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The Journal of Chemical Physics

An operational perspective on the Magnus–Fer conundrum in time-dependent quantum mechanics

Kuntal Mukherjee, Shreyan Ganguly, Ramesh Ramachandran

Published Apr 2, 2025• 0 citations
Publisher

Abstract

The development of analytic methods for studying quantum systems driven by periodic Hamiltonians has remained an active pursuit for gaining insights into physical phenomena in spectroscopy and other related areas in chemical physics. From a theoretical perspective, the success of a given analytic method relies on the operational aspects as well as its exactness in replicating (known) experimental results. While analytic methods built around the Magnus expansion (ME) scheme have been preferred in time-evolution studies, the splitting of the time-propagator into a product of exponential operators in the Fer expansion (FE) scheme has gained wider attention in recent years. To this end, the operational advantages of one scheme over the other have always remained contentious and form the basis for the present report. Employing periodic Hamiltonians as examples, the operational aspects and the relative merits/demerits of the two methods are analyzed, and their long-term/short-term behavior is discussed through analytic expressions in experimentally verifiable systems.

DOI
10.1063/5.0260260

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The development of analytic methods for studying quantum systems driven by periodic Hamiltonians has remained an active pursuit for gaining insights into physical phenomena in spectroscopy and other related areas in chemical physics. From a theoretical perspective, the success of a given analytic method relies on the operational aspects as well as its exactness in replicating (known) experimental results. While analytic methods built around the Magnus expansion (ME) scheme have been preferred in time-evolution studies, the splitting of the time-propagator into a product of exponential operators in the Fer expansion (FE) scheme has gained wider attention in recent years. To this end, the operational advantages of one scheme over the other have always remained contentious and form the basis for the present report. Employing periodic Hamiltonians as examples, the operational aspects and the relative merits/demerits of the two methods are analyzed, and their long-term/short-term behavior is discussed through analytic expressions in experimentally verifiable systems.
The present report examines the utility and exactness of time-propagators derived from Fer expansion (FE). While the mathematical intricacies of the FE scheme are well established, the operational aspects of the same in time-evolution studies remain less explored and authenticated in physical problems of relevance. Through suitable examples, the operational inconsistencies observed in time-evolution studies based on the FE scheme are identified and corroborated through rigorous comparisons with simulations emerging from exact numerical methods. The limitations outlined seriously undermine the advantages associated with the FE scheme over other existing analytic methods.
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.
In this report, an alternate approach to time-evolution in quantum mechanics is discussed using Fer expansion. Specifically, the operational aspects and exactness of the time-propagators in the Schrödinger and Interaction representation is examined in systems modulated by multiple frequencies. Through experimentally relevant examples, the interplay between the natural frequency, driving amplitude and driving frequency in time evolution studies is discussed at both stroboscopic and non-stroboscopic time intervals. Additionally, the factors governing the long time behaviour and convergence properties of the time-propagators near resonance conditions is also discussed through perturbative methods.
In this paper, we revisit, from different perspectives, the applications of Floquet–Magnus expansion (FME) and Fer expansion (FE) in NMR and physics. We use these expansion schemes for the calculation of effective Hamiltonians and propagators. We found that the FME is related to the FE. While both approaches are powerful methods, there are subtle differences between them. To understand the differences, we performed explicit calculation on various problems, namely, Bloch–Siegert shift, Cross Polarization (CP) and Continuous Waves (CW). We put the work in a prospect that is related to spin physics in order to make the calculation more general. As the propagator from the FME takes the form of the evolution operator, which removes the constraint of a stroboscopic observation, we appreciated the effects of time-evolution under Hamiltonians with different orders separately.
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.
In this work, we investigated the orders to which the Floquet-Magnus expansion (FME) and Fer expansion (FE) are equivalent or different for the three-level system. Specifically, we performed the third-order calculations of both approaches based on elegant integrations formalism. We present an important close relationship between the Floquet-Magnus and Fer expansions. As the propagator from the FME takes the form of the evolution operator, which removes the constraint of a stroboscopic observation, we appreciated the effects of time-evolution under Hamiltonians with different orders separately. Our work unifies and generalizes existing results of Floquet-Magnus and Fer approaches and delivers illustrations of novel springs that boost previous applications that are based on the classical information. Due to the lack of an unequivocal relationship between the FME and FE, some disagreements between the results produced by these theories will be found, especially in NMR experiments. Our results can find applications in the optimization of NMR spectroscopy, quantum computation, quantum optical control, and coherence in optics and might bear new awareness in fundamental perusals of quantum spin dynamics. This work is an important theoretical and numerical contribution in the general field of spin dynamics.