Floquet theory in magnetic resonance: Formalism and applications

Konstantin L. Ivanov, Kaustubh R. Mote, Matthias Ernst, Asif Equbal, Perunthiruthy K. Madhu

Research output: Contribution to journalReview articlepeer-review


Floquet theory is an elegant mathematical formalism originally developed to solve time-dependent differential equations. Besides other fields, it has found applications in optical spectroscopy and nuclear magnetic resonance (NMR). This review attempts to give a perspective of the Floquet formalism as applied in NMR and shows how it allows one to solve various problems with a focus on solid-state NMR. We include both matrix- and operator-based approaches. We discuss different problems where the Hamiltonian changes with time in a periodic way. Such situations occur, for example, in solid-state NMR experiments where the time dependence of the Hamiltonian originates either from magic-angle spinning or from the application of amplitude- or phase-modulated radiofrequency fields, or from both. Specific cases include multiple-quantum and multiple-frequency excitation schemes. In all these cases, Floquet analysis allows one to define an effective Hamiltonian and, moreover, to treat cases that cannot be described by the more popularly used and simpler-looking average Hamiltonian theory based on the Magnus expansion. An important example is given by spin dynamics originating from multiple-quantum phenomena (level crossings). We show that the Floquet formalism is a very general approach for solving diverse problems in spectroscopy.

Original languageEnglish (US)
Pages (from-to)17-58
Number of pages42
JournalProgress in Nuclear Magnetic Resonance Spectroscopy
StatePublished - Oct 1 2021


  • Average Hamiltonian theory
  • Dynamic nuclear polarisation
  • Floquet theory
  • Level crossing
  • Magic-angle spinning
  • NMR
  • Solid-state NMR
  • Spin chemistry

ASJC Scopus subject areas

  • Analytical Chemistry
  • Biochemistry
  • Nuclear and High Energy Physics
  • Spectroscopy


Dive into the research topics of 'Floquet theory in magnetic resonance: Formalism and applications'. Together they form a unique fingerprint.

Cite this