Equivalence of the FloquetâMagnus and Fer Expansions to

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On the Equivalence of the Floquet-Magnus and Fer Expansions to Investigate the Dynamics of a Spin System in the Three-Level System Eugene Stephane Mananga J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b01723 • Publication Date (Web): 12 Jul 2017 Downloaded from http://pubs.acs.org on July 25, 2017

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The Journal of Physical Chemistry

On the Equivalence of the Floquet-Magnus and Fer expansions to Investigate the Dynamics of a Spin System in the Three-Level System Eugene Stephane Mananga

1,2,3*

1*

PH.D Program Physics & PH.D Program Chemistry, The Graduate Center, The City University of New York, 365 Fifth Avenue, New York, NY, 10016, USA 2* Department of Applied Physics, New York University, 6 Metrotech Center, Brooklyn, New York 11201, USA 3* The City University of New York, BCC, Department of Engineering, Physics, and Technology, 2155 University Avenue, CPH 118, Bronx, New York 10453, USA

Abstract 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 timeevolution 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, 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. * Corresponding author. Telephone +1 646 345 4613; Fax: +1 718-289-6403 E-mail address: [email protected] or [email protected] (Eugene S. Mananga)

I.

Introduction

The topic of the article opens a way to an infinite number of suggestions. However, it is very important to remember that the considered methods have recently found new major areas of applications such as topological materials. Researchers, dealing with those new applications, are not usually acquainted with the achievements of the magnetic resonance theory, where those methods were developed more than thirty years ago. They repeat the same mistakes that

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were made when the methods of spin dynamics and thermodynamics were developed in the past. This article will be very useful not only for the NMR community but for the new communities in several younger fields. It will be very useful for scientists working in different directions. The attempt to gain a theoretical understanding of the concept of spin dynamics in solid-state nuclear magnetic resonance has initiated important advances towards the quest for more selective transient phenomena. One of such advances consists in the analysis of a class of molecular two-photon processes occurring under transient conditions using density-matrix formalism. The evolution of a quantum system under a time-dependent Hamiltonian is of major interest in various fields of spectroscopy, physics, and chemistry such as atomic and molecular spectroscopies, nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR)1,2. Solving a time-dependent Schrodinger equation is a central problem in quantum physics in general and solid-state NMR in particular. For this purpose, two main perturbation-expansion schemes are available: average Hamiltonian theory (AHT)3-7 and Floquet theory810 . The former was developed by John Waugh 3 from more recent Lie-algebraic concepts while the latter was developed more than a century ago for solving differential equations. These two approaches have been widely used within the NMR spectroscopy for designing sophisticated pulse sequences and understanding of different experiments, thus attracting constant interest in spin dynamics, quantum optimal control, quantum metrology, and quantum information11-17. Interestingly, it has been recognized that Floquet theory also plays an important role for the three-level system18,19. For instance, Ho and Chu20 have shown how to use the SU(3) generators and many-mode Floquet theory in order to obtain perturbative results in absence of dissipation in three-level systems in intense bichromatic fields. Furthermore, Unanyan et al. investigated coherent suppression of tunneling in coherent three-level systems21. The investigation of the spin dynamics of quadrupole nuclei poses a theoretical challenge because the dimension of the density matrix increases with I as (2I+1)(2I+1). The strength of the quadrupolar coupling for such spin systems is often greater or the same order of magnitude than the perturbing RF Hamiltonian. Thus implementing a desired evolution is challenging in both theory and practice15. If a two-level subsystem is connected in a multi-level system by selective irradiation with a week rf field, the two-level subsystem behaves to a good approximation like a spin ½. Irradiation of a spin-1 system with two weak rf fields (for instance modified Rabi oscillation frequency) at the two transitions (1-2; 2-3) is equivalent to applying a radiation field to the total spin without quadrupolar interaction18. It is important to consider these points in the understanding on the equivalence of the two mathematical approaches affected by the number of energy levels of the target system. For the investigation of the three-level systems in this paper, we choose the continuous rf irradiation for simplicity reason.


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The present work compares the two developing approaches that describe the behavior or spin dynamics in solid-state nuclear magnetic resonance: the Floquet-Magnus expansion (FME)22-26 and the Fer expansion (FE)26-29. We investigated the orders to which the FME and Fer expansion approaches are equivalent or different for the three-level system18. Unlike the two-level atom, the three-level atom exhibits a much abundant diverse effects in its interaction with the electromagnetic field, with illustration including laser cooling, population transfer and loss-free pulse propagation19. Our primary motivation for this study is the extension of theories in NMR, from AHT, Floquet theory to FME and Fer expansion. To be specific, we have applied the FME and Fer expansion to investigate the dynamics of spin systems subject to the three-level system. We presented a relationship between the two approaches. While both approaches are powerful methods, there are subtle differences among them. As the propagator from the Floquet-Magnus expansion 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 three-level system is the renowned scheme to describe double quantum transitions and coherence in nuclear magnetic resonance spectroscopy18,30-32 and optics18,33-35. Various authors have recognized the three-level system as favorable candidate to exploit coherent control36 in solid state structures such as semiconductor quantum dots. The three-level system finds its roots in optical spectroscopy with the development of monochromatic tunable laser sources. It offers a vast array of new physics to be investigated such as in NMR where the authors Hatanaka and Hashi31 studied the excitation and detection of coherence between forbidden levels of magnetic dipole transition in three-level spin system by multi-step processes. Furthermore, using the same experimental system (Al in Al2O3), the authors observed transient nutations and spin and stimulated echoes associated with two-quantum transition in multi-levels NMR systems. Also, Brewer and Hahn33 investigated the three-level case and coherent transients for a class of molecular two-photon processes using density-matrix formalism. The problem of double quantum (DQ) coherence is of major interest in quantum physics in general and solid-state NMR in particular. The three level systems can be used for many physical systems in addition to NMR spectroscopy17,18. In the following parts, we investigated the Hamiltonian of a nuclear spin system of noninteracting nuclei with I = 1 using the two approaches FME and FE. The paper is organized as follows. In Sect. II and III, we presented the theoretical formalism of the FME and FE including the background of the calculations of average Hamiltonian and the propagator, respectively. In Sect. IV, we presented the discussion on the convergence of the FME and FE. In Sect. V, we presented the background of the three-level system with the double quantum coherence. In Sect. VI and VII, we presented the applications of the FME and FE to the three level system, respectively. In Sect. VIII, we presented numerical data and In Sect. IX, we presented the comparison between FME and FE. Sect. X is devoted


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to the discussion and conclusion. We also presented an extended appendix containing very useful formula, detailed derivations and calculations of the results.

II.

The average Hamiltonian and the propagator of the FME

Arguably, the most salient difference of the Floquet-Magnus expansion22,23 may be that it expands a propagator in the form of a more general representation of the evolution operators as

U (t )  P(t )e itF P  (0)

(1) which removes the constraint of a stroboscopic observation. P(t) can be seen as the operator that introduces the frame such that the density operator is varying under the time independent Hamiltonian F. The FME is obtained by representing the solution of the time dependent Schrödinger equation

dU (t )  iHU (t ) dt

(2)

in the form of Eq. (1) and using the following exponential ansatz

P(t )  exp  i(t ) (3) where the function (t ) is the argument of the operator P(t ) . Introducing the expansions

(t )    n (t )

(4)

n

and

F   Fn ,

(5)

n

the FME expansion can be summarized as t

 n (t )   n (0)   Gn (u )du  tFn

(6)

0

where the first functions are defined in references22,23. The Eq. (6) includes two operators  n (t ) and Fn independently from each other. Indeed, the periodicity conditions  n ( C )   n (0) (7) where is the period of the modulation H ( C  t )  H (t ) (8) defines as

Fn 

1

C

C

G

n

(u )du

(9)

0

Such that we are free to choose the operators  n (0) i.e. the boundary conditions. At first glance, the choice  n (0)  0 (i.e. P(0)  1 ) appears as the most simple. As was shown in our previous work23, in this case the FME reduces to the


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Magnus expansion (ME) which validity is restricted to stroboscopic observation, as was discussed in numerous paper16,37-47. A much better choice, without any restriction on the observation time, is given by the general rule C



n

(u )du  0

(10)

0

which was shown to simplify higher order terms in the Fn expansion. For example, for the second order F2 , its general form is

F2 

1

C

C

C

C

 G (u)du   2  H (u),  (u)du  2  F ,  (u)du i

i

2

1

1

C 0

0

1

(11)

C 0

The second term cancels, thus yielding

F2  

i 2 C

C

 H (u),  (u)du

(12)

1

0

the well know results23

F1  H 0

(13) t

1 (t )  1 (0)    H m e imt dt  0 m0

F2 

1 H m , H m   2 m0 m

Hm

 im e

imt

(14)

m0

(15)

whereas the choice

 n (0)  0

(16)

gives the ME t

1 (t )  and

F2 

H 0 m0

m

e imt dt 

Hm

 im (e

imt

 1)

(17)

m0

H , H  1 H m , H m   0 m .  2 m0 m m m0

(18)

The difference between the two F2 can be understood from Eq. (1) written as

U ( C , t )  e i ( 0) e iF C e i ( 0)  e i H C

(19)

such that, H , as given by the ME is linked to the MFE through the relationship

H  ei ( 0) Fe i ( 0) .

(20) Thus the second term in F2 in the ME simply arises as the lowest order term in the above transformation. However, this relationship only holds at stroboscopic times. The same average Hamiltonian governs the dynamics between periodic times,

U ( C  t , t )  e i (t ) e iF C ei (t )  e i H (t ) C


(21)

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with

H (t )  ei (t ) Fe i (t )

(22)

In the context of nuclear magnetic resonance, the important characteristics of the operator is to properly relate and explain the modulation sidebands or the decoupling sidebands in the case of heteronuclear decoupling. Let us recall that the regular Magnus expansion as described in the references23,24,48

U ( C )

F,

C

(23)

corresponds to a particular case, (0)  0 , of a more general representation of the FME

U ( C )

C

 e i ( 0) Fe i ( 0) .

(24)

with the choice of, (0)  0 . The calculation of the nth - order average Hamiltonian for,  n (0)  0 , must include the contribution of the term 1 (0) .

III

The Average Hamiltonian and Propagator of the FE

Recently, Takegoshi and co-workers26 contributed an exhaustive description of the Magnus, Floquet, and Fer expansions in solid state nuclear magnetic resonance. In this section we recapitulate the results of Takegoshi for the Fer expansion without going into detail. The propagator expressed in the form of an infinitive expansion for the FE can be written as







U ( C )   exp  i C H Fer , n 0 th

(n)

(25)

and the (n-1) - order average Hamiltonian is found to be

H

( n 1) Fer



1

C

C

H

n 1

(t )dt 

0

i

C

Fn ( C )

(26)

where the function Fn (t ) is given by t

Fn (t )  i  H n1 (t ' )dt '

(27)

0

L

From the left-running Fer expansion ( H n (t ) ), the (n-1)th-order average Hamiltonian is found to be26 L ( n 1)

H Fer



i

C

Fn ( C ) .

(28-a)


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In the following, we used the right-running notation for simplicity. We define t

Fn, j (t )  i  H n1, j (t ' )dt '

(28-b)

0

and the corresponding average Hamiltonian is defined as ( n 1, j )

H Fer



i

C

Fn , j ( C ) .

(28-c)

From the early beginning, the FE27 was formulated in the general form by using the approach

H (t )   H meimt .

(29)

m

However, as soon as dealing the non-stroboscopic point, formula becomes quickly intractable (or reproduce formula to lowest orders in (say ) to have an EXACT comparison between the two scheme). We obtained the following results by calculating the first three order of the Fer expansion with the time-dependent Hamiltonian H (t ) expanded in a Fourier series, ( 0) H FE  H0 , (1, 0 ) H FE 

H m , H m  

H m , H 0 

2m m m 0 H , H m , H 0   H m , H n , H 0   2i  0  3m 2 3mn 2 m 0 m 0 n  0 m0

(1,1) H FE



(30) (31) (32)

It can be noticed, the FME shares some similar characteristics with the FE which are in contrast to the ME where a single exponential factor is preserved. Both the FME and the FE involve more than two exponential factors necessary to allow the expansion in Eq. (21) to clasp at any time.

IV.

Convergence

The applicability of the FME and FE are intimately related to the problem of convergence. This problem has played a crucial role in the field of solid-state NMR and spin dynamics6. These expansions are divergent and the physical nature of their divergences is discussed in the following paragraphs. The authors Casas, Oteo, and Ros investigated a sufficient condition for the absolute convergence of the FME in ref.22. From Casas and co-workers point of view, a sufficient condition on K (t ) is required such that the convergence of the series



n

is guaranteed in the whole interval t  0, T . K (t ) is defined by


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t

K (t )   k ( )d ,

(33)

0

where k (t ) is a non-negative scalar function such that H (t )  k (t ) . Numerical study of the above condition suggests that as n   , the series converges if T



H (t ) dt  K (T )    0.20925 .

(34)

0

Considering the problem of convergence of the Magnus expansion, it is necessary to mention the Magnus paradox75. The divergence of the Magnus expansion leads to paradoxes in the spin dynamics of solid-state NMR and in quantum informatics. Recently, Kuznetsova and co-workers76 investigated the physical factors responsible for the paradoxes stemming from the divergence of the Magnus expansion and the ambiguity of the Floquet Hamiltonian. These paradoxes should be taken into account in the theoretical analysis of the dynamics of spin systems in periodic magnetic field. Blanes and co-workers53 succinctly studied the convergence of the Fer expansion by looking for conditions on H (t ) which ensure Fn  0 as n   . Using a similar approach as above22, Blanes et al.53 derived a convergent radius of the Fer expansion to be 0.8604065, i.e. T



H (t ) dt  K (T )  0.8604065 .

(35)

0

The authors53 point out that, additional properties of H (t ) allow an improvement of this result. This result widens the range K (t )  0.628 (36) 27 originally given by Fer using a slightly different argument. In a similar vein, Zanna52 showed that a similar result holds for the symmetric Fer expansion by proving that the symmetric Fer expansion converges uniformly in the interval 0, T such that





T



H (t ) dt  0.60275 .

(37)

0

The calculated radius of the convergence of the symmetric Fer expansion (   0.60275 ) by Zanna52 is smaller than the calculated radius for the classical Fer expansion (   0.8604065 ) by Blanes53. However, the bounds introduced by Zanna52 are not optimal and could be improved. The fascinating point of the FME approach is that the rate of convergence of the FME is faster than the Fer '

expansion in the sense that, for a prescribed precision, one needs more Fk s (for


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Fer expansion) than  k s (for Floquet-Magnus expansion) even if from the computational point of view, the Fer expansion could require more work than the FME. The convergence of these approaches is extensively discussed in the Literature6,22-24,27, 29,37,43,45-48,52. '

V.

Background of the Three-Level System; DQ Coherence

Let us restrict the investigation to the spin, I=1. The Hamiltonian of this threelevel system is given by18

1   H 0  0 I Z  Q  I Z2  I ( I  1) . 3  

(38)

where Q is the quadrupolar interaction and 0 is the resonance frequency. Working in the fictitious spin-½ operators in the Zeeman basis,  1 , 0 , and  1 , written as 1  a b  b a , 2 i    a b  b a , 2 1   a a  b b , 2

I Xab 

(39-a)

I Yab

(39-b)

I Zab

(39-c)

with

I Xab  I Xba ,

(40-a) (40-b) I  I , (40-c) I  I , We have a, b  1,2,3 for the general three-level sub-system of a multilevel system. The above Eq. (38) can be rewritten in terms of fictitious spin-1/2 operators as a b Y a b Z

b a Y b a Z

H 0  20 I Z13 





2 Q I Z12  I Z23 . 3

(41)

Consider that we irradiated the three-level system by an rf field of strength B1 

1 in a frame rotating with the frequency  around the Z-direction. 

The total Hamiltonian can be written as

H  2(0   ) I Z13 





2 Q I Z12  I Z23  3

21 ( I 1X2  I X23 ),

(42)

where 1 is the applied field and the last term expresses the secular part of the rf Hamiltonian. Working in the transformed Hamiltonian, we have


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~ H (t )  e  i

21tI X2  3

Hei

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21tI X2  3

 I Z12 cos( 21t )   13   I sin( 2  t )   2 Y 1  2 I 1z 3 cos( 21t )  I Y23 sin( 21t )  Q   2 3 3  I cos( 2  t )  Z 1   2 3   I Y sin( 21t ) 





(43)

 I 1X2 cos( 21t )  I Y13 sin( 21t )  I X23 cos( 21t )   21   2 3  I Y sin( 21t )  where the transformations of the operators given in the appendix. Consider the transition 0

 Q .

I Z12 , I Z13 , I Z23 , I 1X2 , I X23 are irradiation

near

a

satellite

Let us assume the action on the 2-3 transition and the

approximation, 1  Q . The terms related to the transition from level 1 to level 2 are ignored. The above Hamiltonian can be written

2 ~ H  2(0   ) I Z23  Q I Z23  3 1  2(  Q ) I Z23  21 I X23 3

21 I X23 (44)

Considering the two-level subsystem (2-3) with the modified Rabi oscillation frequency 21 and we worked with the transformed Hamiltonian i.e. we turned on the rf field for a time t at the (2-3) transition which corresponds to performing a

21t during the transition (2-3), we have

rotation





 ~ H (t )  2(  Q ) I z23 cos( 21t )  I Y23 sin( 21t )  3 23  21 I X cos( 21t )  I Y23 sin( 21t )



VI.



(45)

Application of FME to the Three Level System

For the FME, we can calculate the first order term of the Floquet operator T

1 ~ F1   H (t )dt  0 T 0

(46)

and the associated transformation results from


10

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1 (t )   sin( 21t ) I X23  (1  cos( 21t ))(1  

2

1

( 

Q 3

) sin( 21t ) I

2

1

( 

Q 3

)) I Y23 ,

(47)

2 3 Z

where we choose 1 (0)  0 . High order terms can also be obtained after lengthy calculations and the details of calculations are shown in the appendix. The second order terms are computed to be T 1 ~ F2  H (t ), 1 (t ) dt  2iT 0





1   (  Q )   1 1 1 2 3 3  (  Q ) 1  2 1 ) I Z23  I X  (  Q  3 1 3 2     and t i ~  2 (t )    H ( ), 1 ( ) d  tF2 20



(48)





Q  1 2 (  )  sin( 21t ) I Z23 1  2 1 3 



Q Q 1 2 (  )(1  (  )) sin( 21t ) I X23 3 1 3 21

(49)

The results without Rabi oscillation frequency 21 are presented in the appendix. If we consider the two-level subsystem (2-3) with the modified Rabi

21 , the third order term is computed as

oscillation frequency











i ~ i F2 , 1 (t )dt  1  1 (t ), 1 , (t ), H~ (t ) dt . H (t ),  2 (t ) dt    2T 0 2T 0 2T 0 After lengthy calculation, we obtained the following results for the third Floquet operator F3  F3 A  F3B  F3C where T i ~ F3 A  H (t ),  2 (t ) dt 2T 0 , 2 2  Q  2 3 1 Q   Q  2 3 2  2 2  1 1  (  ) I X  (  ) 1  (  ) I Z 8 3  4 3  1 3   1 T

F3 



T

T

(50) order (51)




(52)

11


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F3 B  

i 2T

Page 12 of 37

T

 F ,  (t )dt 2

1

0 2

2

 Q  2 3 1 Q   Q  2 3  2 2  1 1  (  ) I X  (  ) 1  (  ) I Z 1 3  2 3  1 3  2 2  and T 1 ~ F3C   1 (t ), 1 , (t ), H (t ) dt  2T 0

, (53)

1





31 4





  Q  2 3 2 (  ) I X 1   3 1   2

1  Q   3      2  3   2 2   1 

2

  Q  I Z23      3   (54)

Finally, we add the three terms for the third order of the FME 2

3 1   2  Q  23 F3   1  (1  3 ) 1 1   I 4 3 Q 1  X 2   2

1   2  Q  23  1  2 Q (1  3 ) 1  (1  3 )  I 4 Q  3 Q 1  Z





(55)

The propagator derived from the FME can be written as

U ( C )  exp  i C ( F1( FME )  F2( FME )  F3( FME )  ...)

(56)

where Fn (i ) corresponds to the Floquet operator of order n ( n  1,2,3,... ) and i corresponds to the FME scheme. The first-order term in the FME is 0 ( F1( FME )  0 ) which indicates a good decoupling in the first-order. But, the second-order average Hamiltonian for the FME is different to zero (Eq. (48)) which specifies that the application of the propagator to a density operator must carefully be done. Furthermore, the propagator (Eq. (56) is constructed to comprehend the effects from higher-order to lower-order. The third-order contains two terms and the magnitude of each spin operator can be plot versus the ratio

Q . These terms can be larger in the upper order and thus must be 1

considered. The size of the magnitude of the third-order FME constitute a disadvantage for the FME approach as shown in the numerical analysis (section VIII). The evolution operator allows obtaining the density matrix of the spin system that has evolved from the equilibrium density matrix due to the RF irradiation. Simulations are based on the propagation of the density matrix and only closed form exist for time-independent or self-commuting Hamiltonian.


12

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VII.

Application of FE to the Three Level System

Here, we examine the Fer expansion by using the three level system. Using the Hamiltonian in the rotating frame defined by the Eq. (45), we first calculate the function F1 (t ) as t

~ F1 (t )  i  H (t ' )d '  i sin( 21t ) I X23  i (cos( 21t )  1)( 2 0

 i 2 sin( 21t )

( 

Q

1

3

)

( 

Q

1

3

)

 1) I Y23 (57)

I Z23

and

F1 ( C )  0

(58)

The zero-order average Hamiltonian is calculated as ( 0) H Fer 

i

C

F1 ( C )  0

(59)

The major term in the first-order term is 1 ~ H 1, 0 (t )   F1 (t ), H (t ) 2 Q  (   )  Q 3  (  ) 2  1  (1  3  1 





(  2

1

Q 3

)

  ) cos( 21t ) I X23  

(60)

Q 1 Q        (  ) cos( 21t ) I Z23 3 3 2   and the corresponding term

Q   (  )   3  1)t ( 2   t 1 Q   2 3 F2, 0 (t )  i  H 1, 0 (t ' )d '  i (  ) I X Q 3 0   (  ) 3 ) sin( 2 t )  1 (1  2 1   1 21     1 1 1 1  i (  Q  1 )t  (  Q ) sin( 21t ) I Z23 3 3 2 21   (61) which leads to the calculation of the expression of the first-order average Hamiltonian


13


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(1, 0 ) H Fer i

Page 14 of 37

F2, 0 ( C )

C

  1  (  Q ) 1  3  

1  Q )  1 1 2 3 3 1 ) I Z23  I X  (  Q  1 3 2  

(  2

(62)

After lengthy calculation, we calculate the expression of H1,1 (t ) given in the Appendix, 1 2 (63) H1,1 (t )  F1 (t ),2 H1, 0 (t )   F1 (t ), H1,0 (t ) . 3 3

The corresponding orders F2,1 (t ) and F2,1 ( C ) are also given in the Appendix. After lengthy calculation, we obtain the second-order average Hamiltonian in the Fer expansion. The results without Rabi oscillation frequency 21 are presented in the appendix. If we consider the two-level subsystem (2-3) with the

21 ,

modified Rabi oscillation frequency Hamiltonian in the Fer expansion is (1,1) H Fer i



the

second-order

average

F2,1 ( C )

C

2  2  Q   1  Q  23 1 1  (1  3 ) (1  3 )  1  I 3 3 Q 1   Q 1  X 2 

(64)

2

 1 2  Q   23  Q 1  (1  3 ) )I  (1  3 3 3 Q 1  Q Z  After arrangement, we obtain  2 2  Q (1,1)  H Fer  1 1  3 3  1 

  1  3  Q 

1  2  Q   1 1  3  3  1

   

  1  3  Q 

2

 Q   1

    1  1  Q   2  1  

  1  3  Q 

  1  3  Q 

  2 3  I X   

 2 3 I Z  

(65) The propagator derived from the FE can be written as



 

 



( 0) (1, 0 ) (1,1) U ( C )  exp  i C ( H Fer ) exp  i C ( H Fer ) exp  i C ( H Fer ) ... .


(66)

14

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The first-order term in the FE is 0 ( H Fer  0 ) which indicates a good decoupling in the first-order. But, the second-order average Hamiltonian for the FE is different to zero (Eq. (62)) which specifies that the application of the propagator to a density operator must be done carefully. Furthermore, the propagator (Eq. (66)) is constructed to comprehend the effects from higher-order to lower-order. The third-order (Eq. (65)) contain similar terms compare to the FME (Eq. (55)). (0)

The plot of the magnitude of their respective spin operator versus the ratio

Q 1

can reveals the advantage of one approach over the other. At first glance, the two propagators (Eqs. (56) and (66)) look different. However, the propagator for the FME and that of the FE are identical to the first and second orders. But, the third-order average Hamiltonian for the FE looks different to the FME which specifies that the application of the propagator to a density operator must carefully be done. Furthermore, the two propagators (Eqs. (56) and (66)) are constructed to comprehend the effects from higher-order to lowerorder. The obvious different 3rd – order average Hamiltonians for the FME and FE may invoke the similar discrepancy found for the average Hamiltonian theory and the secular averaging theory (SAT) where Llor2 derived a new formulation of a simple operator for the SAT, allowing the equivalence with the AHT to be explicitly given. The choice between the FME and FE approaches is a matter of convenience depending, for instance, on the complexity of the terms in the upper order average Hamiltonians. Furthermore, the rate of convergence of the FE is faster than that of the FME in the sense that in terms of computational point of view, the FME could requires more work than the FE.

VIII.

Numerical data

In this numerical study, we make use of an important notion of linear algebra and functional analysis called “normed vector space”73. We use the Euclidean norm (n )

also called magnitude of the n-order average Hamiltonian ( H ). Using the most general representation of the spanning property, the average Hamiltonian can be explicitly written as, (n)

H FE , FME  a( , Q , 1 ) I X23  b( , Q , 1 ) I Y23  c( , Q , 1 ) I Z23 , (67) where the numbers

a, b, c are called the coordinates of the nth order average

(n )

Hamiltonian, H , with respect to the three level spin-1 operator basis I i23 , i  X , Y , Z . These coordinates are uniquely determined. The

  basis I  is a finite-dimensional. To handle infinite-dimensional spaces, we 23 i

must generalize the finite basis sets to include infinite basis sets. In this study, H

(n )

, is associate to a vector, hence has a basis. All order of the average


15


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( 0)

(1)

Page 16 of 37

( 2)

Hamiltonian ( H  H  H  H  ... ) can be associated to a vector with a basis. Every order of the average Hamiltonian (i.e. vector) can be expressed as a linear combination of the three level spin-1 operator basis I i23 , i  X , Y , Z in a unique way. If the basis is ordered, then the coefficients in this linear combination

 

(n )

yield coordinates of the n-order of the average Hamiltonian ( H ) relative to the basis. For many reasons, it is opportune to deal with an ordered basis. For instance, when working with a coordinate representation of a vector, it is usual to speak of the first or second coordinate, which makes sense only if an ordering is specified for the basis73. The Euclidean norm is expressed as the square root of the inner product of, H H

(n)

(n )

and itself,

 a2  b2  c2 .

(68)

For analysis convenience, sometimes the magnitude of the square of H

H

(n ) 2

(n )

,

, can be used, to explore the dynamics of a spin system in the three-level

system. The concept of Euclidean norm is very important in linear algebra, functional analysis, and related areas of mathematics. The Euclidean norm has played a crucial role in the field of mathematics and it is used here to address the numerical comparison between the FE and FME. From the above mathematical consideration and analysis, we obtained the following equation for FE and FME,

F3( FME )

1 H

(1,1) Fer

1

1 1  2  1  1  4 3 2  

1 2  1  x  3 3  

2

 x  

2

9  x2

1   x 1 2  x 2  2  2  x 1 3  

(69) 2

(70)

where

x

Q   1 . and we used the approximation 3 Q 1


16

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a)

b)

Fig. 1: a) Plot of the absolute value of the third-order magnitude of the

b)

Plot of the third order magnitude of the ratio FME and the 2

nd

assumption, 

Q FME Vs. FE 1

FME

1

and

FE

1

Vs.

Q 1

.

rd

. The magnitude of the 3 order

order average Hamiltonian FE are given in the Eqs. (69) and (70). We made the

 Q , in generating these data.

a)

b)

Fig. 2: Plot of the absolute value of the amplitudes of the 3rd order FME and the 2nd order average Hamiltonian FE which are given in the Eqs. (55) and (64). We made the assumption, 

 Q , in

generating these data. a) x-components of the spin Operator – I b) z-components of the spin Operator – I


17


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Page 18 of 37

Fig. 3: Plot of the ratio of the amplitudes of the of the x-components of the spin operator-I for the rd

3 order FME to the 2

nd

order average Hamiltonian FE which are given in the Eqs. (55)

and (64). We made the assumption,

IX. IX.1.

  Q , in generating these data.

Comparison between FME and FE Similarities between FME and FE

A quick comparison between both theories shows that their lowest-order terms, (0) F1( FME ) and H Fer are identical. This result corresponds also to the popular

average Hamiltonian ( 0) ( 0) F1( FME )  H Fer  H AHT .

(71)

The second order terms are also identical, (1, 0 ) F2( FME )  H Fer

(72) An important point is to compare the propagators with the first two-order average Hamiltonians. Previous results indicate that the two theories are also equivalent for the CW decoupling and rotary resonance recoupling where the first two orders were found identical24,29. However, for most of the experiments encountered in NMR and physics, the two expansions (FME and FE) are not equivalent. A simple physical illustrative example is a driven harmonic oscillator treated by Casas et al.22,49 or more recently by Goldman and Dalibard50 while exploring methods to generate synthetic spin-orbit couplings and magnetic fields


18

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in cold-atom setups in the context of the Paul trap (a particle moving in a modulated harmonic trap).

IX.2.

Differences between FME and FE

Although the FME and FE are different approaches of the same Schrodinger equation (Eq. (2)), the results they provide are somewhat incompatible in the analysis of various multiple-pulse sequences in NMR. When the first order terms are identical for both theories, more likely the upper order terms (second, third, etc,..) differs largely for both theories. For instance, our recent work shows that the second and third order terms in the FME are different to the first and second order terms, respectively, in the FE for the Bloch-Siegert shift, i.e.

1 (1, 0 ) H Fer  F2( FME )  1 I Y (73) 4 (   RF  0 ) as well as many other examples such as the Cross Polarization (CP) experiment24,29 and the driven harmonic oscillator22,49 where the second-order term in the FME is different to the first-order term in the FE, i.e. (1, 0 ) H Fer  F2( FME )

(74) Due to the difference of approaches, it is important to validate the use of each method for a fine choosing and appropriated specific case in the interpretation of NMR experiments involving sample rotation or pulse crafting. The type of time modulation of the Hamiltonian must be the premium criterion to consider for the choice between both approaches to be adopted to solve each specific problem. The basics exponentiation identity does not hold for these experiments. Hence, the two expansions (FME and FE) are not equivalent unless a strategic modification is made on the basics exponential theorem. The choice of the initial conditions (  n (0) ) will play an important role in making the connection between the FME and the FE. The first important result to be recognized is that the 0thorder average Hamiltonian for the FME and FE are identical ( 0) F1( FME )  H Fer

(75)

Let us discuss two key points envisaged by simple inspection of the above equations. The first key point may be realized if we compare the propagators with the 0th and 1rst - order average Hamiltonians, i.e.

i C F1( FME )  F2 ( FME ) 

U ( C ) FME  e and

U ( C ) Fer  e i C H Fer e i C H Fer (0)

( 1, 0 )



(76) (77)

In general, the above two exponentials are not equivalent unless the experiments present certain physical particularities such as the three-level system, the CW


19


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Page 20 of 37

decoupling and the rotary resonance recoupling which are of particular interest in NMR. In a recent report29, we presented various forms of the Fer expansion such as the Fer-Wilcox expansion51 and the symmetric Fer methods52. Zanna52, recently proposed a self-adjoint version of the Fer factorization in the form

U (t0  t )  ei1 (t )ei 2 (1) ...ei 2 (t )ei1 (t )U 0

(78) which can be implemented as time-symmetric Lie-group numerical integrators with essentially the same computational cost as the conventional Fer methods29,52,53. The symmetric Fer methods can be constructed (up to order 8) by expressing  i (t ) in terms of Magnus operators  k . The second key point is about the relative sizes of the average Hamiltonians in the Fer and FME expansions. The relative sizes of the average Hamiltonians for the FE is studied in reference26. For the FME, the initial conditions (  n (0) ) play a major role in the determination of the relative size of its average Hamiltonians. The function  n (t ) is related to the width of the bands for the spinning sideband signals. The average Hamiltonian for the FME is represented by “ Fn ” which is the intensities of spinning sidebands. An appealing feature of the FME is connecting both expressions,  n (t ) and Fn , into two dependent equations: Eqs. (6) and (9). This unique characteristic is only available in the FME, and neither the average Hamiltonian theory, nor the Fer expansions have such an aspect, or similar relation between the average Hamiltonian and another function.

X.

Discussion and Conclusion

Having presented the expansions, average Hamiltonians and propagators of the FME and FE approaches, we are now in position to discuss both approaches in the three-level system. The discrepancy between the FME and FE approaches arises at higher orders. For instance, we see that the third order expansions of both schemes looks slightly different. Fig. 1. a) shows the plot of the absolute  FME FE value of the third-order magnitude of the and versus Q . The function 1 1 1 FME

is always larger

FE

which indicates a better performance of FE compared 1 1 to FME. Furthermore, the graphical results (Figs. (2) and (3)) show that the absolute value of the amplitude of the x- and z- components of the spin operator I for the 3rd order FME has a magnitude that is greater than the magnitude of the 2nd order average Hamiltonian FE. The third order term of the FME (Eq. (55)) is larger than the 2nd-order term of the Fer expansion (Eq. (64)). Figs. 1. b) and (3) show that under the assumption of strong irradiation ( 1  Q ), the FE is by far perform better than the FME for the spin system in the three-level system. From


20

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the computational effectiveness point of view, this tells us that the rate of convergence of the FE is faster than that of the FME, for a prescribed precision, one needs more

 n ' s (FME) than Fk ' s (FE).

From the computational point of view, in the sense of spin dynamics with fastoscillation Hamiltonian, the FME requires more work than the FE. Hence, the characteristics of the problem at hand might eventually dictate the method to be used. The self-adjoint version of the Fer factorization proposed by Zanna 52 can be written in the form U (t0  t )  ei1 ( t )eiV ( t )ei1 ( t )U 0 (79) with

eV (t )  ei 2 (t )ei 3 (t ) ...ei 3 (t )ei 2 (t )

(80) From the case of the most encountered time-dependent Hamiltonian H (t) which is time periodic, It can be seen that, the second order of the Fer expansion (1,1)

( H FE ) is more similar to the second order expansion of the van Vleck theory5458 . This can be explain by the fact that, the Floquet-Magnus and van Vleck expansions are related with each other by a unitary transformation,

FFME  eiVVT (t0 ) FVVT eiVVT (t0 )

(81)

leading to the following relationship between their respective arguments  i (t ) ,

i  FME ,VVT ,

ei FME (t )  eiVVT (t )eiVVT (t 0 ) .

(82) In revanche to the above formulation of the symmetric Fer expansion, we can state that the FME is a finite " non  symmetric" Fer transformation with three terms expressed as

U (t )  ei1 (t )eitF ei ( 0)

(83) The expansions are consistent with these results. Compared to the FME, the terms appearing in higher-orders of the FE have simpler form and the number of them is smaller. Thus, one can obtain them easily. The high-frequency expansion plays an essential role in explaining the mechanisms of spin dynamics which take place in NMR and spin physics, when using the FME and FE approaches. From the operator propagator point of view, the FME can be considered as a particular case of the FE. This remarkable theoretical finding has a major implication in numerical simulation such as in NMR where the signal obtained with the FME approach will be truncated compared to the full spectra obtained with the aid of the Fer expansion. The study of high-order terms has gained more momentum in the past decade, and we believe that the FE will be more helpful than the FME to analyze such experiments. At a certain point, rather than being competitive, the FME and FE expansions can also be considered as complementary. Generally, the level of success of each theory depends on the type of NMR experiment or physical problem addressed. As mentioning in the


21


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Page 22 of 37

recent physics reports24,29, combining two or more theories such as FE + FME, FE + BW, FME + BW, FE + FME + BW, could lead to a more general and unique framework for treating the time-dependent Hamiltonian of spin dynamics in NMR. This remains one of the most pressing NMR problems of our time. The BrillouinWigner (BW) theory was extensively discussed by Mikami and co-workers56. The advantage of the BW theory is that it is not only efficient to derive higher order terms, but it is even possible to formally write down the whole infinite series expansion, as compared to many other approaches such as the van Vleck degenerate perturbation theory. It is worth noting that the theoretical framework has significantly changed throughout NMR history. Starting from large spin systems using a spin temperature description59, NMR has evolved to small spin systems which can be described by quantum mechanics60-63. Nowadays, the technique of NMR is wellestablished and has been driven by exciting and developing theoretical contributions from quantum physicists and mathematicians. The mathematical structure has progressed substantially over time to explain and modernize the implementation of RF pulses and their effects on spin systems. Therefore, the advances in simulation software for spin dynamics have tremendously stimulated the progress and expansion of NMR theoretical and abstract notions62-74. The finding of a relationship between the Floquet-Magnus and Fer expansions, and its implications, will be of inherent interest to fellow specialists working on this and related topics.

Appendix A

Some Useful Formula

A1.

e i

21tI X2 3 13 i 21tI X2 3 Z

 I Z13 cos( 21t )  I Y23 sin( 21t )

(84-a)

e i

21tI X2 3

I Z12 ei

21tI X2 3

 I Z12 cos( 21t )  I Y13 sin( 21t )

(84-b)

e i

21tI X2 3

I Z23ei

21tI X2 3

 I Z23 cos( 21t )  I Y23 sin( 21t )

(84-c)

e i

21tI X2 3

I 1X2 ei

21tI X2 3

 I 1X2 cos( 21t )  I Y13 sin( 21t )

(84-d)

e i

21tI X2 3

I X23ei

21tI X2 3

 I X23 cos( 21t )  I Y23 sin( 21t )

(84-e)

I e

A2.

I

ab X



, IYab  iI Zab

(85-a)

And cyclic permutation.


22

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


I I I

   0   2i I



a c X

, I Xbc  I Yac , I Ybc 

ac Z

, I Zbc

a c X

, I Ybc

I

a c X

, I Ycb  

I

a c X

, I Zbc

a c Y

, I Zbc

a b r

, I sd c

I I



  

i ab IY 2

(85-b) (85-c)

a b X

, and

(85-d)

i ab IX 2 i   I Yac 2 i  I Xac 2  0 , r,s = X, Y, Z

(85-e) (85-f) (85-g) (85-h)

I Zab  I Zbc  I Zca  0

(85-i)

A3.

I  , I    2I Z

(86-a)

1 1 I  e it  I  e it 2 2 1 1 I X sin(t )  I Y cos(t )  I  eit  I  e it 2i 2i 1 I X cos(t )  I  e it 2 1 I Y sin(t )  I  eit 2 1 I X sin(t )  I  eit 2i 1 IY cos(t )   I  e it 2i I X cos(t )  I Y sin(t ) 

A4.

(86-b) (86-c) (86-d) (86-e) (86-f) (86-g)

Calculation of the Third Order of the FME

The third order term is computed as T T i ~ i F2 , 1 (t )dt  1 F3  H (t ),  2 (t ) dt    2T 0 2T 0 2T





  (t ),  , (t ), H (t )dt .

T

~

1

1

(50)

0

In the following, we are presenting the calculations and the results with

frequency, 1 , while in the text, the results are obtained by using the Rabi


23


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Page 24 of 37

oscillation frequency 21 . After lengthy calculation, we obtained the following results for the third order Floquet operator F3  F3 A  F3B  F3C (51) where

i F3 A  2T

 H (t ), 

T

~

2



(t ) dt

0 2

2

 Q  2 3 1 Q   Q  2 3 2  2 2  1 1  (  ) I X  (  ) 1  (  ) I Z 8  3 4 3  3 1 1     a) The calculation without Rabi oscillation frequency

F3 A

i  2T

 H (t ), 

T

~

2

21

, (52)

gives:



(t ) dt

0

Q  2  2  1 1  (  ) 8  3 1  



2

 sin( 4 2 )  1  I X 4 2   



Q   Q 2  ,  1 2 2  1  cos(4 2 ) 1  (  ) 1  (  )  IY 32 1 3   1 3  

(87)

2

Q  Q  1  sin( 4 2 )  2  1  ) 1  (  ) I Z  (  4 3  1 3  4 2 

F3 B  

i 2T

T

 F ,  2

1

(t )dt

0

, 2 2  Q  2 3 1 Q   Q  2 3  2 2  1 1  (  ) I X  (  ) 1  (  ) I Z 1 3  2 3  1 3  2 2  (53) 1

a) The calculation without Rabi oscillation frequency


21

gives:

24

Page 25 of 37

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


F3 B

i  2T

T

 F ,  (t )dt 2

1

0

Q   2  1 1  (  ) 3  2 2  1

2

1

 sin( 2 2 )  1  I X 2 2   





Q  Q 2 12  ,   2 (  ) cos(2 2 )  1 (  )  1   IY 41  1 3  3 2   1

(88)

2

Q  Q   sin( 2 2 )  1 2  (  ) 1  (  ) 1  IZ 2 3  1 3   2 2  F3C  

1 2T

  (t ),  , (t ), H (t )dt

T

~

1

1

0

   1 ( I 2  I 4  I 6 ) I X  ( I1  I 5  I 8  I10  I12 ) I Y  ( I 3  I 7  I 9  I11 ) I Z  4 The terms parts:

(54)

I1 , I 2 , I 3 ,..., I12 are the integrals calculated in the following

21

b) The calculation without Rabi oscillation frequency

I1 

.

Q   2 21 1  (  )  sin 3 ( 21t )dtI Y  3 1  0

gives:

C





Q  4 2  1  (  ) 1  2 cos 6 ( 2 )  3 cos 4 ( 2 ) I Y 3 1 3 

Q   2 I 2  21 1  (  )  3 1  

2

C

 sin

2



,

(89)



( 21t ) 1  cos( 21t ) dtI X

0

  3 (  2 cos (  2 ) sin(  2 )   2      2 Q  1  (  )  sin( 2 ) cos( 2 )   2 ) I X 1 3     8 3 3  cos ( 2 ) sin ( 2 )   3  C Q I 3  2(  )  sin 2 ( 21t ) cos( 21t )dtI Z 3 0 16 1  (  Q ) cos 3 ( 2 ) sin 3 ( 2 ) I Z 3 3 21

’


,

(90)

(91)

25


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

I4 

2 2

1

( 

Q 3

C

)

2

 sin

2

Page 26 of 37

( 21t ) cos( 21t )dtI X

0

16 1  (  Q ) 2 cos 3 ( 2 ) sin 3 ( 2 ) I X 2 3 31

,

(92)

Q  C sin( 21t ) cos( 21t )   2 I 5   21 1  (  )   dtI X 1 3  0  sin( 21t ) cos 2 ( 21t )  (2 cos 2 ( 2 ) sin 2 ( 2 )    , Q   2  2   1  (  )  (1  4 cos 6 ( 2 )  6 cos 4 ( 2 ) I X   3 1    3  2   3 cos ( 2 ))

Q   2 I 6   21 1  (  )  3 1  

 cos(

2

C

(93)



21t )  2 cos 2 ( 21t )  cos 3 ( 21t ) dtI X

0

cos( 2 ) sin( 2 )    2   ( 2 sin( 2 ) cos 3 ( 2 )  sin( 2 ) cos( 2 )  Q   2 I X  21  (  )  1 4    3   2 )  cos(  2 ) sin(  2 )( 4 cos (  2 ) 1   3    4 cos 2 ( 2 )  3)    I 7  2( 



Q 

2

Q 

) 1  (  ) 3  1 3 

, (94)

cos( 21t )  2 cos 2 ( 21t )  dtI Z 0  cos 3 ( 2 t ) 1  

2

C

  cos( 2 ) sin( 2 )    3  ( 2 sin( 2 ) cos ( 2 )    2 ) cos( 2 ) 2   sin(  Q    4 1 2 IZ (  Q ) 1  (  )   2 )    3  3 21 1   1  cos( 2 ) sin( 2 ) *  3    4 ( 4 cos ( 2 )    2  4 cos ( 2 )  3) 


, (95)

26

Page 27 of 37

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


I8  

2 2

1

( 

Q  C cos( 2t1 ) sin( 21t )   2 ) 2 1  (  )  dtI Y 3  1 3  0  cos 2 ( 21t ) sin( 21t )

Q

cos 2 ( 2 ) sin 2 ( 2 )   1 6  Q   (1  4 cos ( 2 )   2 1 2  IY  ( ) 2 (  Q ) 2 1  (  )  3  1 3 1 3  4   6 cos ( 2 )   3 cos 2 ( 2 ))    C Q  Q  sin( 21t ) cos( 21t )  2 I 9  2(  ) 1  (  )  dtI Z 3  1 3  0  sin( 21t ) cos 2 ( 21t )

,

cos 2 ( 2 ) sin 2 ( 2 )   , 1 6   Q   (1  4 cos ( 2 )  4 1 2 IZ ( )(  Q ) 1  (  )  3   3  3 21 1    6 cos 4 ( 2 )    3 cos 2 ( 2 ))   

I10 

2 2

1 2

( 

Q 3

C

)

2

 sin

2

,

(98)

2

sin 2 ( 21t )  0  sin 2 ( 2 t ) cos( 2 t )dtI Z 1 1     , 3 2 ) sin( 2 )  2 cos ( Q     4 1 2 (   Q ) 1  (  )   sin( 2 ) cos( 2 )   2  I Z 3 1 3   21   4 3 3  cos ( 2 ) sin ( 2 )  3 

I 11  2( 

I12

Q 

Q  2 ) 1  (  ) 3  1 3 

Q

2

C

Q  C 3  2  (  ) 1  (  ) sin ( 21t )dtI Y 1 3  1 3  0 2 2

(99)

2





Q   2 2 1 2  ( ) 2 (  Q ) 2 1  (  ) 1  2 cos 6 ( 2 )  3 cos 4 ( 2 ) I Y 3 1 3  3 1   c) The calculation with the modified Rabi oscillation frequency I1 

(97)

( 21t ) cos( 21t )dtI Y

0

1  2( ) (  Q ) 2 cos 3 ( 2 ) sin 3 ( 2 ) I Y 1 3



(96)

Q  C 3  2 21 1  (  )  sin ( 21t )dtI Y  0  3 1  0


, (100)

21 are: (101)

27


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Q   2 21 1  (  ) 1 3  

I2 

2

C

 sin

2



Page 28 of 37



( 21t ) 1  cos( 21t ) dtI X23

0

(102)

Q   2  1  (  ) I X23 1 3   2

I 3  2(  I4 

2 2

1

Q 3

( 

C

)  sin 2 ( 21t ) cos( 21t )dtI Z  0

(103)

0

Q

C

)

3

2

 sin

2

( 21t ) cos( 21t )dtI X  0

(104)

0

Q  C sin( 21t ) cos( 21t )  23  2 I 5   21 1  (  )   dtI X  0 3  0  sin( 21t ) cos 2 ( 21t )  1 Q   2 I 6   21 1  (  ) 1 3  

cos( 21t )  2 cos 2 ( 21t )   23 dtI X 0 cos 3 ( 2 t )  1 

2

(105)

C

(106)

 Q  2 3  2  2 1  (  ) I X  3 1   2

cos( 21t )    2 I 7  2(  ) 1  (  )   2 cos 2 ( 21t )dtI Z23 3  1 3  0  3  cos ( 21t ) 

Q 

Q 

2

C

(107)

 Q  2 3  4 1 2 (  Q ) 1  (  ) I Z 3 1 3  21  2



I8  

2 2

1

( 

Q  C cos( 2t1 ) sin( 21t )  23  2 ) 2 1  (  )  dtI Y 3  1 3  0  cos 2 ( 21t ) sin( 21t )

Q

0 (108) I 9  2( 

Q  sin( 21t ) cos( 21t )  23 2 ) 1  (  )  dtI Z 3  1 3  0  sin( 21t ) cos 2 ( 21t )

Q 

C

0

(109)

I10 

2 2

1

( 

Q 3

C

) 2  sin 2 ( 21t ) cos( 21t )dtI Y23  0

(110)

0


28

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I 11

sin 2 ( 21t )  2 3  2(  ) 1  (  )   dtI Z 3  1 3  0  sin 2 ( 21t ) cos( 21t )



 Q  2 3  1 2 (  Q ) 1  (  ) I Z 3 1 3  21 

I12

Q 

Q 

2

2

C

2

2

Q  C 3  2  (  ) 1  (  )  sin ( 21t )dtI Y23  0 1 3  1 3 0 2 2

Q

2

(111)

(112)

A5.

Calculation of the Second-order Average Hamiltonian in the FE                i    Q sin( 21t ) I Z23     21    2 2 i   Q  cos( 21t )  1 I Y23  i 21tI X23 , H 1,1 (t )   F1 (t ), H 1, 0 (t )    3 3  21     i   2 3   2 (  Q ) sin( 21t ) I Z     1      1  i (   )(cos( 2 t )  1) I 23    Q 1 Y     2 1    2  2  3 2  3   i 21tI X ,(  Q ) cos( 21t ) I Z        (   ) sin( 2 t ) I 23  2 I 23    Q 1 Y 1 X     (113) 1   Q 2  2  2 cos( 21t )  21t sin( 21t ) I X23  H 1,1 (t )   3 21









    Q  2   sin( 21t )  sin( 21t ) cos( 21t )    2 21   I 2 3      Q    Y 3  2   t  t cos( 2 t )  2 t 2 sin( 2 t )  1 1 1  2 1      Q  2   1  2 cos( 21t )  cos 2 ( 21t )    2 21   I 2 3     Q    Z 3 2 2    t sin( 21t )  21t cos( 21t )  2 1 

















(114)


29


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Page 30 of 37

t

F2,1 (t )  i  H 1,1 (t ' ) dt ' 0 t  1 Q Q  2 ( (   )  1 ) sin( 21t ) dt  (  ) *   3 3 2 1 2 0    iI Y t   3   sin( 21t ) cos( 21t ) dt   0 

Q Q  2  2  iI Z  (  )  1 (  )* 3 3 3  1  2

t t Q  2   t  2 2 2 cos( 2  t ) dt  cos ( 2  t ) dt  dt   iI X  (  )  1 *   1 1   3 0 0  0  3  1 

Q Q 2  1 ( (  )  1)  (cos( 21t )  1) dt  (  )  cos 2 ( 21t )dt   3 3  2 1 0 0 t

t

t Q 2  Q   2 2 2 )  cos( 21t ) dt   iI Y ( )(  ) 1  (  ) * 3 0 1 3 1 3   3  t t  sin( 2  t ) dt  sin( 21t ) cos( 21t ) dt   1  (115) 0 0 

 ( 

Q

a) The calculation without Rabi oscillation frequency F2 ,1 ( C ) 



2 iI Y sin 2 ( 3

2 )(

1

( 

Q 3

are:

)  1)

 2 Q   2 2 )   iI Z 1  (  ) * 1 3    3 1  4  sin( 2 ) cos( 2 )  ( 2 cos 3 ( 2 ) sin( 2 )   Q  21 2  1  (  )  3  2  sin( 2 ) cos( 2 )   2 )    1   



Q 2 (  ) cos 2 ( 3 21

2

21

2 iI X 3

2

2 ) sin 2 (

Q  1  2 (1  (  )) *   1 3  2    2 2 sin( 2 ) cos( 2 )  ) (   21 1    Q   Q  2 1 (  )   (  )(2 cos 3 ( 2 ) sin( 2 )  1  1 3  3  21     sin( 2 ) cos( 2 )   2 )     2 Q   (  ) sin( 2 ) cos( 2 )   3 21     Q

Q  2 2 2  iI Y (  ) 2 1  (  31 3 1 3 

 2 sin 2 ( 2 )    1 ) 2  cos 2 (  2   1 


2 ) sin 2 (

    2 )  

,(116)

30

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and finally we obtain,

(1,1) H Fer

   sin( 2 2 ) 2 1 (  Q  1 )(   ) 3 1 2 21        2 1 1 1  1  (  Q )  1  (  Q )(cos(2 2 ) sin( 2 2 ) I X 3  1 3 3    2 21   sin( 2 2 ) 1    2  2 )  (    ) Q   3 2  1  

    1 1 1 (  Q  )(1  cos(2 2 )   3 21 2        1 1 2 2 Q 2 Q  1  (  Q ) sin 2 ( 2 2 )  (  ) (  (  )) * I Y 3  2 21 3 3 1  2 3  1     1  1  2 ( 2 )(1  cos(2 2 )  2 sin (2 2 ))  1    2  sin( 2 2 ) 2       1 2 2 Q Q IZ  (  )  1 (  )  1 3  1 3 3     2 2 (cos(2 2 ) sin( 2 2 )  2 2 )  2  (117)  

d) The calculation with the modified Rabi oscillation frequency

21 are:

2

Q  Q   2  2  2 F2,1 ( C )   iI Z 1  (  ) (  )   3  3 3 1  1 1    Q   1 Q Q   2 2 2 2   iI X 1  (  )  (1  (  ))( )  (  ) 3 3  2 1 3 1 1 3   1 2  2  Q     1  Q    23   1  3   1  3  i 1  1   I  3  3  1  Q   Q  X 2  1  2 3

 2  Q  i 1  3  1 

  1  3  Q 

   

2

 Q   1

  1  3  Q 

(118)

 2 3 I Z  

and finally we obtain,


31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(1,1) H Fer 

  1 3

i

C

Page 32 of 37

F2,1 ( C )

    1 2 1     Q  23 )  2  (   Q  1 )( 1 3 1 3  I X (   Q )  1  2 2 21    3 1    (2 2 ) 

Q Q  1  2  (  )  1 (  )   2 I Z23  3  1 3 3  2



2  2  Q  1 1  3 3  1 

1  2  Q   1 1  3  3  1

  1  3  Q 

  1  3  Q 

   

2

      1  1  Q 1  3    Q 2  1     Q   1

  1  3  Q 

(119)   2 3  I X   

 2 3 I Z  

Acknowledgments The author acknowledges the support from the CUNY RESEARCH SCHOLAR PROGRAM-2017 and THE NEXT BIIG THING INQUIRY GRANT 2017. He also acknowledges fruitful discussion with Dr. Bingwen Hu. The contents of this paper are solely the responsibility of the author and do not represent the official views of the NIH.

Conflict of Interest The authors confirm that this article content has no conflict of interest.

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Module of FME and FE per ω1 Vs. ωQ/ω1 7000000 6000000 Magnitude of FME and FE per ω1

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Equivalence of the FloquetâMagnus and Fer Expansions to

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Equivalence of the FloquetâMagnus and Fer Expansions to