Motional resonances of three-dimensional dual-species Coulomb crystals

MD simulations are widely used to investigate the motion of atoms and molecules [32]. We use this method to model the dynamics of our mixed Be⁺ and Mg⁺ Coulomb crystals. We developed an MD software written in C++ which uses the fifth-order Runge–Kutta method with adaptive time steps for solving the multi-particle equations of motion.

As sketched in figure 1(a), our trap is a 4-rod linear Paul trap with ring electrodes for axial confinement [33]. To save computation time, in the MD simulation, the time-averaged secular potential is employed

$$\begin{eqnarray}&&{\rm{\Phi }}(x,y,z)=\displaystyle \frac{{e}^{2}{V}^{2}}{4m{{\rm{\Omega }}}^{2}{r}_{0}^{4}}({x}^{2}+{y}^{2})+\displaystyle \frac{\kappa {eU}}{{z}_{0}^{2}}\left({z}^{2}-\displaystyle \frac{1}{2}({x}^{2}+{y}^{2})\right).\end{eqnarray} \tag{ 1 }$$

Here, V = 417.2 V and Ω = 2π × 36.5 MHz are the amplitude and the radio frequency (RF) applied to the rod electrodes, while U = 265 V is the dc voltage applied to the ring electrodes. The distance between the two ring electrodes is $2{z}_{0}=1.85\,\mathrm{cm}$ with κ = 0.021 being a geometrical factor that depends on the diameter of the rings. The distance of the rod electrodes to the trap axis is given by r₀ = 1.475 mm. The values assigned here are the ones used in the experimental set-up and in the simulations. The time-averaged potential given in equation (1) ignores possible effects due to the micromotion. The approximation is justified because the time scale of the secular motion is an order of magnitude slower than that of the micromotion. This is confirmed by comparing the results of some simulations using the full time-dependent potential with results obtained using the time-averaged secular potential (see appendix A).

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To model the laser cooling, we assume an isotropic continuous drag force ${\vec{f}}_{\mathrm{drag}}=-\alpha \vec{v}$ that acts on both ion species. In the experiments we also laser cool both ion species. The linear dependence on velocity $\vec{v}$ is a good approximation when the harmonic oscillations at the secular frequencies ω_j (j = x, y, z) lead to Doppler shifts that are much smaller than the line width of the cooling transition. The drag force model also assumes the weak binding regime, that is, the secular frequencies are much smaller than the line width of the cooling transition. In this regime there are many absorption and emission events per secular oscillation period such that a continuous drag force is a good approximation. This drag force leads to a vanishing temperature. In a real experiment random momentum kicks due to spontaneous emission and collisions with the background gas would give rise to heating effects and lead to a finite temperature even when laser cooling is engaged. We have run some simulations that include heating [17] and find that a finite temperature of a few mK does not affect our qualitative description of motional coupling of mixed ion crystals. Hence, we ignore these heating effects for the results obtained in this work.

As can be seen from equation (1), the radial trapping potential is inversely proportional to the mass of the ions m. Therefore, lighter ions are more strongly bound to the trap center than heavier ions. Secular frequencies for a single ²⁴Mg⁺ ion are approximately ω_z = 2π × 115 kHz along the axial direction and ω_x,y = 2π × 370 kHz along the radial direction. For a single ⁹Be⁺ ion the frequencies are ω_z = 2π × 187 kHz and ω_x,y = 2π × 1.0 MHz. Under this potential, ²⁴Mg⁺ experiences a radial restoring force of 2.1 × 10⁻¹⁹ N at a distance of 1 μm from the trap axis. The drag force constant α was set for both ion species to 2.58 × 10⁻²¹ kg s^–1 in our simulations. This value affects the cooling time and the spectral width of the vibrational resonances, but has little influence on the secular frequencies.

In the simulations the ions' initial positions are randomly chosen, and their initial velocities are set to zero. In the next step the ions are subjected to the trapping and cooling forces until their positions reach a steady state. For the parameters given in this work, 3 ms is enough to crystallize about one hundred ions. An example of a crystal structure obtained in this way is given in figure 1(b). As expected, the Mg⁺ ions form a shell structure surrounding the Be⁺ ions that are located along the trap center.

Once the ions are crystallized by the trap potential and the drag force, we introduce a sinusoidal excitation force along the x direction ${\vec{f}}_{\mathrm{ex}}=\hat{x}{f}_{0}\cos (\omega t)$ to induce motional excitation. Here, $\hat{x}$ is a unit vector along the x direction, f₀ is the amplitude of the excitation force, and $\omega /2\pi$ is the excitation frequency. In the experiment, the force is induced by modulating the voltage of one of the outer electrodes (see figure 1(a)). Since the distance between the electrode and the trap center is sufficiently large compared to the size of the ion crystal, a uniform force is a good approximation. In principle the ion motion is not exactly harmonic, but contains nonlinearities due to the Coulomb interaction between the ions. These nonlinearities become negligible when the driving force is weak enough, such that the ion positions do not deviate too much from their equilibrium positions. In this work, we focus on the fundamental frequencies and therefore employ only small amplitudes of the driving force by setting ${f}_{0}={10}^{-20}$ N in the MD simulation. This is much smaller than the typical radial restoring force of the trap. As a result, the non-harmonic component of the ion motion is negligible in the simulation.

After the excitation force is turned on at t = 0, the ions start oscillating with the frequency ω. The oscillation amplitude first grows and after some time (approximately 300 μs in our simulations) stabilizes by balancing with the cooling force. In this state we determine the amplitude Max(x_i(t)) – Min(x_i(t)) and the mean position $\langle {x}_{i}\rangle$ averaged over one oscillation period. The ions are numbered by the index i. We only consider the motion along the x direction since the motions along y and z are much smaller (about 10% of the motion along x direction). From this data we determine the secular spectrum as the amplitude averaged over all Mg⁺ or all Be⁺ ions. An example is given in figure 2 together with the secular spectra of a single Be⁺ ion and a single Mg⁺ ion for comparison. The Be⁺ spectrum in the mixed crystal is significantly broadened compared with the single ion spectrum and the peak is found to be up-shifted (by 18.5 kHz). A more dramatic change is observed for Mg⁺. Its secular resonance is split into two components, one of which is down-shifted to 276 kHz, and the other is up-shifted to 408 kHz compared with the single ion resonance at 370 kHz.

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To obtain a more detailed picture, we investigate the secular spectra of individual ions within the crystal. We first have a look at the 20 Be⁺ core ions of the crystal in figure 1(b). Apparently all these spectra are different as shown in figure 3. For the case of the core ions located near the center, the secular resonance tends to be up-shifted, while the ions located at the ends experience a down-shift of their secular resonance. These effects yield an inhomogeneous broadening of the averaged spectrum as shown in the bottom plot of figure 3. A qualitative explanation of the position dependence can be given in following way: the Mg⁺ ion shell generates a potential in addition to the trap potential³. For the core ions in the center of the trap, this leads to an enhancement of the confinement and hence to a larger secular frequency. On the other hand, the core ions at the ends are not surrounded by the repelling shell ions and hence experience a de-confinement and therefore a reduction of the secular frequency. Qualitatively this behavior can be seen in figure 3, even though this simple picture neglects the motion of the shell ions while the motion of the core ions is excited. This is a reasonable approximation as can be seen from figure 2 for an excitation close to the single Be⁺ ion secular frequency (1 MHz).

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One noticeable feature is that the secular spectra of individual Be⁺ ions having symmetric positions with respect to the trap center are not identical. The reason for these differences is that the ion crystal is not exactly symmetric when flipping the z-axis. Depending on the exact configuration of their surroundings, ions at symmetric positions experience different local potentials. It should be noted that the asymmetry is not due to the random initial conditions of the simulation. The most stable ion configuration does not necessarily exhibit mirror symmetry with respect to the xy-plane.

Next we investigate the motion of the 80 Mg⁺ shell ions of the ion crystal in figure 1(b) in order to understand the resonance splitting near the single Mg⁺ resonance into two peaks around 276 kHz and around 408 kHz (see figure 2(b)). The excursions along the x-axis of these ions in the time domain are shown in figures 4(a) and (b) for these two frequencies as obtained from our MD simulation. At 276 kHz it appears that ions with smaller magnitude of $\langle {x}_{i}\rangle$ experience a larger excursion than those with larger magnitude of $\langle {x}_{i}\rangle$. For the peak at 408 kHz it is the other way around. The situation can be clearly seen in figures 4(c) and (d), where the correlation between the motional amplitude and averaged position $\langle {x}_{i}\rangle$ is shown. Figures 4(e) and (f) sketch the motion of the shell ions in the axial view represented by the lengths of the green horizontal arrows.

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For a simplified and intuitive picture, we again invoke the additional potential generated by the other group of ions. Figures 4(g) and (h) show how the core ions (dark blue) modify the potential for the shell ions (light red) along the direction of the excitation field. The dashed curves refer to the trap potential without the core ions, while the solid lines sketch the total potential. As can be seen in figures 4(a) and (b), the shell ions close to x = 0 experience a weakening of their potential and hence a decreased resonance frequency, while the ones with larger magnitude of $\langle {x}_{i}\rangle$ have their potential enhanced with a corresponding up-shift of their resonance. The situation is, however, less simple than for the modification of the core ions' potential by the shell ions. This has the following two reasons: The effect on ions with intermediate $\langle {x}_{i}\rangle$ washes out the splitting of the resonance, which may be represented by the broadening of the down-shifted peak in figure 2(b). In addition, the core ions are not as well fixed in space as the shell ions are when driving the resonance of the core ions around 1 MHz.

Since the modified potential due to the core ions is the reason for the resonance splitting, one expects the splitting to increase with the number of core ions. This is indeed the case as is shown in figure 5.

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The hand-waving arguments for the secular spectra of mixed crystals are valid for large enough numbers of both ion species. The spatial distributions of the ions are not precisely reproducible with every MD simulation run because the initial positions are randomized. However, with a decent number of both ion species, the geometry of the crystal can be always characterized by a nested shell and core structure with radial symmetry. As a result, the features described in section 2.2 are also conserved in its secular spectra (see appendix C for details).

When the number of one species gets much larger than the other, such a simple geometric configuration does not always show up, and the ion distribution is not regularly reproduced for different initial conditions. For the extreme example with only one Mg⁺ ion within a large number of Be⁺ ions, the position of the Mg⁺ ion in the Be⁺ crystal turns out to be essentially random. The local potential which the Mg⁺ ion experiences is dependent on its exact position and so is its secular frequency. This is what we see in the experiment and also in the simulations. As a consequence, it is difficult to identify a small number of impurity ions within a larger crystal by their secular frequencies. On the other hand, the majority ions reliably produce secular frequencies close to their single ion resonance.

The experimental setting is sketched in figure 1(a) with the parameters provided in section 2. It is described in more detail in [34]. The ion crystals are imaged with an objective having a focal length of 76 mm at a distance of about 83 mm from the trap center and are observed with an electron multiplying charge-coupled device camera. Due to chromatic aberration, we cannot simultaneously image the Be⁺ and Mg⁺ ions. Instead we move the position of the objective to focus on either of the ion species. An electrical motor stage is employed for this purpose.

To obtain fluorescence from both ion species, we are addressing both cooling transitions, the $2{S}_{1/2}\mbox{--}2{P}_{3/2}$ (natural linewidth γ/2π ≈ 19 MHz) and the $3{S}_{1/2}\mbox{--}3{P}_{3/2}$ (natural linewidth γ/2π ≈ 42 MHz) for Be⁺ and Mg⁺ respectively. The cooling lasers at 313 and 280 nm are both detuned by more than 100 MHz. They run axially in opposite directions through the ion trap with intensities of ≈300 W m⁻² and ≈5 kW m^–2 for Mg⁺ and Be⁺ respectively, determined from the laser powers and beam diameters. We adopted these relatively low intensities in order to reduce distortions of the crystal structure due to radiation pressure.

In addition to the rods forming the quadrupole, there are four outer rod electrodes shown with dashed circles in figure 1(a) (right). Three of these electrodes are utilized to compensate for stray electric fields near the trap center. The excitation force ${\vec{f}}_{\mathrm{ex}}$ is created by applying an ac voltage at frequency ω to the fourth electrode. Laser cooling is sufficiently effective, such that the ion crystals stay intact when the secular vibrations are excited.

A mixed Coulomb crystal is presented in figure 6, where the images of Mg⁺ and Be⁺ are obtained separately by refocusing the imaging system. Due to its strong chromatic aberration, we observe only a small almost homogeneous background from one ion species when we focus on the other. Therefore, the data was taken with both cooling beams on.

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In contrast to our simulations, radiation pressure leads to somewhat asymmetric crystals. This effect could have been avoided by detuning the cooling lasers even further or by reducing their intensity. However, this would require stronger secular excitation due to the reduced fluorescence and hence might eject the ions from the trap. Nonetheless, we believe that it is still reasonable to compare experiment with simulations. By comparing the shape and size of the ion crystal with the simulation, we roughly estimate the number of ions in figure 6 to be 500 for Mg⁺ and 500 for Be⁺ [27, 35].

Several methods have been demonstrated to estimate the motional amplitude of the ions during secular excitation [34]. In this work we measure the increase of the fluorescence of the ions that is obtained with a largely red detuned cooling laser due to the periodic Doppler shift closer to resonance. The secular spectrum is obtained by scanning the excitation frequency ω. The excitation voltages were carefully optimized to be not too large to disturb the crystal structure, and not too small to observe a sufficient amount of change in fluorescence. They ended up to be between 0.1 and 1.5 V. Based on finite element modeling of our geometry, a voltage modulation of 1 V at one of the outer rod electrodes induces an electric field of 0.75 V m^–1 at the trap center. This leads to an excitation force that is an order of magnitude stronger than in the simulations. According to our MD simulations, stronger excitation force should mainly influence the cooling time, rather than the secular spectrum of the crystal that is formed. Figure 7 shows an experimental secular motion spectrum that was recorded for a mixed crystal that roughly corresponds in size and composition to the mixed crystal shown in figure 6.

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As in the simulations, the resonances of the mixed crystals are significantly broadened in comparison with the pure ion crystals. In addition, we find the spectrum near the pure Mg⁺ ion crystal resonance to be split into two components, just as in the simulations and the simplified model described above. The observed splitting of 210 kHz is consistent with the simulation performed with a similar number ratio between Mg⁺ and Be⁺ ions, considering the uncertainty in the estimation of the number of Mg⁺ and Be⁺ ions. We also confirm the broadened resonance near the pure Be⁺ crystal resonance. The down-shift of this resonance is not predicted by our MD simulations but could be due to a significantly stronger motional excitation in the experiment than in the simulations. The influence of the excitation strength on the secular frequencies has been discussed in [16].

It should be pointed out that the images shown in figure 6 are time averaged and do not necessarily reflect the instantaneous positions of the ions. At finite temperature, ions may exchange their positions from one site to another. We expect the hopping have a minor effect on the secular spectrum shown in figure 7 under our experimental conditions.