  We discuss a laser-free, two-qubit geometric phase gate technique for
generating high-fidelity entanglement between two trapped ions. The scheme
works by ramping the spin-dependent force on and off slowly relative to the
gate detunings, which adiabatically eliminates the spin-motion entanglement
(AESE). We show how gates performed with AESE can eliminate spin-motion
entanglement with multiple modes simultaneously, without having to specifically
tune the control field detunings. This is because the spin-motion entanglement
is suppressed by operating the control fields in a certain parametric limit,
rather than by engineering an optimized control sequence. We also discuss
physical implementations that use either electronic or ferromagnetic magnetic
field gradients. In the latter, we show how to ``AESE" the system by smoothly
turning on the \textit{effective} spin-dependent force by shelving from a
magnetic field insensitive state to a magnetic field sensitive state slowly
relative to the gate mode frequencies. We show how to do this with a Rabi or
adiabatic rapid passage transition. Finally, we show how gating with AESE
significantly decreases the gate's sensitivity to common sources of motional
decoherence, making it easier to perform high-fidelity gates at Doppler
temperatures.
