Expansion Appendix B. Time Ordering and the Magnus

Appendix B. Time Ordering and the Magnus Expansion

We consider the evolution of a system under the influence of its Hamiltonian htf. The system is described by its density matrix p(t), which evolves ac- cording to the von Neumann equation

ß(t) = -il*(»,P(t)l. (B-1) If #? is constant with respect to time, Eq. (B-1) may easily be integrated

formally:

p(t)= UWPMU-H», (B-2)

where

U(t) = exp[--ijr/]. (B-3) If 2tf is not constant in the time interval of interest (say, from t = 0 to t — tc),

but is constant in smaller intervals xt,..., xk,..., τη with

*i + ·" + **+ ·*· + ^τη = ' o then by repeated integration we obtain

U(tc) = exp [- « ; τJ · · · exp [- Ufk τJ · · · exp [- ijft τχ], (Β-4) where we denote by J^k the Hamiltonian operative in the &th time interval.

U(tc) as given by Eq. (B-4) equals β χ ρ [ - / ( ^ τΜ+ · · · + ^Tjk+---+Jf1T1)]

only if all J^k commute with each other. Finally, if 2tf varies continuously from t = 0 to t = tc, U(tc) may be expressed formally by

U(tc) = Texp - 1 Γ Jf(t) dt . (B-5)

Equation (B-5) is to be considered the limiting case of Eq. (B-4). T is the 173

Dyson time-ordering operator, which orders operators of greater time arguments in the expanded exponential to the left, e.g.,

(jr(t')Jf(t") if t' > t\

T{^{t')J^{t")} =

\je(t")$e(t') if t" > t\

but also T{Jf2 J f J = Τ{^^2} = J^2^i if JtT2 "acts" at later times on the system than does J^x.

Magnus Expansion

The goal of the Magnus expansion is to find an expression in the form of exp[— iFtc~\ for U(tc) as given by Eq. (B-4) or (B-5). In order to find F we start by expanding Eq. (B-4) in a power series:

U(tc) = [··■(/!)···] x - x [ - ( * ) · · - ] x -

x

r

_ ^

(-^^

^--;

-^

, „ | , (B-6)

3! ⁺ J' where (k) means that the indices of Jt and τ are k.

We consider ^kxk as a small quantity of first order. In case it is not, we simply subdivide the time interval 0, ...,tc further until J^krk is small. By rearranging Eq. (B-6) we get

U(tc) = 1

+ ( - / ) { ^1τ1+ ^2τ2 + · · · + ^ τ „ }

(-i)³

+ - j p « \ ³ + - + 3 K /12T2T12 + /22/1T22T1 + - ] + 6[\?&3 J*i2 J^i t^l^l ~f~ *"j)

+ ···. (B-7a) Note that in all products the operators are ordered in ascending order to the

left! Equation (B-7a) can be rewritten in a compact form as follows:

(-i)² (-i)^m

U(tc) = 1 + (-/){·.·} + ^ T { . . · }² + ... + L - j L T{.-.}- + ..., 2! ml

(B-7b) where {···} = { ^1τ1+ ^2τ2 + · · · + ^ τ/ Ι} .

For example, T{···}² = • • · Τ ( ^1^2 + ^2^Τ1)τ2τ1 = —23Ρ2#ιτ1τ1 as it should according to Eq. (B-7a).

Next we make an ansatz for F:

F = JF + <#⁽¹⁾ + Jf⁽²⁾ + ···. (B-8)

In this ansatz we assume tcJF,...,tcJF^(k\... to be small quantities of first, ...,Ath,... order. We shall confirm this assumption later. Now, as above, we expand exp [— iFtJ:

exp[-/Ffc] = e x p [ - / ic( ^ + J*⁽¹⁾ + i f^{( 2 )}+ · · · ) ]

= 1

[φ <2> (D <φ

+ ^ ^ | ( ^ )² + ^ ^^{( 1 )} + ^^{( 1 )}^ + (^⁽¹⁾)²

2· ( © (D φ <3)

+ J ^^{( 2 )} + jf^{( 2 )}jf + J>f⁽²⁾iF⁽¹⁾ + ···

® @ <D

(~*0³ i(^)³ + (^)²^^{( 1 )} +JP^{( 1 )}(^)² 3!

+ j f ^^{( 1}^ + («#⁽¹⁾)³ + ··

( , / c ) 4{(jpr+·

4!

+ ·.·. (B-9) The encircled numbers indicate of which order the corresponding terms are

small.

We rearrange terms in descending order of magnitude:

exp [ - « * „ ] = 1 ©

-iteJP Φ

_// c^(t) + t|£>!(iP)2 φ

etc.

(B-10)

By equating terms of equal order in Eqs. (B-10) and (B-7a) [or (B-7b)] we find

* = 7 W = 7 { ^ ι^τι + ^ 2 τ2 + - + ^ τπ} , (B-ll)

^^{( 1 )} = ^ [ T { - }²- ( i , ^ )²] = ^ | T { - }²- { · · · }²] (B-12a,b)

= -^-([■*2,Ji'l3T2T1 + [Ji'3,jri]T3T, + [ j r 3 , j r2] T 3 T2+ - ) , (B-12C)

jf^{( 2 )} = - — ( T i - l ^ i / e ^ - S / ^ C i P ^ ^ + ^ ^ i P ] ) (B-13a)

6/c

= ^ ( π · · · } ³ + 2 { . . . }³- ^ [ { . . . } Γ { . . . }² + Τ{...}Μ...}]^ (B-13b)

+ 5 { [ ^ 2 , [ ^ 2 , ^ ι ] ] τ22τ ^ (B-13c)

etc. From Eqs. (B-12) and (B-13) it is clear that icif^{( 1 )} and tc^⁽²⁾ are small quantities of second and third order, if tc i f is a small quantity of first order—

as we actually assumed—and so our ansatz is consistent.

If 3tf(t) jumps only once in the interval 0,..., tc, and ^τ1 = A, J^2τ2 — B, we find

tcJF = A + B, /cJf⁽¹⁾ = -!rlB,Al, and

It follows that

e~^iBe-^iA = cxp(-i(A + B) -\{.B,A~\ + ^ { [ A [ J M ] ] + [ [ * , 4 M ] } + -Λ (B-14) Equation (B-14) is the Baker-Campbell-Hausdorff formula.

To arrive at the Magnus formula we must subdivide the interval 0, ...,tc

more and more finely. The limiting case of Eq. (B-l 1) for all xk -► 0 is

- I f

tc Jo JT(t)dt. (B-15)

Thus «# ("^f-bar") is the average of J^(t) in the interval 0, ...,tc and is therefore called the average Hamiltonian.

Similarly, the limiting case of Eq. (B-12) is

^ ^{(1) =} ΪΓ f ^dh f ^{dh Lje(t2)} ' * ^{( , ι ) ]} · ^(B " ¹⁶⁾

The domain of integration (see Fig. B-l) ensures that everywhere the operator with the greater time argument is on the left—where it should be.

Z(t₂) dt2

FIG. B-l. Domain of integration for ^^{( 1 )}.

The limiting case of Eq. (B-l3) is

1 Γ*° f* Γ<2

&^w = ~7- dt₃ dt2 dh

orc Jo Jo Jo

x {i>r a ₃ ), w ₂ ) , ^(hm+\_w(h\ ^(t ₂ )i *?{t,)-]}.

(B-17) The domain of integration (see Fig. B-2) ensures again that everywhere the operators with greater time arguments are ordered to the left. It also takes care of the factor \ in front of the second line of Eq. (B-l3c).

For the reader who is not satisfied with our claim that Eqs. (B-15)-(B-17) are the limiting cases of Eqs. (B-11)-(B-13) we recommend that he try to recover the latter from the former in a case where 34?(t) jumps n times (n = 5, say) in the interval considered.

m)

X(t3)

FIG. B-2. Domain of integration for ^^{( 2 )}. (Rear, upper left corner of cube).

'' dt, za>

t

r, *i£/^/' Γ / 1

/ y\

/ y

^L

We proceed to indicate proofs for the special cases mentioned in Chapter IV, Section D. By comparing Eqs. (B-10) and (B-7b) for jf^{( k )} = 0, k < «, the validity of

tc<&⁽ⁿ⁾ = Γ - Τ ^^Τ{ · · · Γ^{+ 1} pf^(k) = 0 for k<n) (B-18) becomes immediately evident.

As

A ^ T { . . . r + i ^ ^ ( _ / T Λ l Λ A Λ

x jr(/

) j r ( 0 - JT(r

) jr(ii), (B-i9)

we establish Eq. (4-44), Note there is no denominator (n+\)\ in the right-hand expression of Eq. (B-19). However, the volume of integration is "only"

fe»⁺7(1+1)!.

Before we turn to the special properties of symmetric cycles we prove a lemma. Consider βχρ[/^τ„] ···exp[/^f2τ2] εκρ[}^ι^τϊ\- We call this expres- sion U(— tc) but avoid assigning any physical meaning to it. For U{— tc) we make the ansatz exp[//ic], and for/we make the ansatz

/ = R + RU>+ #*> + .... (B-20) The lemma we want to prove is

Lemma 1:

h™ = (-l)*JF<^k>.

For the sake of compactness of notation we define if⁽⁰⁾ = ^f. Furthermore let us define the symbol Jif^k+1 as the sum of all products of the type

>p<*i)jp<*2)...jp<kn) in Eq. (β-9) that contain n factors and that are small of order m = k+1. Examples are

> 3 (if^{( 0 )})³, ^r23 = jf^{( 0 )}^^{( 1 )} + iF^{( 1 )}iF^{( 0 )}, « ^³ = e^⁽²⁾.

^ 3

Correspondingly we define the symbol h^kn+1; it refers, of course, to an equation analogous to Eq. (B-9) in which the signs of all the /'s are reversed.

As

/! + £ * , = (*! + l) + (Jfc2 + 1) + ··· + (kH+l) = m = * + 1, (B-21)

i

it follows from Lemma 1—if it is correct—that

or

- ( - l )^w^^{f c + 1} = (-1)*Λ*⁺¹. (B-22)

To prove Lemma 1 consider the construction laws of iF^{( k )} and h^(k) [cf. Eqs.

(B-9) and (B-7b), and the corresponding equations with the signs of all I'S reversed]:

(— i)^k 1 (— it Ϋ

3! ³lt+1 v " " / *pk+l 2! "Jf|^{2 + 1}, (B-23)

" "(*+!)!//* ' (£+1)Λ

-⁽- ^ Α * 3^{+ 1}- | * Ι^{+ 1}. (B-24) By taking into account Eq. (B-22) we recognize that «#^(fc) and B^k) differ, term

by term, by ( - 1)\ Therefore, we recover Lemma 1 for k, having made use of it only for k' < k. Recall that the lowest value of« in Eqs. (B-23) and (B-24) is two, and that the largest value of kt occurring in J f |^{+ 1} and h^k2+ x is kmax = k— 1 [see Eq. (B-21)]. As Lemma 1 holds obviously for k = 0 (iF = h) we conclude, by induction, that it holds for all k.

Now consider symmetric cycles,¹⁴⁶ i.e., cycles for which Jif(t) = Jif(tc — t).

An obvious consequence of the symmetry of the cycles is U'Htc) = e x p ^ r j . - e x p ^ v l

= exp [iFs ic] [consequence of Eq. (B-4)]

= exp \i3tfn τ J · · · exp [/«#! τ{] (because the cycle is symmetric)

= U(-tc) (by definition)

= exp [ifs /c] (by definition).

The indices s on F% and/s stand for "symmetric."

We conclude Fs=fs, or

fc = 0

and with the aid of Lemma 1

Σ jr^ik) = Σ ^R(k)>

= 0 k = 0

j p w = (-1)* j?<*>, which requires

jf^(fc) = 0 for A: odd.

From now on we follow closely Wang and Ramshaw.¹

For the sake of completeness we also consider briefly antisymmetric cycles, i.e., cycles for which 3P(t) = -3f(tc-t).

U(tc) = exp[-«feTj exp[-üfB_ t τ„_ J ··· exp[- Uf2τ2] exp[-ί*± τ{\

= expCüfiTi] exp[^2τ2] •••exp[/Jf„_1Tn_1] expftJi^J

(consequence of antisymmetry)

or

exp[-iF./e] = exp[ + iF./J (a stands for antisymmetric). It follows that

Fa = -Fa = 0,

which means that for antisymmetric cycles M vanishes together with all its correction terms. Whereas symmetric cycles are of great practical importance in high-resolution NMR in solids, antisymmetric cycles are not, for obvious reasons.