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Journal of Material Science & Engineering

Equivalence between floquet-magnus and Fer expansions to investigate the dynamics of a spin system in the three-level system

Eugene Stephane Mananga

Published Jan 1, 2017• 0 citations
Publisher
DOI
10.4172/2169-0022-c1-080

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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.

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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.

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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.
The Floquet-Magnus and Fer expansion schemes were introduced in solid-state nuclear magnetic resonance (NMR) in 2011 and 2006, respectively. Key features of the Floquet magnus expansion are its ability to account for the calculations developed in a finite-dimensional Hilbert space instead of an infinite-dimensional space within the Floquet theory as well as its use of its distinguishable function from other theories such as average Hamiltonian theory, Floquet theory, and Fer expansion, which facilitates the evaluation of the spin behavior in between the stroboscopic observation points. This article focuses on revisiting the Floquet-Magnus and Fer expansion approaches and applying both methods to calculate the effective Hamiltonians and propagators, which control the spin system evolution during the Triple Oscillating Field Technique radiation experiment (TOFU). The TOFU pulse sequence is an important technique that was shown to avoid the dipolar truncation problem and form a new basis for accurate distance measurement by solid-state NMR. We take advantage of the interaction frequencies and the time modulation arising from the TOFU pulse sequence allowing selective recoupling of specific terms in the Hamiltonian that fulfill determined specific conditions. The work presented unifies and generalizes existing results of the Floquet-Magnus and Fer expansions and delivers illustrations of novel springs that boost previous applications that are based on the classical information. We believe that the revisited approaches of this work and the derived expressions can serve as useful information and numerical tools for time evolution in time-resolved spectroscopy, quantum control, and open system quantum dynamics.
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.