Controlling the transport of an ion: classical and quantum mechanical solutions

We model a linear Paul trap, assuming confinement of the ion in the radial direction (y and z) due to an oscillating quadrupole potential. The axial confinement is realized by an electrostatic potential along the trap axis x, as illustrated in figure 1. Further details on describing the trap can be found in [27]. The confinement enables us to treat the dynamics only along the x dimension. We consider transport of a single ion with mass m between two neighboring electrodes, which give rise to individual potentials centered at ${{x}_{1}}$ and ${{x}_{2}}$. This may be scaled up to N electrodes and longer transports without any loss of generality. The axial motion of the ion is controlled by a time-dependent electrostatic potential,

$$\begin{eqnarray}&&V\left( x,t \right)={{U}_{1}}\left( t \right){{\phi }_{1}}\left( x \right)+{{U}_{2}}\left( t \right){{\phi }_{2}}\left( x \right),\end{eqnarray} \tag{ 1 }$$

with segment voltages ${{U}_{i}}\left( t \right)$, and normal electrode potentials on the trap axis, ${{\phi }_{i}}\left( x \right)$. They are dimensionless electrostatic potentials obtained with a bias of $+1$ V at electrode i and the remaining electrodes grounded (see figure 1(b)). These potentials are calculated by using a fast multipole boundary element method [27] for the trap geometry used in recent experiments [12] and shown in figure 1. In order to speed up numerics and obtain smooth derivatives, we calculate values for ${{\phi }_{i}}\left( x \right)$ on a mesh and fit rational functions to the resulting data. The spatial derivatives ${{\phi }_{i}}^{\prime} \left( x \right)$ and $\phi _{i}^{^{\prime\prime} }\left( x \right)$ are obtained by differentiation of the fit functions. Previous experiments have shown that the calculated potentials allow for the prediction of ion positions and trap frequencies with an accuracy of one per cent [28, 29] which indicates the precision of the microtrap fabrication process. An increase in the precision can be achieved by calibrating the trapping potentials using resolved sideband spectroscopy. This is sufficient to warrant the application of control techniques as studied here. For the geometry of the trap described in [12], we obtain harmonic trap frequencies of about $\omega =2\pi \cdot 1.3$ MHz with a bias voltage of $-7$ V at a single trapping segment. The individual segments are spaced 280 μm apart. Our goal is to shuttle a single ion along this distance within a time span on the order of the oscillation period by changing the voltages ${{U}_{1}}$ and ${{U}_{2}}$, which are supposed to stay within a predetermined range that is set by experimental constraints. We seek to minimize the amount of motional excitation due to the shuttling process.

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Assuming the ion dynamics to be well described classically, we optimize the time dependent voltages in order to reduce the amount of transferred energy. This corresponds to minimizing the functional J,

$$\begin{eqnarray}&&J={{\left( E\left( T \right)-{{E}_{{\rm T}}} \right)}^{2}}+\mathop{\sum }\limits_{i}\int _{0}^{T}\frac{{{\lambda }_{a}}}{S\left( t \right)}\Delta {{U}_{i}}{{\left( t \right)}^{2}}\;{\rm d}t\;,\end{eqnarray} \tag{ 2 }$$

i.e., to minimize the difference between desired energy ${{E}_{{\rm T}}}$ and the energy E(T) obtained at the final time T. In the above equation, $\Delta {{U}_{i}}\left( t \right)=U_{i}^{n+1}\left( t \right)-U_{i}^{n}\left( t \right)$ is the update of each voltage ramp in an iteration step n, and the second term in equation (2) limits the overall change in the integrated voltages during one iteration. The weight ${{\lambda }_{a}}$ is used to tune the convergence and limit the updates. To suppress updates near t = 0 and t = T the shape function $S\left( t \right)\geqslant 0$ is chosen to be zero at these points in time. For a predominantly harmonic confinement, a measure of the axial energy of the ion in the harmonic approximation of equation (1) is given by

$$\begin{eqnarray}&&E\left( T \right)=\frac{1}{2}m{{\dot{x}}^{2}}\left( T \right)+\frac{1}{2}m{{\omega }^{2}}{{\left( x\left( T \right)-{{x}_{2}} \right)}^{2}}\;.\end{eqnarray} \tag{ 3 }$$

In order to obtain transport without motional excitation, we choose ${{E}_{T}}=0$. Evaluation of equation (3) requires solution of the classical equation of motion. It reads

$$\begin{eqnarray}&&\ddot{x}\left( t \right)=-\frac{1}{m}{{\left. \frac{\partial }{\partial x}V\left( x,t \right) \right|}_{x=x\left( t \right)}}=-\frac{1}{m}\mathop{\sum }\limits_{i=1}^{2}{{U}_{i}}\left( t \right)\phi _{i}^{^{\prime} }\left( x\left( t \right) \right)\end{eqnarray} \tag{ 4 }$$

for a single ion trapped in the potential of equation (1) and is solved numerically using a Dormand–Prince Runge–Kutta integrator [27]. Employing Krotovʼs method for optimal control [30] together with the classical equation of motion, equation (4), we obtain the following iterative update rule:

$$\begin{eqnarray}&&\Delta {{U}_{i}}\left( t \right)=-\frac{S\left( t \right)}{{{\lambda }_{a}}}p_{2}^{\left( n \right)}\left( t \right)\phi _{i}^{^{\prime} }\left( {{x}^{\left( n+1 \right)}}\left( t \right) \right),\end{eqnarray} \tag{ 5 }$$

where n denotes the previous iteration step. ${\bf p}=\left( {{p}_{1}},{{p}_{2}} \right)$ is a costate vector which evolves according to

$$\begin{eqnarray}&&{\bf \dot{p}}\left( t \right)=-\left( \begin{array}{ccccccccccccccc} \frac{{{p}_{2}}}{m}V^{\prime\prime} \left( {{U}_{i}}\left( t \right),x\left( t \right) \right) \\ {{p}_{1}} \\ \end{array} \right),\end{eqnarray} \tag{ 6 }$$

with its 'initial' condition defined at the final time T:

$$\begin{eqnarray}&&{\bf p}\left( T \right)=-2\;m\left( E\left( T \right)-{{E}_{{\rm T}}} \right)\left( \begin{array}{ccccccccccccccc} {{\omega }^{2}}\left( x\left( T \right)-{{x}_{2}} \right) \\ \dot{x}\left( T \right) \\ \end{array} \right).\end{eqnarray} \tag{ 7 }$$

The algorithm works by propagating x(t) forward in time, solving equation (4) with an initial guess for ${{U}_{i}}\left( t \right)$ and iterating the following steps until the desired value of J is achieved:

• (i)  
  Obtain p(T) according to equation (7) and propagate p(t) backwards in time using equation (6).
• (ii)  
  Update the voltages according to equation (5) at each time step while propagating x(t) forward in time with the immediately updated voltages.

The optimization algorithm shows rapid convergence and brings the final excitation energy E(T) as close to zero as desired. An example of an optimized voltage ramp is shown in figure 2(a). The voltages obtained are not symmetric under time reversal in contrast to the initial guess. This is rationalized by the voltage updates occurring only during forward propagation which breaks the time reversal symmetry. We find this behavior to be typical for the Krotov algorithm combined with the classical equation of motion.

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When quantum effects are expected to influence the transport, the ion has to be described by a wave function $\Psi \left( x,t \right)$. The control target is then to perfectly transfer the initial wavefunction, typically the ground state of the trapping potential centered around position ${{x}_{1}}$, to a target wavefunction, i.e., the ground state of the trapping potential centered around position ${{x}_{2}}$. This is achieved by minimizing the functional

$$\begin{eqnarray}&&J=1-{{\left| \int _{-\infty }^{\infty }\Psi {{\left( x,T \right)}^{*}}{{\Psi }^{{\rm tgt}}}\left( x \right)\;{\rm d}x \right|}^{2}}+\int _{0}^{T}\frac{{{\lambda }_{a}}}{S\left( t \right)}\mathop{\sum }\limits_{i}\Delta {{U}_{i}}{{\left( t \right)}^{2}}\;{\rm d}t\;.\end{eqnarray} \tag{ 8 }$$

Here, $\Psi \left( x,T \right)$ denotes the wave function of the single ion propagated with the set of voltages ${{U}_{i}}\left( t \right)$, and ${{\Psi }^{{\rm tgt}}}\left( x \right)$ is the target wave function. The voltage updates $\Delta {{U}_{i}}\left( t \right)$, scaling factor ${{\lambda }_{a}}$ and shape function S(t) have identical meanings as in section 2.2. $\Psi \left( x,T \right)$ is obtained by solving the time-dependent Schrödinger equation (TDSE),

$$\begin{eqnarray}&&i\hbar \frac{\partial }{\partial t}\Psi \left( x,t \right)=\hat{\mathsf H}\left( t \right)\Psi \left( x,t \right)=\left( -\frac{{{\hbar }^{2}}}{2\;m}\frac{{{{\rm d}}^{2}}}{{\rm d}{{x}^{2}}}+\mathop{\sum }\limits_{i=1}^{N}{{U}_{i}}\left( t \right){{\phi }_{i}}\left( x \right) \right)\Psi \left( x,t \right).\end{eqnarray} \tag{ 9 }$$

As in the classical case, optimization of the transport problem is tackled using Krotovʼs method [14, 16]. The update equation derived from equation (8) is given by

$$\begin{eqnarray}&&\Delta {{U}_{i}}\left( t \right)=\frac{S\left( t \right)}{{{\lambda }_{a}}}\Im {\rm m}\int _{-\infty }^{\infty }{{\chi }^{n}}{{\left( x,t \right)}^{*}}\;{{\phi }_{i}}\left( x \right)\;{{\Psi }^{n+1}}\left( x,t \right)\;{\rm d}x\;,\end{eqnarray} \tag{ 10 }$$

with n denoting the iteration step. $\chi \left( x,t \right)$ is a costate wave function obeying the TDSE with 'initial' condition

$$\begin{eqnarray}&&\chi \left( x,T \right)=\left[ \int _{-\infty }^{\infty }{{\left( \Psi \left( x,T \right) \right)}^{*}}{{\Psi }^{{\rm tgt}}}\left( x \right)\;{\rm d}x\ \right]{{\Psi }^{{\rm tgt}}}\left( x,T \right).\end{eqnarray} \tag{ 11 }$$

Optimized voltages ${{U}_{i}}\left( t \right)$ are obtained similarly to section 2.2, i.e., one starts with the ground state, propagates $\Psi \left( x,t \right)$ forward in time according to equation (9), using an initial guess for the voltage ramps, and iterates the following steps until the desired value of J is achieved:

• (i)  
  Compute the costate wave function at the final time T according to equation (11) and propagate $\chi \left( x,t \right)$ backwards in time, storing $\chi \left( x,t \right)$ at each timestep.
• (ii)  
  Update the control voltages according to equation (10) using the stored $\chi \left( x,t \right)$, while propagating $\Psi \left( x,t \right)$ forward using the immediately updated control voltages.

Equations (10) and (11) imply a sufficiently large initial overlap between the wave function, which is forward propagated under the initial guess, and the target state in order to obtain a reasonable voltage update. This emphasizes the need for good initial guess ramps and illustrates the difficulty of the control problem when large phase space volumes need to be covered. The limitation of a sufficiently good initial guess may possibly be overcome using gradient-free optimization tools such as those of [31, 32], based on the simplex method. To solve the TDSE numerically, we use the Chebshev propagator [33] in conjunction with a Fourier grid [34, 35] for efficient and accurate application of the kinetic energy part of the Hamiltonian. Denoting the transport time by T and the inter-electrode spacing by d, the average momentum during the shuttling is given by $\bar{p}=md/T$. Typical values of these parameters yield a phase space volume of $d\cdot \bar{p}/h\approx {{10}^{7}}$. This requires the numerical integration to be extremely stable. In order to ease the numerical treatment, we can exploit the fact that the wavefunctionʼs spatial extent is much smaller than d and most excess energy occurs in the form of classical oscillations. This allows for propagating the wave function on a small moving grid that extends around the instantaneous position and momentum expectation values [27]. The details of our implementation combining the Fourier representation and a moving grid are described in appendix A.

Any optimization, no matter whether it employs classical or quantum equations of motion, starts from an initial guess. For many optimization problems, and in particular when using gradient-based methods for optimization, a physically motivated initial guess is crucial for success of the optimization [36]. Here, we design the initial guess for the voltage ramps such that the ion is dragged from position ${{x}_{1}}$ to ${{x}_{2}}$ in a smooth fashion. The details on how to obtain the initial guess voltages are explained in appendix B. An example is shown in figure 2. If the electrode potentials have translational symmetry, i.e., ${{\phi }_{j}}\left( x \right)={{\phi }_{i}}\left( x+d \right)$, then $U_{1}^{0}\left( t \right)=U_{2}^{0}\left( T-t \right)$. This condition is approximately met for sufficiently homogeneous trap architectures.

Most current ion traps are fairly well described by a simple harmonic model,

$$\begin{eqnarray}&&V\left( x,t \right)=-{{u}_{1}}\left( t \right)\frac{1}{2}m\omega _{0}^{2}{{\left( x-{{x}_{1}} \right)}^{2}}-{{u}_{2}}\left( t \right)\frac{1}{2}m\omega _{0}^{2}{{\left( x-{{x}_{2}} \right)}^{2}}\;,\end{eqnarray} \tag{ 12 }$$

where ${{\omega }_{0}}$ is the trap frequency and ${{u}_{i}}$ are dimensionless control parameters which correspond to the electrode voltages. Since the equations of motion can be solved analytically, one can also hope to solve the control problem analytically. One option is given by Pontryaginʼs maximum principle [18, 37] which allows to determine time-optimal controls. Compared to numerical optimization which always yields local optima, Pontryaginʼs maximum principle guarantees the optimum to be global.

In general, a cost functional is minimized which depends on the equations of motion and the control. Here we seek to minimize the transport time T. The cost functional then is given by

$$\begin{eqnarray}&&J\left[ {{u}_{1}},{{u}_{2}} \right]=\int _{0}^{{{T}_{{\rm min} }}}\;{\rm d}t={{T}_{{\rm min} }}\;,\end{eqnarray} \tag{ 13 }$$

which is independent of the controls ${{u}_{1}}$, ${{u}_{2}}$. The optimization problem is formally equivalent to finding a classical trajectory by the principle of least action. It is captured in terms of a classical control Hamiltonian which depends on the target functional and equations of motion. In our example, it becomes

$$\begin{eqnarray}&&{{H}_{c}}\left( {{p}_{1}},{{p}_{2}},x,v,{{u}_{1}},{{u}_{2}} \right)={{p}_{1}}v+{{p}_{2}}\left( {{u}_{1}}\cdot \left( x-{{x}_{1}} \right)+{{u}_{2}}\cdot \left( x-{{x}_{2}} \right) \right)\omega _{0}^{2}\;,\end{eqnarray} \tag{ 14 }$$

where the costates ${{p}_{1}}$, ${{p}_{2}}$ obey the Hamiltonian equations,

$$\begin{eqnarray}&&{{\dot{p}}_{1}}=-\frac{\partial {{H}_{c}}}{\partial x}\;,\quad {{\dot{p}}_{2}}=-\frac{\partial {{H}_{c}}}{\partial v}\;.\end{eqnarray} \tag{ 15 }$$

We bound the controls ${{u}_{1}}$ and ${{u}_{2}}$ by ${{u}_{{\rm max} }}$ which corresponds to the experimental voltage limit.

Pontryaginʼs principle states that ${{H}_{c}}$ in general becomes maximal for the optimal choice of the control [18, 37]. Since ${{H}_{c}}$ is linear in ${{u}_{i}}$ and ${{x}_{1}}\leqslant x\leqslant {{x}_{2}}$, ${{H}_{c}}$ becomes maximal depending on the sign of ${{p}_{2}}$,

$$\begin{eqnarray}&&{{u}_{1}}\left( t \right)=-{{u}_{2}}\left( t \right)={\rm sign}({{p}_{2}}){{u}_{{\rm max} }}\;.\end{eqnarray} \tag{ 16 }$$

Evaluating equation (15) for ${{H}_{c}}$ of equation (14) leads to

$$\begin{eqnarray}&&\mathop{{{p}_{1}}}\limits^{.}=-{{p}_{2}}\omega _{0}^{2}\left( {{u}_{1}}+{{u}_{2}} \right)\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&\mathop{{{p}_{2}}}\limits^{.}=-{{p}_{1}}.\end{eqnarray} \tag{ 18 }$$

In view of equation (16), the only useful choice is ${{p}_{2}}\left( 0 \right)>0$. Otherwise the second electrode would be biased to a positive voltage, leading to a repulsive instead of an attractive potential acting on the ion. The equations of motion for the costate thus become

$$\begin{eqnarray}&&\mathop{{{p}_{1}}}\limits^{.}=0\Rightarrow {{p}_{1}}\left( t \right)={{c}_{1}}\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\mathop{{{p}_{2}}}\limits^{.}=-{{p}_{1}}\Rightarrow {{p}_{2}}\left( t \right)={{p}_{2}}\left( 0 \right)-{{c}_{1}}t.\end{eqnarray} \tag{ 20 }$$

For a negative constant ${{c}_{1}}$, ${{p}_{2}}$ is never going to cross zero. This implies that there will not be a switch in voltages leading to continuous acceleration. For positive ${{c}_{1}}$ there will be a zero crossing at time ${{t}_{{\rm sw}}}={{p}_{2}}\left( 0 \right)/{{c}_{1}}$. The optimal solution thus corresponds to a single switch of the voltages. We will analyze this solution and compare it to the solutions obtained by numerical optimization below in section 3.

For quantum mechanical equations of motion, geometric optimal control is limited to very simple dynamics such as that of three- or four-level systems, see e.g. [38]. A second analytical approach that is perfectly adapted to the quantum harmonic oscillator utilizes the Lewis–Riesenfeld theory which introduces dynamical invariants and their eigenstates [39]. This invariant based inverse engineering approach (IEA) has recently been applied to the transport problem [40, 41]. The basic idea is to compensate the inertial force occurring during the transport sequence. To this end, the potential is written in the following form:

$$\begin{eqnarray}&&V\left( x,t \right)=-F\left( t \right)x+\mathcal{U}\left( x-\alpha \left( t \right) \right),\end{eqnarray} \tag{ 21 }$$

with $\mathcal{U}$ an arbitrary function and F and α fulfilling the constraint

$$\begin{eqnarray}&&\ddot{\alpha }\left( t \right)=F\left( t \right)/m.\end{eqnarray} \tag{ 22 }$$

We choose $\alpha \left( t \right)$ to be the transport function of section 2.4 and identify $\mathcal{U}$ in equation (21) with the trapping potential of equation (1), such that F(t) compensates the inertial force given by the acceleration of the trap center. Forced to follow the minimum of the trapping potential instantaneously, an ion in its motional ground state will not be displaced at all and therefore not gain coherent excitation during the transport. For the potential of equation (21), the Hermitian operator

$$\begin{eqnarray}&&\hat{\mathsf I}=\frac{1}{2\;m}{{\left( p-m\dot{\alpha } \right)}^{2}}+\mathcal{U}\left( x-\alpha \right)\end{eqnarray} \tag{ 23 }$$

fulfills the invariance condition for all conceivable quantum states $|\Psi \left( t \right)\rangle$:

$$\begin{eqnarray}&&\frac{{\rm d}}{{\rm d}t}\langle \Psi \left( t \right)|\hat{\mathsf I}\left( t \right)|\Psi \left( t \right)\rangle =0\quad \Leftrightarrow \quad \frac{{\rm d}\hat{\mathsf I}}{{\rm d}t}=\frac{\partial \hat{\mathsf I}}{\partial t}+\frac{1}{{\rm i}\hbar }\left[ \hat{\mathsf I}\left( t \right),\hat{\mathsf H}\left( t \right) \right]=0\end{eqnarray} \tag{ 24 }$$

with $\hat{\mathsf H}$ the Hamiltonian of the ion. Note that a more general invariant with further parameters can be derived [42], which allows for additional terms in the potential, used to describe e.g. trap expansions [40]. The requirement for transporting the initial ground state to the ground state of the trap at the final time corresponds to $\hat{\mathsf H}$ and $\hat{\mathsf I}$ having a common set of eigenfunctions at initial and final time. This is the case for $\dot{\alpha }\left( 0 \right)=\dot{\alpha }\left( T \right)=0$ [40, 43]. The additional compensating force is generated using the same trap electrodes as for the trapping potential of equation (1) by applying an additional voltage $\delta {{U}_{i}}$. For a given transport function $\alpha \left( t \right)$ we therefore have to solve the underdetermined equation,

$$\begin{eqnarray}&&m\ddot{\alpha }\left( t \right)=-\phi _{1}^{^{\prime} }\left( x\left( t \right) \right)\delta {{U}_{1}}\left( t \right)-\phi _{2}^{^{\prime} }\left( x\left( t \right) \right)\delta {{U}_{2}}\left( t \right),\end{eqnarray} \tag{ 25 }$$

where x(t) is given by the classical trajectory. Since the ion is forced to follow the center of the trap we can set $x\left( t \right)=\alpha \left( t \right)$. The compensating force is supposed to be a function of time only, cf equation (22), whereas changing the electrode voltages by $\delta {{U}_{i}}$ will, via the ${{\phi }_{i}}\left( x \right)$, in general yield a position-dependent force. This leads to a modified second derivative of the actual potential:

$$\begin{eqnarray}&&m{{\omega }_{c}}{{\left( t \right)}^{2}}=\mathop{\sum }\limits_{i=1}^{2}\phi _{i}^{^{\prime\prime} }\left( \alpha \left( t \right) \right)\left( U_{i}^{0}\left( t \right)+\delta {{U}_{i}}\left( t \right) \right)=m\left( {{\omega }^{2}}+\delta \omega {{\left( t \right)}^{2}} \right),\end{eqnarray} \tag{ 26 }$$

where $\delta \omega {{\left( t \right)}^{2}}$ denotes the change in trap frequency due to the compensation voltages $\delta {{U}_{i}}$, ω is the initially desired trap frequency, and $U_{i}^{0}\left( t \right)$ is found in equation (B.3). A time-varying actual frequency ${{\omega }_{c}}\left( t \right)$ might lead to wavepacket squeezing. However, since equation (25) is underdetermined, we can set $\delta \omega {{\left( t \right)}^{2}}=0$ leading to ${{\omega }_{c}}\left( t \right)=\omega$ as desired. With this condition we can solve equation (25) and obtain

$$\begin{eqnarray}&&\delta {{U}_{i}}\left( t \right)=\frac{\ddot{\alpha }\left( t \right){{\left( -1 \right)}^{i}}\;m\;\phi _{j}^{^{\prime\prime} }\left( \alpha \left( t \right) \right)}{\phi _{2}^{^{\prime\prime} }\left( \alpha \left( t \right) \right)\phi _{1}^{^{\prime} }\left( \alpha \left( t \right) \right)-\phi _{2}^{^{\prime} }\left( \alpha \left( t \right) \right)\phi _{1}^{^{\prime\prime} }\left( \alpha \left( t \right) \right)}\;,i,j\in \{1,2\},j\ne i.\quad \end{eqnarray} \tag{ 27 }$$

Note that equation (27) depends only on the trap geometry. The transport duration T enters merely as a scaling parameter via $\ddot{\alpha }\left( t \right)=\alpha ^{\prime\prime} \left( s \right)/{{T}^{2}}$. An example of a voltage sequence obtained by this method is shown in figure 2(b). The voltage curves are symmetric under time inversion like the guess voltages, that are derived from the same potential functions ${{\phi }_{i}}\left( x \right)$.

In any experiment, there is an upper limit to the electrode voltages that can be applied. It is the range of electrode voltages that limits the maximum transport speed. Typically this range is given by ±10 V for technical reasons. It could be increased by the development of better voltage supplies. We define the minimum possible transport time ${{T}_{{\rm min} }}$ to be the smallest time T for which less than 0.01 phonons are excited due to the total transport. To examine how ${{T}_{{\rm min} }}$ scales as a function of the maximum electrode voltages ${{U}_{{\rm max} }}$, we have carried out numerical optimization combined with classical equations of motion. The initial guess voltages, cf equations (B.1) and (B.3), were taken to preserve a constant trap frequency of $\omega =2\pi \cdot 1.3$ MHz for a $^{40}{\rm C}{{{\rm a}}^{+}}$ ion. The transport ramps were optimized for a range of maximum voltages between 10–150 V and transport times between 10 ns and 300 ns with voltages truncated to $\pm \;{{U}_{{\rm max} }}$ during the updates. The results are shown in figures 3 and 4. Figure 3 depicts the final excitation energy versus transport time, comparing the initial guess (black) to an optimized ramp with ${{U}_{{\rm max} }}=10$ V (blue) in figure 3(a). Note that we measure the final energy in units of $\hbar \omega$, i.e., in terms of the trap excitation, in both classical and quantum cases. This allows for convenient comparison with the requirements for light–matter interaction, e.g. qubit readout or entangling gates. For the initial guess, the final energy displays an oscillatory behavior with respect to the trap period (${{T}_{{\rm per}}}=0.769\;\mu {\rm s}$ for $\omega =2\pi \cdot 1.3$ MHz) as it has been experimentally observed in [12], and an overall decrease of the final energy for longer transport times. The optimized transport with ${{U}_{{\rm max} }}=10$ V (blue line in figure 3(a)) shows a clear speed up of energy neutral transport: an excitation energy of less than 0.01 phonons is obtained for $T_{{\rm min} }^{{\rm opt}}=0.284\;\mu {\rm s}$ compared to $T_{{\rm min} }^{{\rm guess}}=1.391\;\mu {\rm s}$. The speedup increases with maximum voltage as shown in figure 3(b). The variation of $T_{{\rm min} }^{{\rm opt}}$ on ${{U}_{{\rm max} }}$ is studied in figure 4(a). We find a functional dependence of

$$\begin{eqnarray}&&T_{{\rm min} }^{{\rm opt}}\left( {{U}_{{\rm max} }} \right)\approx a{{\left( \frac{{{U}_{{\rm max} }}}{1\;{\rm V}} \right)}^{-b}}\end{eqnarray} \tag{ 28 }$$

with $a=0.880\left( 15 \right)\;\mu {\rm s}$ and $b=0.487\left( 5 \right)$. Optimized voltages are shown in figure 4(b) for the left electrode with ${{U}_{{\rm max} }}=10$ V. As the transport time decreases, the voltage ramp approaches a square shape. A bang-bang-like solution is attained at $T=280$ ns. However, for such a short transport time, classical control of energy neutral transport breaks down due to an insufficient voltage range and the final excitation amounts to 5703 mean phonons.

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In the following we show that for purely harmonic potentials, the exponent b in equation (28) is universal, i.e., it does not depend on trap frequency nor ion mass. It is solely determined by the bang-bang like optimized voltage sequences, where instantaneous switching between maximum acceleration and deceleration guarantees shuttling within minimum time. The technical feasibility of bang-bang shuttling is thoroughly analyzed in [45]. The solution is obtained by the application of Pontryaginʼs maximum principle [18, 37] as discussed in section 2.5 and assumes instantaneous switches. Employing equations (16) and (20) the equation of motion becomes

$$\begin{eqnarray}\ddot{x}=\omega _{0}^{2}{{u}_{{\rm max} }}\cdot \left\{ \begin{array}{ccccccccccccccc} d, & t<{{t}_{{\rm sw}}} \\ -d, & t>{{t}_{{\rm sw}}} \\ \end{array} \right..\end{eqnarray} \tag{ 29 }$$

This can be integrated to

$$\begin{eqnarray}x\left( t \right)=\left\{ \begin{array}{ccccccccccccccc} {{x}_{1}}+{{u}_{{\rm max} }}{\rm d}\omega _{0}^{2}{{t}^{2}}, & 0\leqslant t\leqslant {{t}_{{\rm sw}}} \\ {{x}_{1}}+d-{{u}_{{\rm max} }}{\rm d}\omega _{0}^{2}{{\left( t-{{T}_{{\rm min} }} \right)}^{2}}, & {{t}_{{\rm sw}}}\leqslant t\leqslant {{T}_{{\rm min} }} \\ \end{array} \right.\end{eqnarray} \tag{ 30 }$$

with the boundary conditions $x\left( 0 \right)={{x}_{1}}$, $x\left( {{T}_{{\rm min} }} \right)={{x}_{2}}$ and $\dot{x}\left( 0 \right)=\dot{x}\left( {{T}_{{\rm min} }} \right)=0$. Using the continuity of $\dot{x}$ and x at $t={{t}_{{\rm sw}}}$, we obtain

$$\begin{eqnarray}&&{{t}_{{\rm sw}}}=\frac{T}{2},\quad {{T}_{{\rm min} }}=\frac{\sqrt{2}}{{{\omega }_{0}}}\sqrt{\frac{1}{{{u}_{{\rm max} }}}}.\end{eqnarray} \tag{ 31 }$$

Notably, the minimum transport time is proportional to $u_{{\rm max} }^{-1/2}$ which explains the behavior of the numerical data shown in figure 4. This scaling law can be understood intuitively by considering that in the bang-bang control approach, the minimum shuttling time is given by the shortest attainable trap period, which scales as $u_{{\rm max} }^{-1/2}$. Assuming a trap frequency of ${{\omega }_{0}}=2\pi \cdot 0.55$ MHz in equation (31), corresponding to a trapping voltage of $-1$ V for our trap geometry, we find a prefactor $\sqrt{2}/{{\omega }_{0}}=0.41\;\mu {\rm s}$. This is smaller than $a=0.880\left( 15 \right)\;\mu {\rm s}$ obtained by numerical optimization for realistic trap potentials. The difference can be rationalized in terms of the average acceleration provided by the potentials. For realistic trap geometries, the force exerted by the electrodes is inhomogeneous along the transport path. Mutual shielding of the electrodes reduces the electric field feedthrough of an electrode to the neighboring ones. Thus, the magnitude of the accelerating force that a real electrode can exert on the ion when it is located at a neighboring electrode is reduced with respect to a constant force generating harmonic potential with the same trap frequency.

The minimum transport time of $T_{{\rm min} }^{{\rm opt}}=0.284\;\mu {\rm s}$ identified here for ${{U}_{{\rm max} }}=10$ V, cf the blue line in figure 3(a), is significantly shorter than operation times realized experimentally. For comparison, an ion has recently been shuttled within $3.6\;\mu {\rm s}$, leading to a final excitation of $0.10\pm 0.01$ motional quanta [12]. Optimization may not only improve the transport time but also the stability with respect to uncertainties in the time. This is in contrast to the extremely narrow minima of the final excitation energy for the guess voltage ramps shown in black in figure 3(a), implying a very high sensitivity to uncertainties in the transport time. For example, for the fourth minimum of the black curve, located at $3.795\;\mu {\rm s}$ and close to the operation time of [12] (not shown in figure 3(a)), final excitation energies of less than 0.1 phonons are observed only within a window of 3 ns. Optimization of the voltage ramps for $T=3.351\;\mu {\rm s}$ increases the stability against variations in transport time to more than 60 ns.

In conclusion we find that optimizing the classical motion of an ion allows us to identify the minimum operation time for a given maximum voltage and improve the stability with respect to timing uncertainies for longer operation times. The analytical solution derived from Pontryaginʼs maximum principle is helpful to understand the minimum time control strategy. Numerical optimization accounts for all typical features of realistic voltage ramps. It allows for identifying the minimum transport time, predicting $36.9\%$ of the oscillation period for current maximum voltages and a trap frequency of $\omega =2\pi \cdot 1.3$ MHz. This number can be reduced to $12.2\%$ when increasing the maximum voltage by one order of magnitude.

However, these predictions may be rendered invalid by a breakdown of the classical approximation.

We now employ quantum wavepacket dynamics to test the classical solutions, obtained in section 3.1. Provided the trap frequency is constant and the trap is perfectly harmonic, the wave function will only be displaced during the transport. For a time-varying trap frequency, however, squeezing may occur [44]. Significant squeezing is also expected due to anharmonicities of the potential if a very rapid switching of trap voltages is applied [45]. Note that this effect, caused by the ionʼs displacement to regions of different local trap frequencies far away from the minimum can be avoided in the IEA per se by ensuring a constant trap frequency for the ionʼs trajectory. In extreme cases, anharmonicities might also lead to wavepacket dispersion. Since these two effects are not accounted for by numerical optimization of classical dynamics, we discuss in the following at which timescales such genuine quantum effects become significant. To this end, we have employed the optimized voltages shown in figure 4(b) in the propagation of a quantum wavepacket. We compare the results of classical and quantum mechanical motion in figure 5(a), cf the red and lightblue lines. A clear deviation is observed. Also, as can be seen in figure 5(b), the wavefunction fails to reach the target wavefunction for transport times close to the classical limit $T_{{\rm min} }^{{\rm opt}}$. This is exclusively caused by squeezing and can be verified by inspecting the time evolution of the wavepacket in the final potential: we find the width of the wavepacket to oscillate, indicating a squeezed state. No wavepacket dispersion effects are observed; i.e., the final wavepackets are still minimum uncertainty states, with ${\rm min} \left( \Delta x\cdot \Delta p \right)=\hbar /2$. This means that no effect of anharmonicities in the potential is observed. An impact of anharmonicities is expected once the size of the wavefunction becomes comparable to segment distance d (see figure 1). Then the wavefunction extends over spatial regions in which the potentials deviate substantially from harmonic potentials. For the ion shuttling problem, this effect does not play a role over the relevant parameter regime. The effects of anharmonicities in the quantum regime for trapped ions were thoroughly analyzed in [46]. Squeezing increases ${{T}_{{\rm min} }}$ from $0.28\;\mu {\rm s}$ to $0.86\;\mu {\rm s}$ for the limit of exciting less than 0.01 phonons, see the red curve in figure 5(a), i.e., it only triples the minimum transport time. We show that squeezing can be suppressed altogether in the following section.

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In the invariant based IEA, the minimal transport time is determined by the maximum voltages that are required for attaining zero motional excitation. The total voltage that needs to be applied is given by ${{U}_{i}}\left( t \right)=U_{i}^{0}\left( t \right)+\delta {{U}_{i}}\left( t \right)$ with $U_{i}^{0}\left( t \right)$ and $\delta {{U}_{i}}\left( t \right)$ found in equations (B.3) and (27). The maximum of ${{U}_{i}}\left( t \right)$, and thus the mininum in T, is strictly related to the acceleration of the ion provided by the transport function $\alpha \left( t \right)$, cf equation (27). If the acceleration is too high, the voltages will exceed the feasibility limit ${{U}_{{\rm max} }}$. At this point it can also be understood why the acceleration should be zero at the beginning and end of the transport: for $\ddot{\alpha }\left( 0 \right)\ne 0$ a non-vanishing correction voltage $\delta {{U}_{i}}\ne 0$ is obtained from equation (27). This implies that the voltages do not match the initial trap conditions, where the ion should be located at the center of the initial potential.

We can derive a transport function $\alpha \left( t \right)$ compliant with the boundary conditions using equation (B.1). For this case, figure 6 shows the transport time $T_{{\rm min} }^{{\rm IEA}}$ versus the maximum voltage ${{U}_{{\rm max} }}$ that is applied to the electrodes during the transport sequence. For large transport times, the initial guess voltages $U_{i}^{0}\left( t \right)\propto {{\omega }^{2}}$ dominate the compensation voltages $\delta {{U}_{i}}\left( t \right)\propto \ddot{\alpha }\left( t \right)=\alpha ^{\prime\prime} \left( s \right)/{{T}^{2}}$. This leads to the bend of the red curve. When the trap frequency ω is lowered, the bend decreases. For the limiting case of no confining potential $\omega =U_{i}^{0}\left( t \right)=0$, $T_{{\rm min} }^{{\rm IEA}}$ is solely determined by the compensation voltages. In this case the same scaling of $T_{{\rm min} }^{{\rm IEA}}$ with ${{U}_{{\rm max} }}$ as for the optimization of classical dynamics is observed, cf black and blue lines in figure 6. For large ${{U}_{{\rm max} }}$, this scaling also applies to the case of non-zero trap frequency, cf red line in figure 6.

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We have tested the performance of the compensating force by employing it in the time evolution of the wavefunction. It leads to near-perfect overlap with the target state with an infidelity of less than ${{10}^{-9}}$. The final excitation energy of the propagated wave function is shown in figure 5 (green line) for a maximum voltage of ${{U}_{{\rm max} }}=10$ V. For the corresponding minimum transport time, $T_{{\rm min} }^{{\rm IEA}}\left( 10\;{\rm V} \right)=418$ ns, a final excitation energy six orders of magnitude below that found by optimization of the classical dynamics is obtained. This demonstrates that the invariant based IEA is capable of avoiding the wavepacket squeezing that was observed in section 3.2 when employing classically optimized controls in quantum dynamics. It also confirms that anharmonicities do not play a role since these would not be accounted for by the IEA-variant employed here. Note that an adaptation of the invariant based IEA to anharmonic traps is found in [41]. Similarly to numerical optimization of classical dynamics, IEA is capable of improving the stability against variations in transport time T. The final excitation energy obtained for $T=3.351\;\mu {\rm s}$ stays below 0.1 phonons within a window of more than $13$ ns.

A further reduction of the minimum transport time may be achieved due to the freedom of choice in the transport function $\alpha \left( t \right)$, by employing higher polynomial orders in order to reduce the compensation voltages $\delta {{U}_{i}}\left( t \right)$, cf equation (27). However, the fastest quantum mechanically valid transport has to be slower than the solutions obtained for classical ion motion. This follows from the bang-bang control being the time-optimal solution for a given voltage limit and the IEA solutions requiring additional voltage to compensate the wavepacket squeezing. We can thus conclude that the time-optimal quantum solution will be inbetween the blue and black curves of figure 6.

Numerical optimization of the wavepacket motion is expected to become necessary once the dynamics explores spatial regions in which the potential is strongly anharmonic or subject to strongly anharmonic fluctuations. This can be expected, for example, when the spatial extent of the wavefunction is not too different from that of the trap. Such a regime can be reached with trapped electrons in nanoscale Paul traps [47] where the wavefunction is of comparable size to that of a trapped ion but the trap dimension is of the order of a few 100 nm. In the following, we introduce the parameter $\xi ={{\sigma }_{0}}/d$, which is the wavefunction size normalized to the transport distance. While for current trap architectures, such a scenario is rather unlikely, further miniaturization might lead to this regime. Also, it is currently encountered in the transport of neutral atoms in tailored optical dipole potentials [48, 49].

Gradient-based quantum OCT requires an initial guess voltage that ensures a finite overlap of the propagated wave function $\Psi \left( T \right)$ with the target state ${{\Psi }^{{\rm tgt}}}$, see equation (11). Otherwise, the amplitude of the co-state χ vanishes. The overlap can also be analyzed in terms of phase space volume. For a typical ion trap setting with parameters as in figure 1, the total covered phase space volume in units of Planckʼs constant is $m{{d}^{2}}\omega /2\pi h\approx {{10}^{7}}$. This leads to very slow convergence of the optimization algorithm, unless an extremely good initial guess is available.

We utilize the results of the optimization for classical dynamics of section 3.1 as initial guess ramps for optimizing the wavepacket dynamics and investigate the convergence rate as a function of the system dimension, i.e., of ξ. The results are shown in figure 7(a), plotting the mean improvement per optimization step, $\Delta J$, averaged over 100 iterations, versus the scale parameter ξ. We computed the convergence rate $\overline{\Delta J}$ for different, fixed optimization weights ${{\lambda }_{a}}$ in equation (10). The curves in figure 7(a) are truncated for large values of $\overline{\Delta J}$, where the algorithm becomes numerically unstable. Values below $\overline{\Delta J}={{10}^{-6}}$ (dashed grey line in figure 7(a)) indicate an insufficient convergence rate for which no significant gain of fidelity is obtained with reasonable computational resources. In this case the potentials are insufficiently anharmonic to provide quantum control of the wavefunction.

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Numerical optimization of the wavepacket dynamics is applicable and useful for scale parameters of $\xi \approx 0.05$ and larger, indicated by arrows (2) and (3) in figure 7(a). Then the wavefunction size becomes comparable to the transport distance, leading for example to a phase space volume of around $10\;h$ for arrow (2). At this scale the force becomes inhomogeneous across the wavepacket. This leads to a breakdown of the IEA, as illustrated for $\xi =0.4$ in figures 7(b) and 8. The fidelity ${{\mathfrak{F}}_{{\rm IEA}}}$ for the IEA drops below $94.6\%$, whereas ${{\mathfrak{F}}_{{\rm qOCT}}}=0.999$ is achieved by numerical optimization of the quantum dynamics.

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