In trapped-ion chains, multipartite Greenberger-Horne-Zeilinger (GHZ) states can be prepared natively with the help of a single bichromatic laser pulse. However, higher-order terms in the expansion in the Lamb-Dicke parameter $\eta$ limit the GHZ state preparation infidelity for rectangular and bell-like pulses to the order of ${\eta }^4$. For tens of ions, the infidelity caused by out-of-Lamb-Dicke effects can reach several percents. We propose an amplitude and phase-modulated pulse shape, an “echoed lemniscate pulse,” which cancels this contribution into error in the leading order. For the proposed pulse, the infidelity scales as ${\eta }^6$. The improved scaling is achieved because of a special phase trajectory of a collective motional mode following the figure-eight curve (lemniscate). We demonstrate that the lemniscate pulse allows achieving lower infidelity than bell-like pulses, which can be as low as ${10}^{-4}$ for 20-ion chains.

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• Pulse shaping
• Quantum gates
• Quantum information with trapped ions
• Quantum state engineering
• Trapped ions