Parallel execution of quantum gates in a long linear ion chain via Rydberg mode shaping

We present a mechanism that permits the parallel execution of multiple quantum gate operations within a single long linear ion chain. Our approach is based on large coherent forces that occur when ions are electronically excited to long-lived Rydberg states. The presence of Rydberg ions drastically affects the vibrational mode structure of the ion crystal, giving rise to modes that are spatially localized on isolated subcrystals which can be individually and independently manipulated. We theoretically discuss this Rydberg mode shaping in an experimentally realistic setup and illustrate its power by analyzing the fidelity of two conditional phase flip gates executed in parallel. The ability to dynamically shape vibrational modes on the single-ion level might find applications in quantum simulators and quantum computation architectures.

Figure 1

(a) Level structure of Ca${}^{+}$ and schematics of the envisioned setup. Red and blue symbols refer to ions in Rydberg states and ions in electronically low-lying (ELL) states, respectively. Subcrystals of ion pairs are isolated within a linear crystal formed by 100 ions by the excitation of selected ions to the Rydberg $n{P}_{1/2}$ state (here the 45th, 48th and 53rd, 56th). Using laser-induced spin-dependent forces, quantum gates can be executed on the two subcrystals in parallel. (b) Vibrational modes of a crystal formed by 100 ions in ELL states. Depicted is the modulus of the normal mode matrix ${\mathbf{B}}_m^{(j,x)}$ where $j$ ($m$) refers to the mode (ion) index (see text for further detail). (c) Vibrational modes in the presence of four Rydberg ions. The black dashed lines delimit the region corresponding to the ions that are shown in panel (a). The Rydberg ions drastically reshape the vibrational mode structure, leading to the emergence of modes that are localized on the two subcrystals (see inset).

Figure 2

(a) Modulus of the eigenvector ${\mathbf{B}}_m^{(j,x)}$ when ions 45 and 56 are in the Rydberg state. Localized modes reside on ions in ELL states forming the subcrystal delimited by the two Rydberg ions. Panel (b) shows the corresponding eigenenergy of the localized modes obtained from the full (circles) and truncated (diamonds) calculations. The largest discrepancy between these calculations is about $0.3\%$. The dots show the (quasicontinuous) energy spectrum of the ion chain without mode shaping. In all the calculations we use ${\omega }_{\mathrm{ELL}}/{\omega }_s=150$ and ${\omega }_{\mathrm{Ryd}}/{\omega }_s=198.5$.

Figure 3

(a) Fidelity of the two CPF gates. The solid (dotted) curve is the result calculated using all (only the four localized) modes. The dashed curve corresponds to the gate fidelity without mode shaping, where the bus mode is the highest energy mode [Fig. 2b], whose average phonon number is $3.25$. In the calculations, we assume that all phonon modes when the ions are in the ELL states have the same temperature. (b) Maximal fidelity vs delay time. $F_{\max }$ is found by maximizing the fidelity over $\nu$ within the range shown in (a). The solid (dashed) curve stands for the calculation with (without) mode shaping.

Figure 4

Level scheme used in the Rydberg excitation. A strong MW field results in a large Autler-Townes splitting [30] between the two dressed states.

Figure 5

(a) Carrier resonant transition of the $|-\rangle$ state. The black (dotted) curves are solutions of the Hamiltonian Eq. (A2) [Hamiltonian Eq. (A4)]. The gray and dark-gray curve correspond to the probability of the $|P\rangle$ and $|S\rangle$ state. The parameters are ${\Omega }_{-}=2\pi \times 1$ MHz, ${\Omega }_{\mathrm{MW}}=2\pi \times 400$ MHz, ${\Delta }_S=2\pi \times 136.074$ MHz, and ${\Delta }_P=2\pi \times 293.957$ MHz. These parameters lead to a vanishing polarizability of the dressed $|-\rangle$ state. (b) Adiabatic evolution of the population in the $P$ and $S$ state. The change rate is $c={\Omega }_{\mathrm{MW}}/4.7$.

Figure 6

$\mathbf{T}$ matrix elements corresponding to one of the localized mode [mode index 56; see Fig. 1c] and the bare mode. The mode vector is shown in Fig. 1b (bare mode) and the inset of Fig. 1c (localized mode).