Qubit dynamics driven by smooth pulses of finite duration

In this study, we investigate the dynamics of a qubit driven by a pulsed field of finite duration, focusing on smooth pulse shapes that start and end linearly in time. All of these pulse shapes, such as a sine function sampled between two adjacent nodes, provide smooth alternatives to the commonly used rectangular pulse. Owing to its continuity, the sine pulse exhibits weaker power broadening and leads to the rapid vanishing of the wings in the excitation line profile, significantly reducing the sidebands. Additionally, these pulse shapes avert the spurious truncation effects that arise with infinite-duration shapes, such as the Gaussian, where truncation is inevitable. We derive two approximate analytic solutions which describe the ensuing quantum dynamics. Both approximations assume that the field changes linearly at the beginning and end of the driving pulse, and adiabatically in between. The first approximation matches the linear and adiabatic parts at an appropriate instant of time and is expressed in terms of Weber's parabolic cylinder functions. The second, much simpler, approximation uses asymptotics of the Weber function to replace it with simpler functions, along with additional transformations. Both approximations were shown to be highly accurate, with near-complete agreement between theory and observations on two IBM Quantum processors, further validated by measurements of the predicted weaker power broadening and highly suppressed sidebands.