Effective Evolution of a Driven Qubit Beyond the Rotating Wave Approximation

Fast quantum gates require high-amplitude pulses. The applicability of the rotating wave approximation is thus questionable, and one needs to take into account effects of the fast-oscillating terms in the Hamiltonian. These terms generate intricate features in the state evolution of the qubit on a time scale of 1/ω, where ω is the qubit resonance frequency. These features complicate the task to track the state evolution. For a constant drive envelope, an effective Hamiltonian includes the Bloch-Siegert term [1], a shifted resonance condition, and a term which renormalizes the Rabi frequency. Using the Magnus expansion [2] we obtain an effective Hamiltonian for time-dependent pulses of amplitude comparable to the qubit frequency, and the time dependence of this Hamiltonian is solely determined by the drive. This Hamiltonian yields a family of smooth trajectories in the rotating frame, each of which agrees stroboscopically with the actual state trajectory. Our approach has the potential to cost-effectively design accurate pulse shapes for quantum gates.

[1] F Bloch and A Siegert, Physical Review 57, 522 (1940)
[2] R R Ernst, G Bodenhausen, A Wokaun, Clarendon Press, Oxford (1987)