Motion-Insensitive Time-Optimal Control of Optical Qubits

In trapped-atom quantum computers, high-fidelity control of optical qubits is challenging due to the motion of atoms in the trap. If not corrected, the atom motion gets entangled with the qubit degrees of freedom through two fundamental mechanisms, (i) photon recoil and (ii) thermal motion, both leading to a reduction of the gate fidelity. We develop motion-insensitive pulses that suppress both sources of infidelity by modulating the phase of the driving laser field in time. To eliminate photon recoil, we use bang-bang pulses$-$derived using time-optimal control$-$which shorten the gate duration by about 20 times compared to conventional pulses. However, even when photon recoil is eliminated, we find that the gate error does not vanish, but is rather limited by a bound arising from thermal motion-induced entanglement. Remarkably, this bound is independent of the Rabi frequency, meaning that, unlike for photon recoil, operating in the resolved sideband regime does not mitigate this source of infidelity. To overcome this bound, we derive smooth-phase pulses, which allow for a further reduction of the gate error by more than an order of magnitude for typical thermal atoms. Motion-insensitive pulses can be refined to compensate for laser inhomogeneities, enhancing the gate performance in practical situations. Our results are validated through simulations of one-qubit gates operating on the optical clock transition of ${}^{88}$Sr atoms trapped in an optical tweezers array.