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Description
In trapped-ion systems, the majority of entangling operations are implemented in the adiabatic regime [1,2]. Adiabatic in this context means that we can selectively excite a single set of terms in the Lamb-Dicke expansion and are able to neglect the remaining off-resonant terms. Then only a single motional mode (with secular frequency $\nu$) participates in the interaction. This is possible if the interaction is slow compared to the frequency gap between adjacent motional modes. Restricting operations to a small subspace of the Hilbert space makes high-fidelity control easier yet poses severe limitations on the achievable gate speed.
Operating in the non-adiabatic regime allows for a significant gate speed-up. Dominating error contributions are shifted towards a few coherent effects which are potentially controllable while incoherent errors, which dominate in the adiabatic regime, become less significant.
The non-adiabatic regime is rich in novel physics. For example, as the gate time becomes comparable to the timescale of the mediating ion crystal we can study the origin of entanglement creation [3].
Recent advances have increased gate speeds by around an order of magnitude (resulting in gate durations of ~1 us) using Rydberg interactions [4], or by operating in the non-adiabatic regime [5], where several additional effects must be accounted for in order to retain high fidelity.
We investigate a Mølmer–Sørensen (MS) gate [6] in a beam configuration where the unwanted carrier coupling is nulled, allowing its use in the non-adiabatic regime for the first time. More precisely, we choose an optical addressing beam configuration in which the carrier term and all higher order even contributions are cancelled by interference. This configuration, realised by two counter-propagating beams, is a standing-wave optical lattice. Phase stabilisation of an optical lattice in free space has been demonstrated [7].
We present the theoretical treatment and in-depth simulations of the standing-wave MS-gate in both the adiabatic and the non-adiabatic regime. In the adiabatic regime we identify operating ranges where the standing-wave MS-gate outperforms a conventional MS-gate. In the non-adiabatic case we introduce amplitude modulation techniques [8] to ensure phase-space loop closure for all motional modes. We present gate solutions for up to five-ion crystals with sub-microsecond entangling times with fidelities $\mathcal{F}\geq 99.9\%$, and analyse the impact of experimental imperfections on the entanglement fidelity.
[1]: Physical Review Letters, 117(6):60504, 2016.
[2]: Physical review letters, 117(6):60505, 2016.
[3]: Physical Review Letters, 94(5), 50504, 2005.
[4]: Nature, 580(7803), 2020.
[5]: Nature, 555(7694), 2018.
[6]: Physical Review A, 62(2):11, 2000.
[7]: Physical review letters, 116(3):33002, 2016.
[8]: New Journal of Physics, 16(5):53049, 2014.
