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Theoretical and Mathematical Physics

Higher approximations in the theory of the average Hamiltonian

L. L. Buǐshvili, E. B. Volzhan, Merab Menabde

Published Feb 1, 1981• 24 citations
Publisher
DOI
10.1007/bf01030852

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The Magnus expansion underpins average Hamiltonian theory (AHT), a central tool in quantum control and sensing that is used to generate efficient approximations of system dynamics. Here, we demonstrate that the conditions for Magnus expansion convergence are violated under typical operating regimes for sensing using solid-state spin systems. We also establish that certain symmetries, such as those found in rapid echo pulse sequences, cancel all higher-order Magnus terms outside standard convergence frameworks. We introduce an exact method to compute the sensor response beyond the AHT regime, and apply it to both broadband and narrow-band magnetometry protocols. Our results clarify AHT limits and enable opportunities in pulse sequence design through symmetry-driven convergence, with implications for entanglement-enhanced sensing and quantum Fisher information estimation in control protocols.
We present a quantum averaging theory (QAT) for analytically modeling unitary gate dynamics in driven quantum systems beyond the rotating-wave approximation. QAT addresses the simultaneous presence of distinct timescales by generating a rotating frame with a dynamical phase operator that toggles with the high-frequency dynamics and yields an effective Hamiltonian for the slow degree of freedom. By accounting for the fast-varying effects, we demonstrate that high-fidelity two-qubit gates in strongly driven systems are achievable by going beyond the validity of first-order approximations. The QAT results rapidly converge with numerical calculations of a fast-entangling Mølmer-Sørensen trapped-ion-qubit gate in the strong coupling regime, illustrating QAT's ability to simultaneously provide both an intuitive, effective-Hamiltonian model and high accuracy.
Evolution of quantum mechanical systems under time-dependent Hamiltonians has remained a challenging problem of interest across all disciplines. Through suitable approximations, different averaging methods have emerged in the past for modeling the time-evolution under time-dependent Hamiltonians. To this end, the development of analytic methods in the form of time-averaged effective Hamiltonians has gained prominence over other methods. In particular, the advancement of spectroscopic methods for probing molecular structures has benefited enormously from such theoretical pursuits. Nonetheless, the validity of the approximations and the exactness of the proposed effective Hamiltonians have always remained a contentious issue. Here, in this report, we reexamine the equivalence between the effective Hamiltonians derived from the Magnus formula and Floquet theory through suitable examples in magnetic resonance.