Laser-free trapped ion entangling gates with AESE: adiabatic elimination of spin-motion entanglement

We can factor $\hat{U}_\mathrm{g}$ into two unitaries that represent the 1st and 2nd order terms of the time propagator because the two terms commute. We therefore consider the 1st order terms in the Magnus expansion separately:

$$\begin{eqnarray} &&\hat{U}_{1} = \exp\left(-i\Omega_{\mathrm{g}}^{^{\prime}}\hat{S}_{z}\int^{t_\mathrm{g}}_{0}\mathrm{d}t^{^{\prime}}\gamma\left(t^{^{\prime}}\right) \left[\hat{a}^{\dagger}\mathrm{e}^{\mathrm{i}\delta t^{^{\prime}}} + \hat{a}\mathrm{e}^{-\mathrm{i}\delta t^{^{\prime}}}\right]\right), \end{eqnarray} \tag{ 4 }$$

where $\delta\equiv \omega\pm\omega_{\mathrm{g}}$ represents the field detuning from a 'gating' motional mode and $\Omega_{\mathrm{g}}^{^{\prime}}\equiv \Omega_{\mathrm{g}}/2$ or $\Omega_{\mathrm{g}}^{^{\prime}}\equiv\Omega_{\mathrm{g}}$, depending on if the gradient is DC ($\omega_{\mathrm{g}} = 0$) or RF ($\omega_{\mathrm{g}}\neq 0$), respectively. We can simplify this equation via integration by parts:

$$\begin{align} \hat{U}_{1} & = \exp\!\left(\frac{\Omega_{\mathrm{g}}^{^{\prime}}}{\delta}\hat{S}_{z}\left[\int^{\tau}_{0}\!\!\!\mathrm{d}t^{^{\prime}}\dot{\gamma}\left(t^{^{\prime}}\right)\left(\hat{a}^{\dagger}\mathrm{e}^{\mathrm{i}\delta t^{^{\prime}}} - \hat{a}\mathrm{e}^{-\mathrm{i}\delta t^{^{\prime}}}\right)\!+ \!\!\int^{t_\mathrm{g}}_{t_\mathrm{g}-\tau}\!\!\!\!\!\!\!\!\mathrm{d}t^{^{\prime}}\dot{\gamma}\left(t^{^{\prime}}\right)\left(\hat{a}^{\dagger}\mathrm{e}^{\mathrm{i}\delta t^{^{\prime}}} - \hat{a}\mathrm{e}^{-\mathrm{i}\delta t^{^{\prime}}}\right)\right] \right) \nonumber \\ & = \exp\!\left(\frac{i\Omega_{\mathrm{g}}^{^{\prime}}}{\delta^{2}}\hat{S}_{z}\left[\int^{\tau}_{0}\!\!\!\mathrm{d}t^{^{\prime}}\ddot{\gamma}\left(t^{^{\prime}}\right)\left(\hat{a}^{\dagger}\mathrm{e}^{\mathrm{i}\delta t^{^{\prime}}} + \hat{a}\mathrm{e}^{-\mathrm{i}\delta t^{^{\prime}}}\right) \!+ \!\!\int^{t_\mathrm{g}}_{t_\mathrm{g}-\tau}\!\!\!\!\!\!\!\!\mathrm{d}t^{^{\prime}}\ddot{\gamma}\left(t^{^{\prime}}\right)\left(\hat{a}^{\dagger}\mathrm{e}^{\mathrm{i}\delta t^{^{\prime}}} + \hat{a}\mathrm{e}^{-\mathrm{i}\delta t^{^{\prime}}}\right)\right] \right), \end{align} \tag{ 5 }$$

where we have assumed that $\gamma(0) = \dot{\gamma}(0) = \dot{\gamma}(\tau) = \dot{\gamma}(t_\mathrm{g}-\tau) = \gamma(t_\mathrm{g}) = \dot{\gamma}(t_\mathrm{g}) = 0$, and $\dot{\gamma}(t) = 0$ when $t\in[\tau,t_\mathrm{g}-\tau]$. Under these assumptions, the off-resonant couplings in equation (5) will be suppressed by pulse shaping $\gamma(t)$, similar to the way this is done to suppress off-resonant Rabi flopping on the carrier transition [27, 39]. In general, the magnitude of $\Omega_{\mathrm{g}}^{^{\prime}}\delta^{-2}\int^{t}_{t_{0}}\ddot\gamma(t)\mathrm{e}^{\pm \mathrm{i}\delta t}~\propto~\Omega_{\mathrm{g}}^{^{\prime}}/(\delta^{3}\tau^{2})$, meaning the infidelity from residual spin-motion entanglement (per mode) will scale like $\Omega_{\mathrm{g}}^{^{\prime} 2}/(\delta^{6}\tau^{4})$. Choosing these properties of $\gamma(t)$ AESEs, suppressing its effect by roughly $(\delta\tau)^4$ compared to when $\tau\rightarrow 0$ and $\gamma(t)\rightarrow 1$. The ions must complete several phase space loops over $t_\mathrm{g}$, but this additional suppression marks a crucial distinction between using AESE and simply operating in the 'far-detuned' limit; the latter would require a significantly larger δ to achieve the same degree of spin-motion disentanglement, resulting in a much slower gate in comparison. We can factor $\hat{U}_{1}$ from the (assumed to be ideal) gate $\hat{U}_{2} = \hat{U}_\mathrm{g}$, and evaluate the infidelity of a pair of qubits in the state $\vert \psi(0)\rangle$, initialized to the nth Fock state:

$$\begin{eqnarray} \mathcal{I}_{n} & = & 1-\sum_{n^{^{\prime}}}|\langle \psi\left(0\right)\vert\langle n^{^{\prime}}\vert\hat{U}_{\mathrm{1}}\vert \psi\left(0\right)\rangle\vert n\rangle|^{2}. \end{eqnarray} \tag{ 6 }$$

For the specific examples in this work, we set $\gamma(t)\equiv \sin^{2}(\pi t/\tau)$. Plugging this expression into equation (5), we get a spin-dependent displacement operator:

$$\begin{eqnarray} \hat{U}_{1} & = & \exp\left(\hat{S}_{z}\frac{-i\pi^{2}\Omega_{\mathrm{g}}^{^{\prime}}}{\delta\left(\pi^{2}-\tau^{2}\delta^{2}\right)}\sin\left[\delta\left[t_\mathrm{g}-\tau\right]\right]\left[\hat{a}^{\dagger}\mathrm{e}^{\mathrm{i}\delta t_\mathrm{g}/2} + \hat{a}\mathrm{e}^{-\mathrm{i}\delta t_\mathrm{g}/2}\right] \right). \end{eqnarray} \tag{ 7 }$$

Inserting this into equation (6), assuming the limit $\tau\delta \gg 1$ has been reached, the time-averaged integrals are:

$$\begin{align} \mathcal{I}_{n} &\simeq \frac{\pi^{4}\Omega_{\mathrm{g}}^{^{\prime} 2}}{2\tau^{4}\delta^{6}}\left(2n+1\right)\lambda_{\hat{S}_{z}}^{2} \nonumber \\ & = \frac{4\pi^{4}\Omega_{\mathrm{g}}^{^{\prime} 2}}{5\tau^{4}\delta^{6}}\left(2n+1\right) \end{align} \tag{ 8 }$$

where $\lambda_{\hat{S}_{z}}^{2}$ is the variance of $\hat{S}_{z}$ at t = 0, which we take to be $8/5$ in the second line upon averaging over $\mathrm{SU}(4)$.

Traditionally, a geometric phase gate ensures spin-motion disentanglement by setting $t_\mathrm{g}$ to be an integer multiple of $2\pi/\delta$. With AESE, however, we reach the same end by operating in a parametric limit, independent of set relationship between δ and $t_\mathrm{g}$. One immediate result of this is that the mode frequencies can fluctuate significantly without inducing residual spin-motion entanglement, negating the need to increase experimental complexity with dynamical decoupling sequences [11, 25, 29, 35, 40] and reducing temperature requirements (see section 5.2). Further, the gradient can now generate spin-motion coupling with many modes at once without the need for numerically optimized pulse sequences that ensure each qubit becomes disentangled with each mode at $t_\mathrm{g}$ [30, 31, 41].

Similar to how shaping $\gamma(t)$ suppresses off-resonant 1st order transitions in the $\delta\tau\gg 1$ limit, it also suppresses off-resonant 2nd order transitions. To show this, we note equation (1) comprises a linear sum of terms, each with a time dependence $\propto~\gamma(t)\mathrm{e}^{\pm \mathrm{i}\delta t}$ for some angular frequency $\delta \equiv \omega\pm \omega_\mathrm{g}$. We assume $|\delta|\tau\gg 1$ and that $|\delta^{^{\prime}}-\delta^{^{\prime\prime}}|\tau\gg 1$, where the primes indicate different sidebands. Because of its $\propto [\hat{H}_\mathrm{g}(t^{^{\prime}}),\hat{H}_\mathrm{g}(t^{^{\prime\prime}})]$ dependence, the exponent of $\hat{U}_{2}$ is a linear sum of terms that are proportional to the time integral:

$$\begin{align} \beta\left(t_\mathrm{g}\right) & = \int^{t_\mathrm{g}}_{0}\int^{t^{\prime}}_{0}\mathrm{d}t^{\prime}\mathrm{d}t^{\prime\prime}\gamma\left(t^{\prime}\right)\gamma\left(t^{\prime\prime}\right)\sin\left(\delta^{\prime}t^{\prime}+\delta^{\prime\prime}t^{\prime\prime}\right) \nonumber \\ & = -\int^{t_\mathrm{g}}_{0}\mathrm{d}t^{\prime}\gamma\left(t^{\prime}\right)\left\{\frac{\gamma\left(t^{\prime}\right)}{\delta^{\prime\prime}}\cos\left(\left[\delta^{\prime}+\delta^{\prime\prime}\right]t^{\prime}\right)-\frac{1}{\delta^{\prime\prime}}\int^{t^{\prime}}_{0}\mathrm{d}t^{\prime\prime}\dot{\gamma}\left(t^{\prime\prime}\right)\cos\left(\delta^{\prime}t^{\prime}+\delta^{\prime\prime}t^{\prime\prime}\right)\right\},\nonumber \\ & = -\int^{t_\mathrm{g}}_{0}\mathrm{d}t^{\prime}\gamma\left(t^{\prime}\right)\Bigg\{\frac{\gamma\left(t^{\prime}\right)}{\delta^{\prime\prime}}\cos\left(\left[\delta^{\prime}+\delta^{\prime\prime}\right]t^{\prime}\right) - \frac{\dot{\gamma}\left(t^{\prime}\right)}{\delta^{^{\prime\prime} 2}}\sin\left(\left[\delta^{\prime}+\delta^{\prime\prime}\right]t^{\prime} \right) \nonumber \\ & \quad + \frac{1}{\delta^{^{\prime\prime} 2}}\int^{t^{\prime}}_{0}\mathrm{d}t^{\prime\prime}\ddot\gamma\left(t^{\prime\prime}\right)\sin\left(\delta^{\prime} t^{\prime}+\delta^{\prime\prime}t^{\prime\prime} \right)\Bigg\} \end{align} \tag{ 9 }$$

where we have integrated by parts and used the fact that $\gamma(0) = \dot{\gamma}(0) = 0$. When $|\delta^{^{\prime}}-\delta^{^{\prime\prime}}|\tau \gg 1$, every term in this equation scales $\propto~ \delta^{^{\prime\prime} 2}\tau$, their effect on the gate fidelity (again) scaling $\propto ~\Omega_{\mathrm{g}}^{^{\prime} 2}/(\delta^{6}\tau^{4})$. For this reason, we assume all terms such that $\delta^{^{\prime}}\neq \delta^{^{\prime\prime}}$ are suppressed, which gives:

$$\begin{align} \beta\left(t_\mathrm{g}\right) & = -\frac{1}{\delta^{^{\prime}}}\int^{t_\mathrm{g}}_{0}\mathrm{d}t^{^{\prime}}\gamma^{2}\left(t^{^{\prime}}\right) \nonumber \\ \beta\left(t_\mathrm{g}\right)& = -\frac{t_\mathrm{g}\langle \gamma^{2}\rangle}{\delta^{^{\prime}}}, \end{align} \tag{ 10 }$$

where $\langle \gamma^{2}\rangle$ is the time averaged value of $\gamma^{2}(t)$. Every term in $\hat{U}_{2}$ is an integral similar to equation (9). This, when considered with the arguments of section 2.1, allows us to analytically express the time propagator for the gate. For systems driven by DC ($\omega_\mathrm{g} = 0$) gradients this gives:

$$\begin{eqnarray} &&\hat{U}_\mathrm{g} \simeq \exp\left(i\langle \gamma^{2}\rangle t_\mathrm{g}\sum_{j}\frac{\Omega_{j}^{2}}{\delta_{j}}\hat{S}_{z,j}^{2} \right), \end{eqnarray} \tag{ 11 }$$

and for RF gradients ($\omega_\mathrm{g}\neq 0$), we get:

$$\begin{eqnarray} &&\hat{U}_\mathrm{g} \simeq \exp\left(\frac{i}{2}\langle \gamma^{2}\rangle t_\mathrm{g}\sum_{j}\frac{\omega_{j}\Omega_{j}^{2}}{\omega_{j}^{2}-\omega_\mathrm{g}^{2}}\hat{S}_{z,j}^{2} \right). \end{eqnarray} \tag{ 12 }$$

The results of this section apply to mixed-species as well as same-species crystals. This indicates AESE could be used to entangle two different ion species in a quantum logic spectroscopy experiment [42], or two same species ions in the presence of one or more sympathetic cooling ions. If we assume the sympathetic cooling ions are initialized to a specific internal state, equation (12) indicates their effect will be to add a $\propto~ \hat{\sigma}_{z,j}$ shift to each ion j. Since these are single qubit shifts that commute with $\hat{U}_\mathrm{g}$, they can be tracked or coherently canceled with a spin-echo (a spin echo has the added benefit that it would remove the impact of spectator ions even in their state preparation is not perfect).

For DC gradients, the effective detunings from the gating modes are the mode frequencies themselves, reducing equation (1) to:

$$\begin{eqnarray} &&\hat{H}_{\mathrm{DC}} = \hbar\gamma\left(t\right)\left(\hat{S}_{z,\mathrm{c}}\Omega_\mathrm{c}\left[\hat{a}_\mathrm{c}^{\dagger}\mathrm{e}^{\mathrm{i}\omega_\mathrm{c}t} +\hat{a}_\mathrm{c}\mathrm{e}^{-\mathrm{i}\omega_\mathrm{c}t}\right]+\hat{S}_{z,\mathrm{s}}\Omega_\mathrm{s}\left[\hat{a}_\mathrm{s}^{\dagger}\mathrm{e}^{\mathrm{i}\omega_\mathrm{s}t} +\hat{a}_\mathrm{s}\mathrm{e}^{-\mathrm{i}\omega_\mathrm{s}t}\right]\right). \end{eqnarray} \tag{ 13 }$$

We AESE the system by choosing parameters such that $2\pi/\omega_{\alpha}\ll \tau$. When this condition is met, we can write down the propagator for the system:

$$\begin{align} \hat{U}_{\mathrm{DC}} & \simeq \exp\left(i\langle \gamma^{2}\rangle t_\mathrm{g}\left[\frac{\Omega_\mathrm{c}^{2}}{\omega_\mathrm{c}} \hat{S}_{z,\mathrm{c}}^{2} + \frac{\Omega_\mathrm{s}^{2}}{\omega_\mathrm{s}}\hat{S}_{z,\mathrm{s}}^{2}\right]\right) \nonumber \\ & = \exp\left(-i\Omega_{\mathrm{DC}}t_\mathrm{g}\hat{\sigma}_{z,1}\hat{\sigma}_{z,2}\right), \end{align} \tag{ 14 }$$

giving an effective gate frequency of:

$$\begin{align} \Omega_{\mathrm{DC}}\equiv \frac{3}{4}\left(\frac{\Omega_\mathrm{s}^{2}}{\omega_\mathrm{s}}-\frac{\Omega_\mathrm{c}^{2}}{\omega_\mathrm{c}}\right), \end{align} \tag{ 15 }$$

where we have substituted $\langle \gamma^{2}\rangle = 3/8$. For this case, the entanglement generated by the COM mode coherently cancels the entanglement generated by the STR mode. Noting that for a given gradient $\Omega_{\alpha}^{2}~\propto ~ 1/\omega_{\alpha}$, which means $\Omega_{\mathrm{DC}}~\propto~ \omega_\mathrm{s}^{-2}-\omega_\mathrm{c}^{-2}$, we see that performing the DC version of the scheme in a regime where $\omega_\mathrm{s}$ is at least a factor of two or three smaller than $\omega_\mathrm{c}$ is desirable; for example, if $\omega_\mathrm{s}/2\pi = 1\,\mathrm{MHz}$, $\Omega_{\mathrm{DC}}$ is $8/9$ of what it would be in the absence of the COM mode. Further, the STR mode's symmetry typically makes its heating rate negligible compared to the COM mode [11, 23] potentially allowing operation in low frequency (LF) regimes that have larger Lamb–Dicke factors, and adding another benefit to performing the gate in a regime where $\omega_\mathrm{s}$ is small compared to $\omega_\mathrm{c}$, i.e. near the 'zig-zag' instability.

When the gradient generating current oscillates at RF frequencies ($\omega_\mathrm{g}\sim \mathrm{MHz}$) we can choose the gradient's detuning $\delta_{\alpha}\equiv \pm(\omega-\omega_\mathrm{g})$ from at least one of the gating modes—a technique used in almost every geometric phase gate scheme. This allows us to enhance the gate speed and choose the primary gating mode we entangle the spins with, the latter being advantageous because it allows us to operate on modes that heat slowly due to symmetry (see section 5.1 for more details).

The Hamiltonian for this system is given by:

$$\begin{eqnarray} &&\hat{H}_{\mathrm{RF}} = \hbar\gamma\left(t\right)\cos\left(\omega_\mathrm{g}t\right)\left(\hat{S}_{z,\mathrm{c}}\Omega_\mathrm{c}\left[\hat{a}_\mathrm{c}^{\dagger}\mathrm{e}^{\mathrm{i}\omega_\mathrm{c}t} +\hat{a}_\mathrm{c}\mathrm{e}^{-\mathrm{i}\omega_\mathrm{c}t}\right]+\hat{S}_{z,\mathrm{s}}\Omega_\mathrm{s}\left[\hat{a}_\mathrm{s}^{\dagger}\mathrm{e}^{\mathrm{i}\omega_\mathrm{s}t} +\hat{a}_\mathrm{s}\mathrm{e}^{-\mathrm{i}\omega_\mathrm{s}t}\right]\right), \end{eqnarray} \tag{ 16 }$$

which gives an effective gate speed of:

$$\begin{eqnarray} &&\Omega_{\mathrm{RF}} = \frac{3}{16}\Big|\frac{\omega_\mathrm{s}\Omega_\mathrm{s}^{2}}{\omega_\mathrm{s}^{2}-\omega_\mathrm{g}^{2}} - \frac{\omega_\mathrm{c}\Omega_\mathrm{c}^{2}}{\omega_\mathrm{c}^{2}-\omega_\mathrm{g}^{2}}\Big|. \end{eqnarray} \tag{ 17 }$$

While splitting the gradient into $\pm\omega_\mathrm{g}$ frequency components ostensibly results in an immediate factor of 4 reduction in speed, we can more than compensate for this reduction by tuning $\omega_\mathrm{g}$ such that $\omega_\mathrm{s}/|\omega_\mathrm{s}^{2}-\omega_\mathrm{g}^{2}|\gg 1/\omega_\mathrm{s}$. Further, we can also operate with an $\omega_\mathrm{g}$ that is blue detuned from $\omega_\mathrm{s}$, i.e. $\omega_\mathrm{g}\gt\omega_\mathrm{s}$. The ability to control the sign of δ_α allows us to ensure the entanglement generated by each mode coherently adds instead of cancels. Therefore, using an RF gradient not only lets us further enhance the speed of the gate, but also relax the restriction that we operate in a regime where the gating mode frequencies are significantly split; this reduces the speed penalty for operating in a regime where $\omega_\mathrm{s}\sim \omega_\mathrm{c}$.

In figure 1, we provide numerical calculations of the RF version of this scheme's infidelity, showing how AESE eliminates errors from residual spin-motion entanglement—even without ground-state cooling—for realistic experimental parameters. In each example, we pulse shape over the entire interaction, setting $\gamma(t) = \sin^{2}(\pi t/t_\mathrm{g})$ (i.e. $\tau = t_\mathrm{g}/2$). This allows us to minimize the value of $\delta_\mathrm{s}$ needed for AESE at a given $t_\mathrm{g}$. For a target qubit state $\vert T\rangle$, the infidelity is:

$$\begin{eqnarray} &&\mathcal{I}_{n} = 1-\sum_{n^{^{\prime}}}|\langle T\vert\langle n^{^{\prime}}|\psi\left(t_\mathrm{g}\right)\rangle\vert n\rangle|^{2}, \end{eqnarray} \tag{ 18 }$$

where we are tracing over the motion, and assuming a system initialized to $\vert \psi(0)\rangle\vert n\rangle$. We increase the value of $t_\mathrm{g}$ by increasing the value of $\omega_\mathrm{g}$. Increasing $\omega_\mathrm{g}$ increases the magnitude of $\delta_\mathrm{s}$ and τ, both of which parametrically suppress residual spin-motion entanglement; this indicates a clear trade-off between gate speed and motional insensitivity, the optimal parameters depending on the specific situation. In figure 1 we take the relevant gating modes to be the radial COM $\omega_\mathrm{c}/2\pi = 3\,\mathrm{MHz}$, and the radial STR, where $\omega_\mathrm{s}/2\pi = 250\,\mathrm{kHz}$ (green left plot) and $\omega_\mathrm{s}/2\pi = 1\,\mathrm{MHz}$ (red center and blue right plots). The green and red plots use the same gradient strength, differing only in the stretch mode frequency, highlighting that one can achieve a given error with a faster gate by operating close to the zig-zag instability (due to the $\propto~\omega_\mathrm{s}^{-1/2}$ dependence of $\Omega_\mathrm{s}$). We also show the dependence of $\mathcal{I}_{n}(t_\mathrm{g})$ on the size of the magnetic field gradient, using both $300\,\mathrm{T\,m}^{-1}$ (green left, red center) and 150 T m⁻¹ (blue right). Further, we calculate $\mathcal{I}_{n}$ for systems initialized to the ground state of motion n = 0 (solid), and the state where n = 10 (dashed) to model the gate when the crystal is ground state cooled and at Doppler temperatures, respectively. We can see that both n = 0 and n = 10 periodically converge to one another when $\hat{U}_{1}\rightarrow \hat{I}$, while both plots continue to improve with $t_\mathrm{g}$ as the off-resonant terms in $\hat{U}_{2}$ are AESEd. Figure 1 demonstrates that AESE should enable high fidelity gates even at the Doppler temperature, and while achieving relatively small values of $t_\mathrm{g}$.

Zoom In Zoom Out Reset image size

We now consider a system evolving under equation (19) with an additional Rabi transition term, which will map the system between the $\vert 1\mathrm{z}\rangle$ and $\vert 1\mathrm{c}\rangle$ states; one can do this with either lasers or microwaves. The full gating Hamiltonian is then:

$$\begin{eqnarray} &&\hat{H}_{\mathrm{r}} = 2\hbar\Omega_{z}\hat{P}_{z}\left(\hat{a}^{\dagger}\mathrm{e}^{\mathrm{i}\omega t} + \hat{a}\mathrm{e}^{-\mathrm{i}\omega t}\right) + \frac{\hbar\Omega_{\mathrm{R}}\left(t\right)}{2}\hat{S}_{x}^{\mathrm{cz}}, \end{eqnarray} \tag{ 23 }$$

where we have introduced $\Omega_{\mathrm{R}}(t)$ to represent the time-dependent Rabi frequency. Here, $\Omega_{\mathrm{R}}(t)$ is constrained such that it is only non-zero during the shelving times τ, and that $\int^{t_{0}+\tau}_{t_{0}}\mathrm{d}t^{^{\prime}}\Omega_{\mathrm{R}}(t^{^{\prime}}) = \pm\pi$ for each shelving period. Transforming into the interaction picture with respect to this term gives equation (21) where:

$$\begin{eqnarray} &&\phi\left(t\right) = \frac{1}{2}\int^{t}_{0}\mathrm{d}t^{^{\prime}}\Omega_{\mathrm{R}}\left(t^{^{\prime}}\right), \end{eqnarray} \tag{ 24 }$$

showing the desired ramping of the effective spin-motion coupling of the system. If we choose $\Omega_{\mathrm{R}}(t)$ such that it has the opposite sign when shelving from $\vert 1\mathrm{z}\rangle\rightarrow\vert 1\mathrm{c}\rangle$ compared to when shelving from $\vert 1\mathrm{z}\rangle\leftarrow\vert 1\mathrm{c}\rangle$, then $\phi(t_\mathrm{g}) = 0$, and the state of the interaction picture wavefunction corresponds exactly to that of the lab frame wavefunction [27]. Therefore, we can safely think of the dynamics of the system as resulting from equation (21). If we assume the initial state is within the clock state manifold and ignore the term $\propto~ \hat{S}_{y}^\textrm{cz}$, this Hamiltonian implements an entangling gate with AESE.

In figure 2 we explore the efficacy of this procedure for a sequence such that $\Omega_{\mathrm{R}}(t) = 2\pi \sin^{2}(\pi t/\tau)/\tau$ when $t\lt\tau$ and $\Omega_{\mathrm{R}}(t) = -2\pi \sin^{2}(\pi t/\tau)/\tau$ when $t\gt t_\mathrm{g}-\tau$, including the effects of $H_\textrm{e}$. We choose a smooth envelope (as opposed to a square one) for $\Omega_{\mathrm{R}}(t)$ in order to suppress off-resonant 1st order effects from $\hat{H}_{\mathrm{e}}$ by ensuring $\dot{\hat{H}}_{\mathrm{e}}(0) = \dot{\hat{H}}_{\mathrm{e}}(\tau) = \dot{\hat{H}}_{\mathrm{e}}(t_\mathrm{g}-\tau) = \dot{\hat{H}}_{\mathrm{e}}(t_\mathrm{g}) = 0$. In the figure, we simulate chains of two $^{40}\mathrm{Ca}^{+}$ ions experiencing a $50\,\mathrm{T\,m}^{-1}$ gradient. We show gate infidelity versus τ for mode frequencies of $\omega_\mathrm{s}/2\pi = 250\,\mathrm{kHz}$ ($t_\mathrm{g}\simeq 170\,\mu\mathrm{s}$), $\omega/2\pi = 500\,\mathrm{kHz}$ ($t_\mathrm{g}\simeq 650\,\mu\mathrm{s}$), and $\omega/2\pi = 1\,\mathrm{MHz}$ ($t_\mathrm{g}\simeq 2.5\,\mathrm{ms}$). In each plot, we can see that, initially, increasing τ decreases $\mathcal{I}_{n}$ because we are reaching the limit $\omega\gg 1/\tau$ and achieving AESE. For larger values of τ, however, we see that $\mathcal{I}_{n}$ begins to increase simply because the system is exposed to $\hat{H}_{\mathrm{e}}$ for longer. Further, we see that choosing larger values of ω decreases the minimum achievable value of $\mathcal{I}_{n}$ because the motional component of $\hat{H}_{\mathrm{e}}$ becomes more and more off-resonant. This indicates that, for a given gradient and value of ω, we can suppress the infidelity caused by $\hat{H}_{\mathrm{e}}$ by increasing ω, and, unfortunately, the value of $t_\mathrm{g}$. In the next section, we discuss a way of suppressing the effects of $\hat{H}_{\mathrm{e}}$ without sacrificing gate speed.

Zoom In Zoom Out Reset image size

We can also induce $\vert 1\mathrm{z}\rangle\leftrightarrow\vert 1\mathrm{c}\rangle$ shelving with an ARP transition. To show this, we consider equation (19) with an additional set of single-qubit interactions:

$$\begin{eqnarray} &&\hat{H}_{\mathrm{arp}} = 2\hbar\Omega_{z}\hat{P}_{z}\left(\hat{a}^{\dagger}\mathrm{e}^{\mathrm{i}\omega t} + \hat{a}\mathrm{e}^{-\mathrm{i}\omega t}\right) + \frac{\hbar\Omega_{y}\left(t\right)}{2}\hat{S}_{y}^{\mathrm{cz}} + \frac{\hbar\Delta\left(t\right)}{2}\hat{S}_{z}^{\mathrm{cz}}. \end{eqnarray} \tag{ 25 }$$

The $\hat{U}_{\mathrm{sh}}$ operator given by equation (20) diagonalizes these added terms when $\phi = \frac{1}{2}\arctan[\Omega_{y}(t)/\Delta(t)]$. Transforming $\hat{H}_{\mathrm{arp}}$ in this way gives:

$$\begin{align} \tilde{H}_{\mathrm{arp}} & = \hat{U}^{\dagger}_{\mathrm{sh}}\left(t\right)\hat{H}_{\mathrm{arp}}\hat{U}_{\mathrm{sh}}\left(t\right) + i\hbar\dot{\hat{U}}^{\dagger}_{\mathrm{sh}}\left(t\right)\hat{U}_{\mathrm{sh}}\left(t\right) \nonumber \\ & = \tilde{H}_{z} + \frac{\Delta_{\mathrm{d}}\left(t\right)}{2}\hat{S}_{z}^{\mathrm{cz}} - \hbar\dot{\phi}\hat{S}_{x}^{\mathrm{cz}}, \end{align} \tag{ 26 }$$

where $\Delta_{\mathrm{d}}(t)\equiv [\Omega_{y}^{2}(t)+\Delta^{2}(t)]^{1/2}$ is the splitting of the $\vert \pm\rangle$ dressed states defined by $\hat{U}_{\mathrm{sh}}$. To see how we can use this to shelve, consider a system initialized to the $\{\vert 1\mathrm{c}\rangle,\vert 0\rangle\}$ manifold at a time t₀, with boundary conditions: $\Delta(t_{0}) = -\Delta(t_{0}+\tau) = \Delta_{0}$, $\Omega_{y}(t_{0}) = \Omega_{y}(t_{0}+\tau) = \Delta(t_{0}+\tau/2) = 0$, and $\Omega_{y}(t_{0}+\tau/2) = \pm\Delta_{0}$. If we choose functional forms of $\Omega_{y}(t)$ and $\Delta(t)$ that smoothly transition between these boundary conditions, we can induce the $\vert 1\mathrm{c}\rangle\leftrightarrow\vert 1\mathrm{z}\rangle$ shelving we want; then, if we set $\tau\gg 2\pi /\omega$, we also perform the gate with AESE. We can see that the additional $\propto~\hat{S}^{\mathrm{cz}}_{z}$ commutes with every term in equation (19) except the $\propto~\hat{S}_{y}^{\mathrm{cz}}$ component of $\hat{H}_{\mathrm{e}}$ and the $\propto~\dot{\phi}(t)$ diabatic term, both of which are rotated in the xy-plane of the Bloch sphere at a frequency $\Delta_{\mathrm{d}}$. In the limit $\Delta_{\mathrm{d}}(t)\gg \Omega_{z}$, the effect of $\hat{H}_{\mathrm{e}}$ averages to zero. This allows us to increase τ to AESE the gate, removing infidelities from residual spin-motion coupling, without increasing the infidelity due to $\hat{H}_{\mathrm{e}}$. At the same time $\dot{\phi}(t)~\propto~ 1/\tau$, meaning diabaticity's effect on the gate fidelity is also suppressed for large values of τ.

In figure 3 we show $\mathcal{I}_{n}$ versus τ for a chain of two $^{40}\mathrm{Ca}^{+}$ ions coupled to their motion with a $50\,\mathrm{T\,m}^{-1}$ gradient. We choose a gating mode frequency of $\omega/2\pi = 250\,\mathrm{kHz}$ for systems initialized to n = 0 (solid lines) and n = 10 (dashed lines). For each run, we set $\Delta(t^{^{\prime}}) = \Delta_{0}\cos(\pi t^{^{\prime}}/\tau)$ and $\Omega_{y}(t^{^{\prime}}) = \Delta_{0}\sin^{2}(\pi t^{^{\prime}}/\tau)$, where $t^{^{\prime}} = t$ when $t\lt\tau$, $t^{^{\prime}} = \tau$ when $\tau\unicode{x2A7D} t \unicode{x2A7D} t_\mathrm{g}-\tau$, and $t^{^{\prime}} = t+2\tau-t_\mathrm{g}$ when $t\gt t_\mathrm{g}-\tau$; this makes the dressed state splitting $\Delta_{\mathrm{d}}(t) = \Delta_{0}[\cos^{2}(\pi t^{^{\prime}}/\tau)+\sin^{4}(\pi t^{^{\prime}}/\tau)]^{1/2}$. We chose these functions for $\Delta(t)$ and $\Omega_{y}(t)$ because they meet the boundary conditions described above, while ensuring $\Delta(t)$ and $\Omega_{y}(t)$ vanish at the end of each shelving sequence to suppress unwanted off-resonant effects. While we could also suppress effect of $\hat{H}_{\mathrm{e}}$ by increasing ω and slowing the gate, we can now decrease the infidelities due to $\hat{H}_{\mathrm{e}}$ by increasing $\Delta_{\mathrm{d}}$, which does not slow the gate. Further, we can now, in theory, suppress the infidelity from $\hat{H}_{\mathrm{e}}$ arbitrarily by continuing to increasing the shelving fields strengths, and thus the values of $\Delta_\mathrm{d}(t)$, during the sequence.

Zoom In Zoom Out Reset image size

We now discuss the infidelities from heating when performing a two-qubit gate with AESE. By 'heating', we here mean motional excitations due to a bath of extraneous electric fields that perturb the crystal. When qubits experience a spin-dependent force, the displacements from this E-field bath induce a shift proportional to the spin component of the gate [32]. While we are proposing a new parameter regime, this scheme is still a geometric phase gate [15, 16, 39, 45, 46] and previous derivations of heating-induced infidelities apply. Following the prescription outlined in [32], we model this effect with a time-dependent E-field oscillating at frequency $\omega_\mathrm{e}$, which gives:

$$\begin{eqnarray} &&\hat{H}_\mathrm{g} = \hbar\gamma\left(t\right)\sum_{\alpha = \mathrm{s,c}}\left(\hat{S}_{z,\alpha}\Omega_{\alpha}^{^{\prime}}\left[\hat{a}_{\alpha}^{\dagger}\mathrm{e}^{\mathrm{i}\delta_{\alpha}t} +\hat{a}_{\alpha}\mathrm{e}^{-\mathrm{i}\delta_{\alpha}t}\right] + 2\hbar \cos\left(\omega_\mathrm{e}t\right) g_{\alpha}\left[\hat{a}_{\alpha}^{\dagger}\mathrm{e}^{\mathrm{i}\omega_{\alpha}t} +\hat{a}_{\alpha}\mathrm{e}^{-\mathrm{i}\omega_{\alpha}t}\right]\right), \end{eqnarray} \tag{ 27 }$$

where g_α is the Rabi frequency of the E-field projected onto mode α. Following [32], we transform into the interaction picture with respect to the E-field terms, giving:

$$\begin{eqnarray} &&\hat{H}_{\mathrm{g},I} = \hbar\gamma\left(t\right)\sum_{\alpha = \mathrm{s,c}}\hat{S}_{z,\alpha}\Omega_{\alpha}^{^{\prime}}\left[\left(\hat{a}_{\alpha}^{\dagger}+d_{\alpha}^{*}\right)\mathrm{e}^{\mathrm{i}\delta_{\alpha}t} +\left(\hat{a}_{\alpha}+d_{\alpha}\right)\mathrm{e}^{-\mathrm{i}\delta_{\alpha}t}\right], \end{eqnarray} \tag{ 28 }$$

where:

$$\begin{eqnarray} &&d_{\alpha}\equiv g_{\alpha}\left(\frac{1-\mathrm{e}^{\mathrm{i[}\omega_{\alpha}+\omega_\mathrm{e}]t}}{\omega_{\alpha}+\omega_\mathrm{e}} + \frac{1-\mathrm{e}^{\mathrm{i}[\omega_{\alpha}-\omega_\mathrm{e}]t}}{\omega_{\alpha}-\omega_\mathrm{e}}\right), \end{eqnarray} \tag{ 29 }$$

is the displacement due to the E-field force. There are two spectral regimes where E-field noise has the potential to cause significant errors: when $2\pi/\omega_\mathrm{e} \gtrsim t_\mathrm{g}$ and when $\omega_\mathrm{e}\sim \omega_{\alpha}$, for any α.

5.1.1. LF E-field noise

For a DC gradient, i.e. when $\delta_\mathrm{c(s)} = \omega_\mathrm{c(s)}$, $\omega_\mathrm{g} = 0$ , and $\Omega_\mathrm{c(s)}^{^{\prime}} = \Omega_\mathrm{c(s)}$, equation (28) has a near-resonant term when $\omega_\mathrm{e}\ll \Omega_{\alpha}^{^{\prime}}$, giving a significant contribution from LF E-field noise. Using this to drop counter-rotating terms in equation (28) gives:

$$\begin{eqnarray} &&\hat{H}_\mathrm{LF} = \hbar\gamma\left(t\right)\sum_{\alpha = \mathrm{s,c}}\hat{S}_{z,\alpha}\Omega_{\alpha}\left[\hat{a}_{\alpha}^{\dagger}\mathrm{e}^{\mathrm{i}\omega_{\alpha}}t +\hat{a}_{\alpha}\mathrm{e}^{-\mathrm{i}\omega_{\alpha}t}\right] - \frac{4\hbar\Omega_{\alpha}g_{\alpha}\omega_{\alpha}}{\omega_{\alpha}^{2}-\omega_\mathrm{e}^{2}}\cos\left(\omega_\mathrm{e}t\right)\hat{S}_{z,\alpha}. \end{eqnarray} \tag{ 30 }$$

Since E-field noise tends to decrease with $\omega_\mathrm{e}$ [5], sensitivity to small $\omega_\mathrm{e}$ could be a significant source of infidelity for DC gradient schemes—if we do not contain the damage. Fortunately, the error term in equation (30) commutes with the gate at all times, and the resulting infidelity can be strongly decreased with dynamical decoupling sequences [29]. If we use the RF gradient explored scheme explored in section 3.2, however, the error term in equation (30) no longer contains a near-resonant term at $\omega_\mathrm{e}\simeq 0$, making the gate insensitive to LF E-field noise.

5.1.2. White noise heating

One of the main advantages of using the RF gradient scheme (discussed in section 3.2) is that the effect of low-frequency E-field noise is averaged to zero. If this is the case, the main effect of heating on the gate's error budget is from 'Markovian' E-field noise near the gating mode frequencies, i.e. $\omega_\mathrm{e}\sim \omega_{\alpha}$, which we assume to be a white noise bath. Sutherland et al [32] derived an analytic formula for the effect of this on a single-mode two-qubit gate. Here, the infidelity due to heating of multiple modes will be additive because the E-field terms in equation (27) commute when $\alpha \neq \alpha^{^{\prime}}$. If a system has multiple modes, this gives:

$$\begin{align} \mathcal{I}_{\mathrm{h}} & = 4t_\mathrm{g}\sum_{\alpha}\frac{\dot{n}_{\alpha}\Omega_{\alpha}^{^{\prime} 2}}{\delta_{\alpha}^{2}}\lambda_{\hat{S}_{z,\alpha}}^{2}\nonumber \\ & = \frac{32 t_\mathrm{g}}{5}\sum_{\alpha}\frac{\dot{n}_{\alpha}\Omega_{\alpha}^{^{\prime} 2}}{\delta_{\alpha}^{2}}, \end{align} \tag{ 31 }$$

where, in the second line, we have averaged over all initial states, giving $\lambda_{\hat{S}_{z,\mathrm{s(c)}}}^{2} = 8/5$. This equation indicates another advantage of operating when tuned significantly closer to the STR mode than the COM mode. For same-species two-ion crystals, typically $\dot{n}_\mathrm{s}\ll\dot{n}_\mathrm{c}$ [11, 23, 47] due to the fact that an external E-field couples to the COM mode directly but the STR differentially; this usually leads the STR mode to heat well over an order of magnitude more slowly than the COM mode. Using the numbers from the middle red plot in figure 1, i.e. two $^{40}\mathrm{Ca}^{+}$ ions being driven by a $150\,\mathrm{T\,m}^{-1}$ RF gradient, where $\omega_\mathrm{s}/2\pi = 1\,\mathrm{MHz}$ and $\omega_\mathrm{c}/2\pi = 3\,\mathrm{MHz}$, we can estimate $\mathcal{I}_{\mathrm{h}}$ in near-term experiments. Choosing $\omega_\mathrm{g}/2\pi = 1.1\,\mathrm{MHz}$ makes $\delta_\mathrm{s}/2\pi = 100\,\mathrm{kHz}$ and $\delta_\mathrm{c}/2\pi = 1.9\,\mathrm{MHz}$. Thus if we assume heating rates of $\dot{n}_\mathrm{s} = 1$ quanta s⁻¹ and $\dot{n}_\mathrm{c} = 100$ quanta s⁻¹, we get $\mathcal{I}_{\mathrm{h}}\simeq 1\times 10^{-5}$.

If a gate's mode frequencies fluctuate during operation its phase space trajectory distorts, causing two distinct error-inducing mechanisms. The first is from residual spin-motion entanglement due to the system's phase space trajectory not closing. The infidelity from this mechanism scales linearly with initial phonon number, making it sensitive to temperature [32]. The second mechanism is the ion chain's phase space trajectory encompassing the wrong area, which results in the wrong entanglement angle for the gate. The infidelity from this error mechanism does not scale with initial phonon number, meaning its effect on gate infidelity is independent of temperature [32]. Here, we show that AESEing can suppress the former, temperature dependent mechanism arbitrarily by increasing the values of $\delta_{\alpha}\tau$. Since mode frequency fluctuations were the largest temperature-dependent sources of error in [11], AESE could reduce the need for sub-Doppler cooling. In current QCCD algorithms, we spend over an order of magnitude more time on ground state cooling than on two-qubit gates [13, 28]. Performing gates with AESE could, therefore, lead to reductions in algorithm time even if it increases gate time.

To exemplify this, we examine the effect of a static shift ε_α , which gives:

$$\begin{eqnarray} &&\hat{H}_\mathrm{g} = \hbar\gamma\left(t\right)\sum_{\alpha = \mathrm{s,c}}\hat{S}_{z,\alpha}\Omega_{\alpha}^{^{\prime}}\left[\hat{a}_{\alpha}^{\dagger}\mathrm{e}^{\mathrm{i}\delta_{\alpha}t} +\hat{a}_{\alpha}\mathrm{e}^{-\mathrm{i}\delta_{\alpha}t}\right] + \hbar\varepsilon_{\alpha}\hat{a}_{\alpha}^{\dagger}\hat{a}_{\alpha}. \end{eqnarray} \tag{ 32 }$$

Following the derivation given in [32], again, we transform this equation into a frame that rotates with $\hbar\varepsilon_{\alpha}\hat{a}_{\alpha}^{\dagger}\hat{a}_{\alpha}$:

$$\begin{eqnarray*} &&\hat{H}_\mathrm{g} = \hbar\gamma\left(t\right)\sum_{\alpha = \mathrm{s,c}}\hat{S}_{z,\alpha}\Omega_{\alpha}^{^{\prime}}\left[\hat{a}_{\alpha}^{\dagger}\mathrm{e}^{\mathrm{i}\left(\delta_{\alpha}+\varepsilon_{\alpha}\right)t} +\hat{a}_{\alpha}\mathrm{e}^{-\mathrm{i}\left(\delta_{\alpha}+\varepsilon_{\alpha}\right)t}\right]. \nonumber \end{eqnarray*}$$

Without AESE, when $t_\mathrm{g}$ is not an integer multiple of $2\pi /[\delta_{\alpha}+\varepsilon_{\alpha}]$, the 1st order terms in equation (3) no longer vanish, and contribute an amount:

$$\begin{align} \mathcal{I}_{1} & = \frac{\pi^{2}}{64}\sum_{\alpha}\frac{\varepsilon^{2}_{\alpha}}{\Omega_{\alpha}^{^{\prime} 2}}\left(2n_{\alpha}+1\right)\lambda_{\hat{S}_{z,\alpha}}^{2} \nonumber \\ & = \frac{\pi^{2}}{40}\sum_{\alpha}\frac{\varepsilon^{2}_{\alpha}}{\Omega_{\alpha}^{^{\prime} 2}}\left(2n_{\alpha}+1\right) \end{align} \tag{ 33 }$$

to the infidelity, where we have assumed $\gamma(t) = 1$, and, in the second line, averaged over SU(4) to get $\lambda_{\hat{S}_{z,\alpha}}^{2} = 8/5$. With AESE, $\mathcal{I}_{1}$ vanishes when the limit $1/\tau\gg \delta_{\alpha}$ is satisfied, independent of the particular value of δ_α ; as long as this condition remains true, we can assume the contribution from residual spin-motion entanglement to be negligible.

With or without AESE, drifts of the motional frequencies of the gating modes cause the qubits to accumulate entanglement at the wrong rate. This is because the gate detunings and the effective Lamb–Dicke factors both depend on the mode frequency. This causes a temperature independent error. Assuming the errors from each mode α are additive, we can insert $\omega_{\alpha}\rightarrow \omega_{\alpha}+\varepsilon_{\alpha}$ and $\delta_{\alpha}\rightarrow \delta_{\alpha}+\varepsilon_{\alpha}$, into the time propagator for the gate:

$$\begin{align} \hat{U}_\mathrm{t}=\prod_{\alpha}\exp\left(i\langle \gamma^{2}\rangle t_\mathrm{g}\hat{S}_{z,\alpha}^{2}\frac{\Omega_{\alpha}^{^{\prime} 2}\omega_{\alpha}}{\left[\delta_{\alpha}+\varepsilon_{\alpha}\right]\left[\omega_{\alpha}+\varepsilon_{\alpha}\right]} \right), \end{align} \tag{ 34 }$$

noting that we have factored out the $\propto (\omega_{\alpha}+\varepsilon_{\alpha})^{-1/2}$ dependence of $\Omega_{\alpha}^{^{\prime}}$. Keeping only leading-order terms in $\varepsilon_{\alpha}/\delta_{\alpha}$ and $\varepsilon_{\alpha}/\omega_{\alpha}$, we can factor $\hat{U}_\mathrm{t} = \hat{U}_\mathrm{g}\hat{U}_{\mathrm{e}}$, where $\hat{U}_\mathrm{g}$ the 'ideal' gate operator, and:

$$\begin{align} \hat{U}_{\mathrm{e}}=\prod_{\alpha}\exp\left(-i\langle \gamma^{2}\rangle\hat{S}_{z,\alpha}^{2}\frac{\varepsilon_{\alpha}t_\mathrm{g}\Omega_{\alpha}^{^{\prime} 2}\left[\omega_{\alpha}+\delta_{\alpha}\right]}{\omega_{\alpha}\delta_{\alpha}^{2}}\right), \end{align} \tag{ 35 }$$

is the error Hamiltonian. Again following the prescription outlined in [32], we can immediately compute the expected infidelity:

$$\begin{align} \mathcal{I}_{\mathrm{m}} & = \sum_{\alpha}\frac{\langle \gamma^{2}\rangle^{2}\Omega_{\alpha}^{^{\prime} 4}\left[\omega_{\alpha}+\delta_{\alpha}\right]^{2}}{\omega_{\alpha}^{2}\delta_{\alpha}^{4}}\varepsilon_{\alpha}^{2}t_\mathrm{g}^{2}\lambda^{2}_{\hat{S}_{z,\alpha}^{2}}\nonumber \\ & = \sum_{\alpha}\frac{16\langle \gamma^{2}\rangle^{2}\Omega_{\alpha}^{^{\prime} 4}\left[\omega_{\alpha}+\delta_{\alpha}\right]^{2}}{5\omega_{\alpha}^{2}\delta_{\alpha}^{4}}\varepsilon_{\alpha}^{2}t_\mathrm{g}^{2}, \end{align} \tag{ 36 }$$

where, in the second line, we have inserted $\lambda_{\hat{S}_{z}^{2}}^{2} = 16/5$ via averaging over SU(4) for $\vert \psi(0)\rangle$. Importantly, the infidelity due to this error mechanism is independent of the motional state of the ions, which relaxes the requirement for ground state cooling. Plugging in the the parameters for the RF-gradient two-qubit gate exemplified in section 5.1 to equation (36), we get $\mathcal{I}_{\mathrm{m}}\simeq 10^{-7}$ if we assume $\varepsilon_{\alpha}/2\pi = 50\,\mathrm{Hz}$; this is over two orders-of-magnitude smaller than the value of $3\times 10^{-5}$ that was estimated for a ground state cooled crystal experiencing that shift in [11].

In addition to drifts of ω_α , there is also a static shift from the Kerr-like interaction between the radial and axial STR modes [48, 49]. The size of this shift is proportional to the phonon occupation of that spectator mode, and is typically larger when $\omega_\mathrm{s}$ is smaller; this means the infidelity associated with it is a potential concern, especially when operating at Doppler temperatures. Fortunately, AESE renders the crystal significantly less sensitive to static motional shifts. Plugging in parameters for the example two-qubit gate discussed in section 5.1, the formula derived in [49] predicts this shift will be $\varepsilon_\mathrm{s}/2\pi \simeq -31.5\times n_\mathrm{a}\,\mathrm{Hz}$, where $n_\mathrm{a}$ is the number of phonons in the axial STR mode. To analyze this effect at Doppler temperatures, we set $n_{a} = 10$, making $\varepsilon_{\alpha}/2\pi \equiv -315\,\mathrm{Hz}$, which gives $\mathcal{I}_{\mathrm{m}}\simeq 5\times 10^{-6}$. Finally, we note that the infidelity due to static motional shifts can be suppressed arbitrarily by increasing the values of δ_α , giving experimentalists the ability to exchange the size of this error for an increase in $t_\mathrm{g}$.

In this section, we anticipate three potential sources of noise that laser-free gates performed with AESE would not necessarily be insensitive to: memory errors, the non-repeatability of gradient field—both of which can be dynamically decoupled—and motional frequency fluctuations near-resonant to δ_α .

The scheme we are describing is a direct $\hat{\sigma}_{z,1}\otimes \hat{\sigma}_{z,2}$ geometric phase gate [17], which cannot be performed on a magnetic field insensitive 'clock' state, meaning the gate is sensitive to memory errors during operation. We take the Hamiltonian representing this error mechanism as:

$$\begin{eqnarray} &&\hat{H}_{\mathrm{mem}} = \frac{\hbar}{2}\sum_{j}\Delta_{j}\left(t\right)\hat{\sigma}_{z,j}, \end{eqnarray} \tag{ 37 }$$

where $\Delta_{j}\equiv B_{j}(t)\partial \omega_{j}/\partial B_{j}$ is the energy shift of each ion due to an unaccounted for magnetic field acting on the qubit. Crucially, equation (37) commutes with equation (1) at all times, meaning we can dynamically decouple the effects of slow $\dot{\Delta}_{j}(t) \lt 2\pi/t_\mathrm{g}$ with a spin-echo, or higher-order Walsh sequence [29]. Srinivas et al [11] was able to suppress the effect of higher-frequency magnetic field noise on their gate by over an order of magnitude using 'Intrinsic Dynamical Decoupling' (IDD) [27, 44] via tuning the bichromatic carrier interaction already present in the gate. It is possible to perform gates with AESE and IDD using the $\propto~ J_{2}$ scheme described in [27], but it would require additional control fields, add sources of infidelity due to higher-order interactions with the carrier [39], and slow the gate due to the $\propto~ J_{2}$ dependence of the effective spin-dependent force.

In contrast to previous laser-free gating schemes that drive the qubit transition with the gradient [21–23, 35] or an added microwave field [11, 24, 25, 27, 37] equation (1) only contains $\propto~ \hat{S}_{z,\alpha}$ terms. This means that the scheme does not necessitate nulling the magnetic field at the point of the ions when it is performed with a spin-echo. This could simplify experiments and allow for larger gradients. For the schemes described above that generate spin-motion coupling with a DC gradient, the gate will be sensitive to the non-repeatability of the gradient during the two halves of the spin-echo. Fortunately, there are several ways of avoiding this problem. First, if we use the RF gradient technique described in section 3.2, non-repeatability is a non-issue because the gradient will effectively dynamically decouple the effect on a timescale set by $\omega_\mathrm{g}$. For a DC gradient, this sensitivity could be coherently suppressed with Walsh sequences [29], or by nulling the magnetic field with highly-correlated electrodes such that their fluctuations cancel [50].

5.5.1. Time-dependent fluctuations of the gate mode frequencies

Section 5.2 showed that performing gates with AESE significantly reduces sensitivity to static shifts of the gate mode frequencies $\omega_{\alpha}\rightarrow \omega_{\alpha}+\varepsilon_{\alpha}$. This decrease in sensitivity is general so long as the limit $\delta_{\alpha}/2\pi \gg 1/\tau$ remains satisfied. For a time-dependent shift $\omega_{\alpha}\rightarrow \omega_{\alpha}+\varepsilon_{\alpha}(t)$, if the noise-spectral density (NSD) of $\varepsilon_{\alpha}(t)$ has a non-negligible component near δ_α there will be spin-motion coupling terms that do not vanish with AESE, causing residual spin-motion coupling and errors that are sensitive to temperature.

To illustrate this potential issue qualitatively, we examine the infidelity from 'motional dephasing' described in [32]. This derivation assumes that the motional frequency fluctuations $\varepsilon_{\alpha}(t)$ follow a white noise NSD, an assumption that nullifies the coherent suppression of spin-motion entanglement from AESE. Importantly, this assumption is typically chosen to simplify calculations, not because it is an accurate description of the NSD. The NSD for $\varepsilon_{\alpha}(t)$ is usually better described as $\propto~ 1/f_{\alpha}$ noise, meaning lower-frequency drifts—that enable AESE to suppress spin-motion entanglement—will be much more likely than spectral components near δ_α . Therefore, in order to represent a (plausible but unlikely) worst-case-scenario, we assume white noise and apply the derivation exactly as described in [32]. This gives:

$$\begin{align} \mathcal{I}_{\mathrm{d}} &\simeq \sum_{\alpha}\frac{2\eta_{\alpha}\Omega_{\alpha}^{^{\prime} 2}t_\mathrm{g}}{\delta_{\alpha}^{2}}\left(\left[2n_{\alpha}+1\right]\lambda_{\hat{S}_{z,\alpha}}^{2} + \frac{3\Omega_{\alpha}^{^{\prime} 2}}{\delta_{\alpha}^{2}}\lambda_{\hat{S}_{z,\alpha}^{2}}^{2} \right) \nonumber \\ & = \sum_{\alpha}\frac{16\eta_{\alpha}\Omega_{\alpha}^{^{\prime} 2}t_\mathrm{g}}{5\delta_{\alpha}^{2}}\left(\left[2n_{\alpha}+1\right] + \frac{6\Omega_{\alpha}^{^{\prime} 2}}{\delta_{\alpha}^{2}} \right) \end{align} \tag{ 38 }$$

where $\eta_{\alpha} = 2/t_{\mathrm{d},\alpha}$, and $t_{\mathrm{d},\alpha}$ is the coherence time between neighboring Fock states of mode α. We have, again, averaged over SU(4) in the second line. For the system we introduced in section 5.1, assuming $n_\mathrm{s} = n_\mathrm{c} = 10$ to represent Doppler temperatures, we get an infidelity of $\mathcal{I_{\mathrm{d}}}\simeq 3\times 10^{-3}$. This indicates that, although unlikely, if there is a non-trivial component of the motional shift NSD, then it may become necessary to cool the system below the Doppler limit.