Dynamics and control of fast ion crystal splitting in segmented Paul traps

We desire to split a two-ion crystal residing at center segment C along the trap axis x, to obtain two ions stored in separated potential wells at the positions of the splitting segments S neighboring C, see figure 1. Note that we consider only the spatial dimension along the trap axis, as we assume that tight radial confinement persists throughout the process and the ions are always located on the rf node of the trap. Typical distances between segments range between 50 and 500 μm, while the initial ion distance is 2–4 μm. The total external electrostatic potential along the trap axis can be written as

$$\begin{eqnarray}&&\Phi (x)\approx \beta {{x}^{4}}+\alpha {{x}^{2}}+\gamma x,\end{eqnarray} \tag{ 1 }$$

where the coefficients $\alpha ,\beta ,\gamma$ are given by the the trap geometry and the voltages applied to the trap segments. This Taylor approximation is valid as long as the the ions are located sufficiently close to x = 0, which is the center of the C segment. Throughout the splitting process, the external potential is changing from a single well potential ${{\alpha }_{i}}>0$ to a double well potential ${{\alpha }_{f}}<0$, crossing the critical point (CP) at $\alpha =0$. Note that $\beta >0$ is required to guarantee confinement at $\alpha \leqslant 0$. The approximation of equation (1) holds for $\alpha \geqslant 0$ and for $\alpha \lesssim 0$ as long as the separation of the two potential wells is small compared to the width of segment C. When the distance of the ions from the center of the C segment becomes comparable to the width of the segment, anharmonic terms of order $>4$ contribute significantly to the total potential. These are not taken into account here since the outcome of the splitting process is determined around the CP, as will be pointed out in the following sections.

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Beyond the CP, the equilibrium distance depends significantly on higher order terms of the potentials. However, the distance of the separated wells is still increasing monotonically for decreasing $\alpha$ as long as the variation $\beta$ is sufficiently small, and the corresponding trap frequencies in these wells are monotonically increasing. Thus, the motion beyond the CP corresponds to an ordinary shuttling process of a stiff harmonic trap. This suggests that the separation outcome should essentially not depend on higher order terms, a finding which is supported by numerical calculations in section 6.

For studies that require precision beyond the CP, the higher order terms can be taken into account numerically. A cubic term does not contribute to the potential if the trap is sufficiently symmetric along the trap axis.

Including Coulomb repulsion, the total electrostatic potential of a two-ion crystal at a center-of-mass position ${{x}_{0}}$ and distance d is given by

$$\begin{eqnarray}&&{{\Phi }_{tot}}\left( {{x}_{0}},d \right)=\Phi \left( {{x}_{0}}+d/2 \right)+\Phi \left( {{x}_{0}}-d/2 \right)+\frac{\kappa }{d},\end{eqnarray} \tag{ 2 }$$

with $\kappa =e/4\pi {{\epsilon }_{0}}$. At the CP, the harmonic confinement vanishes, and a weak residual confinement is maintained by the interplay between Coulomb repulsion and quartic part of the external potential. It is therefore desirable to maximize β at the CP. For a given trap geometry, the attainable β is limited by the voltage range that can be applied to the trap electrodes ¹ . The coefficients of the potential equation (1) are given by the segment bias voltages and the electrostatic properties of the trap:

$$\begin{eqnarray}&&\alpha ={{U}_{C}}{{\alpha }_{C}}+{{U}_{S}}{{\alpha }_{S}}+{{U}_{O}}{{\alpha }_{O}}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\beta ={{U}_{C}}{{\beta }_{C}}+{{U}_{S}}{{\beta }_{S}}+{{U}_{O}}{{\beta }_{O}}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\gamma =\Delta {{U}_{S}}{{\gamma }_{S}}+\Delta {{U}_{O}}{{\gamma }_{O}}+\gamma ^{\prime} .\end{eqnarray} \tag{ 5 }$$

An offset parameter $\gamma ^{\prime}$ is introduced for taking trap non-idealities—leading to a symmetry breaking force along the trap axis—into account, see section 4.2. In contrast to the symmetric quadratic and quartic contributions, the asymmetric tilt potential is controlled by the differential voltages $\Delta {{U}_{S,O}}$ between the corresponding left and right electrodes of the respective pair. The segment coefficients are given by Taylor expansions of the standard potentials ${{\phi }_{n}}\left( x \right)$, which are the dimensionless electrostatic potentials along the trap axis if a +1 V bias is applied to segment n and all other segments are grounded [13, 14]:

$$\begin{eqnarray}&&{{\phi }_{n,m}}(x)={{\phi }_{n}}{{\left| _{x_{0}^{(m)}}+{{\phi }_{n}}^{\prime} \right|}_{x_{0}^{(m)}}}\delta x+\frac{1}{2}\phi _{n}^{^{\prime\prime} }{{\left| _{x_{0}^{(m)}}\delta {{x}^{2}}+\frac{1}{24}\phi _{n}^{(4)} \right|}_{x_{0}^{(m)}}}\delta {{x}^{4}}+\mathcal{O}\left( \delta {{x}^{6}} \right),\end{eqnarray} \tag{ 6 }$$

with $\delta x=x-x_{0}^{\left( m \right)}$, i.e. the Taylor expansions are carried out at center of segment m, $x_{0}^{\left( m \right)}$. The coefficients for equations (3), (4), (5) are obtained for $m=C,n=C,S,O$:

$$\begin{eqnarray}&&{{\alpha }_{n}}=\frac{1}{2}{{f}_{n}}\phi _{n,C}^{^{\prime\prime} }(0),\qquad {{\beta }_{n}}=\frac{1}{24}{{f}_{n}}\phi _{n,C}^{(4)}(0),\qquad {{\gamma }_{n}}={{f}_{n}}\phi _{n,C}^{^{\prime} }(0),\end{eqnarray} \tag{ 7 }$$

with ${{f}_{C}}=1$ and ${{f}_{S,O}}=2$ accounting for two $S,O$ segments acting symmetrically at x = 0. Note that ${{\gamma }_{C}}=0$ by definition.

In the following, for numerical calculations, we use the specific geometry parameters of a three dimensional microstructured segmented ion trap A as detailed in section 3. There, other traps and their geometry parameters are listed and analyzed as well.

We consider two ions of mass m and charge e, with their equilibrium positions given by the center-of-mass ${{x}_{0}}$ and the equilibrium distance d:

$$\begin{eqnarray}&&{{x}_{L,R}}={{x}_{0}}\pm d/2,\end{eqnarray} \tag{ 8 }$$

determined by minimizing the total electrostatic potential equation (2). The confinement is characterized by the local trap frequency, which is given by the curvature of the external potential at the ion positions:

$$\begin{eqnarray}&&\omega =\sqrt{\frac{e}{m}{{\Phi }^{^{\prime\prime} }}({{x}_{L,R}})}.\end{eqnarray} \tag{ 9 }$$

The extremal points of the external potential equation (1) are given by

$$\begin{eqnarray}&&x_{0}^{\left( 0 \right)}=\frac{\alpha }{{{3}^{1/3}}\zeta }-\frac{\zeta }{2\cdot {{3}^{2/3}}\beta }\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&x_{0}^{(\pm )}=\frac{(i\sqrt{3}\pm 1)\alpha }{2\cdot {{3}^{1/3}}\zeta }+\frac{(1\mp i\sqrt{3})\zeta }{4\cdot {{3}^{2/3}}\beta }\end{eqnarray} \tag{ 11 }$$

where

$$\begin{eqnarray}&&\zeta (\alpha ,\beta ,\gamma )={{\left( 9{{\beta }^{2}}\gamma +\sqrt{3}\sqrt{8{{\alpha }^{3}}{{\beta }^{3}}+27{{\beta }^{4}}{{\gamma }^{2}}} \right)}^{1/3}}.\end{eqnarray} \tag{ 12 }$$

Initially, at $\alpha ={{\alpha }_{i}}$, the confining harmonic part of the external potential and the Coulomb repulsion are dominant, thus we can neglect the quartic potential. The trap frequency is then given by ${{\omega }^{2}}=2\alpha e/m$ at an ion distance of $d={{\left( \kappa /\alpha \right)}^{1/3}}$. At the CP, $\alpha =0$, and without tilt, $\gamma =0$, the ion distance is determined by quartic confinement and Coulomb repulsion:

$$\begin{eqnarray}&&{{d}_{{\rm CP}}}={{\left( 2\kappa /\beta \right)}^{1/5}}.\end{eqnarray} \tag{ 13 }$$

The Coulomb repulsion pushes the ions away from the trap center (where the curvature of the external potential vanishes), such that a residual harmonic confinement persists because of the quartic term. The minimum trap frequency during the splitting process is thus given by [8]

$$\begin{eqnarray}&&{{\omega }_{{\rm CP}}}={{\beta }^{3/10}}{{\left( 3e/m \right)}^{1/2}}{{\left( 2\kappa \right)}^{1/5}}.\end{eqnarray} \tag{ 14 }$$

Near the CP, the equilibrium distance can be computed from a perturbative expression up to second order:

$$\begin{eqnarray}&&d(\alpha )\approx {{d}_{{\rm CP}}}-\frac{1}{5}{{\left( \frac{16}{{{\beta }^{4}}\kappa } \right)}^{1/5}}\alpha +\frac{2}{25}{{\left( \frac{4}{{{\beta }^{7}}{{\kappa }^{3}}} \right)}^{1/5}}{{\alpha }^{2}},\end{eqnarray} \tag{ 15 }$$

for $|\alpha |\ll \beta d_{{\rm CP}}^{2}$ and $|\alpha |\ll \kappa d_{{\rm CP}}^{-3}$.

The center-of-mass position of the ion crystal near the CP to first order in the tilt parameter γ is:

$$\begin{eqnarray}&&{{x}_{0}}(\alpha ,\gamma )\approx \gamma \left( -\frac{1}{3\cdot {{2}^{2/5}}{{\beta }^{3/5}}{{\kappa }^{2/5}}}-\frac{{{2}^{1/5}}}{45\cdot {{\beta }^{6/5}}{{\kappa }^{4/5}}}\alpha +\frac{26\cdot {{2}^{4/5}}}{675{{\beta }^{9/5}}{{\kappa }^{6/5}}}{{\alpha }^{2}} \right)\end{eqnarray} \tag{ 16 }$$

If the ions are sufficiently separated, $\alpha \ll 0$, the Coulomb repulsion can be neglected and the equilibrium positions approximately coincide with the extrema of the external potential:

$$\begin{eqnarray}&&{{d}_{f}}=\sqrt{-2{{\alpha }_{f}}/\beta }\end{eqnarray} \tag{ 17 }$$

and the final trap frequency is given by $\omega _{f}^{2}=-4{{\alpha }_{f}}e/m$.

A static background force along the trap axis can tilt the external potential and thus keep the ions confined in one common potential well throughout the splitting process. We make use of the external potential minima equation (11) to obtain an estimate for the tilt parameter $\tilde{\gamma }$, beyond which the splitting ceases to work. In the following, we assume $\gamma >0$.

In the presence of a non-zero potential tilt, an imperfect bifurcation occurs, i.e. the second potential well opens up at $\tilde{\alpha }<0$, see figure 2(c). We obtain a scaling law for $\tilde{\gamma }$ by calculating at which tilt parameter the original potential well is deep enough to keep both mutually repelling ions confined, see figure 3. The saddle point where the second potential well opens can be found by solving $x_{0,c}^{\left( 0 \right)}=x_{0,c}^{\left( + \right)}$ for $\tilde{\alpha }$, yielding $\tilde{\alpha }=-\frac{3}{2}{{\beta }^{1/3}}{{\left| \gamma \right|}^{2/3}}$. From this we obtain its position ² to be $x_{0,c}^{\left( +,0 \right)}=\frac{1}{2}{{\left( \gamma /\beta \right)}^{1/3}}$. At $\tilde{\alpha }$, the left potential minimum is located at twice the distance from the origin $x_{0,c}^{\left( - \right)}=-{{\left( \gamma /\beta \right)}^{1/3}}$. The potential attains the same value as on the saddle point $V(x_{0,c}^{\left( +,0 \right)})$ at the position $\tilde{x}_{c}^{\left( + \right)}=-\frac{3}{2}{{\left( \gamma /\beta \right)}^{1/3}}$. The depth of the potential well defined by the saddle point when the right well opens is therefore

$$\begin{eqnarray}&&\Delta {{V}_{c}}=V\left( x_{0,c}^{(-)} \right)-V\left( x_{0,c}^{(+,0)} \right)=\frac{27}{16}{{\left( \frac{{{\gamma }^{4}}}{\beta } \right)}^{1/3}}.\end{eqnarray} \tag{ 18 }$$

We can now define a criterion that determines whether the ions are actually separated by comparing the Coulomb potential to the depth of the initial well at the CP, equation (18). If the Coulomb repulsion pushes the right ion beyond the saddle point $x_{0,c}^{\left( +,0 \right)}$, it will end up in the right potential well, otherwise the two ions will stay in the left well. Thus, the Coulomb energy at an ion distance of $x_{0,c}^{\left( +,0 \right)}-\tilde{x}_{c}^{\left( + \right)}$ has to be larger than the well depth $\Delta {{V}_{c}}$. These considerations lead to a critical tilt value of

$$\begin{eqnarray}&&\tilde{\gamma }<\pm {{C}_{\gamma }}{{\left( {{\kappa }^{3}}{{\beta }^{2}} \right)}^{1/5}}.\end{eqnarray} \tag{ 19 }$$

Despite the fact that the situation depicted in figure 3 does not actually occur, as the external force at the saddle point vanishes and therefore cannot balance the Coulomb force, the obtained scaling behavior is confirmed by numerical calculations, revealing a prefactor of ${{C}_{\gamma }}$ = 1.06.

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The result of equation (19) enables us to determine the required degree of precision by which the background axial field has to be corrected. For this calculation, only the geometry parameters ${{\beta }_{C,S,O}}$ are needed. Furthermore, the sensitivity decreases as ${{\beta }^{2/5}}$, which directly characterizes the gain in robustness when the accessible voltage range is enhanced. For trap A (section 3), we derive a value of $\tilde{\gamma }\approx 3\;{\rm V}\;{{{\rm m}}^{-1}}$, corresponding to the requirement to set $\Delta {{U}_{O}}$ more accurately than about 9 mV.

In this section, we show that the energy transfer can be quantitatively described by a simple model, which interpolates between impulsive acceleration for short times and adiabatic behavior for long times. We first derive the impulsive approximation, and then refine the model to include the onset of adiabaticity. We also confirm the validity of our model by simulations.

A naive approach towards crystal splitting is the linear interpolation between two voltage sets pertaining to a single well and a double well, leading to a constant variation rate of the harmonic coefficient α. As this does not involve dedicated control of the ion distance, it is equivalent to a rapid sweep across a structural transition of the ion crystal. This leads to an unfavorable power-law scaling of the energy transfer with respect to the sweep time [18], which prevents attaining adiabaticity.

In the following, we derive an approximation for the energy transfer, assuming the variation of α around the CP to be uniform. We consider the energy transfer to be caused by impulsive displacement: at the CP, the equilibrium distance changes most rapidly, while the confinement—and therefore the restoring forces—are reduced. Figure 5(a) shows that the situation corresponds to a harmonic oscillator that is suddenly dragged at uniform speed, causing displacement and therefore a gain in potential energy. Within the characteristic timescale set by half the trap oscillation cycle ${{\tau }_{{\rm CP}}}/2=\pi /{{\omega }_{{\rm CP}}}$, this yields the displacement:

$$\begin{eqnarray}&&\delta {{d}_{{\rm CP}}}\approx {{\dot{d}}_{{\rm CP}}}\;\xi \;{{\tau }_{{\rm CP}}}/2\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\qquad \approx \ {{\left. \frac{\partial d}{\partial \alpha } \right|}_{{\rm CP}}}{{\dot{\alpha }}_{{\rm CP}}}\;\xi \;{{\tau }_{{\rm CP}}}/2\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&\qquad \approx \ {{\left( \beta _{{\rm CP}}^{4}\kappa \right)}^{-1/5}}{{\dot{\alpha }}_{{\rm CP}}}\;\xi \;{{\tau }_{{\rm CP}}}/2,\end{eqnarray} \tag{ 22 }$$

where equation (15) was used in the last line. The factor $\xi <1$ accounts for a reduction of the displacement, which is due to the fact that the trap frequency rapidly increases beyond the CP. Thus, the restoring forces set in before ${{\tau }_{{\rm CP}}}/2$ and the resulting displacement is reduced with respect to the dragged oscillator at constant frequency. This sudden displacement mechanism is sketched in figure 5(a). The potential energy of an ion is consequently increased by

$$\begin{eqnarray}&&\delta E=\frac{1}{2}m\omega _{{\rm CP}}^{2}{{\left( \delta {{d}_{{\rm CP}}}/2 \right)}^{2}}\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&\quad \ \;=\ \frac{{{\pi }^{2}}}{8}\;{{\xi }^{2}}\;m{{\left( \beta _{{\rm CP}}^{4}\kappa \right)}^{-2/5}}\dot{\alpha }_{{\rm CP}}^{2},\end{eqnarray} \tag{ 24 }$$

which serves as an approximation of the final energy transfer.

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For a sufficiently small ${{\left| {\dot{\alpha }} \right|}_{{\rm CP}}}$, adiabaticity sets in and the energy transfer scales exponentially with the splitting time. The reason for this is that the Coulomb repulsion serves to push the ions outwards, providing smooth variation of the equilibrium distance as compared to discontinuous behavior of the minima of the external potential, see figure 2(b). It therefore leads to rapid, but continuous variation of the equilibrium positions with α. The onset of the adiabatic regime is identified by comparing displacement $\delta {{d}_{{\rm CP}}}$ to the change of the equilibrium distance within ${{\tau }_{{\rm CP}}}$ below the CP (see figure 5(a)), which means that the ion acceleration around the CP is sufficiently slow to prevent sudden displacement. We therefore compare the acceleration ${{\ddot{d}}_{{\rm CP}}}$ to the reference acceleration ${{d}_{{\rm CP}}}\omega _{{\rm CP}}^{2}$. Note that

$$\begin{eqnarray}&&{{\ddot{d}}_{{\rm CP}}}={{\left. \frac{{{\partial }^{2}}d}{\partial {{\alpha }^{2}}} \right|}_{{\rm CP}}}\dot{\alpha }_{{\rm CP}}^{2}+{{\left. \frac{\partial d}{\partial \alpha } \right|}_{{\rm CP}}}{{\ddot{\alpha }}_{{\rm CP}}}.\end{eqnarray} \tag{ 25 }$$

For sufficiently uniform variation of α, the second term can generally be neglected, such that by using equation (15), we obtain

$$\begin{eqnarray}&&{{\ddot{d}}_{{\rm CP}}}=\frac{2}{25}{{\left( \frac{4}{\beta _{{\rm CP}}^{7}{{\kappa }^{3}}} \right)}^{1/5}}\dot{\alpha }_{{\rm CP}}^{2}.\end{eqnarray} \tag{ 26 }$$

This yields the adiabaticity parameter

$$\begin{eqnarray}&&\chi =\frac{{{\ddot{d}}_{{\rm CP}}}}{{{d}_{{\rm CP}}}\omega _{{\rm CP}}^{2}}\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&\quad =\ \frac{4}{25}\frac{m}{3e}{{2}^{-1/5}}\beta _{{\rm CP}}^{-9/5}{{\kappa }^{-6/5}}\dot{\alpha }_{{\rm CP}}^{2}.\end{eqnarray} \tag{ 28 }$$

We empirically find the following model, which describes the exponential decrease of the energy transfer in the adiabatic regime $\chi <1$:

$$\begin{eqnarray}&&\delta E^{\prime} \approx \delta E{\rm exp} \left[ c\frac{\chi -1}{\chi } \right],\end{eqnarray} \tag{ 29 }$$

where c is a scaling constant. Numerical simulations are carried out for different constant values for β and a linear variation of α around the CP. The results are shown in figure 5(b). It can be seen that the approximations (24), (29) hold over a wide range of splitting times and quartic coefficients, and that large energy transfers in the regime of 10⁴–10⁶ phonons are readily obtained. The simulations yield values of the model parameters of ${{\xi }^{2}}\approx$ 0.1 and ${{c}^{2}}\approx 0.1$. We conclude that in this regime, the energy transfer depends only on the ion mass, the variation rate of α and the quartic confinement at the CP. As can be seen from the simulation results, still large energy transfers are obtained at the onset of adiabaticity, such that splitting at energy transfers on the single phonon level would require splitting times on the order of several hundreds of μs.

As we will show in further sections, this problem can be overcome using ramps that ensure a small ion acceleration ${{\ddot{d}}_{{\rm CP}}}$ at the CP.

Thus, the energy transfer can be reduced by ensuring a small variation rate of α at the CP.

A residual static force along the trap axis, expressed by the coefficient $\gamma ^{\prime}$ in equation (5), can originate from stray charges, laser induced charging of the trap [19], trap geometry imperfections or residual ponderomotive forces along the trap axis. The behavior of the equilibrium positions in the presence of an imperfectly compensated tilt, shown in figure 2, reveals a discontinuity for the critical $\tilde{\gamma }$, leading to diverging acceleration. The divergence of the acceleration impedes us from performing the splitting process adiabatically for $|\gamma |\lesssim \tilde{\gamma }$, i.e. the voltages cannot be changed sufficiently slow to suppress motional excitation. Thus, one might encounter the situation that the tilt is sufficiently well compensated to allow for splitting, but sufficiently low excitations cannot be obtained irrespective of the splitting time and other control parameters. For small tilt parameters, $|\gamma |\ll \tilde{\gamma }$, we can employ the perturbative expressions (15), (16) of the equilibrium positions to obtain

$$\begin{eqnarray}&&\frac{{{\partial }^{2}}{{x}_{R,L}}}{\partial {{\alpha }^{2}}}=\frac{{{\partial }^{2}}{{x}_{0}}}{\partial {{\alpha }^{2}}}\pm \frac{1}{2}\frac{{{\partial }^{2}}d}{\partial {{\alpha }^{2}}}=\gamma \frac{52\cdot {{2}^{4/5}}}{675{{\beta }^{9/5}}{{\kappa }^{6/5}}}\pm \frac{2}{25}{{\left( \frac{4}{{{\beta }^{7}}{{\kappa }^{3}}} \right)}^{1/5}}.\end{eqnarray} \tag{ 30 }$$

We can estimate the tilt parameter at which the acceleration of one of the ions is twice as large as the tilt-free case determined by equation (26) to be about 67% of the critical tilt $\tilde{\gamma }$. Due to the divergence of the acceleration at $\tilde{\gamma }$, we can expect the actual acceleration at this tilt value to be substantially larger, we thus conclude that a residual tilt $|\gamma |\ll \tilde{\gamma }$ is required to realize crystal splitting at low motional excitation. A possible experimental scheme for this has been demonstrated in [10]: the separation process is performed on a slow (second) timescale under continuous Doppler cooling and detection. The ion positions are extracted from the camera image, and a deviation of the center-of-mass from the initial value is restored by automatic adjustment of the outer electrode differential voltage $\Delta {{U}_{O}}$.

Microstructured ion traps exhibit anomalous heating, i.e. the mean phonon number increases due to thermalization with the electrodes at a timescale much faster than predicted by the assumption that only Johnson–Nyquist noise is present [17]. This process can be modeled as $\mathop{{\bar{n}}}\limits^{.}={{\Gamma }_{h}}$, with the heating rate ${{\Gamma }_{h}}\left( \omega \right)={{S}_{E}}\left( \omega \right){{e}^{2}}/4\;m\hbar \omega$, where the spectral electric-field-noise density ${{S}_{E}}$ depends on the trap frequency ω. A polynomial decrease ${{S}_{E}}\propto {{\omega }^{-a}}$ is often assumed, where experimentally determined values for the exponent a range from 0.5–2.5. Additionally, peaked features might arise in the noise spectrum that are caused by technical sources. Moreover, the absolute values of the heating rates strongly depend on the properties of the electrode surfaces. Typical values at trap frequencies in the 1 MHz regime range from 0.1 to tens of phonons per millisecond. As the trap frequency is strongly decreased around the CP, we can expect a significant amount of excess energy after the splitting caused by anomalous heating, increasing for longer splitting durations [9]. We model this contribution by integrating over a time dependent heating rate:

$$\begin{eqnarray}&&\Delta {{\bar{n}}_{{\rm th}}}=\int _{0}^{T}{{\Gamma }_{h}}\left( \omega (t) \right){\rm d}t.\end{eqnarray} \tag{ 31 }$$

For the simulations that follow we will employ an experimentally determined relation for trap A (section 3) which is ${{\Gamma }_{h}}\left( \omega \right)\approx 6.3\cdot {{\left( \omega /2\pi \times {\rm MHz} \right)}^{-1.81}}\;m{\rm }{{s}^{-1}}$. This does not depend on the geometry of trap A but on the properties of our trap apparatus.

In the case of imperfect control of the ion distance around the CP, section 4.1, or in the presence of an uncompensated tilt, section 4.2, one will attempt to reduce the motional excitation by splitting very slowly. This might, however, be unsuccessful as anomalous heating will strongly contribute to the energy gain at large splitting times. Experimental procedures for ensuring a sufficient degree of control are therefore ultimately required.

The calculation of suitable voltage ramps relies on the signs and on the magnitude ordering of the geometry parameters. In table 1 we list values for several different microstructured traps. We assume that any reasonable segmented trap geometry will exhibit similar characteristics. From the results of section 4, it is clear that we desire a large positive value of ${{\beta }_{{\rm CP}}}$. We assume that the voltages that can be applied to the segments are limited by hardware constraints to the symmetric maximum/minimum values $\pm {{U}_{{\rm lim} }}$. To achieve the largest possible β at the CP, we begin the splitting protocol by ramping the O segments to $+{{U}_{{\rm lim} }}$, keep them at constant bias around the CP, and ramp them back to zero bias after the splitting.

The CP is defined by the condition $\alpha =0$, which is accomplished by suitable voltages ${{U}_{C,S}}$. This leaves one degree of freedom, which can be eliminated by maximizing ${{\beta }_{{\rm CP}}}$. We solve equation (3) for ${{U}_{C}}$ :

$$\begin{eqnarray}&&{{U}_{C}}=\frac{1}{{{\alpha }_{C}}}\left( \alpha -{{\alpha }_{O}}{{U}_{O}}-{{\alpha }_{S}}{{U}_{S}} \right).\end{eqnarray} \tag{ 32 }$$

The largest possible ${{\beta }_{{\rm CP}}}$ is then given by inserting this result into equation (4) and setting $U_{O}^{\left( {\rm CP} \right)}=+{{U}_{{\rm lim} }},U_{S}^{\left( CP \right)}=-{{U}_{{\rm lim} }}$:

$$\begin{eqnarray}&&\mathop{{\rm max} }\limits_{{{U}_{C}},{{U}_{S}}}{{\beta }_{{\rm CP}}}=\left( {{\beta }_{O}}+\frac{{{\beta }_{C}}}{{{\alpha }_{C}}}{{\alpha }_{S}}-{{\beta }_{S}}-\frac{{{\beta }_{C}}}{{{\alpha }_{C}}}{{\alpha }_{O}} \right){{U}_{{\rm lim} }}.\end{eqnarray} \tag{ 33 }$$

Static splitting voltage sets are obtained by fixing the initial, CP and final voltage configurations and interpolating between these. The procedure consists of the following steps:

• (i)  
  Determine the initial ${{\alpha }_{i}}>0$ from equation (3) using the initial voltages $U_{C}^{\left( i \right)}<0$ V, $U_{S}^{\left( i \right)}=U_{O}^{\left( i \right)}=0$ V.
• (ii)  
  Choose the voltages at the CP such that the maximum ${{\beta }_{{\rm CP}}}$ is attained, by setting $U_{O}^{\left( CP \right)}=+{{U}_{{\rm lim} }},U_{S}^{\left( CP \right)}=-{{U}_{{\rm lim} }}$ and $U_{C}^{\left( CP \right)}$ from equation (32) for $\alpha =0$. If the geometry parameters are such that $U_{C}^{\left( CP \right)}$ exceeds $\pm {{U}_{{\rm lim} }}$, set $U_{C}^{\left( CP \right)}=-{{U}_{{\rm lim} }}$ and obtain $U_{S}^{\left( CP \right)}$ solving equation (3) for ${{U}_{S}}$ rather than ${{U}_{C}}$. A variable offset $\Delta U_{C}^{\left( CP \right)}$ is added to $U_{C}^{\left( CP \right)}$, which serves to guarantee that the CP voltage set actually corresponds to $\alpha =0$. It is therefore a tuning parameter that allows for compensation of imperfections. A similar technique has been employed experimentally in [4] ³ .
• (iii)  
  Determine the desired final voltages. We choose $U_{C}^{\left( f \right)}=0$ V, $U_{S}^{\left( f \right)}=U_{S}^{\left( CP \right)}=-{{U}_{{\rm lim} }}$ and $U_{O}^{\left( f \right)}=0$ V. This choice is convenient when $U_{C}^{\left( i \right)}\approx -{{U}_{{\rm lim} }}$ and ensures that the ions are finally kept close to the respective centers of the S segments with a trap frequency similar to the initial one. Obtain ${{\alpha }_{f}}$ from equation (3).
• (iv)  
  For approaching the CP, ${{\alpha }_{i}}\geqslant \alpha >0$, set

  $$\begin{eqnarray}&&{{U}_{S}}(\alpha )=\left( 1-\frac{\alpha }{{{\alpha }_{i}}} \right)U_{S}^{(CP)}\end{eqnarray} \tag{ 34 }$$

  and

  $$\begin{eqnarray}&&{{U}_{O}}(\alpha )=\left\{ \begin{array}{ccccccccccccccc} 2\left( 1-\frac{\alpha }{{{\alpha }_{i}}} \right){{U}_{{\rm lim} }}\qquad \alpha >\frac{{{\alpha }_{i}}}{2} \\ {{U}_{{\rm lim} }}\qquad \qquad \qquad \ \;\alpha \leqslant \frac{{{\alpha }_{i}}}{2} \\ \end{array} \right.\end{eqnarray} \tag{ 35 }$$

  and obtain ${{U}_{C}}\left( \alpha \right)$ from equation (32).
• (v)  
  Beyond the CP, $0\geqslant \alpha \geqslant {{\alpha }_{f}}$, set

  $$\begin{eqnarray}&&{{U}_{S}}(\alpha )=-{{U}_{{\rm lim} }}\end{eqnarray} \tag{ 36 }$$

  and

  $$\begin{eqnarray}&&{{U}_{O}}\left( \alpha \right)=\left\{ \begin{array}{ccccccccccccccc} {{U}_{{\rm lim} }}\qquad \qquad \qquad \quad \ \ \alpha >\frac{{{\alpha }_{f}}}{2} \\ 2\left( 1-\frac{\alpha }{{{\alpha }_{f}}} \right){{U}_{{\rm lim} }}\qquad \quad \alpha \leqslant \frac{{{\alpha }_{f}}}{2} \\ \end{array} \right.\end{eqnarray} \tag{ 37 }$$

  and obtain ${{U}_{C}}\left( \alpha \right)$ from equation (32).

We now show how to design suitable time-domain voltage ramps ${{U}_{n}}\left( t \right)$ that will assure well-controlled splitting. It has been shown in section 4.1 that a small value of the acceleration at the CP, ${{\ddot{d}}_{{\rm CP}}}$, is required for achieving a low energy transfer. This in turn is guaranteed by well-controlled variation of the distance d(t) throughout the splitting process. As $d\left( \alpha \right)$ is monotonically decreasing with α, it can be inverted to obtain $\alpha \left( d \right)$ which is used to compute the final voltage ramp as ${{U}_{n}}\left( \alpha \left( d\left( t \right) \right) \right)$ (see figure 6).

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Possible choices for d(t) are a sine-squared ramp

$$\begin{eqnarray}&&d(t)={{d}_{i}}+\left( {{d}_{f}}-{{d}_{i}} \right){{{\rm sin} }^{2}}\left( \frac{\pi t}{2\;T} \right)\end{eqnarray} \tag{ 38 }$$

or a polynomial ramp

$$\begin{eqnarray}&&d(t)={{d}_{i}}+\left( {{d}_{i}}-{{d}_{f}} \right)\left( -10\frac{{{t}^{3}}}{{{T}^{3}}}+15\frac{{{t}^{4}}}{{{T}^{4}}}-6\frac{{{t}^{5}}}{{{T}^{5}}} \right)\end{eqnarray} \tag{ 39 }$$

Both ramps fulfil $d\left( 0 \right)={{d}_{i}},d\left( T \right)={{d}_{f}},\dot{d}\left( 0 \right)=\dot{d}\left( T \right)=0$. The polynomial ramp, used in the following, additionally fulfils $\ddot{d}\left( 0 \right)=\ddot{d}\left( T \right)=0$, while the second derivative of the sine-squared ramp displays discontinuities. However, these features presumably play no role in experiments, as the voltage ramps are generally subject to discretization and filtering. Different methods can be employed for the determination of $d\left( \alpha \right)$:

• The equilibrium distance can be computed by employing realistic trap potentials from simulation data, using the voltage configuration pertaining to a given α as determined by the static voltage sets ${{U}_{n}}\left( \alpha \right)$. This method requires the simulated potentials to match the actual trap potential with great precision.
• The equilibrium distance can be computed using values from calibration measurements for the coefficients ${{\alpha }_{n}},{{\beta }_{n}}$. This circumvents the need for simulations and accounts for parameter drifts. It only yields valid values for distances that are small compared to the electrode width, however we will show in section 6 that this procedure yields useful voltage ramps.
• Ion distances can be measured by imaging the ion crystal on a camera, while voltage configurations for decreasing α values are applied. This is the most direct method, as the ion distance in image pixels can be gauged by measuring the trap frequency from resolved sideband spectroscopy [20]. The imaging magnification is determined from the trap frequency by using $d={{\left( \frac{2e\kappa }{m{{\omega }^{2}}} \right)}^{1/3}}$. This method benefits from the accuracy of resolved sideband spectroscopy, which is typically between 10 kHz and 100 Hz.

We first analyze the dependence of the energy transfer on the duration of the splitting process T, the result is shown in figure 7. The calculation is carried out for the ideal case of perfectly compensated potential tilt. We see that the final excitation becomes sufficiently low to remain in the Lamb–Dicke regime for typical laser–ion interaction settings at times larger than about $40\;\mu$s, which clearly outperforms the naive approach of voltage interpolation from section 4.1.

We also take into account increased anomalous heating around the CP by employing the averaged heating rate according to equation (31). We see that for our specific heating rates, the limit of about one phonon per ion cannot be overcome, but as the anomalous heating contribution is scaling as T, the splitting result becomes rather insensitive with respect to the precise choice of the T for $T\gtrsim 50$ μs.

The simulation results verify our approach of calculating the voltage ramps using the Taylor approximated potentials. One recognizes that the resulting energy transfer in this case is larger by a factor of about two throughout the entire range of splitting durations. This is due to the fact that the Taylor expansion leads to deviating voltages pertaining to the CP, which are sufficiently strong to increase the acceleration as explained in section 4.1. The discrepancy becomes irrelevant for splitting times larger than T = 60 μs. At around 60–70 μs the oscillatory excitation becomes smaller than $\bar{n}=0.1$, corresponding to the limit we can currently resolve in our experiment. The slight inaccuracy for low phonon numbers is due to numerical artifacts. Even lower energy transfers at shorter T could possibly be achieved by ramp engineering, i.e. by the application of shortcut-to-adiabaticity approaches [21, 23].

Two crucial parameters for the splitting operation are the offset voltage at the CP $\Delta U_{C}^{\left( CP \right)}$ and the potential tilt γ. Small variations of these parameters lead to strong coherent excitations as shown in figure 8.

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The CP voltage offset $\Delta U_{C}^{\left( CP \right)}$ serves both for modeling and compensation of inaccuracies of the trap potentials, leading to a wrongly determined CP voltage configuration and therefore to increased acceleration. It is implemented into the simulations by just adding it to $U_{C}^{\left( CP \right)}$ as determined by equation (32) in the calculation of the static voltage sets. We see that even for sufficiently slow splitting, the Lamb–Dicke regime can only be attained if this voltage offset, and therefore the CP voltages in general, are correct within a window of about 20 mV, on the other hand it becomes clear that this voltage serves as convenient fine tuning parameter. The minimum excitation does not occur at $\Delta U_{C}^{\left( CP \right)}$ = 0, but is slightly shifted to positive values.

This can be understood by considering that ${{\left| {\dot{\alpha }} \right|}_{{\rm CP}}}$ is increased for any $\Delta U_{C}^{\left( CP \right)}\ne 0$, but ${{\ddot{\alpha }}_{{\rm CP}}}$ is decreased for $\Delta U_{C}^{\left( CP \right)}>0$. With $\partial d/\partial \alpha$, the second term in equation (25) leads to a reduced total acceleration for small positive $\Delta U_{C}^{\left( CP \right)}$. Larger values again lead to increased acceleration because of a smaller ${{\beta }_{{\rm CP}}}$ value. All other calculations in this work are done using $\Delta U_{C}^{\left( CP \right)}$ = 0.

For the case of an uncompensated tilt $\gamma ^{\prime}$, we observe an even stronger dependence of the energy transfer. Fine tuning of the voltage difference on the outer segments $\Delta {{U}_{O}}$ on the sub-mV level is required to reach the single phonon regime. Moreover, we observe that moderate uncompensated potential tilts reduce the energy transfer to one of the ions, as its CP acceleration is reduced by a more smooth $x\left( \alpha \right)$ dependence. This might be of interest for specific applications where only the energy transfer to one of the ions is of importance.

Finally we study the dependence of the energy transfer on the limiting voltage ${{U}_{{\rm lim} }}$. We find that by increasing the voltage limit, beyond ${{U}_{{\rm lim} }}=10$ V used so far, we can obtain lower coherent excitations as shown in figure 9. For this simulation, only the maximum voltage on the outer segments (max ${{U}_{O}}$) is increased and all other limits remain unchanged. We infer that by increasing the voltage limit on these electrodes up to about 50 V, one can reduce the mean phonon number by a factor of $\approx \;8$ for T = 60 μs. For lower splitting durations the enhancing factor becomes slightly smaller.

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