Shuttling of Rydberg ions for fast entangling operations

TL;DR

This paper proposes a rapid entangling scheme for Rydberg ions in a linear trap using ion shuttling and electric pulses, achieving sub-microsecond operation times and high fidelity.

Contribution

It introduces a novel fast entangling protocol for Rydberg ions utilizing ion shuttling and voltage pulses, surpassing traditional laser-driven gate speeds.

Findings

Entanglement can be achieved in under 1 microsecond.

Fidelity exceeds 99.9% with realistic parameters.

The method outperforms conventional laser-based gates by over two orders of magnitude.

Abstract

We introduce a scheme to entangle Rydberg ions in a linear ion crystal, using the high electric polarizability of the Rydberg electronic states in combination with mutual Coulomb coupling of ions that establishes common modes of motion. After laser-initialization of ions to a superposition of ground- and Rydberg-state, the entanglement operation is driven purely by applying a voltage pulse that shuttles the ion crystal back and forth. This operation can achieve entanglement on a sub-$\mu$s timescale, more than two orders of magnitude faster than typical gate operations driven by continuous-wave lasers. Our analysis shows that the fidelity achieved with this protocol can exceed $99.9\%$ with experimentally achievable parameters.

Figures

Equations

$${\left({\omega }_j^{\alpha \beta }\right)}^2$$

$$+(J^{\alpha \beta }{)}^2\left[1+(-1{)}^j\sin (2{\theta }^{\alpha \beta })\right]$$

$$(J^{\alpha \beta }{)}^2$$

$${\theta }^{\alpha \beta }$$

$$H_{\text{p}}=\sum _{\alpha \beta =\uparrow ,\downarrow }\left(\sum _{j=1}^2{\omega }_j^{\alpha \beta }{\tilde{a}}_j^{†}{\tilde{a}}_j+V_0^{\alpha \beta }\right){\Pi }^{\alpha \beta }$$

$$\text{mspace}-10.0mu{H}_{\text{d}}(t)$$

$$U_{\text{I}}(t)$$

$$\times \exp \left[i\sum _{j=1}^2{\phi }_j^{\alpha \beta }(t)-i{\Phi }_{\text{e}}^{\alpha \beta }(t)\right]{\Pi }^{\alpha \beta }$$

$$A_j^{\alpha \beta }(f_0,{\omega }_j^{\alpha \beta },T)$$

$$\times \left[\cos {\theta }^{\alpha \beta }-(-1{)}^j\sin {\theta }^{\alpha \beta }\right],$$

$${\phi }_j^{\alpha \beta }(f_0,{\omega }_j^{\alpha \beta },T)$$

$$\times {\left[\cos {\theta }^{\alpha \beta }-(-1{)}^j\sin {\theta }^{\alpha \beta }\right]}^2,$$

$${\Phi }_{\text{e}}^{\alpha \beta }(f_0,{\omega }_j^{\alpha \beta },T)$$

$$V^{\alpha \beta }$$

$$-4e\gamma \left(Z_1{r}_1+Z_2{r}_2\right),$$

$$V^{\alpha \beta }$$

$$+z_2\left[m{\left({\omega }^{\beta }\right)}^2{\bar{Z}}_2+\frac{\text{C}}{({\bar{Z}}_1-{\bar{Z}}_2{)}^2}\right]$$

$$+V_{\text{trap}}+V_{\text{DC}}+V_0^{\alpha \beta }+\mathcal{O}(z_1^n{z}_2^l),$$

$$V_{\text{trap}}$$

$$+z_2^2\left[\frac{m{\left({\omega }^{\beta }\right)}^2}{2}+\frac{\text{C}}{|{\bar{Z}}_1-{\bar{Z}}_2{|}^3}\right]-\frac{2\text{C}{z}_1{z}_2}{|{\bar{Z}}_1-{\bar{Z}}_2{|}^3}$$

$$V_{\text{DC}}$$

$$+z_2\left[-4e\gamma r_2-\frac{2\text{C}(r_1-r_2)}{|{\bar{Z}}_1-{\bar{Z}}_2{|}^3}\right]$$

$$V_0^{\alpha \beta }=\frac{m{\left({\omega }^{\alpha }\right)}^2{\bar{Z}}_1^2}{2}+\frac{m{\left({\omega }^{\beta }\right)}^2{\bar{Z}}_2^2}{2}+\frac{\text{C}}{|{\bar{Z}}_1-{\bar{Z}}_2|}$$

$$\frac{\text{C}}{|{\bar{Z}}_1-{\bar{Z}}_2{|}^3}=2e\gamma$$

$$V_{\text{DC}}\approx -4e\gamma (r_2{z}_1+r_1{z}_2)$$

$${\bar{Z}}_1^{\alpha \beta }$$

$${\bar{Z}}_2^{\alpha \beta }$$

$$Z_{\text{c}}^{\alpha \beta }$$

$$V_0^{\alpha \beta }=\frac{3}{2}{\left[\frac{m{\text{C}}^2{\left({\omega }^{\alpha }\right)}^2{\left({\omega }^{\beta }\right)}^2}{{\left({\omega }^{\alpha }\right)}^2+{\left({\omega }^{\beta }\right)}^2}\right]}^{\frac{1}{3}}$$

$$V^{\alpha \beta }$$

$${\left(J^{\alpha \beta }\right)}^2=\frac{2{\left({\omega }^{\alpha }{\omega }^{\beta }\right)}^2}{{\left({\omega }^{\alpha }\right)}^2+{\left({\omega }^{\beta }\right)}^2}$$

$$\left(\begin{array}{c}{z}_1\\ z_2\end{array}\right)=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)\left(\begin{array}{c}{q}_1\\ q_2\end{array}\right),$$

$$V$$

$$+\frac{m}{2}\left[\left({\left({\omega }^{\alpha }\right)}^2-{\left({\omega }^{\beta }\right)}^2\right)\sin 2\theta -2{\left(J^{\alpha \beta }\right)}^2\cos 2\theta \right]q_1{q}_2$$

$${\left({\omega }_j^{\alpha \beta }\right)}^2$$

$$+(J^{\alpha \beta }{)}^2\left[1+(-1{)}^j\sin (2{\theta }^{\alpha \beta })\right].$$

$${\theta }^{\alpha \beta }=\frac{\pi }{4}-\frac{1}{2}\arctan \frac{{\left({\omega }^{\alpha }\right)}^2-{\left({\omega }^{\beta }\right)}^2}{2{\left(J^{\alpha \beta }\right)}^2}.$$

$$H^{\alpha \beta }$$

$$H_{\text{p}}=\sum _{\alpha \beta =\uparrow ,\downarrow }\left(H^{\alpha \beta }\right){\Pi }^{\alpha \beta }$$

Full text

In Rydberg states of an atom a valence electron is excited to a state with a high principal quantum number, leading to extraordinary large polarizabilities (Gallagher, 1994) and making them extremely susceptible to electric fields. Such high electric field susceptibility was employed for electric field sensing Osterwalder and Merkt (1999); Facon et al. (2016); Penasa et al. (2016) and quantum information processing Haroche and Raimond (2006). For pairs, or arrays of atoms, a mutual electrical dipolar interaction of Rydberg states may lead to a blockade mechanism, which was proposed for generating entanglement Jaksch et al. (2000). Pioneering experiments realized blockade driven entanglement with pairs of Rydberg atoms in optical tweezers (Isenhower et al., 2010; Wilk et al., 2010). Lately, arrays of Rydberg atoms (Gross and Bloch, 2017) or atoms in reconfigurable optical tweezer potentials Bernien et al. (2017); Labuhn et al. (2016) have been used and allowed for remarkable progress in quantum simulation (Saffman et al., 2010).

More recently, trapped ions excited to Rydberg states Feldker et al. (2015); Higgins et al. (2017a, b) have been investigated for exploring their unique features. The large electric polarizability has been characterized by spectroscopy and electric field mapping was exploited to position a single ion precisely inside the electric trap Feldker et al. (2015); Mokhberi et al. (2019). Moreover, it has been shown that transitions to Rydberg states can be driven coherently from low-lying electronic states Higgins et al. (2019). A gate operation to entangle trapped Rydberg ions via a dipole-dipole interaction has been proposed, but requires microwave dressing of Rydberg states to cancel their polarizability Müller et al. (2008); Li et al. (2013).

Here, we propose a scheme for entangling a pair of trapped ions, where we utilize unique features of this Coulomb crystals in Rydberg states: the electric polarizability and the corresponding energy shift of Rydberg states by an impulsive electric field. We design electric field waveforms that kick the two-ion crystal and impose a state-dependent force on common modes of motion. The shuttling of the crystal Walther et al. (2012); Bowler et al. (2012) leads to a geometric phase, which can be controlled using the Rydberg principle quantum number $n$, the trap parameters and the shape of the kick. This entanglement operation is driven solely electrically and its duration may be as short as a few hundred ns, much faster than typical light-driven gates for ions Kirchmair et al. (2009); Benhelm et al. (2008); Ballance et al. (2016); Gaebler et al. (2016) and competing with gate operation times driven by pulsed laser sources Wong-Campos et al. (2017). It resembles laser-less ion entanglement operations driven by either static Khromova et al. (2012) or dynamic magnetic gradients Warring et al. (2013); Weidt et al. (2016); Harty et al. (2016) on the spin states of ions, however, driving large electric field gradients and performing strong electric kicks is an established technology in Paul traps.

In the following, we sketch the state-dependent force for a single kicked ion and fully discuss the case of a two-ion crystal. We continue with the description of an entangling operation for a two-ion crystal. Furthermore, we describe the dominating sources of imperfections and optimize the shape of the electric kick. We conclude with a feasibility study, taking into account typical experimental parameters.

Spin-dependent electric kick. We consider a single ion in a linear Paul trap, where a combination of radio-frequency and static electric fields generate a three-dimensional harmonic confinement. We are interested in the motion of the ion along the trap axis, the axis of weakest confinement, which is described by a harmonic oscillator with frequency $\omega$. Exciting an ion to a high-lying Rydberg state modifies the effective confinement due to the high polarizability Higgins et al. (2019) - one may think of modifying its effective mass - such that the trap frequency $\omega^{\alpha}$ becomes state-dependent, where $\alpha=\{\uparrow,\downarrow\}$ denotes Rydberg state or ground state, respectively. Applying an electric kick displaces the ion out of its equilibrium position, introduces contributions from the induced electric dipole force but also drives the harmonic oscillator into vibrational excitation. A state-dependent phase is accumulated, see Fig. 1. The coherently excited motion can be reduced to the initial state by properly choosing the pulse amplitude $f(t)$ and pulse duration $T$. A phase difference between Rydberg state and ground state is acquired.

Entanglement operation. For two ions we control the phase of the electronic basis states $\ket{\alpha\beta}$ = $\{\ket{\mathbin{\downarrow}\downarrow}$, $\ket{\mathbin{\downarrow}\uparrow}$, $\ket{\mathbin{\uparrow}\downarrow}$, $\ket{\mathbin{\uparrow}\uparrow}\}$. The Coulomb interaction between the ions leads to state-dependent collective frequencies $\omega_{j}^{\alpha\beta}$ with the mode index $j=1,2$ where $\alpha\beta$ denotes the internal states of both individual ions, either in ground state or Rydberg state (see suppl. information):

[TABLE]

Rydberg excitations in the ion crystal will affect the electric potential on neighboring ions, which leads to asymmetrical vibration around the center-of-mass due to a difference of effective masses Morigi and Walther (2001). The potential energy is expressed in terms of the state-dependent creation $\tilde{a}_{j}^{\dagger}=(a_{j}^{\alpha\beta})^{\dagger}$ and annihilation $\tilde{a}_{j}=a_{j}^{\alpha\beta}$ operators ($\hbar=1$).

[TABLE]

$V_{0}^{\alpha\beta}$ depends on the equilibrium positions of the ions, $\Pi^{\alpha\beta}=\ket{\alpha}_{1}\bra{\alpha}_{1}\otimes\ket{\beta}_{2}\bra{\beta}_{2}$ is the projection operator.

Fast switching of an additional electric field $f(t)$ kicks the ions out of their equilibrium positions and drives the harmonic oscillator. The interaction of the electric field with the ion crystal can be described by a state-dependent kick $F_{j}^{\alpha\beta}(t)=f(t)l_{j}^{\alpha\beta}[\cos\theta^{\alpha\beta}-(-1)^{j}\sin\theta^{\alpha\beta}]$ acting on the vibrational mode with oscillator length $l_{j}^{\alpha\beta}=\sqrt{\hbar/(2m\omega_{j}^{\alpha\beta})}$ (Cirac and Zoller, 2000; García-Ripoll et al., 2003, 2005). Specifically, for ion crystals containing Rydberg excitations, we obtain the driving Hamiltonian

[TABLE]

The second term of Eq. (3) is proportional to the crystal center $Z_{\text{c}}^{\alpha\beta}$ and only affects the phase evolution of ion crystals with one Rydberg excitation, see suppl. information. The analytic time evolution operator $U_{\text{I}}$ for the driven harmonic oscillator is obtained using a Magnus expansion (Blanes et al., 2010).

[TABLE]

The first term describes coherently generated vibrational mode excitation $A_{j}^{\alpha\beta}$ using the displacement operator $\mathcal{D}(A_{j}^{\alpha\beta})=\exp(A_{j}^{\alpha\beta}\tilde{a}_{j}+\text{h.c.})$. The total phase $\phi^{\alpha\beta}:=\varphi_{1}^{\alpha\beta}+\varphi_{2}^{\alpha\beta}+\Phi_{\text{e}}^{\alpha\beta}$ that is accumulated by each of the four basis states contains contributions from the vibrational modes and the crystal center displacement, respectively.

Assuming a constant driving $f(t)=f_{0}$ for time $t\in[0,T]$, we obtain quantities from Eq. (4):

[TABLE]

The significance of the analytical equations (5-7) is that the entanglement operation is controlled only by the kick shape ($f_{0},T$) and the common mode frequencies $\omega_{j}^{\alpha\beta}$. Therefore, arbitrary phase rotations and entanglement generation can be realized. For a controlled phase gate with two ions, we consider a phase difference $\phi^{\uparrow\uparrow}-\phi^{\downarrow\downarrow}=\pi$ while $\phi^{\downarrow\downarrow}=\phi^{\uparrow\downarrow}=\phi^{\downarrow\uparrow}$ and no residual excitation in phonon modes, thus $A_{j}^{\alpha\beta}=0$.

Case study and experimental feasibility for 40Ca+ ions. In the case study, we consider Rydberg $n$P1/2 states with a scalar polarizability $\mathcal{P}\propto n^{7}$. The state-dependent trap frequency is $\omega^{\uparrow}=\sqrt{\left(\omega^{\downarrow}\right)^{2}+\Delta\omega^{2}}$ with $\Delta\omega^{2}=-16\gamma^{2}\mathcal{P}/m$ (Li and Lesanovsky, 2012; Li et al., 2013) and $\omega^{\downarrow}=2\sqrt{e\gamma/m}$, where $e$ is the electric unit charge and $\gamma$ the field gradient of the Paul trap. The trap frequencies are modified by the interaction between the highly excited electron with the core charge of the other ion and the large field sensitivity of Rydberg ions. Note that the relative frequency differences are $\leq 10^{-4}$, such that the excitation of the center-of-mass modes dominates, with a small excitation of the stretching mode for state $\ket{\uparrow\downarrow}$. For each vibrational mode, $A_{j}^{\alpha\beta}$ is periodic and can be minimized by choosing a gate time $\tau=2\pi/\omega_{j}^{\alpha\beta}$, see Fig. 2. Taking $n=64$ and $\omega_{1}^{\uparrow\uparrow}=2\pi\cdot 0.71\,$MHz we realize a controlled phase gate at $\tau$=1.4 $\upmu$s and $3\tau=$4.2 $\upmu$s (indicated by black arrows) with mitigated coherent excitation in the center-of-mass mode for state $\ket{\mathbin{\uparrow}\uparrow}$ and correct relative phases.

Varying the vibrational frequency by the field gradient of the Paul trap, the relative phase of $\pi$ between states $\ket{\mathbin{\uparrow}\uparrow}$ and $\ket{\mathbin{\uparrow}\downarrow}$ is modified, see Fig. 3(a). With a specific combination of electric kick time and amplitude, we achieve a phase difference $\phi^{\uparrow\uparrow}-\phi^{\downarrow\downarrow}=\pi$, and minimize coherent excitation of modes. Rydberg states with higher principal quantum number and larger polarizability require smaller electric kick amplitude $f_{0}$ to acomplish the desired phase evolution.

To characterize the entanglement operation, we analyze the state fidelity $F$ as the square of the overlap between a superposition $\ket{\Psi(0)}=1/2\left[(\ket{\downarrow}+\ket{\uparrow})\otimes(\ket{\downarrow}+\ket{\uparrow})\right]$, initialized in the motional ground state and evolved under Eq. (4), with the ideal target state. Fulfilling the phase conditions as indicated by the vertical dashed lines, the fidelity is limited by residual phonons due to the chosen electric kick. Optimizing the kick strength, kick duration and the ion confinement, a fidelity of $99.9\%$ can be achieved see Fig. 3(b). For the 36P state the required electric kick strength is $E(t)=\frac{\hbar}{e}f(t)=28.75\,$V/m with a field gradient of the Paul trap of $\gamma=1.32\cdot 10^{6}\,$V/m2 and a vibrational mode frequency of $\omega_{1}^{\uparrow\uparrow}=2\pi\cdot 0.57\,$MHz, experimentally feasible with trapped Rydberg ions Feldker et al. (2015); Higgins et al. (2017a). Thereby, the ion crystal would be displaced along the trap axis by $10.9\,\upmu$m for a total operation time of $\tau$ = 1.76 $\upmu$s Walther et al. (2012); Bowler et al. (2012). The method comes with the advantage, that effects due to micromotion are mitigated, as the ion crystal moves along the trap axis. In principle, one might also employ a combination of axial and radial displacements for the gate, however, this will require synchronizing the electric kicking and oscillating radio-frequency field for the Paul trap as experimentally demonstrated in Ref.  Jacob et al. (2016). The parasitic Stark shift from the applied electric field is estimated to be about four orders of magnitude smaller as compared to the separation of Rydberg energy levels, thus we do not anticipate state mixing and fidelity reduction.

Lifetime limitation and optimized kicks. A significant reduction of fidelity will arise from the finite Rydberg state lifetime, about 30$\,\upmu$s to 100$\,\upmu$s Glukhov et al. (2013), see dashed lines in Fig. 4(a). Operation times above 1 $\upmu$s limit the fidelity to $90\%-99\%$ depending on the Rydberg state. However, a bang-bang interaction by three consecutive kicks $f(t)=\{f_{0},-f_{0},f_{0}\}$ at times $t=\{0,T/4,3T/4\}$ leads to a fidelity of $99.9\%$ , see Fig. 4(a-c). We emphasize the importance of multi-kick sequences, in this example composed out of three kicks, as compared to the single constant pulse, see Fig. 4(c). Note, that electric bang-bang control of single ions has been demonstrated with up to 10000 phonons and displacement pulses of sub-ns resolution (Alonso et al., 2016), experimental parameters that even exceed the requirements for our proposed operation. Aiming for faster operations with higher fidelity, we will explore more complex phase trajectories of the wavepacket. The additional benefit of such schemes is a robustness against imperfections of the driving kick waveform. Such approaches have been discussed, however, in the context of the laser-driven Mølmer-Sørensen interaction Schäfer et al. (2018); Shapira et al. (2018) and might readily be adapted to our electric scheme. Alternatively optimal control theory may deliver optimized electric field waveforms (Caneva et al., 2011; Fürst et al., 2014; Rach et al., 2015).

Conclusion and outlook. In this work, we proposed a new scheme for fast entanglement operations based on electric kicks applied to trapped Rydberg ions in a linear Paul trap. Crucial to the successful implementation of this scheme is the high polarizability of $n$P Rydberg states that leads to a modification of the ion confinement and state-dependent vibrational modes. By tuning the field gradient of the Paul trap and shaping the electric kick, we optimize the scheme for entanglement operations of two ions. The parameter values required are well within regimes accessible by state-of-the-art experiments.

In future we may extend the scheme to linear ion crystals entangling more than two ions, or investigate spin-spin interactions in two-dimensional ion crystals Nath et al. (2015) by state-dependent electric forces. In this context, we will study operations at finite temperature of the ion crystal, motional dephasing and heating by electric noise. The presented scheme may be adapted to the platform of neutral Rydberg atoms trapped in arrays of optical tweezers Bernien et al. (2017); Labuhn et al. (2016); Keesling et al. (2019). A set of common motions, analogous to the normal modes of vibration for the ion crystal, is established by the dipole-dipole interaction. State dependent forces between different Rydberg states can be implemented by a fast shuttling of the tweezer centers Ohl de Mello et al. (2019) such that the trapped Rydberg atoms explore the AC-Stark shift from the tweezer potential, similar as the ion crystal explores the axial kick via its polarizability. In the array of Rydberg atoms the emerging collective energy-shifts may then be exploited to generate entanglement. We believe that experimental and theoretical work building on our ideas will be of relevance across a broad set of fields, such as multi-particle quantum systems with collective spin-motion coupling, quantum simulation and quantum information.

We thank Rene Gerritsma for comments. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) within the SPP 1929 Giant interactions in Rydberg Systems (GiRyd), the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) [ERC Grant Agreement No. 335266 (ESCQUMA)], within QuantERA by ERyQSenS, and by the EPSRC [Grant No. EP/R04340X/1]. A. M. acknowledges funding by Marie-Skłodowska-Curie grant agreement No. 796866 (Rydion). I. L. gratefully acknowledges funding through the Royal Society Wolfson Research Merit Award.

Appendix A Supplementary information

A.1 Equilibrium positions and static potential energy

The trap potential of two ions with state-dependent trap frequencies $\omega^{\alpha}$ with $\alpha=\{\uparrow,\downarrow\}$ resembling two possible internal states, reads

[TABLE]

where $Z_{j}$ is the position of the $j$-th ion and $\text{C}=e^{2}/4\pi\epsilon_{0}$ is the Coulomb constant with $e$ the elementary charge, $\epsilon_{0}$ the vacuum permittivity, $\gamma$ the trap field gradient and $r_{j}$ the relative coordinate of the outer electron of each ion. We have introduced the interaction between the excited electron of a Rydberg state with the ion core by the last term. For small vibrations $z_{j}$ of the ionic core around the equilibrium position $\bar{Z}_{j}$, we expand the potential

[TABLE]

where we kept up to quadratic expansions in terms of small quantities $r_{j}$ and $z_{j}$. $V_{0}^{\alpha\beta}$ gives a static potential, which depends on the equilibrium positions,

[TABLE]

As the change in equilibrium positions by the additional static potential $V_{\text{DC}}$ is typically small, we calculate the ion distances between two ground state ions:

[TABLE]

and obtain:

[TABLE]

We identify this term as a cross-coupling between the negatively singly charged electron of one ion with the positively double charged core of the other ion. For two ions in the Rydberg state both ions see an additional trap potential. If only one ion is in the Rydberg state, the neighboring ground state ion will see a modified trap potential. The term $V_{\text{DC}}$ describes the electron dynamics, which is typically much faster than the ion core dynamics, and can be treated via second order perturbation. Thereby, we obtain modified trap frequencies for Rydberg ions compared to ground state ions (see main text). From the force balance condition (linear orders) the state-dependent equilibrium positions are obtained

[TABLE]

and we find the geometric center of the two ions:

[TABLE]

With the equilibrium positions, we can calculate the static potential

[TABLE]

We should note that the state-dependent difference of the static potential $V_{0}^{\alpha\beta}$ will be canceled by proper laser detuning. This means that the laser frequency for excitation of a single ion and two ions to the Rydberg state will be different.

A.2 Phonon modes

Using the equilibrium positions, we obtain

[TABLE]

with

[TABLE]

We introduce a transformation, which mixes position coordinates by an angle $\theta$:

[TABLE]

and the potential becomes

[TABLE]

with

[TABLE]

The potential is diagonal when the mixing angle becomes

[TABLE]

At low temperatures, the vibrations are described by vibrational quanta acting on the collective coordinates $q_{j}=l_{j}^{\alpha\beta}\left(\tilde{a}_{j}^{\dagger}+\tilde{a}_{j}\right)$ with the oscillator length $l_{j}^{\alpha\beta}=\sqrt{\hbar/\left(2m\omega_{j}^{\alpha\beta}\right)}$ and the phonon Hamiltonian is expressed in terms of the state-depending creation $\tilde{a}_{j}^{\dagger}=(a_{j}^{\alpha\beta})^{\dagger}$ and annihilation $\tilde{a}_{j}=a_{j}^{\alpha\beta}$ operators ($\hbar=1$).

[TABLE]

Summing up all basis states we obtain the phonon Hamiltonian of the full system:

[TABLE]

with the state projection operator

[TABLE]

A.3 Electric kick

The Hamiltonian for the fast electric pulse driving the ions is given by

[TABLE]

The driving Hamiltonian is given by:

[TABLE]

A.4 Time evolution operator

We obtain the interaction Hamiltonian by:

[TABLE]

As the Hamiltonian in the interaction picture is time dependent we use a Magnus expansion for time ordered systems:

[TABLE]

Using the commutator relations $\left[\tilde{a}_{i},\tilde{a}_{j}^{\dagger}\right]=\delta_{ij}$ and $\left[\tilde{a}_{i},\tilde{a}_{j}\right]=\left[\tilde{a}_{i}^{\dagger},\tilde{a}_{j}^{\dagger}\right]=0$ of creation and annihilation operators we find

[TABLE]

The complete time evolution operator is thereby:

[TABLE]

with

[TABLE]

A.5 Gate fidelity

We analyze the time evolution of the electronic basis states $\ket{\alpha\beta}$ = $\{\ket{\mathbin{\downarrow}\downarrow}$, $\ket{\mathbin{\downarrow}\uparrow}$, $\ket{\mathbin{\uparrow}\downarrow}$, $\ket{\mathbin{\uparrow}\uparrow}\}$. As the states $\ket{\mathbin{\downarrow}\uparrow}$ and $\ket{\mathbin{\uparrow}\downarrow}$ are symmetric, we only consider state $\ket{\mathbin{\uparrow}\downarrow}$. An ideal controlled phase gate has the evolution operator

[TABLE]

The state fidelity without lifetime limit is defined as the overlap $F=|\bra{\Psi(0)}U_{\text{CP}}U_{I}(T)\ket{\Psi(0)}|^{2}$, which we evaluate here explicitly for the two ion superposition state $\ket{\Psi(0)}=1/2\left[(\ket{\downarrow}+\ket{\uparrow})\otimes(\ket{\downarrow}+\ket{\uparrow})\right]$ with the ion crystal initially in the motional ground state. For a constant driving field we obtain:

[TABLE]