven and damped by the oscillating voltage of the resonator U_RLC at a corresponding force ${F}_{\text{RLC}}=\frac{q}{{D}_{\text{res,p/Be}}}{U}_{\text{RLC}}=\frac{q}{{D}_{\text{res,p/Be}}}L{\dot{I}}_{\text{L}}$. The resonator itself is described by a second-order differential equation. In addition to the ion-induced currents, the resonator is charged by the effective thermal Johnson–Nyquist noise current I_noise. We describe the resonator in terms of the current through the coil I_L. The full set of coupled differential equations, expressed in terms of input parameters of the simulation, is then:

$$\begin{equation}{m}_{\text{p}}{\ddot{z}}_{\text{p}}+2{q}_{\text{p}}{C}_{2\text{,p}}{U}_{0,\text{p}}{z}_{\text{p}}-\frac{{q}_{\text{p}}}{{D}_{\text{res,p}}}L{\dot{I}}_{\text{L}}-\frac{{q}_{\text{p}}}{{D}_{\text{com,p}}}{U}_{\text{com}}=0\end{equation} \tag{ 10 }$$

$$\begin{equation}{m}_{\text{Be}}{\ddot{z}}_{\text{Be}}+2{q}_{\text{Be}}{C}_{2\text{,Be}}{U}_{\text{0,Be}}{z}_{\text{Be}}-\frac{{q}_{\text{Be}}}{{D}_{\text{res,Be}}}L{\dot{I}}_{\text{L}}+\frac{{q}_{\text{Be}}}{{D}_{\text{com,Be}}}{U}_{\text{com}}=0\end{equation} \tag{ 11 }$$

$$\begin{equation}L{C}_{\text{res}}{\ddot{I}}_{\text{L}}+\frac{L}{R}{\dot{I}}_{\text{L}}+{I}_{\text{L}}+{I}_{\text{noise}}+\frac{{q}_{\text{p}}}{{D}_{\text{res,}\;\text{p}}}{\dot{z}}_{\text{p}}+\frac{{q}_{\text{Be}}}{{D}_{\text{res,Be}}}{\dot{z}}_{\text{Be}}=0\end{equation} \tag{ 12 }$$

$$\begin{equation}{U}_{\text{com}}=\frac{1}{{C}_{\text{com}}}\int \left(\frac{{q}_{\text{p}}}{{D}_{\text{com,p}}}{\dot{z}}_{\text{p}}-\frac{{q}_{\text{Be}}}{{D}_{\text{com,Be}}}{\dot{z}}_{\text{Be}}\right)\mathrm{d}t.\end{equation} \tag{ 13 }$$

Here, the first two equations describe the axial motion of the particles with external forces F_RLC and F_com and the latter two the resonator and common endcap capacitance. The subscripts p and Be denote the quantities related to the proton and the ⁹Be⁺ ion(s), respectively. In the case where no common endcap electrode is present (figure 2(b)) we set C_com → ∞.

We neglect the simulation of the radial modes since residual magnetic and electric field inhomogeneities which would couple the axial and radial modes are sufficiently small in our experiment. Furthermore, we neglect Coulomb-interaction between the beryllium ions since only the common mode of the beryllium cloud interacts with the resonator, as we will outline in section 6.2.

We solve equations (10)–(13) numerically with typical time steps of Δt = 1–5 ns compared to $\approx 600\text{kHz}$ oscillator frequency. The time steps are chosen such that effects which can be attributed to the finite time step are negligibly small and do not affect the coupling studies while still ensuring a reasonable computation time. If the motion of more than one beryllium ion is simulated, either an effective cloud or an array of individual beryllium ions is calculated, depending on whether the single particle behaviour is of interest. A key ingredient of our simulation code is the proper implementation of I_noise. Equation (12) is a type of Langevin equation where the resistance R determines, according to the fluctuation–dissipation-theorem, both the dissipation and noise terms. In this case I_noise obeys a delta-correlated Gaussian probability distribution with zero mean. The mean squared value of the single-sided power spectral density is given by $\left\langle {I}_{\text{noise}}^{2}\right\rangle =4{k}_{\text{B}}{T}_{\text{res}}{\Delta}f/R$. In the time domain representation, the bandwidth Δf must be substituted by the Nyquist frequency, which is equal to half of the sampling frequency f_s = 1/Δt according to the Nyquist–Shannon sampling theorem, yielding Δf = 1/(2Δt). Furthermore, in order to achieve a delta-correlation of two subsequent noise samples, each noise value is multiplied with a random value drawn from a Gaussian distribution with a mean of 0 and a standard deviation of 1, abbreviated as ${\mathcal{N}}_{\text{n}}(0,1)$. In total, in each step n the equation for the noise sample then reads

$$\begin{equation}{I}_{\text{noise,n}}=\sqrt{2{k}_{\text{B}}{T}_{\text{res}}/(R{\Delta}t)}\cdot {\mathcal{N}}_{\text{n}}(0,1).\end{equation} \tag{ 14 }$$

Each equation in equations (10)–(13) is solved individually by employing a fourth-order symplectic integrator [43]. This method conserves a Hamiltonian for conservative forces and, although the resonator noise damps and excites the particles, avoids long-term drifts and reproduces energy conservation in the limit R → ∞. While the integrator can be applied directly to the equations of motion of the particles, the integrator for the resonator equation requires a modification due to the dissipative term: following the notation in reference [43], one step for the modified discretized RLC circuit equation then reads

$$\begin{equation}\begin{aligned}\hfill {I}_{\text{n}}^{(i)}& ={I}_{\text{n}}^{(i-1)}+{c}_{i}{\dot{I}}_{\text{n}}^{(i-1)}{\Delta}t\hfill \\ \hfill {\dot{I}}_{\text{n}}^{(i)}& =\frac{1}{1+\frac{{d}_{i}{\Delta}t}{RC}}{\dot{I}}_{\text{n}}^{(i-1)}-\frac{1}{1+\frac{{d}_{i}{\Delta}t}{RC}}\frac{{d}_{i}{\Delta}t}{LC}\left({I}_{\text{n}}^{(i-1)}+{I}_{\text{ext}}\right).\hfill \end{aligned}\end{equation} \tag{ 15 }$$

Here i ∈ {1, 2, 3, 4} denotes the sub-steps where ${I}_{\text{n}}^{(0)}\equiv {I}_{\text{n}}$, ${\dot{I}}_{\text{n}}^{(0)}\equiv {\dot{I}}_{\text{n}}$, ${I}_{\text{n}}^{(4)}\equiv {I}_{n+1}$ and ${\dot{I}}_{\text{n}}^{(4)}={\dot{I}}_{\text{n}}^{(3)}\equiv {\dot{I}}_{n+1}$. Besides, I_ext = I_noise + I_res,p + I_res,Be is the external current. In the case of R → ∞ the factor $\left.1/(1+\frac{{d}_{i}{\Delta}t}{RC}\right)$ is 1 and the energy-conserving case is restored. In order for the algorithm to be symplectic, the dimensionless coefficients c_i and d_i , which define the length of the interleaved time steps, must be chosen appropriately. While two out of four coefficients are negative, i.e. correspond to a step backwards in time, their sum obeys ∑c_i = ∑d_i = 1.

Laser cooling of the ⁹Be⁺ ions is modeled by assuming that a narrow linewidth laser with frequency f_L, detuned by δ_L relative to the transition frequency f₀ = f_L − δ_L, interacts with the ⁹Be⁺ ions. In each simulation step in which a ⁹Be⁺ ion is in the ground state it can absorb a photon with probability

$$\begin{equation}{P}_{\text{abs}}={B}_{12}({\dot{z}}_{\text{Be}}){\Delta}t=\frac{{I}_{\text{laser}}}{{I}_{\text{sat}}}\frac{{\Gamma}{\Delta}t/2}{1+4{\left(\frac{2\pi {\delta }_{\text{L}}+{k}_{\text{L}}{\dot{z}}_{\text{Be}}}{{\Gamma}}\right)}^{2}}.\end{equation} \tag{ 16 }$$

B₁₂ and B₂₁ are the Einstein rate coefficients for absorption and emission, respectively. Further, k_L is the laser wave vector in axial direction, Γ the natural linewidth of the transition ($\approx 2\pi \times 20$ MHz in our case), I_laser the laser intensity and ${I}_{\text{sat}}=\frac{\pi h{\Gamma}{f}_{\text{0}}^{3}}{3{c}^{2}}$ the saturation intensity [44]. The fraction $\frac{{I}_{\text{laser}}}{{I}_{\text{sat}}}$ is a free parameter and is the simulation equivalent to the laser intensity in the experiment. If an absorption occurs, the ion's velocity is reduced by ℏk_L/m_Be and it enters the excited internal electronic state. In the excited state the ion cannot absorb photons but can spontaneously decay to the electronic ground state with probability ΓΔt. Upon spontaneous decay, the ion receives an axial velocity kick, ℏk/m_Be cos(φ), where φ is randomly distributed from 0 to 2π to account for the randomness in the direction of the spontaneous emission. We note that this is a simplified model as it does not account for the anisotropy of spontaneous emission in a magnetic field [45], which is however negligible for the studies in this paper. Additionally, the ion can emit photons by stimulated emission with probability ${B}_{21}({\dot{z}}_{\text{Be}}){\Delta}t={B}_{12}({\dot{z}}_{\text{Be}}){\Delta}t$ where the ion's velocity is increased by +ℏk_L/m_Be.

The above considerations enable us to accurately simulate the dynamics of the trapped particles and resonator including laser cooling of the beryllium ions under ideal conditions. However, in practice, several experimental uncertainties can occur. While long-term axial frequency drifts can easily be compensated, drifts on the order of the duration of one axial frequency measurement, typically about 60 s, lead to frequency fluctuations that affect the energy exchange between the particles. The sources of these fluctuations are various, for example trapping voltage instabilities and drifts of the power supplies, finite pressure and temperature stabilities of the liquid helium bath, drifts due to the Coulomb-interaction between the ⁹Be⁺ ions, or residual inhomogeneities of the electric potential or magnetic field. A realistic estimate for the axial frequency fluctuation of the proton is σ(ν_z ) ≈ 50–100 mHz or σ(ν_z )/ν_z ≈ 1 × 10⁻⁷ at 60 s for our experiment [24]. In the following, we summarize the individual contributions by introducing an effective dynamical axial frequency variation by adding a modification ΔU to the trapping potential, where ΔU performs a random walk:

$$\begin{equation}{\Delta}{U}_{n+1}={\Delta}{U}_{\text{n}}+\delta U\cdot {\mathcal{N}}_{\text{n}}(0,1).\end{equation} \tag{ 17 }$$

The index n indicates the step number and ${\mathcal{N}}_{\text{n}}(0,1)$ is another number drawn from a Gaussian distribution. δU is a scaling factor determining the expectation value of the standard deviation of the random walk σ(U₀) and is related to the frequency stability by σ(U₀)/U₀ = 2σ(ν_z )/ν_z . In order to express δU in terms of practical experimental quantities, we define ξ_stab as the relative voltage stability for frequency measurements of length 60 s and obtain

$$\begin{equation}\delta {U}_{0}={\xi }_{\text{stab}}\cdot {U}_{0}\cdot \sqrt{\frac{1}{n(t=60\,\,\,\mathrm{s})}}.\end{equation} \tag{ 18 }$$

Here n(t = 60 s) indicates the number of steps required for 60 s of simulated time. It should be noted that ΔU_n does not necessarily correspond to the actual behavior of the power supplies used in the experiment but is rather only a convenient tool to implement dynamical frequency fluctuations within one simulation run.

Data preparation and analysis is performed in the programming language R [46]. Since typical simulated time spans are on the order of 30–60 s, time steps of Δt = 1–5 ns imply about 10¹⁰ simulation steps, each with several dozens of operations. In order to minimize the computation time, the calculation-intensive part is outsourced to a compiled C++ program via the Rcpp-package [47]. The main R script can call several instances of the C++ program in parallel via the doParallel- and foreach-package [48, 49], allowing threads up to the number of virtual cores to run in parallel.

In summary, we emphasize that the physics in our simulation is implemented from first principles. We only use experimentally established model assumptions and no heuristic scaling factors.

Next, we demonstrate that the simulation results agree well with theory and experiments, in particular those in reference [12]. We simulate the ⁹Be⁺ ions individually and consider no additional heating effects besides the one stemming from the resonator [50]. Figure 3 shows an FFT spectrum recorded with a common resonator involving the detector resonance, a proton in the PT detuned by −35 Hz from the detector's resonance frequency ν_res and several ⁹Be⁺ ions with ν_z,Be = ν_res in the ST. Compared to the simulated spectrum in figure 3(b), the absolute signal level of the experimental spectrum in figure 3(a) is boosted by several amplifier stages. The broad resonance with full-width at half maximum (FWHM) γ_res can be matched to the unperturbed lineshape of the noise-charged parallel RLC-circuit and the two dips stem from the proton and ⁹Be⁺ ions. From such a dip spectrum several properties can be derived. First, the dip position yields the axial eigenfrequency of the particle. Here we confirm that the simulations produce the dip at the correct frequency at ${\nu }_{\mathit{\text{z}}}=1/(2\pi )\sqrt{2q{C}_{2}{U}_{0}/m}$. Second, the dip width γ_z , defined as FWHM of the dip signal, is a measure of the resonator-to-ion coupling strength. Assuming γ_z ≪ γ_res, it can be calculated as

$$\begin{equation}{\gamma }_{z}=\frac{1}{2\pi }\frac{R}{m}\frac{{q}^{2}}{{D}_{\text{res}}^{2}}N,\end{equation} \tag{ 19 }$$

where N is the number of particles with mass m [28].The dip widths of the simulated dip spectra match with equation (19). Similarly, the simulated noise resonance has the same FWHM as we expect from theory. Next, since the resonator is a noisy thermal bath, the particle energies follow a Boltzmann distribution with the mean temperature defined by the resonator. We confirm that both the absolute resonator temperature, which is a free input parameter in our simulations, as well as the Boltzmann distributions of the single particle energies are correctly reproduced. Regarding laser cooling, we reproduce the theoretical Doppler limit of ${T}_{\text{D}}=\frac{\hslash {\Gamma}}{2{k}_{\text{B}}}\approx 0.5$ mK as minimal temperature of the ⁹Be⁺ ions. Furthermore, the scattering rate of photons γ_s for a single ⁹Be⁺ ion artificially held at a constant velocity matches the theoretical one [44]:

$$\begin{equation}{\gamma }_{\text{s}}=\frac{{I}_{\text{laser}}}{{I}_{\text{sat}}}\frac{{\Gamma}/2}{1+\frac{{I}_{\text{laser}}}{{I}_{\text{sat}}}+4{\left(\frac{2\pi {\delta }_{\text{L}}+{k}_{\text{L}}{\dot{z}}_{\text{Be}}}{{\Gamma}}\right)}^{2}}.\end{equation} \tag{ 20 }$$

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While the previously mentioned properties must be met for each oscillator individually and are straightforward to check, the coupling between them, especially between the proton and the ⁹Be⁺ ions, is more subtle. In order to reach our goal of $\approx 10\,\text{mK}$ axial proton temperature, it is crucial to understand the coupling mechanism in detail. In the following, we will compare the experimental data of reference [12] with FFT spectra of simulation data for three different temperature regions. Additionally, since the simulations provide us access to the time domain data, we can demonstrate that our experimental signature is truly caused by a temperature manipulation.

Reference [12] employed the coupling scheme with a common RLC resonator. First, the coupling between the proton and ⁹Be⁺ ions was demonstrated above the resonator temperature by bringing the proton to resonance with excited beryllium ions. Therefore, the first temperature scale at which we compare simulations against experiment is at particle temperatures above the resonator temperature, depicted in figure 4. Here we simulate a single proton and a cloud of 10 ⁹Be⁺ ions. The beryllium ions are excited by a parametric drive with frequency 2ν_z,Be, which is implemented as an additional force F_dr (t) = F_0,dr  cos(4πν_z,Be t)in equation (11). In the experiment, special care was taken to ensure that the proton was not directly affected by the heating drive but only via the coupling to the ⁹Be⁺ ions. Since the axial frequencies of both particles are matched, not only the ⁹Be⁺ ions but also the proton significantly gains energy, demonstrating resonator-mediated energy exchange between the particles. Since γ_z,p < γ_z,Be < γ_res, the proton reaches a temperature higher than the one of the resonator. This results in a sharp peak above the resonator signal in the FFT spectrum in both the experimental and simulation FFTs, see figures 4(a) and (b), respectively. The slight asymmetry in the experimental FFT spectrum stems from a small anharmonic component of the axial potential in the proton trap. Since this is a well-understood effect [12, 51], we have included such an anharmonicity in the simulations as well to reproduce the experimental signal. The time domain data of the simulations are depicted in figure 4(c). Here, the heating drive is switched on at t = 10 s. Before that, the energies of all three oscillators perform a random walk through a Boltzmann distribution with the same temperature. With the parametric drive switched on, both the proton and ⁹Be⁺ ions gain energy equivalent to several hundred kelvin. The average energy per ion is plotted for the beryllium cloud.

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Second, the coupling at the equilibrium temperature of the resonator was demonstrated. To this end, the characteristic response of the coupled three-oscillator system is recorded, shown in figure 5. We employ a single proton and a single ⁹Be⁺ ion, whose frequencies are both detuned 130 Hz from the resonator eigenfrequency. We then sweep the beryllium frequency over the proton frequency and display the signal power of the corresponding frequency spectra as colour-coded columns in a heat map. Both the experimental and simulated data in figures 5(a) and (b) exhibit the characteristic response of a coupled three-oscillator system. In addition, we plot the energy exchange between the proton, the ⁹Be⁺ ion and the resonator at 130 Hz detuning in the time domain for the simulated data in figure 5(c). To emphasize the coupling between the proton and ⁹Be⁺ ions, noise and dissipation were turned off by setting R = ∞ and T_res = 0 for this plot. Since the resonator FWHM γ_res is $\approx 30\,\text{Hz}$, the resonator is only weakly excited by the particles. The exact behavior of the energy exchange in figure 5(c) strongly depends on the initial relative particle phase.

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Last, we establish common-resonator coupling between a single proton and 50 laser-cooled ⁹Be⁺ ions. The frequency domain data in both experiment and simulation, figures 6(a) and (b) respectively, show a broad dip with reduced signal-to-noise and a superimposed narrow dip below. The broad dip corresponds to the beryllium cloud which, due to the laser cooling, cannot fully compensate the resonator noise current and hence exhibits a reduced signal strength. The narrow dip corresponds to the proton, which is coupled to both the resonator and the ⁹Be⁺ cloud. In the time-domain plot, figure 6(c), both particles have an initial energy of 10 K ⋅ k_B. After turning on the laser at t = 7.5 s, not only the beryllium ions but also the proton is significantly cooled within a few seconds. This is exactly the effect we desire and this specific cooling scheme will be more thoroughly investigated in section 6.2.

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In this section, we have shown that our simulations are able to unambiguously reproduce the observed experimental signatures [12]. By giving access to the time domain data, the results provide another layer of understanding. The laser cooling simulations match the theoretical expectations down to the Doppler limit, and are therefore proven to be a practical and credible tool to investigate cooling schemes that achieve particle temperatures in the mK-regime.

Next, we will theoretically investigate the three coupling scenarios whose experimental setups were described in section 3, namely common-endcap coupling, resonant coupling via a common RLC circuit, and off-resonant coupling via a common RLC circuit.

We simulate individual beryllium ions in section 6.2 and effective clouds otherwise since only in section 6.2 absolute laser intensities are of interest. The initialization of the particle modes is randomized and has no influence on the coupling schemes. If not explicitly stated otherwise, we use the following simulation parameters for the resonator: R = 236 × 10⁶Ω, L = 3 mH, C_res = 21.4 pF, yielding a resonator Q-factor of 20 000 and a resonance frequency of 626.9 kHz. Regarding the trap parameters, we adjust the product C₂ U₀ such that the particle oscillates at the desired frequency. The effective electrode distances are D_res,p = 10.0 mm for the PT, D_res,Be = 5.7 mm for the ST and D_com,p = D_com,Be = 3.2 mm for the CT and BT (compare figure 2) [24]. The common endcap capacitance is C_com = 5.5 pF for the common-endcap coupling in section 6.1 and C_com → ∞ for the resonator coupling in sections 6.2 and 6.3. Additionally, we assume that the resonator has an effective noise temperature of 10 K, accounting for the fundamental Johnson–Nyquist noise at 4.2 K (the physical temperature of the experiment) and the input noise of our cryogenic amplifiers [13, 30]. We use 80 ⁹Be⁺ ions as this is approximately the largest number of ⁹Be⁺ ions that were used in reference [12]. Finally, a time step of Δt = 4 ns was used for all subsequent simulations.

6.1. Common-endcap coupling

In order to simulate the common-endcap coupling in our trap system, we detune the resonator eigenfrequency by 200 kHz so that the energy exchange is dominated by the common-endcap coupling. Since the resonator terms in equations (10)–(13) are then negligible for the ion–ion-coupling with no laser applied, the system of coupled differential equations can be solved analytically. In order to derive a general solution we allow for more than one particle in each cloud by setting q_p → N_p q_p, m_p → N_p m_p, q_Be → N_Be q_Be and m_Be → N_Be m_Be in equations (12) and (13), where N_p,Be is the number of protons and beryllium ions, respectively. For the sake of a compact notation we abbreviate D_com,p ≡ D_p and D_com,Be ≡ D_Be in the following calculations. Furthermore, we assume that the particle eigenfrequencies are exactly matched, i.e. $\frac{2{q}_{\text{p}}{C}_{2\text{,}\;\text{p}}{U}_{\text{0,p}}}{{m}_{\text{p}}}=\frac{2{q}_{\text{Be}}{C}_{2\text{,Be}}{U}_{\text{0,Be}}}{{m}_{\text{Be}}}={\omega }_{0}^{2}$. Solving the coupled differential equation for the proton motion yields

$$\begin{equation}{z}_{\text{p}}(t)={c}_{1}\,{\text{e}}^{\text{i}{\lambda }_{1}t}+{c}_{2}\,{\text{e}}^{\text{i}{\lambda }_{2}t}+{c}_{3}\,{\text{e}}^{\text{i}{\lambda }_{3}t}+{c}_{4}\,{\text{e}}^{\text{i}{\lambda }_{4}t},\end{equation} \tag{ 21 }$$

where c_1,2,3,4 are free coefficients determined by the initial conditions and λ_1,2,3,4 are the corresponding eigenfrequencies:

$$\begin{equation}{\lambda }_{1,2}=\pm {\omega }_{0}\end{equation} \tag{ 22 }$$

$$\begin{equation}{\lambda }_{3,4}=\pm \sqrt{{\omega }_{0}^{2}-\frac{{m}_{\text{p}}{D}_{\text{p}}^{2}{N}_{\text{Be}}{q}_{\text{Be}}^{2}+{m}_{\text{Be}}{D}_{\text{Be}}^{2}{N}_{\text{p}}{q}_{\text{p}}^{2}}{{C}_{\text{com}}{D}_{\text{p}}^{2}{D}_{\text{Be}}^{2}{m}_{\text{p}}{m}_{\text{Be}}}}\end{equation} \tag{ 23 }$$

$$\begin{equation}\approx \pm ({\omega }_{0}-{\Omega}).\end{equation} \tag{ 24 }$$

Here we abbreviated

$$\begin{equation}{\Omega}=\frac{1}{2}\frac{{m}_{\text{p}}{D}_{\text{p}}^{2}{N}_{\text{Be}}{q}_{\text{Be}}^{2}+{m}_{\text{Be}}{D}_{\text{Be}}^{2}{N}_{\text{p}}{q}_{\text{p}}^{2}}{{\omega }_{0}{C}_{\text{com}}{D}_{\text{p}}^{2}{D}_{\text{Be}}^{2}{m}_{\text{p}}{m}_{\text{Be}}}.\end{equation} \tag{ 25 }$$

Assuming the initial conditions z_p = z₀, ${\dot{z}}_{\text{p}}=0$, z_Be = 0 and ${\dot{z}}_{\text{Be}}=0$, the temporal evolution of the energy of the proton is

$$\begin{align}\hfill {E}_{\text{p}}(t)=& \frac{1}{2}\frac{{m}_{\text{p}}{z}_{0}^{2}}{{({m}_{\text{p}}{D}_{\text{p}}^{2}{N}_{\text{Be}}{q}_{\text{Be}}^{2}+{m}_{\text{Be}}{D}_{\text{Be}}^{2}{N}_{\text{p}}{q}_{\text{p}}^{2})}^{2}}\left(\right.{\omega }^{2}\left[{({m}_{\text{p}}{D}_{\text{p}}^{2}{N}_{\text{Be}}{q}_{\text{Be}}^{2})}^{2}\right.\hfill \\ \hfill & \left.\left.+{({m}_{\text{Be}}{D}_{\text{Be}}^{2}{N}_{\text{p}}{q}_{\text{p}}^{2})}^{2}+2{m}_{\text{p}}{m}_{\text{Be}}{N}_{\text{p}}{N}_{\text{Be}}{q}_{\text{p}}^{2}{q}_{\text{Be}}^{2}{D}_{\text{p}}^{2}{D}_{\text{Be}}^{2}\,\mathrm{cos}({\Omega}t)\right]+\mathcal{O}({\Omega}\omega )\right).\hfill \end{align} \tag{ 26 }$$

Since Ω ≪ ω, the terms with $\mathcal{O}({\Omega}\omega )$ can be neglegted. From equation (26) $\frac{{\Omega}}{2\pi }$ can be identified as the Rabi frequency of the coupled proton-beryllium system. The difference between equation (25) and previously reported versions [2, 3] arises from the fact that in this work, up to equation (24) no approximations were used. In order for the energy to be exchanged completely, i.e. E_p(t = π/Ω) = 0, the condition

$$\begin{equation}\frac{{N}_{\text{p}}{q}_{\text{p}}^{2}}{{m}_{\text{p}}{D}_{\text{p}}^{2}}\stackrel{!}{=}\frac{{N}_{\text{Be}}{q}_{\text{Be}}^{2}}{{m}_{\text{Be}}{D}_{\text{Be}}^{2}}\end{equation} \tag{ 27 }$$

must be satisfied. Inserting this condition into equation (25) reproduces the previously reported formula [2, 3] for the Rabi oscillation frequency. Equation (27) reflects the simultaneous conservation of energy and conservation of charge on the common-endcap capacitance within one Rabi cycle. We note that the studied system of coupled oscillators is symmetric in many experiments and equation (27) is automatically fulfilled. However, we emphasize that if the condition in equation (27) is not met, the energy is exchanged only partially and with a faster frequency. As demonstrated in the following, this has considerable implications for the realization of the cooling schemes in the experiment. Figure 7(a) shows an exemplary energy exchange between a single proton and 9 or 80 ⁹Be⁺ ions, top and bottom plot, respectively. Since D_p = D_Be and m_Be ≈ 9 m_p in our case, the energy exchange is complete for 9 beryllium ions. While the Rabi frequency for 80 beryllium ions is higher, only a fraction of the proton's energy is exchanged. This would be a severe limitation of the common-endcap coupling technique as one would need to choose between a complete energy exchange (N_Be = 9) or a fast coupling time (N_Be ≫ 9). However, the behavior of the Rabi oscillations when varying the number of beryllium ions is equivalent to introducing a small relative frequency detuning between the two species. Consequently, one can reverse the effect of using N_Be ≫ 9 by introducing a small relative detuning. The analytical calculations are laborious, thus we make use of the numerical simulations here. Figure 7(b) shows the Rabi oscillation amplitude (peak-to-peak) for 80 ⁹Be⁺ ions as a function of relative particle detuning. While for an exact frequency matching only $\approx 4\enspace \text{K}\cdot {k}_{\text{B}}$ are exchanged per Rabi cycle, at +40 mHz relative detuning the full Rabi amplitude is recovered. In the case of nonzero detunings equation (25) does not hold anymore. However, if the detuning is chosen such that the energy is exchanged completely, the previously reported ${\Omega}\propto \sqrt{{N}_{\text{Be}}}$ is preserved, i.e. ${\Omega}(40\text{mHz},80\,\,\text{Be})=\sqrt{\frac{80}{9}}{\Omega}(\text{0 mHz},9\,\,\text{Be})$. The parameters presented here yield Ω(0 mHz, 80 Be)/(2π) = 53 mHz and Ω(40 mHz, 80 Be)/(2π) = 34 mHz. Although matching two species to a few ten mHz is experimentally challenging, it is crucial to account for this in the experimental realization of the coupling scheme as it impacts the cooling times and the achieved temperature limits.

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The initial proposals [2, 42] envisaged cooling close to the Doppler limit by coupling the target ion for exactly half a Rabi cycle (one π-pulse) to the subsequently laser-cooled ⁹Be⁺-cloud. However, figure 7(b) shows that a frequency mismatch of only $\approx 10$ mHz limits the residual proton energy after one π-evolution to ≳1 K ⋅ k_B, far higher than the 10 mK required for our experiment. As a result, we investigate a cooling scheme that is based on repetitively turning on the cooling laser and performing several imperfect energy exchanges. Compared to continuous laser operation, chopping the laser achieves orders of magnitude faster cooling. The reason for this can be understood in the Rabi oscillation picture: in the continuous case the ⁹Be⁺ cloud is artificially held at its minimal temperature. The cooling rate for the proton is given by the local slope of the energy exchange curve, i.e. it vanishes to first order for continuously cooled ⁹Be⁺ ions [3].

To determine a realistic cooling time constant, we incorporate the experimental frequency uncertainties in this section and the following ones twofold: first, we add an additional detuning δν to the proton frequency due to non-ideal matching to the ⁹Be⁺ frequency. δν is drawn uniformly from the interval δν ∈ [−Δ_unc, + Δ_unc] with Δ_unc = 100 mHz and randomly for every simulation. Second, we assume a relative voltage stability as defined in section 4 of ξ_stab = 1 × 10⁻⁷ for both the proton and beryllium trapping voltages. In a first step we determine the ideal laser cycle time for the chopped laser scheme. The ⁹Be⁺ ions are initialized at an energy of 1 mK ⋅ k_B and the proton at an energy of 10 K ⋅ k_B. As figure of merit we use the average temperature gain of the beryllium cloud over time, ΔT_Be/Δt, displayed in figure 8(a). By switching on the cooling laser at the time of maximum ΔT_Be/Δt, in this simulated case at 6.0 s, to first order the fastest cooling time constant is achieved. We note that more complex laser schemes with varying cycle times could potentially lead to an even better cooling performance, however, in this paper we will limit the discussion to schemes with constant laser cycle times. The on-time of the laser is set to 2% of the laser cycle time. Figure 8(b) shows a representative cooling run with the cooling laser turned on every 6.0 s. Averaging several of such runs with the uncertainties described above allows us to perform a least-square fit with an exponential function

$$\begin{equation}{T}_{\text{p}}={T}_{\text{p},0}\,{\text{e}}^{-t/{\tau }_{\text{CC}}}+{T}_{\text{p},\mathrm{m}\mathrm{i}\mathrm{n}}\end{equation} \tag{ 28 }$$

and extract a cooling time constant τ_CC as it is done in figure 9(a). We obtain τ_CC = 18.6(1.4) s for the simulated case with 80 ⁹Be⁺ ions. The uncertainty is estimated by fitting the single runs and calculating the standard deviation of the mean τ_CC.

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In addition to the cooling time constant, the other parameter of interest is the lower temperature limit, determined by the equilibrium of the competing cooling and heating rate. In case of the common-endcap coupling, the heating by the resonator noise is strongly suppressed due to the resonator detuning of several hundred kHz. Additionally, other sources of heating are negligibly small [50]. We investigate the lower limit that can be achieved in the simulations by repeating the procedure for the cooling time constant, but initializing the proton at 1.5 mK, shown in figure 9(b). The cooling time constant is within uncertainty indistinguishable from the one with 10 K initial proton temperature. More importantly, the lower temperature limit is dictated by the temperature of the ⁹Be⁺ cloud, which itself is cooled to the Doppler limit of T_D ≈ 0.5 mK.

The last quantity of practical interest is the sensitivity of the cooling performance to relative axial frequency detunings between the proton and the ⁹Be⁺ cloud. The cooling efficiency for a fixed relative detuning depends on the laser cycle time. In the worst case, even for a rather small relative detuning of $\approx 10$ mHz no cooling can occur if the laser cycle time is an integer multiple of the Rabi cycle. The relevant quantity in the experiment is the precision with which the particle frequencies can be matched. Hence we vary the frequency matching precision Δ_unc as defined above instead of introducing a fixed detuning. Furthermore, since for large detuning intervals fitting the temperature evolution with equation (28) becomes an erroneous measure of cooling effectiveness, we will use the mean proton temperature at 30 s after initialization instead. Figure 10(a) depicts the sensitivity of the proton temperature at 30 s as a function of the detuning uncertainty. The initial proton energy was 10 K ⋅ k_B. The proton temperature at 30 s nearly doubles when moving from 100 mHz to 200 mHz, indicating that absolute detunings of $> 100$ mHz contribute little to the cooling process, which is a result of the comparably low Rabi frequency of 34 mHz. Figure 10(b) shows the simulated proton temperature after initializing the proton at 0.5 mK ⋅ k_B and letting it evolve for 30 s while applying the chopped laser to the ⁹Be⁺ ions. Since for the common-endcap coupling the resonator is detuned, no heating process competes with the cooling process. As a result, the lower temperature limit is not affected and the proton temperature can still reach the Doppler limit of T_D ≈ 0.5 mK.

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6.2. Resonant coupling mediated by an RLC resonator

A sketch of the experimental setup for the resonant coupling scheme is shown in figure 2(b) and the experimental proof of principle was recently demonstrated in reference [12]. This approach is the easiest to observe experimentally as the proton and the ⁹Be⁺ ions are continuously being detected in resonance with the RLC resonator, which is the standard operation mode of our Penning-trap experiment.

In the case of the resonant coupling with an RLC circuit, the resonator noise causes the energy exchange between the proton and ⁹Be⁺ ions to be highly incoherent on a ms-timescale. As a result, a cooling scheme that relies on single π-pulses will not be able to consistently achieve the desired 10 mK. Although a chopped laser scheme might offer an improved cooling performance compared to a continuous laser operation, we will investigate the continuous case in this work since it has already been experimentally demonstrated [12].

As an introductive theoretical analysis, we first consider two identical particles in two separate but identical traps connected to the same resonator. The motion of these two particles can be described by a symmetric and antisymmetric mode, where the individual oscillations are in phase or shifted by a phase of π, respectively. The individual image currents induced into the resonator interfere destructively for the antisymmetric mode. As a result, the resonator only couples to the symmetric mode and only this mode is thermalized to the detector temperature. The same argument applies to two identical ions in the same trap. The experimental evidence for this is provided by the linear scaling of the dip width with the number of ions in the trap, which arises from substituting q → N_i ⋅ q and m → N_i ⋅ m in equation (19). However, this substitution is only valid if the particles' motions are in phase. On the other hand, by introducing an additional damping constant to one of the particles, energy is transferred between the symmetric and antisymmetric mode. In our experiment, the cooling laser constitutes this additional damping constant. The fact that only the symmetric mode is heated by the resonator and the damping of the ⁹Be⁺ ions by the laser continuously transfers energy between all modes implies that even a weak intensity laser is sufficient to eventually cool all modes except the symmetric one down to the Doppler limit. In the case of negligible energy of the non-symmetric modes, the final proton temperature is determined by the proton's relative component of the symmetric mode temperature. The relative components are given by the relative dip widths, γ_p and γ_Be, respectively:

$$\begin{equation}{T}_{\text{res}}={T}_{\text{p}}+{T}_{\text{Be}}\end{equation} \tag{ 29 }$$

$$\begin{equation}{T}_{\text{p}}=\frac{{\gamma }_{\text{p}}}{{\gamma }_{\text{Be}}}{T}_{\text{Be}}.\end{equation} \tag{ 30 }$$

These equations result in the T_p ∝ 1/N_Be-scaling initially proposed in reference [42] and measured in reference [12]:

$$\begin{equation}{T}_{\text{p}}=\frac{1}{1+\frac{{\gamma }_{\text{Be}}}{{\gamma }_{\text{p}}}}{T}_{\text{res}}\underset{\sim }{\propto }\frac{1}{{N}_{\text{Be}}}.\end{equation} \tag{ 31 }$$

However, in this form equation (31) is only true for weak laser intensities, i.e. if the damping rate of the ⁹Be⁺ ions due to the laser, γ_L, is much smaller than γ_Be. At higher laser intensities, the beryllium cloud is decoupled from the resonator. This effect can be accounted for by a reduction of the effective dip width, expressed by the substitution ${\gamma }_{\text{Be}}\to {\tilde{\gamma }}_{\text{Be}}=k{\gamma }_{\text{Be}}$ with dimensionless scaling parameter 0 ⩽ k ⩽ 1 in reference [12].

Our simulation results show that, in contrast to the lower limit of the proton temperature, a higher laser intensity is beneficial for the cooling time constant as it increases the rate of energy dissipation. In order to study these relations quantitatively in terms of experimental parameters, we simulate the proton and a cloud of 80 continuously laser-cooled ⁹Be⁺ ions tuned on resonance with the detector. Figure 11 shows the cooling time constant τ_RC and the lower temperature limit T_p,min as a function of laser intensity for two exemplary laser detunings of −20 MHz and −200 MHz, depicted in subfigures (a) and (b), respectively. The cooling time constant is determined by least-squares fits with equation (28) for an initial proton energy of 10 K ⋅ k_B. The lower temperature limit is determined by initializing the proton at 0 K ⋅ k_B and averaging the proton energy after it has reached equilibrium.

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While a much stronger laser intensity is required for −200 MHz laser detuning, the general behavior of τ and T_p,min are the same: the cooling time constant drops with larger laser intensity as the ⁹Be⁺ ions are damped more strongly and reaches values down to a few hundred ms. However, at such high laser intensities the beryllium ions are decoupled too strongly from the resonator, leading to a suboptimal proton temperature limit. Instead, the proton temperature saturates in the region where γ_L ≪ γ_Be. The simulated values in the weak intensity limit agree well with the theoretical prediction of T_p,min = 350 mK by equation (31), indicated by the blue dashed line.

According to equation (19), the beryllium dip width can be increased not only by loading more beryllium ions, but also by reducing the effective electrode distance D_res. Since the cooling time constants are a function of both the laser intensity and D_res,Be, we use T_p,min in the weak intensity limit as a measure of improved cooling performance. In figure 12(a), D_res,Be is decreased starting from the currently manufactured trap with D_res,Be = 5.7 mm. The red line is the prediction of equation (31) and we observe excellent agreement between simulation and theory. However, even for a trap with D_res,Be = 2.0 mm, which is challenging to manufacture and difficult to operate with 80 ⁹Be⁺ ions, the lower temperature limit is only about 50 mK, a factor of 5 higher than our desired temperature.

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While the resonant coupling scheme with continuous laser operation has a comparably high temperature limit, it is the most robust technique against experimental uncertainties. Figure 12(b) shows the final proton temperature as a function of the detuning of the ⁹Be⁺ cloud. The fact that the minimal T_p,min is shifted from 0 Hz detuning is caused by the effect of maximizing the coupling strength between the two particle species by introducing a relative detuning as described in section 6.1. However, since the laser is not chopped here, the effect on the cooling performance is negligible. Since the particles are continuously detected on resonance, matching and stabilizing them to $\approx 100$ mHz is comparably straightforward and routinely done in our experiment. As figure 12(b) shows, the increase in T_p,min is negligible for ≈100 mHz detuning.

6.3. Off-resonant coupling mediated by an RLC resonator

The off-resonant resonator coupling approach combines the ideas of the previous two cooling schemes by detuning the particle frequencies by δν_res from the resonator eigenfrequency, typically several kHz [3]. Similar to the common-endcap coupling, the particles then perform Rabi oscillations mediated by a common effective capacitance

$$\begin{equation}{C}_{\text{eff}}=\frac{\mathfrak{I}({Z}_{\text{RLC}}^{-1})}{\omega }\,\underset{\sim }{\propto }\,\delta {\nu }_{\text{res}},\end{equation} \tag{ 32 }$$

where Z_RLC is the impedance of the RLC detection circuit. Combined with equation (25) this implies that the Rabi frequency decreases approximately linear with the resonator detuning. With this scheme, the application of π-pulses is not only inhibited by the required frequency matching precision, but also by the residual heating due to the resonator noise. As a result, we employ a chopped laser scheme again.

In order to pinpoint the optimal laser cycle time and cooling time constant, we follow the same procedure as for the common-endcap coupling for different resonator detunings. However, due to the residual particle heating by the resonator noise, the lower temperature limit is higher than the Doppler limit and is a function of δν_res. Moreover, the incoherence introduced by the noise causes the ideal laser cycle time for the cooling time constant to become suboptimal for reaching the lowest final temperatures. Figure 13(a) shows the simulated proton temperature as a function of laser cycle time at 8 kHz resonator detuning. The proton was initialized at 50 mK and the system was evolved for 30 s to allow it to come into equilibrium. The displayed temperature is calculated from the mean energy of an additional 30 s-interval afterwards. While the laser cycle time for the cooling time constant is optimal at 3.5 s, about a factor of 2 in the lower temperature limit is gained by reducing it to 1.0 s. Equivalent scans were repeated for each resonator detuning in the following data.

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The simulated performance of the off-resonant coupling scheme in terms of cooling time constants and lower temperature limits is summarized in figure 13(b). The cooling time constants rise roughly linearly with the detuning, as expected from equations (25) and (32). The lower temperature limits saturate at slightly below 10 mK, which results from the two opposing effects of the decline of resonator noise and the reduction of the Rabi frequency.

We investigate the effect of the frequency matching uncertainty for 8 kHz and 20 kHz resonator detuning in the same manner as for the common-endcap coupling. Figure 14(a) shows the temperature at 30 s after initializing the proton at an energy of 10 K ⋅ k_B. If the particle frequency matching precision Δ_unc is known experimentally, one can optimize the cooling time by adjusting the laser cycle time accordingly. For example, if the frequency matching precision is worse than the 100 mHz assumed in this work, one can decrease the laser cycle time to account for the faster (but incomplete) Rabi oscillations. However, this requires Δ_unc to be smaller than or at least comparable to the Rabi frequency. This condition is, in contrast to the common-endcap coupling, fulfilled for the coupling scheme investigated in this section. The procedure in context of figure 8(a) was repeated for every frequency matching precision for the blue and green data points labelled 'corr.' in figure 14(a), leading to a significantly better cooling performance compared to the uncorrected ones. Especially in the case of 8 kHz the sensitivity of the proton temperature to Δ_unc is significantly smaller compared to the common-endcap coupling. Additionally, figure 14(b) shows the equilibrium proton temperature as a function of the frequency matching uncertainty. An uncertainty of 1 Hz causes the temperature to be roughly a factor of 2 higher than for 100 mHz for both 8 kHz and 20 kHz resonator detuning.

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We have theoretically studied the sympathetic cooling of the axial mode of a single proton with laser-cooled ⁹Be⁺ ions. A summary of the results is given in table 1. Of the three studied cooling techniques, the common-endcap coupling scheme has a comparably long cooling time constant, but still outperforms the established method of resistive cooling. Due to the absence of resonator heating, the proton temperature is only limited by the Doppler limit of the beryllium ions. However, the method is experimentally challenging since the particles cannot be detected during the coupling process. Additionally, the low Rabi frequency of $\approx 30$ mHz makes it the most sensitive scheme to relative frequency mismatches between the particles. In contrast, the RLC resonator mediated resonant coupling scheme can be tuned to have cooling time constants in the 100 ms-regime. It requires no additional experimental instrumentation to conduct since all three oscillators are being detected on resonance, which is the standard operation mode of our Penning trap. However, it is limited to $\approx 350$ mK axial proton temperature with the current experimental setup. Nevertheless, it will certainly be used as a pre-cooling mechanism or as an alternative to feedback cooling. The off-resonant RLC resonator coupling technique combines both approaches and yields a fast cooling time constant while capable of achieving $\approx 10$ mK proton temperature, which suffices as cooling scheme to prepare cold (anti-)protons for g-factor measurements [2, 18, 52]. Furthermore, it is more robust than the common-endcap coupling against relative frequency mismatches. Its main advantage is that the ratio between cooling time constant and lower temperature limit can be adjusted to the needs of the particular application.

Table 1. Summary of the three different coupling schemes in terms of the cooling time constant τ and the proton temperature limit T_p,min. The sensitivity to detuning is defined as the frequency matching precision where the cooling performance is a factor of 2 worse than the reference at 100 mHz.

Method	τ(s)	Tp,min (K)	Sensitivity to detuning
Common-endcap coupling	20 s	0.5 mK	Off-resonance, $\approx 200$ mHz
Resonant RLC coupling	0.1–30 s	350 mK	On-resonance, $\approx 1$ Hz
Off-resonant RLC coupling	4–10 s	10–30 mK	Off-resonance, $\approx 200$–500 mHz

Overall, the studies presented here establish the cooling methods for the next generation of precision measurements at BASE. While the simulations already supported the proof-of-principle measurement in reference [12], they allow a quantitative assessment of the performance of future techniques and an identification of their experimental challenges, which would otherwise require dedicated measurement campaigns. The next step is to apply these schemes experimentally and validate the conclusions regarding temperature limits and cooling times drawn in this work. For this purpose an improved experimental setup is currently being commissioned [41]. It extends the current version by a dedicated trap for temperature measurements with $\approx 1$ mK resolution and a cryogenic switch to detune the resonator for the application of the chopped laser schemes.

Once cooling to 10 mK is routinely achieved experimentally, the cooling techniques described here will allow magnetic moment measurements to be conducted at improved sampling rates and reduced systematic shifts and will thus have considerable impact on future tests of CPT invariance with protons and antiprotons. In addition, the two-trap sympathetic cooling techniques are currently being developed for tests of quantum electrodynamics with highly charged ions [19] or the determination of the magnetic moment of ³He²⁺ [53].

Since the techniques are practically independent of the target ion and the species can be macroscopically separated, applications beyond these specific cases are conceivable. For example, high-precision mass measurements of stable ions [54] or radionuclides with moderate lifetimes at online facilities [55, 56], studies with molecular ions in radiofrequency traps [57] as well as trapped-ion quantum computing [58] could benefit from the techniques presented in this paper. Moreover, sub-Doppler cooling techniques such as sideband cooling [59, 60] may be employed in the future to reach even lower temperatures.

This paper comprises parts of the PhD thesis work of C Will at the Ruprecht Karl University of Heidelberg.

We thank the ALPHATRAP-group at the Max-Planck-Institute for Nuclear Physics for helpful discussions regarding the cooling method presented here and acknowledge similar developments toward cooling highly charged ions in the ALPHATRAP collaboration.

We acknowledge financial support from the Max-Planck-Society, RIKEN Chief Scientist Program, RIKEN Pioneering Project Funding, the RIKEN JRA Program, the Helmholtz-Gemeinschaft, the DFG through SFB 1227 'DQ-mat', the European Union (Marie Sklodowska-Curie Grant Agreement Number 721559), the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement Nos. 832848-FunI and 852818-STEP), the DAAD RISE program and the Max-Planck-RIKEN-PTB Center for Time, Constants and Fundamental Symmetries.

The authors declare no competing interests.

The data that support the findings of this study are available upon reasonable request from the authors.