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We discuss a laser-free, two-qubit geometric phase gate technique forgenerating high-fidelity entanglement between two trapped ions. The schemeworks by ramping the spin-dependent force on and off slowly relative to thegate detunings, which adiabatically eliminates the spin-motion entanglement(AESE). We show how gates performed with AESE can eliminate spin-motionentanglement with multiple modes simultaneously, without having to specificallytune the control field detunings. This is because the spin-motion entanglementis suppressed by operating the control fields in a certain parametric limit,rather than by engineering an optimized control sequence. We also discussphysical implementations that use either electronic or ferromagnetic magneticfield gradients. In the latter, we show how to ``AESE" the system by smoothlyturning on the \textit{effective} spin-dependent force by shelving from amagnetic field insensitive state to a magnetic field sensitive state slowlyrelative to the gate mode frequencies. We show how to do this with a Rabi oradiabatic rapid passage transition. Finally, we show how gating with AESEsignificantly decreases the gate's sensitivity to common sources of motionaldecoherence, making it easier to perform high-fidelity gates at Dopplertemperatures.

Laser-free trapped ion entangling gates with AESE: Adiabatic Elimination of Spin-motion Entanglement