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arXiv:2410.04296 · Physical Review Research

Beyond average Hamiltonian theory for quantum sensing

Jner Tzern Oon, Sebastian Camilo Carrasco, Connor Hart, George A. Witt, Vladimir S. Malinovsky, Ronald Lee Walsworth

Published Jan 15, 2026• 1 citation
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Abstract

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.

DOI
10.1103/w2sf-zxdk

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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.
The Floquet-Magnus expansion is a widely used tool to derive effective descriptions of time-periodic quantum systems by approximating their dynamics with a time-independent Hamiltonian. However, its standard formulation is, strictly speaking, restricted to bounded Hamiltonians. In this work, we extend its definition and analysis to a broad class of time-periodic unbounded Hamiltonians. Our approach is based on an a priori distinct nonperturbative framework for the construction of effective Hamiltonians, which we show to reproduce the Floquet-Magnus expansion. A particular strength of our framework is that it allows us to prove that the resulting effective dynamics approximates the original time evolution propagators to arbitrary order in the high-frequency limit without requiring convergence of the Floquet-Magnus expansion, a condition that is already highly restrictive even in the bounded setting. We illustrate the scope of the method on representative models: the quantum Rabi Hamiltonian in the interaction picture, and the periodically driven quantum harmonic oscillator.
The evolution of a quantum system under time-dependent driving exhibits phenomena that are absent in its stationary counterpart. However, the high dimensionality and non-commutative nature of quantum dynamics make this a challenging problem. The Magnus expansion provides an analytic framework to approximate the effective dynamics on short time-scales, but computing high-order terms with existing methods is computationally expensive. We introduce a scalable approach that reduces the computational effort to depend only on the degrees of freedom defining the time-dependent control function. We focus specifically on Hamiltonians consisting of a constant drift term and a controllable term. Our method provides a polynomial expression for the Magnus expansion which can be evaluated several orders of magnitude faster than previous techniques, enabling broad applications in the realms of quantum simulation and quantum optimal control. We showcase an application of the method by designing control pulses for the 5-qubit phase gate on a neutral-atom platform utilizing Rydberg atoms.
We introduce a framework for designing Hamiltonian engineering pulse sequences that systematically accounts for the effects of higher-order contributions to the Floquet-Magnus expansion. Our techniques result in simple, intuitive decoupling rules, despite the higher-order contributions naively involving complicated, nonlocal-in-time commutators. We illustrate how these rules can be used to efficiently design improved Hamiltonian engineering pulse sequences for a wide variety of tasks such as dynamical decoupling, quantum sensing, and quantum simulation.
While quantum devices rely on interactions between constituent subsystems and with their environment to operate, native interactions alone often fail to deliver targeted performance. Coherent pulsed control provides the ability to tailor effective interactions, known as Hamiltonian engineering. We propose a Hamiltonian engineering method that maximizes desired interactions while mitigating deleterious ones by conducting a pulse sequence search using constrained optimization. The optimization formulation incorporates pulse sequence length and cardinality penalties consistent with linear or integer programming. We apply the general technique to magnetometry with solid state spin ensembles in which inhomogeneous interactions between sensing spins limit coherence. Defining figures of merit for broadband Ramsey magnetometry, we present novel pulse sequences which outperform known techniques for homonuclear spin decoupling in both spin-1/2 and spin-1 systems. When applied to nitrogen vacancy (NV) centers in diamond, this scheme partially preserves the Zeeman interaction while zeroing dipolar coupling between negatively charged NV − centers. Such a scheme is of interest for NV − magnetometers which have reached the NV − –NV − coupling limit. We discuss experimental implementation in NV ensembles, as well as applicability of the current approach to more general spin bath decoupling and superconducting qubit control.
Quantum state processing is one of the main tools of quantum technologies. While real systems are complicated and/or may be driven by non-ideal control, they may nevertheless exhibit simple dynamics approximately confined to a low-energy Hilbert subspace. Adiabatic elimination is the simplest approximation scheme allowing us to derive in certain cases an effective Hamiltonian operating in a low-dimensional Hilbert subspace. However, these approximations may present ambiguities and difficulties, hindering a systematic improvement of their accuracy in larger and larger systems. Here, we use the Magnus expansion as a systematic tool to derive ambiguity-free effective Hamiltonians. We show that the validity of the approximations ultimately leverages only on a proper coarse-graining in time of the exact dynamics. We validate the accuracy of the obtained effective Hamiltonians with suitably tailored fidelities of quantum operations.
We present a general, nonperturbative method for deriving effective Hamiltonians of arbitrary order for periodically driven systems based on an iterated integration-by-parts technique. The resulting family of effective Hamiltonians reproduces the well-known Floquet-Magnus expansion, now enhanced with explicit error bounds that quantify the distance between the exact and approximate dynamics at each order, even in cases where the Floquet-Magnus series fails to converge. We apply the method to the semiclassical Rabi model and provide explicit error bounds for both the Bloch-Siegert Hamiltonian and its third-order refinement. Our analysis shows that, while the rotating-wave approximation more accurately captures the true dynamics than the Bloch-Siegert Hamiltonian in most regimes, the third-order approximation ultimately outperforms both.