Introduction of the Floquet-Magnus expansion in solid-state nuclearmagnetic resonance spectroscopyEugène S. Mananga and Thibault Charpentier Citation: J. Chem. Phys. 135, 044109 (2011); doi: 10.1063/1.3610943 View online: http://dx.doi.org/10.1063/1.3610943 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i4 Published by the American Institute of Physics. Related ArticlesSecond-order dipolar order in magic-angle spinning nuclear magnetic resonance J. Chem. Phys. 135, 154507 (2011) Single crystal nuclear magnetic resonance in spinning powders J. Chem. Phys. 135, 144201 (2011) Resistive detection of optically pumped nuclear polarization with spin phase transition peak at Landau level fillingfactor 2/3 Appl. Phys. Lett. 99, 112106 (2011) High-resolution 13C nuclear magnetic resonance evidence of phase transition of Rb,Cs-intercalated single-walled nanotubes J. Appl. Phys. 110, 054306 (2011) Distribution of non-uniform demagnetization fields in paramagnetic bulk solids J. Appl. Phys. 110, 013902 (2011) Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 135, 044109 (2011)

Introduction of the Floquet-Magnus expansion in solid-state nuclearmagnetic resonance spectroscopy

Eugène S. Mananga1 and Thibault Charpentier2,a)

1CEA, Neurospin/I2BM, Laboratoire de Résonance Magnétique Nucléaire, F-91191Gif-sur-Yvette cedex, France2CEA, IRAMIS, Service Interdisciplinaire sur les Systèmes Moléculaires et Matériaux, LSDRM, UMRCEA/CNRS 3299, F-91191 Gif-sur-Yvette cedex, France

(Received 20 August 2010; accepted 27 June 2011; published online 25 July 2011)

In this article, we present an alternative expansion scheme called Floquet-Magnus expansion (FME)used to solve a time-dependent linear differential equation which is a central problem in quantumphysics in general and solid-state nuclear magnetic resonance (NMR) in particular. The commonlyused methods to treat theoretical problems in solid-state NMR are the average Hamiltonian theory(AHT) and the Floquet theory (FT), which have been successful for designing sophisticated pulsesequences and understanding of different experiments. To the best of our knowledge, this is the firstreport of the FME scheme in the context of solid state NMR and we compare this approach with otherseries expansions. We present a modified FME scheme highlighting the importance of the (time-periodic) boundary conditions. This modified scheme greatly simplifies the calculation of higherorder terms and shown to be equivalent to the Floquet theory (single or multimode time-dependence)but allows one to derive the effective Hamiltonian in the Hilbert space. Basic applications of the FMEscheme are described and compared to previous treatments based on AHT, FT, and static perturbationtheory. We discuss also the convergence aspects of the three schemes (AHT, FT, and FME) andpresent the relevant references. © 2011 American Institute of Physics. [doi:10.1063/1.3610943]

I. INTRODUCTION

Much progress has been made in the application of solid-state nuclear magnetic resonance (NMR) to elucidate molec-ular structure and dynamics in systems not amenable to char-acterization by any other way. The importance of solid-statenuclear magnetic resonance stands in its ability to determineaccurately intermolecular distances1, 2 and molecular torsionangles.3, 4 The methods have been used to systems includingboth microscopically ordered preparations such as membraneproteins,5–8 nanocrystalline proteins,9–11 amyloid fibrils,12–16

and also disordered or amorphous systems such as glasses.17

Site-specific resolution can be obtained either by uniaxialorientation of the sample with respect to the static mag-netic field18 or, through magic-angle sample spinning (MAS).Nowadays, MAS is widely used to obtain high resolutionNMR spectroscopy of solids because of its effect of averag-ing out the orientation-dependent component of nuclear spininteractions, principally chemical shifts anisotropic and mag-netic dipolar couplings. This technique can be combined withcross polarization to increase the spectral sensitivity of rareand low-gamma nuclei such as 13C , 15 N (Ref. 19) in biopoly-mers or other organic solids. Therefore, MAS NMR tech-niques have improved to the point where complete structuredetermination is possible.10, 12, 13

As the technique of MAS spreads to the field of solid stateNMR, the concept of average Hamiltonian theory (AHT),20

which is the main theoretical tool to describe the effect of

a)Electronic mail: thibault.charpentier@cea.fr.

time-dependent interactions, was found to be less descriptiveto rotating systems, such as sample-spinning experiments.21

However, these types of experiments were found to bemore conveniently described using Floquet theory.22–24 Inthis work, we introduce the fusion of AHT and FT as pro-vided by the Floquet-Magnus expansion (FME) that can bevery useful in simplifying calculations and also for pro-viding a more intuitive understanding of spin dynamicsprocesses.

The purpose of this article is to introduce the FMEscheme to solid-state NMR, to give a general and coherentframework of the scheme, and to compare its use in solid-state NMR with other averaging approaches. Similarly to theAHT and FT theory, the primary aim of the FME approach isto provide a scheme to build an approximation of the Hamil-tonian describing the stroboscopic evolution of the systemover several periods. The FME approach is essentially dis-tinguished from AHT with its function �(t) which providesan easy and alternative way for evaluating the spin behaviorin between the stroboscopic observation points. The FME ap-proach is in fact the fusion of the two major methods usedto describe the spin dynamics in solid-state NMR: the aver-age Hamiltonian theory based on the Magnus expansion (ME)and the Floquet theory based on the Fourier expansion. Thefirst method (AHT) was developed by Haeberlen and Waughin 196820 and is appropriate for stroboscopic sampling. TheAHT technique does not satisfactorily describe the case ofMAS spectra because in this case, the signal is usually ob-served continuously with a time resolution much shorter thanthe rotor period.25, 26 Nevertheless, in a variety of cases, MAS

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044109-2 E. S. Mananga and T. Charpentier J. Chem. Phys. 135, 044109 (2011)

experiments have been successfully analyzed using AHT,21, 27

which yields an effective Hamiltonian averaged over somecycle time of a periodic pulse sequence. From its naturalformulation, AHT has been gradually applied to almost ev-ery kind of situation, sometimes abusively.21 Some examplesthat lend to the application of AHT include the use of a pre-requisite of the approach to design a frequency-modulatedanalog of TPPM and explored other possibilities involv-ing simultaneous phase and frequency modulation.28, 29 Sim-ilarly, Eden and Levitt utilized symmetry arguments30 basedon AHT to develop optimized heteronuclear decoupling se-quences involving rotor-synchronized pulse sequences whosefundamental element “C” is a 2π pulse. Therefore, despitethe emergence of alternative approaches such as the Floquettheory,22, 23 the exact effective Hamiltonian theory,31, 32 andthe Fer expansion,33 which have advantages in some circum-stances, the average Hamiltonian theory still remains of cen-tral importance in theory of multiple-pulse NMR. All ap-proaches are equivalent to first order. With the increase ofthe level of sophistication of NMR experiments, second or-der terms are of increasing importance, such as in diffusionexperiment.34

The second method (FT), which was first proposed byShirley22 and introduced to NMR by Vega23, 24 and Maricq,35

provides a more universal approach for the description ofthe full time dependence of the response of a periodicallytime-dependent system.36, 37 It allows the computation of thefull spinning sideband pattern that is of importance in manyMAS experimental circumstances to obtain information onanisotropic sample.

The Floquet theory is based on the transformation ofthe N-dimensional Hilbert space of the original probleminto an infinite-dimensional Floquet-Hilbert space, wherethe periodically time-dependent Hamiltonian becomes time-independent Floquet Hamiltonian.22, 38, 39 For numerical com-putations, this requires truncation of the infinite dimension ofthe Floquet-Hilbert space. Some aspects of a formal Floquettheory have been discussed by Sambe,36 Yajima and Kitada,40

Moore,41 and Dzyublik.42 Also, a generalized Floquet Hamil-tonian based on the quasi-energy operator have been exploredby Howland,43–45 Bunimovich et al.46 and Blekher et al.47

Hilbert space-Floquet theory was extended to Liouville space,using a Floquet Liouville supermatrix approach by Chu, Ho,Wang, and Jiang48–51 and applied by Kavenaugh and Silbey.52

The Floquet theory was further simplified by the use of themultipole basis proposed by Sanctuary,53 which exploits therotational invariance properties of irreducible tensor operatorsusing a multispin basis.29 Providing the basis for a formal de-scription of FT or AHT is not the goal of this paper, but, in-stead, the relevance and potential of the Floquet-Magnus ex-pansion scheme in solid-state NMR over the other averagingschemes such as AHT and FT.

The AHT and FT result respectively in average and ef-fective Hamiltonians that are expanded in a set of terms of in-creasing orders. These Hamiltonians are in general connectedwith stroboscopic detection schemes. In the case of AHT, thetime evolution between the detection points is not described.In single-mode Floquet theory, the stroboscopic Hamilto-nian is again connected to stroboscopic detection schemes.

However, this theory provides the option to evaluate the spinevolution between the time points of detection. In contrastto common approaches of AHT and FT, the main advan-tage of the FME scheme is to overcome the limitations ofthe stroboscopic detection schemes. In the Floquet-Magnusapproach, even when the first and second order F1 and F2

of the effective Hamiltonian are identical to their counter-parts in AHT and FT, the �1,2(t) functions provide an easyway for evaluating the spin evolution during ‘‘the time in be-tween’’ through the Magnus expansion of the operator con-nected to this part of the evolution. �1(t) and �2(t) are con-nected to the appearance of features like spinning sidebandsin MAS. The evaluation of �1(t) and �2(t) is useful espe-cially for the analysis of the non-stroboscopic evolution. Forexample, in the case of C7,54–56 for non-stroboscopic detec-tion scheme, they can be used to estimate the intensity ofthe spinning sidebands manifold in the double-quantum di-mension. Higher order effects (F3, �3(t)) can also be eval-uated using the FME approach, in a way easier than in thecase of AHT or FT. The FME scheme can also be extendedto multimode Hamiltonians for Hilbert space analysis espe-cially in the incommensurate case. To the best of our knowl-edge, we present here the first report highlighting the basicsof the FME scheme and compare this approach with the otherseries expansions. We present a generalized FME scheme,based on the importance of the boundary conditions (at theorigin of time), which provides a natural choice for the op-erators �n(0) to simplify calculation of higher order termsand allows FT to be managed in the Hilbert space. Applica-tions of the FME scheme are described and compared to pre-vious treatments based on AHT, FT, and static perturbationtheory (SPT). We also discuss the convergence aspects of thethree schemes (AHT, FT, and FME) and present the relevantreferences.

An outline of the paper is as follows. In Sec. II, we de-scribe the FME with a brief illustration of the ME and the cel-ebrated Floquet theorem (FT), which ensures the factorizationof the solution in a periodic part and a purely exponential fac-tor. We explicitly give the first contributions to the Floquet-Magnus expansion with the addition of higher order effects(F3, �3(t)) as an improvement to the two first-order terms F1,2

and �1,2(t). In Sec. III, we discuss the convergence aspects ofthe three schemes (ME, FT, and FME) and present the rel-evant references. In Sec. IV, we compare the three schemesand we extend the comparison to the Floquet-Van Vleck ap-proach and the static perturbation theory. Special emphasis isput on the choice of the periodic boundary conditions �n(0).The application of the approach to the case of a multimodalHamiltonian is also briefly discussed. Section V of the papersummarizes our conclusions

II. THEORY

A. The Floquet theorem

Floquet theory is a branch of the theory of ordinary dif-ferential equations relating to the class of solutions to lin-ear differential equations of the form (in this paper we use a

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044109-3 The Floquet-Magnus expansion J. Chem. Phys. 135, 044109 (2011)

formulation in connection with quantum mechanics but with-out loss of generality)

idU

dt= H (t)U (t), (1)

with U (0) = I as initial condition. H (t) is a complex n by nmatrix-valued function and its matrix elements are integrableperiodic functions of time t with period denoted T. The Flo-quet theorem allows one to write

U (t) = P(t) exp(−i Ft), (2)

where F and P(t) are n by n matrices. P(t) is a periodic func-tion of time with period T, i.e., P(t) = P(t + T ) and F is con-stant. In NMR, this structure is exploited in many situationsincluding time-dependent periodic magnetic fields or samplespinning, which is the focus of this paper. The structure ofU (t) Eq. (2) was exploited in two ways by Levante et al.38

The first one consists in performing a Fourier expansion ofthe formal solution, leading to an infinite system of lineardifferential equations with constant coefficients. The price ofthis approach is to handle an infinite dimension that can onlybe resolved numerically by truncation. The second approachis of perturbative nature and applied directly to the form ofEq. (2) as following:

P(t) =∞∑

n=1

Pn(t), (3)

F =∞∑

n=1

Fn. (4)

Each term Fn in Eq. (3) is fixed such that Pn(t) = Pn(t + T )to ensure the Floquet structure Eq. (2) at any order of approx-imation.

B. The Magnus expansion

The ME provides an exponential representation of the so-lution of a first order linear homogeneous differential equationfor a linear operator. It is introduced in Eq. (1) by the follow-ing form of the propagator:

U (t) = e−i�(t). (5)

With the help of the Wilcox formula,57 Eq. (1) can be rewrit-ten as

idU

dt= i

d

dt{e−i�(t)} =

{∫ 1

0e−is�(t) d�

dte+is�(t)ds

}e−i�(t).

(6)

Introducing the adjoint operator defined in terms of thecommutator as ad�Y = [�, Y ] (also known as the Liouvilleoperator), we obtain

e−is�(t) d�

dte+is�(t) = exp {−isad�} d�

dt. (7)

The integration in Eq. (6) can then be formally written as∫ 1

0exp {−isad�} ds = e−iad� − 1

−iad�

= φ(−iad�), (8)

where

φ(x) = ex − 1

x. (9)

Inserting Eqs. (6)–(9) in Eq. (1), we arrive at

φ(−iad�)d�

dt= H (t). (10)

Or, equivalently,

d�

dt= φ−1 (−iad�) H (t). (11)

Equation (11) is at the basis of the ME which is simplyobtained by introducing the expansion of φ−1(x)

φ−1(x) = x

ex − 1=

∑k

Bk xk

k!, (12)

where Bk are the Bernoulli numbers (B0 = 1, B1 = −1/2,B2 = 1/6, B3 = 0, B4 = −1/30. . . ) and the perturbative ex-pansion � (t) = ∑∞

n=1 �n (t). Then Eq. (11) reads

d�

dt=

∑k

Bk

k!(−i)kadk

� {H (t)} . (13)

Equation (13) justifies the name of exponential pertur-bation theory also used for the ME in some contexts. TheME is used in NMR spectroscopy with the AHT whichis built up on the basis of the conventional ME to findF in Eq. (2) by computing �(t = T ) from the propagatorU (T ) = exp {−i FT } = exp {−i� (T )}. Given the relevanceof the ME, criteria for the existence and the convergence havebeen extensively developed in the literature since Magnusproposal in 1954.58 In fact, Magnus was well aware that ifthe function �(t) is assumed to be differentiable, it may notexist everywhere.

C. The Floquet-Magnus expansion

Using Eqs. (1) and (2), we obtain the following inhomo-geneous first-order differential equation for P(t) :

id P

dt= H (t) P (t) − P (t) F, (14)

where P(0) = I can be assumed, but this is non-mandatoryas will be discussed later. F is an unknown constant ma-trix which is determined from the time periodicity condi-tion P(t + T ) = P(t). Using the exponential ansatz P(t)= e−i�(t) with �(t + T ) = �(t) and the same procedure forME derivation (see Appendix A), one can obtain the follow-ing equation:

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044109-4 E. S. Mananga and T. Charpentier J. Chem. Phys. 135, 044109 (2011)

d�

dt=

∞∑k=0

Bk

k!(−i)k (ad�)k

{H (t) + (−1)k+1 F

}. (15)

Note the similarity between Eqs. (15) and (13), but withthe additional constant term (−1)k+1 F. It is worth noticingthat Eq. (15) is independent of �(0) as well as Eq. (14)(derivation is given in Appendix A). As will be shown be-low, the advantage of the FME approach is its ability tomake a choice for �(0) different from the generally assumed�(0) = 0 (i.e., P(0) = I ), allowing a simplification of theperturbative calculation of �(t) and F . A choice of �(0) �= 0is equivalent to the use of a more general representation of theevolution operator (and therefore less restrictive) for Eq. (2)as

U (t) = P(t)e−i tF P+(0), (16)

which removes the constraint of a stroboscopic observation.P(t) can then be seen as the operator that introduces the framesuch that the density operator is varying under the time in-dependent Hamiltonian F. Indeed, if ρ(t) denotes the densityoperator of the system, we have

ρ(t) = P(t)e−i tF P+(0)ρ(0)P(0)eitF P+(t), (17)

which can be rewritten as

P+(t)ρ(t)P(t) = e−i tF P+(0)ρ(0)P(0)eitF (18)

or, introducing ρs(t) = P+(t)ρ(t)P(t),

ρs(t) = e−i tFρs(0)eitF . (19)

Stroboscopic observation at times t0 + nT is governed bythe effective Hamiltonian (and thus the AHT) P(t0)FP+(t0)according to

ρ (t0 + nT ) = P (t0 + nT ) e−inTF P+ (t0) ρ (t0) P (t0)

× einTF P+(t0 + nT ). (20)

Using P(t0 + nT ) = P(t0) this can be rewritten as

ρ(t0 + nT ) = e−inTP(t0)FP+(t0)ρ(t0)einTP(t0)FP+(t0). (21)

Thus, the representation Eq. (16) of the evolution opera-tor will allow us to work with a more general framework deal-ing both with the stroboscopic observation (AHT, �(0) = 0)and the effective Hamiltonian (as described by the Floquettheory, �(0) �= 0), from an unified point of view. Knowl-edge of both P(t) and F yields the stroboscopic Hamiltonian(AHT) HAH T = P(0)FP+(0) or can be obtained directly us-ing P(0) = I . As Eq. (14) is independent of �(0), Eq. (15)can serve as the basis to construct the exponential perturba-tion expansion for both �(t) and F

The relationship with the regular Magnus expansion canbe obtained from

U (T, 0) = exp(−i�(T )) = exp{−iT e−i�(0) Fei�(0)}, (22)

such that

�(T )

T= e−i�(0) Fei�(0). (23)

Again, Eq. (23) points out that it is only in the case�(0) = 0 that the FME gives the AHT as provided by theME. However the ME is limited to the construction of theAHT, whereas the FME also constructs the operator �(t) giv-ing the new opportunity to obtain the evolution of the systemin between the stroboscopic detection points.

At this stage, we follow the procedure describedin Ref. 59 introducing the perturbation expansions � (t)= ∑∞

n=1 �k (t) and F = ∑∞n=1 Fk (t) in Eq. (15). Succes-

sive order Fk terms are determined from the boundary con-ditions �k(t + T ) = �k(t). A recursive generation scheme59

can then be built as follows (n ≥ 1):

�̇n(t) =n−1∑j=0

(−i) j BJ

j!

{W ( j)

n (t) + (−1) j+1T ( j)n (t)

}(24)

The termsW ( j)n (t) and T ( j)

n (t) are given by the same re-currence:

W ( j)n (t) =

n− j∑m=1

[�m(t), W ( j−1)

n−m (t)]

(1 ≤ j ≤ n − 1) (25)

and

T ( j)n (t) =

n− j∑m=1

[�m (t) , T ( j−1)

n−m (t)]

(1 ≤ j ≤ n − 1) , (26)

but with the initial conditions

W (0)1 (t) = H (t) , W (0)

n>1 (t) = 0, (27)

T (0)n (t) = T (0)

n = Fn. (28)

Taking into account Eqs. (27) and (28) for substitut-ing the terms j = 0 in Eq. (24), we can obtain the generalformula

�n(t) = �n(0) +∫ t

0Gn(τ )dτ − tFn, (29)

with

Fn = 1

T

∫ T

0Gn(τ )dτ . (30)

The first order contributions to the Floquet-Magnus ex-pansion give (see Appendix B) explicitly

G1(τ ) = H (τ ), (31)

G2(τ ) = − i

2[H (τ ) + F1,�1(τ )], (32)

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044109-5 The Floquet-Magnus expansion J. Chem. Phys. 135, 044109 (2011)

G3(τ ) = − i

2[H (τ ) + F1,�2(τ )] − i

2[F2,�1(τ )]

− 1

12[�1(τ ), [�1(τ ), H (τ ) − F1]] . (33)

If higher order terms can be computed numerically quiteeasily with the help of Eqs. (24)–(28), we believe that sym-bolic calculations software can enable formal derivation ofhigher order terms.

III. DISCUSSION OF CONVERGENCE

A. The Magnus expansion

In general, the Magnus series does not converge unlessH (t) is small in a suitable sense.60 Indeed, the convergenceof the Magnus expansion is generally discussed in terms of aradius of convergence rc defined as the number for which thefollowing statement holds:

if∫ t

0||H (s) ||ds < rc then the Magnus expansion

converges.

Several results on the radius of convergence rc in termsof H (t) have been obtained in the literature. Pechukas andLight61 and Karasev and Mosolova62 obtained rc = log2= 0.693..., whereas Chacon and Fomenko60 got rc = 0.577.

Blanes et al.63 obtained the improved bound of rc = 1.086.

Recently, a new method was developed64 to enlarge thelargest domain of convergence of the Magnus expansion (rc

= 1.086...) previously obtained. An analytic estimate of thenew domain of convergence found was almost twice thepreceding one65 (rc = 2) and this new analytic bound wasin agreement with the numerical estimate of the convergenceradius such as no accuracy was lost in the bound. Therefore,there are more than three different convergence estimated inthe literature of Magnus expansion. These convergence esti-mates are given with their respective proofs in the referencestherein.

The latest improved bound rc = π was derived byMoan66 but in the context of the conventional Magnus expan-sion for real matrices A (t) = −i H (t). This important resultswas then generalized to matrices in the Hilbert space (thus forcomplex matrices) by Casas.67

A new version of Magnus expansion was reported re-cently by Butcher et al.68 The new scheme grows on treesand forests to reorder the terms of Magnus expansion formore efficient computation. While this scheme did not pro-vide any substantial new result to the convergence of the ME,it provides a new mean to compute Magnus expansion to thedesired order.

B. The Floquet theory

Recoupling schemes have all been extensively treatedwith Floquet theory in conjonction with the Van VleckTransformation.69–76 The Floquet theory approach has alsobeen used successfully to the study of decoupling of dipo-lar interactions.77–84 The discussion of the convergence of theFloquet theory was presented by Maricq.35 In this article, an

indirect method that took advantage of the periodicity of P(t)and the method of Picard approximations was used to producea convergent sequence for the propagator U(t). This was nec-essary to determine the convergence of the two interdependentseries P (t) = ∑

n Pn(t) and F = ∑n Fn .

C. The Floquet-Magnus expansion

The Floquet-Magnus scheme is the fusion of the Magnusexpansion and the Floquet theory. Therefore, its solution hasthe required structure and evolves in the desired group (Liegroup). Because the Floquet scheme is a convergent sequence,once the convergence is fulfilled in one period, it is assuredfor any value of time. On the contrary, in the general Magnuscase, the bound always gives a running condition. The latestimproved bound in the conventional Magnus expansion wasfound to be rc(M E) = π .67 Blanes et al.63 and Casas et al.59

showed that absolute convergence of the Floquet-Magnus se-ries is guaranteed at least if∫ T

0||H (t)||dt < rc(FME) ≡ 0.20925. (34)

This bound rc(FME) in the periodic Floquet case turnsout to be smaller than the corresponding bound rc (ME) = π

in the conventional Magnus expansion. At first glance, thiscould be understood as a failure of the result. Certainly, themethod has been given an algorithmic formulation which al-ready allows direct implementation.59, 63, 66

IV. COMPARISON BETWEEN FLOQUET-MAGNUS ANDOTHER THEORIES

We consider the most encountered case of a time-dependent Hamiltonian H (t) which is time periodic with pe-riod T. It can then be expanded in a Fourier series as

H (t) =∑

m

Hmexp(imωt) =∑m �=0

Hmexp(imωt) + H0.

(35)

With ω = 2π/T . Substituting Eq. (35) into the first contribu-tions to the Floquet-Magnus expansion Eq. (30) to Eq. (33)gives

�1(t) =∑m �=0

Hm

imωfm(t) + �1(0), (36)

with

fm(t) = exp(imωt) − 1, (37)

F1 = H0, (38)

�2(t) = −1

2

∑m �=0

[Hm,�1(0)]

mωfm(t) + i

∑m �=0

[H0, Hm]

m2ω2fm(t)

+ i

2

∑m �=0,n �=0

(1 − δm+n)[Hm, Hn]

m(m + n)ω2fm+n(t)

− i

2

∑m �=0,n �=0

[Hm, Hn]

mnω2fm(t) + �2(0), (39)

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044109-6 E. S. Mananga and T. Charpentier J. Chem. Phys. 135, 044109 (2011)

and

F2 = −i [H0,�1 (0)] + 1

2

∑m �=0

[Hm, H−m]

mω

+∑m �=0

[H0, Hm]

mω. (40)

At this stage, the features of the FME scheme dependmainly on the choice of the initial conditions. Let us con-sider the following two cases corresponding to � (0) = 0 and�(0) �= 0, respectively.

A. Case � (0) = 0

In this case, we obtain the following explicit results:

F1 = H0, (41)

�1(t) =∑m �=0

Hm

imωfm(t), (42)

F2 = +1

2

∑m �=0

1

mω[Hm, H−m] +

∑m �=0

1

mω[H0, Hm], (43)

and

�2 (t) = i

2

∑m �=0

∑n �=0

(1 − δn+m)[Hm, Hn]

m (m + n) ω2fm+n(t)

− i

2

∑m �=0

∑n �=0

[Hm, Hn]

mnω2fm (t)

+ i∑m �=0

[H0, Hm]

m2ω2fm(t). (44)

The choice of �(0) = 0 reproduces the two first orderterms of AHT,F1 and F2, and therefore corresponds to thestroboscopic detection scheme. However, the FME also pro-vides the operators �1(t) and �2(t) allowing for the evalua-tion of the evolution in between the stroboscopic points. Thirdorder terms were too cumbersome to be calculated and repro-duced here.

B. Case �(0) �= 0

As in the previous case, we first obtain

F1 = H0. (45)

As �n+1 (t) depends upon �n (t), we seek for a suitablechoice so that any increase in the number of terms should beavoided. Making the following choice of �1 (0)

�1 (0) =∑m �=0

Hm

imω, (46)

seems natural from the examination of Eq. (36). It gives atime-dependent evolution operator as

�1 (t) =∑m �=0

Hm

imωeimωt . (47)

Accordingly, at each further step of the calculation, thechoice of the contributions to �n(0) is made according to thesimple rule

∫ t

0eipωudu = eipωt

i pω︸︷︷︸�n (t)

− 1

i pω︸︷︷︸�n (0)

. (48)

Then the second order contribution G2 (t) (Eq. (32)) reads

G2 (t) = −1

2

∑m,n �=0

[Hm, Hn]

nωei(n+m)ωt

−∑m �=0

[H0, Hm]

mωeimωt . (49)

Time integration yields much simpler expressions thanEqs. (43) and (44):

�2 (t) = i

2

∑m,n �=0

(1 − δm+n)[Hm, Hn]

n (n + m) ω2ei(n+m)ωt

+ i∑m �=0

[H0, Hm]

m2ω2eimωt , (50)

F2 = 1

2

∑m �=0

[Hm, H−m]

mω. (51)

It clearly shows that the application of Eq. (48) hasgreatly simplified the expressions of �2(t) and F2. In thiscase, this allows us to pursue with the same approach for�3(t) (see Appendix B) yielding

�3(t) = 1

4

∑m,n,k �=0

(1 − δm+n)(1 − δm+n+k)

× [Hk, [Hm, Hn]]

in(n + m)(k + m + n)ω3ei(k+n+m)ωt

+ 1

2

∑m,n �=0

(1 − δm+n)[H0, [Hm, Hn]]

in(n + m)2ω3ei(n+m)ωt

+ 1

2

∑m �=0

(1 − δm+k)[Hk, [H0, Hm]]

im2(m + k)ω3ei(k+m)ωt

+∑m �=0

[H0, [H0, Hm]]

im3ω3eimωt

+ 1

4

∑m,n �=0

[Hn, [Hm, H−m]]

imn2ω3einωt + 1

12

∑m,n,k �=0

× (1 − δm+n+k)[Hm, [Hn, Hk]]

inm(k + m + n)ω3ei(k+n+m)ωt ,

(52)

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044109-7 The Floquet-Magnus expansion J. Chem. Phys. 135, 044109 (2011)

and F3 as

F3 = +1

3

∑m,n �=0

[H−m, [Hm−n, Hn]]

nmω2

+ 1

2

∑m,n �=0

[Hm, [H0, H−m]]

m2ω2. (53)

Equation (53) nicely reproduces the result of Ernstet al.71, 77

C. Similarities between theories (AHT, FT, and FME)

A quick comparison between all theories developed so farshows that the lowest-order term F1 as provided by AHT,20

Floquet theory,35, 69, 71, 85, 86 or FME are all identical. This isthe popular average Hamiltonian

H (0)AH T = H (1)

e f f (FT ) = F1(F M E) = H0. (54)

Next, the first-order term in the AHT (Ref. 69) is identicalto the first-order term in the stroboscopic detection.86 Thesetwo Hamiltonians are also identical to Eq. (43) of the FMEwith the stroboscopic detection condition �(0) = 0:

H (1)AH T = H (1)

S(stroboscopic) = F2(F M E,�(0)=0). (55)

Similarly, Maricq35 has shown the equivalence betweenthe full, non-transformed, Floquet Hamiltonian, and the AHTHamiltonian by calculating a propagator at multiples of theperiod of the Hamiltonian. In the article, it used a perturbativescheme to show that the two expansions (AHT, FT) are equalfor each of the first two orders. For the Floquet Hamiltonian,the following second order term was found by Maricq

H (2)FT (Maricq) =

∞∑k=1

1

kω{[Hk, H−k] + [H0, Hk] − [H0, H−k]}

(56)

which can be shown, after straightforward algebra, to be iden-tical to Eq. (43). Higher order terms of the Floquet Hamil-tonian by Maricq become very tricky to manipulate ratherquickly.

The highlighted similarities of FME scheme with the Flo-quet theory stem here from a judicious choice of the ini-tial conditions. As demonstrated in Subsection IV B, with�1(0) = ∑

m �=0Hm

imω, the FME approach retrieves identical re-

sults at least for the first three orders F1,2,3 than the FT andVan Vleck transformation.21, 69, 87 The FME approach is how-ever unique though it provided expressions for �1,2,3(t). InFT,35 the zeroth-order term for the Pn(t) is P0 = 1. Next,the first order term P1(t) in the FT is similar to �1(t),Eq. (42) in the FME for the stroboscopic case (�1(0) = 0).However, the direct comparison of higher order terms includ-ing �2 (t) (Eq. (44)) and P2(t) becomes impossible, sinceP2(t) is not available easily. Obviously, the expressions of�2,3(t) are obtained more easily in the FME than P2(t) fromthe FT. Therefore the FME provides a more simplistic ap-

proach for higher orders comparatively to other averagingschemes.

D. Differences between theories (AHT, FT, FME)

Although Floquet theory, also known as secular averag-ing theory, and AHT are different approaches of the sameSchrodinger equation, the results they provide are somewhatincompatible in the analysis of various multiple-pulse se-quences in NMR. In discussing the equivalence between FTand AHT, Llor87 found a supplementary non-secular term inthe first-order of AHT which is identical of F2(F M E,�(0)=0),i.e., H (1)

AH T = H (1)S(stroboscopic) = F2(F M E,�(0)=0). The answer to

that is to invalidate the use of AHT in the interpretation ofmany NMR experiments, such as sample rotation and pulsecrafting. An important point arises in the differences betweenAHT and the FT schemes: these approaches are respectivelygiven in terms of time integrals and Fourier coefficient com-binations, so, the type of time modulation of the Hamiltoniandetermines the choice between the methods to be adopted tosolve a specific problem. FME gives a unified view which en-compasses both the time-integral and Fourier expansion ap-proach. One of the appealing features of the FME scheme isits expressions for �1,2,3(t) which are not present in other av-eraging approaches, and additional terms in the scheme couldarise from the initial conditions (�1(0),�2(0),�3(0), ....).

E. Static perturbation theory versus FME

Here, we wish to revisit the static perturbation theory21

which has been shown to yield the correct form of Zeemantruncated NMR interactions without the limit of stroboscopicobservation of the AHT. This will give us the opportunity toshed a new light on the FME scheme and the derivation of acriterion for the two theories being compatible.

For sake of simplicity (but without loss of generality), weconsider the Hamiltonian

H = ω0 IZ + λ∑

m

(−1)m R2,−m T2,+m . (57)

This is a common form of Hamiltonian in solid-stateNMR. ω0 IZ is the Zeeman interaction, R2,m are the latticeparts of the internal interaction which encode its orientationaldependence with respect to the magnetic field, T2,m are sec-ond rank m-order spherical tensor describing the spin systemas defined by [IZ , T2,m] = mT2,m . The SPT in terms of theirreducible tensor operators gives the diagonal Hamiltonian(with respect to ω0 IZ )21

HS PT = ω0 IZ + λR2,0T2,0 + λ2

2ω0

∑m �=0

R2,m R2,−m

m

× [T2,m, T2,−m]. (58)

As discussed in the seminal work,21 discrepancies be-tween AHT and FT appear in the rotating frame representa-tion (or more generally in the interaction frame) where the

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044109-8 E. S. Mananga and T. Charpentier J. Chem. Phys. 135, 044109 (2011)

Hamiltonian becomes time-dependent:

H (t) = e+iω0 IZ t He−iω0 IZ t

= λ∑

m

(−1)m R2,−m T2,+meimω0t . (59)

The FME expansion Eqs. (45)–(47) yields as first orderterms:

F1 = λR2,0T2,0, (60)

�1 (t) = λ∑m �=0

(−1)m R2,−m T2,+m

imω0eimω0t , (61)

whereas the AHT (stroboscopic detection, Eqs. (41)–(43))yields

�1 (t) = λ∑m �=0

(−1)m R2,−m T2,+m

imω0(eimω0t − 1). (62)

The FME scheme Eq. (51) provides the same second or-der term as in SPT theory

F2 = λ2

2ω0

∑m �=0

R2,m R2,−m

m[T2,m, T2,−m]. (63)

This shows that FME provides an expansion in the rotat-ing frame which is in agreement with the static perturbationtheory and Van Vleck transformations. This is not the caseof the Magnus expansion. This agreement, as obtained fromEq. (46), can be easily explained by the connection that existsbetween the FME and SPT propagators as follows.

The propagator in SPT can be written as

U (t) = e−i Ht = e−i Se−i HS PT t ei S, (64)

where S is the diagonalizing matrix defined asH = e−i S HS PT ei S . When transformed back into thelaboratory frame, the FME yields the propagator

U (t) = e−iω0 IZ t × e−i�(t)e−i Ft ei�(0), (65)

which can be rewritten as

U (t) = exp(−ie−iω0 IZ t� (t) eiω0 IZ t )e−iω0 IZ t e−i Ft ei�(0).

(66)

Comparison between Eqs. (66) and (64) shows that thetwo conditions

� (0) = e−iω0 IZ t� (t) eiω0 IZ t , (67)

[IZ , F] = 0, (68)

yield the following final form of the propagator:

U (t) = e−i�(0)e−i(ω0 IZ +F)t ei�(0), (69)

which is equivalent to Eq. (64). This means that the condition

S = � (0) = e−iω0 IZ t�(t)eiω0 IZ t , (70)

has to be fulfilled for propagators Eqs. (64) and (65) describ-ing the same evolution at any time. The condition Eq. (67) isindeed satisfied by Eqs. (46), (47), (50), and (52), justifyinga posteriori our choice for � (0). In the case � (0) = 0, Eq.(67) is only true at stroboscopic point ω0τ = 1 for Eqs. (42)and (44).

F. Extension to multimode Hamiltonian

Application of FME to multimode Hamiltonian withfrequencies ω = (ω1, . . . , ωN ) is straightforward. Consider-ing the generalized Fourier expansion of the Hamiltonian(m = (m1, . . . , m N ) represents the frequency indices)

H (t) =∑

m

Hmexp(−im · ωt), (71)

we obtain

�1 (t) =∑

m ·ω �=0

Hm

i m ·ω e−im·ωt (72)

and

F1 =∑

m ·ω=0

Hm . (73)

Similarly, calculation of second order terms is straight-forward using Eqs. (50) and (51). These expressions highlightthe fact that the multimode Hamiltonian case can be easilytreated in Hilbert space with the FME.

V. CONCLUSION

In this work, we have presented and generalized theFloquet-Magnus expansion that has been useful to shed newlights on AHT, Floquet Theory, and the static perturbationtheory. The theory is based on two operators: �(t) that de-scribes the evolution within the period and F which is theHamiltonian governing the evolution at multiple of the pe-riod. A crucial parameter has been shown to be the periodicboundary condition �(0). The FME theory can be directlyconnected to the AHT method (based on Magnus expansion)for �(0) = 0 yielding F = HAH T . But, in contrast to theMagnus expansion, the knowledge of the operator �(t) allowsthe evolution in-between the stroboscopic points to be evalu-ated. Equivalence with the Floquet theory is obtained from aspecial choice of �(0) �= 0 leading also to an expansion thatis equivalent with the static perturbation theory. The generalconditions for such an equivalence have been derived, and theFME provides a more concise and intuitive approach than Flo-quet theory.

The FME provides a quick means to calculate higherorder term (here third order terms could be easily derived)allowing the disentanglement of the stroboscopic observa-tion �(t) and effective Hamiltonian F that will be useful todescribe spin dynamics in solid-state NMR and understanddifferent synchronized or non-synchronized experiments. Wemade an attempt to sketch out the mean features of the tech-nique, and hope that this paper will be helpful to describe thetime evolution of the spin system at all times. The FME offers

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044109-9 The Floquet-Magnus expansion J. Chem. Phys. 135, 044109 (2011)

a simple way to handle multiple incommensurate frequen-cies and thus open the perspectives to deal with multi-modeHamiltonian in the Hilbert space. For instance, some inter-esting problems that are amenable to the similar treatmentpresented here are Hamiltonian as involved in many of theNMR pulse sequences (rotational-resonance, dipolar recou-pling, proton decoupling, . . . ). The FME approach in solid-state NMR spectroscopy can provide new aspects not presentin AHT and FT such as recursive expansion scheme in Hilbertspace that can facilitate the devise or improvement of pulsesequence.

We summarized the convergence properties of the threeapproaches (AHT, FT, FME) that are already extensively dis-cussed in the literature and we gave the relevant references. Adifferent and extremely important question we did not tacklein this article is the possibility of enhanced performance of theFME approach which certainly deserves further attention. Thevarious fundamental questions that arise when dealing withthis approach were also not considered in this work. Morequantitative work is necessary to bring out the salient featuresof the Floquet-Magnus expansion and explore its use in solidstate NMR and in many other theoretical areas.

ACKNOWLEDGMENTS

This work was supported by the French National Re-search Agency (ANR) under the “DESIRE” project.

APPENDIX A: DERIVATION OF THE FME EXPANSION

Using the following representation of the propagator:

U (t) = P(t)e−i Ft P+(0), (A1)

in Eq. (1) we obtain

idU

dt= i

d P

dte−i tF P+(0) + P(t)Fe−i tF P+(0)

= H (t)P(t)e−i tF P+(0). (A2)

Multiplying all terms on the right hand side by P(0)leads to

id P

dt= H (t)P(t) − P(t)F. (A3)

Applying the Wilcox formula Eq. (6) to P(t)= e−i�(t)gives

id P

dt= i

d

dt(e−i�(t)) = φ(−iad�(t))

d�

dte−i�(t), (A4)

which inserted in Eq. (A3) yields

φ(−iad�(t))d�

dte−i�(t) = H (t)e−i�(t) − e−i�(t) F, (A5)

which can be transformed intod�

dt= φ−1 (−iad�) (H (t) − e−iad� F). (A6)

Using φ−1(x)ex = x + φ−1(x), we finally arrive at

d�

dt=

∞∑k=0

Bk

k!(−iad�)k (H (t) + (−1)k+1 F). (A7)

APPENDIX B: THE FME EXPANSION TO THIRDORDER

From the recursive generation formula Eqs. (24)–(28),we have the following results for the second order terms:

W (0)2 = 0, T (0)

2 = F2,

W (1)2 = [�1, H ] , T (1)

2 = [�1, F1].(B1)

The third order terms read as follows:

�̇3(t) =2∑

j=0

(−i) j B j

j!

× {W ( j)

3 + (−1) j+1T ( j)3

}W (0)

3︸︷︷︸=0

− T (0)3︸︷︷︸

=F3

+ i

2

{W (1)

3 + T (1)3

} − 1

12

{W (2)

3 − T (2)(3)

}, (B2)

W (1)3 =

2∑m=1

[�m, W (0)

3−m

] =

⎡⎢⎣�1, W (0)

2︸︷︷︸=0

⎤⎥⎦ +

⎡⎢⎣�2, W (0)

1︸︷︷︸=H (t)

⎤⎥⎦

= [�2, H ] , (B3)

T (1)3 =

2∑m=1

[�m, T (0)

3−m

] =

⎡⎢⎣�1, T (0)

2︸︷︷︸=F2

⎤⎥⎦ +

⎡⎢⎣�2, T (0)

1︸︷︷︸=F1

⎤⎥⎦

= [�1, F2] + [�2, F1] , (B4)

W (2)3 =

1∑m=1

[�m, W (1)

3−m

] = [�1, W (1)

2

]

= [�1, [�1, H]] , (B5)

T (2)3 =

1∑m=1

[�m, T (1)

3−m

] = [�1, T (1)

2

] = [�1, [�1, F1]] ,

(B6)

�̇3 (t) = −F3 + i

2{[�2, H ] + [�1, F2] + [�2, F1]}

− 1

12{[�1, [�1, H]] − [�1, [�1, F1]]} . (B7)

After time integration of Eq. (B7), we obtain the finalexpression

�3 (t) = −tF3 + i

2

∫ t

0{[�2, H + F1] + [�1, F2]} dτ

− 1

12

∫ t

0{[�1, [�1, H − F1]]} dτ + �3 (0) .

(B8)

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044109-10 E. S. Mananga and T. Charpentier J. Chem. Phys. 135, 044109 (2011)

F3 can be deducted from the above equation with T = t and�3(T ) = �3(0):

F3 = i

2T

∫ T

0{[�2, H + F1] + [�1, F2]} dτ

− 1

12T

∫ T

0{[�1, [�1, H − F1]]} dτ. (B9)

APPENDIX C: CALCULATION OF �n=1,2,3(t) AND F1,2,3FOR A PERIODIC TIME-DEPENDENT HAMILTONIAN

Our starting point is the Fourier expansion of the Hamil-tonian as

H (t) =∑

m

Hmexp(imωt) = H0 +∑m �=0

Hmexp(imωt)

= H0 + H, (C1)

with ω = 2πT . Derivation of first and second order terms is

straightforward and not reproduced here. G3(τ ) (Eq. (33)) canbe decomposed in three terms

G3 (t) = − i

2[H (t) + F1,�2 (t)] − i

2[F2,�1 (t)]

− 1

12[�1 (t) , [�1 (t) , H (t) − F1]] = A+B+C.

(C2)

After straightforward but lengthy calculations, we obtain

A = 1

4

∑m,n,k �=0

[Hk, [Hm, Hn]]

n(n + m)ω2(1 − δm+n)ei(m+n+k)ωt

+ 1

2

∑m,k �=0

[Hk, [H0, Hm]]

m2ω2ei(m+k)ωt − i[H0,�2(t)],

(C3)

B = −1

4

∑m,n �=0

[[Hm, H−m], Hn]

mnω2einωt , (C4)

C = 1

12

∑m,n,k �=0

[Hm, [Hn, Hk]]

mnω2ei(m+n+k)ωt . (C5)

Integration gives Eq. (52) for �3(τ ). Only terms A and Ccontribute to F3 as

F3 = 1

4

∑m,n �=0

(1 − δm+n)[H−(m+n), [Hm, Hn]]

n(n + m)ω2

+ 1

2

∑m,n �=0

[H−m, [H0, Hm]]

m2ω2

+ 1

12

∑m,n �=0

[Hm, [Hn, H−(m+n)]]

mnω2. (C6)

Notably, summation indices can be reordered so that thefirst term contributes to the third term of the right-hand sideof Eq. (C6). Final result is Eq. (53).

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