Employing Variational Principles to Define the Effective Hamiltonian of a Periodically Driven Qubit

The trajectory of a linearly driven qubit in the rotating frame can be determined using the effective Hamiltonian introduced in [1], which improves on the rotating wave approximation. By its definition, this Hamiltonian (i) is analytic in time and, (ii), yields an effective driven-qubit trajectory that coincides with the exact trajectory at equally-spaced points in time. In general, these effective trajectories are significantly smoother than the exact ones. The effective Hamiltonian is determined by an infinite sum whose convergence, however, is not always guaranteed [1]. As the requirements (i) and (ii) are fulfilled by a continuum of Hamiltonians, here we hypothesize that the effective Hamiltonian additionally satisfies, (iii), a variational principle that is motivated by the smoothed trajectories mentioned above. Our variational principles state that the integral of a certain function of the Hamiltonian, e.g., its positive eigenvalue, over the full pulse duration is minimized by the effective Hamiltonian of [1]. To find evidence for or against our hypotheses, we carry out variational calculations aimed at numerically minimizing the corresponding integrals.

[1] D. Zeuch et al., arXiv preprint: 1807.02858 (2018).