Two-qubit quantum gates with minimal pulse sequences

Working with trapped atoms at a close distance to each other, we show that one can implement entangling gates based on nonindependent qubits using a single pulse per qubit, or a single structured pulse. The optimal parameters depend on approximate solutions of Diophantine equations, causing the fidelity to never be exactly one, even under ideal conditions, although the errors can be made arbitrarily smaller at the cost of stronger fields. We fully characterize the mechanism by which the gates operate and study the effects of thermal motion and intensity fluctuations in the laser beams for different physical implementations of the gates. If instead of one pulse, we control the system with a two-pulse sequence, a plethora of mechanisms become possible where one can choose the optimal parameters from a wide range of values to achieve high-fidelity gates that are more protected from the effects of laser intensity fluctuations.