WKB calculation of adiabatic spin dynamics

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WKB calculation of adiabatic spin dynamics

N. Papanicolaou

To cite this version:

N. Papanicolaou. WKB calculation of adiabatic spin dynamics. Journal de Physique, 1988, 49 (9),

pp.1493-1505. �10.1051/jphys:019880049090149300�. �jpa-00210830�

WKB calculation of adiabatic spin dynamics

N. Papanicolaou (*)

Department ^of Physics, University ^{of Crete,} and Research Center of Crete, 714 09 Iraklion, Greece

(Requ le 11 avril 1988, accepte le 18 mai 1988)

Résumé.

Nous employons ^la méthode BKW pour calculer directement ^la limite adiabatique ^{et inclure} systématiquement ^des corrections non adiabatiques. ^Nous présentons des résultats détaillés pour la précession

de spin ^{dans un} champ magnétique arbitraire lentement variable. Dans le cas particulier ^d’un champ périodique, ^nous retrouvons le résultat bien connu de Berry. ^{Nous le} généralisons ^en incluant deux corrections

non adiabatiques.

Abstract.

The WKB method is employed ^{for a} direct calculation of the adiabatic limit and a systematic

inclusion of nonadiabatic corrections. Detailed results are presented ^for spin precession ^{in an} arbitrary slowly varying magnetic ^{field. In the} special ^{case of a} periodic ^field we recover the well-known result of Berry ^and

extend it to include two nonadiabatic corrections.

Classification

Physics ^Abstracts

03.65

42.50

1. Introduction.

Adiabatic processes have played ^an important ^role

in several problems of classical and quantum physics.

Nevertheless, ^some special features of the adiabatic limit became apparent only ^{after the recent work of} Berry [1] ^{and its} subsequent applications ^{to a} variety

of physical situations. The aim of the present paper is to discuss a simple method for the calculation of the adiabatic limit which allows for a systematic

inclusion of nonadiabatic corrections.

We consider spin precession ^{in a} time-dependent magnetic field described by the Hamiltonian

where the components ^{of S are} the usual spin operators corresponding ^to arbitrary ^total spin S2

s (s ⁺ 1). ^{The field B is} conveniently paramet- rized by spherical coordinates with respect ^{to a fixed} reference frame, ^{as is indicated} schematically ⁱⁿ figure 1. Both the magnitude B ^{and the} angular

variables 0 and _cp may vary with time. By ^conven- tion, ^{the field at t}

0 is contained in the xz plane.

We also assume in figure 1 that the field is ^a periodic

function of time with period ^T.

(*) ^{Also at the} Department ^of Physics, Washington University, ^St. Louis, MO 63130, U.S.A.

Fig. ^1.

Conventions concerning the time evolution of a

periodic magnetic field. The field at t

0 is contained in the xz plane.

The approach followed in the present paper ^is based on a well-known variant of the WKB expan- sion [2, 3] ^{and was} outlined in a recent short communication [4]. Ordinarily, ^{WKB is used to}

obtain ^a semiclassical approximation ^{to a} quantum theory ^{as an} expansion in powers of the Planck constant h. Here we are not expanding ^{in h but}

rather in inverse powers of ^a long time scale which could be loosely identified with the period ^{T. In} fact,

the method applies ^{to a} wide class of slowly varying magnetic fields which are not necessarily periodic.

WKB as it is used here provides ^a direct method for the calculation of the adiabatic limit and the deri- vation of nonadiabatic corrections.

In the interim, ^the author became aware of a more

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049090149300

recent paper of Berry [5], ^{of some} work of Garrison

quoted in the above reference, ^{and of a} paper of

Nakagawa [6], where the issue of nonadiabatic corrections is studied by different methods. Here

we elaborate the WKB calculation described in reference [4] ^{in order to} provide complete results for

spin precession ^{in a} time-dependent magnetic ^field.

These results could also serve as a prototype ^for analogous calculations in other systems ^{of current} interest. We should warn the reader that some of the conventions adopted ^{in the} present paper differ from those of reference [4].

The problem is formulated in section 2 where it is shown that the study ^of arbitrary spin s is reduced to the spin-1/2 ^case by group-theoretical arguments.

Actually, ^{such a} reduction has been known to be

possible ^{for a} long ^time, but the method of section 2 is based on a holomorphic realization of the spin algebra ^{which is} interesting ^{in its own} right. ^The

main point of this work is contained in section 3 where we carry ^{out a} detailed WKB calculation for

an essentially arbitrary slowly varying magnetic

field. Section 4 summarizes ^our conclusions together

with some suggestions ^for applications ^{to related} problems.

2. Formulation of the problem.

Because the Hamiltonian is linear in the spin operators, the solution for arbitrary spin s may be reduced to the spin-1/2 ^case by group-theoretical arguments [7-9]. ^The simplest derivation is based on

coherent spin ^states and may be found in the work of Perelomov [10]. ^Yet, the actual simplicity ^{of that} approach ^is somewhat obscured in the above work because of ^a certain ambivalence between the use of coherent states or a holomorphic realization of the

spin algebra. Therefore, ^{it seems} appropriate ^to present ⁱⁿ this section ^a brief discussion of the

holomorphic realization as it applies ^to spin preces- sion in ^a magnetic ^field.

As usual, the elements of the canonical basis for total spin s ^{are denoted} by Is, J.L > ^with

J.L = - s, ..., s. Coherent states are then defined from

where ^z is an arbitrary complex variable and z its conjugate. ^An important property ^{of the} coherent states is the resolution of unity

where the integration ^{extends over} the entire com-

plex plane. Now, let I t/1> ^{be an} arbitrary spin ^state’

and

its realization as a function of the complex ^variable

z, which will be referred to as the holomorphic

realization. The scalar product between any ^two

wavefunctions cp (z ) and 4/ (z) may be inferred from the completeness ^relation (2.2) :

The action of some spin operator Q ^{on a wavefunc-}

tion t/1 (z) may be determined from

Applying this relation for Q

S3 ^{and Q}

^{S± -} Sl ^± i S2, ^and using well-known properties ^{of the}

coherent states, ^we find that the spin operators ^are realized in terms of the differential operators

At first sight, S3 appears ^{not to} be hermitean and S- not to be the hermitean conjugate ^{of S+.}

However, ^{the correct} hermiticity properties ^are

recovered by invoking the scalar product (2.4). ^One

may verify directly ^{that the} operators (2.6) ^{close the} spin algebra and that the Casimir invariant (S3 )2 ⁺

(1/2) (S+ ^{S- + S-} S+ ) ^is identically equal ^to s (s + 1 ). Therefore, (2.6) provides ^a restriction of the spin algebra ^{to a} subspace with definite total

spin. It is also worth noting ^that (2.6) ^{reduces to the} Dyson-Maleev realization of the spin algebra ^with

the identifications a/az = a ^{and z = a * where}

a and a * are Bose operators.

To complete this construction ^we consider the action on a wavefunction qi ^{of a} group element of

SU(2) specified by

Denoting by Rg ^{the operator acting on w we find}

that

a relation that summarizes all information concern-

ing ^the representation theory of the rotation group.

As an illustration, consider the elements s, 03BC > ^of

the canonical basis. It is not difficult to see that the

coherent state (2.1) may be expanded ^as

Therefore, ^the holomorphic realization of s, J.L) ^is

the simple ^monomial

and the irreducible representations ^{of the} spin algebra ^{are realized} by polynomials ^{in z with} degree

smaller than 2 s + 1. The action of a group ^trans- formation on t/J SJL ^is given by (2.8), ^i.e.

The right-hand ^{side of} (2.11) ^{is a} polynomial ⁱⁿ

z of degree ^{2 s.} Reexpressing ^this polynomial ^{as a}

linear superposition of the monomials .p S#-, ^{leads to}

the Wigner formula for the computation ^{of the}

rotation matrices in an arbitrary irreducible repre- sentation.

The preceding formalism is especially well suited for the study of Hamiltonians which are linear in the

spin operators. Using spherical variables for the

magnetic ^field, Hamiltonian (1.1) is written as

In view of the holomorphic realization of the spin operators (2.6) ^the corresponding Schrodinger equation ^is given by ^the first-order partial differential

equation

In practice, (2.13) ^{must be} solved for the wavefunc-

tion w = qf (z, t ) with initial condition

where 1/10 (z) ^{is some} known , function. ^{We should}

mention that the Planck constant has been consis-

tently suppressed, but it would have droped ^{out of} (2.13) anyway because the Hamiltonian is linear in the spin operators.

The mathematical problem posed by (2.13) ^and (2.14) ^can be solved with the method of character- istics. The characteristic equation ^is

to be supplemented by

These are ordinary differential equations ^whose general solution is obtained as follows. Equation (2.15a) ^{is solved} with initial condition z (t

0)

zo to yield ^a solution of the form

which is then inserted in (2.15b) leading ^{to an} ordinary differential equation for qf ^{at fixed} zo. Let

qi (zo, t ) ^be the solution of the latter equation satisfying ^{the initial} condition w (zo, t

0) = t/J 0 (zo). The solution of the original ^{initial value} problem ^is given by ip (zo (z, t ), t ) where zo= zo (z, t )

is calculated by inverting (2.16).

The characteristic equation (2.15a) ^is independent

of the specific spin value s, ^{a fact} already ^known

from the early ^{work of} Majorana [7]. ^{Also note that} (2.15a) ^{is a Riccati} equation which becomes more

transparent when it is viewed as a system ^{of two} linear equations :

A simple calculation shows that the complex ^function

solves (2.15a). Evidently, ^the system (2.17) ^{is ident-}

ical to the spin-1/2 equation ^with C+ ^{and C_}

representing ^the amplitudes along ^the positive ^and negative ^{third direction.}

To implement specific initial conditions the formal solution of (2.17) is written as

where the functions a (t ) ^and b (t ) satisfy ^{the initial}

conditions a (t = 0 ) = 1, b (t = 0 ) = 0 and the unit-

arity ^{relation aa + bb}

^{1. In other} words, ^{the 2 x 2}

matrix appearing ⁱⁿ (2.19) ^{is a} group element of

SU(2) ^{which we} will denote by g ^{as in} (2.7). ^We

write symbolically

where the matrix g ^{is now a} function of time calculated by solving (2.17) in the form (2.19).

However, ^{the norm} of C is time independent:

Employing (2.19) ⁱⁿ (2.18) ^we express ^z

z (t) =

-

C_ (t)/C+ (t) ^{in terms of} its initial value

z0=-C_(0)/C+(0): ¹

We may then solve (2.15b) using ^as input (2.22). ^An explicit calculation shows that the solution is given

by ^{w = Cst.} C+-2s = ^Cst.

[aC+ (0) + bC_ (0)]-ZS ^{= Cst.}

C +2s(o) (a - bzO)-2S. Imposing the initial condition

41 (t

0) = f/I 0 (zo) ^{we find that} 4, = q, 0 (zo) (a - bzO)-2s. ^On expressing zo in terms of z from (2.22) the final solution is given by

Comparing this result with (2.8) ^{we see that the time}

evolution of the wavefunction is equivalent ^{to a} time-dependent rotation of its initial value t/1 0 (z ).

The important feature of (2.23) is that it provides ^a

solution for arbitrary spin ^{in terms} of the functions

a and b calculated from the spin-1/2 problem.

In order to make this result ^more explicit ^the

initial wavefunction t/1o(z) is ^{written as a linear} superposition ^{of the monomials} t/1 SJL (z ) ^introduced

earlier in (2.10) :

The 2 s + 1 amplitudes C, (0 ) ^{specify the spin state}

at t

0. Using (2.24) ⁱⁿ (2.23) yields

Regrouping ^the polynomials ⁱⁿ (2.25) ^{as a linear} superposition ^{of the} t/1 SJ.L ^gives

where the D(’) (a, b ) ^are the elements of the

(2 s ⁺ 1 ) ^x (2 s + 1 ) rotation matrix corresponding

to the SU(2) parameters a ^{and b. As} expected (2.26)

reduces to (2.19) ^{for the} special spin ^{value s}

^1/2.

The spin-1/2 problem (2.17-19) may also be used to construct the solution for a classical spin preces-

sing ^{in a} magnetic field. The classical vector M

(Ml, M2, M3) defined from

where z is the complex variable of (2.18) ^and

,u is the time-independent ^{constant of} (2.21), ^{can be}

shown to satisfy the familiar equations for classical

spin precession :

One can further show that the formal solution of

(2.28) ^is given by Mi (t) = Rij (t) Mj (0) ^where

R

(Rij) is the usual rotation matrix

It is often useful to view the solution from a

reference frame whose z-axis coincides with the initial direction of the magnetic ^field Bo. Taking ^into

account the conventions of figure 1 the transformed solution is obtained by the substitution

where g is the matrix of (2.20) ^and

Carrying ^out the matrix multiplications ⁱⁿ (2.30)

gives

The rotated solution is thus calculated by making ^the

substitution (a, b) --+ (ar, br) ^{in all} previous ^for-

mulas.

An analytical solution for a and b, or ar and

br, ^is possible only ^for special choices of the time-

dependent magnetic field. For example, ^an explicit

solution is known for a field of constant magnitude

B which precesses around the z-axis at constant

angle 0 ^{with constant} frequency úJ (q:> = - úJ t ) :

In general, ^{one must} consider either a direct numeri- cal solution or some sort of a systematic perturbative expansion ^{such as the WKB} expansion of section 3.

The present section will be concluded with ^a

preliminary discussion of the adiabatic limit on the basis of the explicit ^solution (2.33) ; ^that is, ^{the limit}

in which co IB ^{1, so}

The full simplicity of the adiabatic limit is revealed

by restricting (2.34) ^{to a} complete ^revolution (t = T = 2 7T / ú) ) :

and by viewing the solution (2.35) from the rotated frame (2.32). Inserting (2.35) ⁱⁿ (2.32), ^and noting

that in the present ^{case 9}

6 0, yields ^the simple

result

Using the ar ^and br calculated above in place ^{of the}

a and b in equations (2.25) ^and (2.26) ^{one obtains}

the adiabatic approximation ^{of the} wavefunction in the rotated frame :

from which we abstract the evolution of the coef- ficients after a complete cycle :

Therefore, ^{we recover} Berry’s result in a special

case ; namely, ^{after a} complete ^adiabatic revolution,

states with definite azimuthal spin J.L = - s, ..., s

along ^{the axis} Bo acquire ^a phase factor exp (i J.L y )

with

The first ^term in (2.39) is the usual dynamical phase

and the second term is the Berry phase ^{which is} equal ^{to the solid} angle ^{that the contour C subtends}

at the origin of the reference frame. Applying (2.39)

for B

const, 0

^const and qJ

^{w t we recover}

the phase ^{y of} (2.35).

Concerning ^{the relative} significance ^{of the two}

terms in (2.39), it would appear that the Berry phase

is ^a subleading correction to the dynamical phase

which grows with T. However, given ^that integer multiples ^{of 2 7r are} irrelevant in the calculation of the phase ^{y, the two terms in} (2.39) ^are equally significant.

One can finally show that during ^{an adiabatic} cycle the classical spin ^{M of} (2.27) ^and (2.28)

advances by ^{a total azimuthal} angle equal ^to

y, while it precesses ^{at a constant} angle ^{from the}

instantaneous direction of the magnetic ^field.

3. WKB solution.

The approximate ^result summarized in equations (2.36) ^and (2.39) ^{is valid as} long ^{as the} period ^of

revolution of the magnetic ^{field is} large compared

with the period ^of spin precession (BT > 1 ).

Clearly, this result should be corrected to account for finite-T effects which may include transitions between spin ^{states. In this} respect, ^{it is useful to} realize that the Berry phase appears ^to be a sublead-

ing correction to the dynamical phase ^which suggests that (2.39) ^contains only the first two terms of a

systematic expansion. ^{Of course, a} calculation of nonadiabatic corrections to (2.39) ^{must be} supplemented by ^a prescription ^{for the} calculation of the corresponding transition probabilities.

Roughly speaking, ^{an adiabatic} perturbation ^ex- pansion ^{can be} envisaged ^{as an} expansion in powers of 1 / T. ^{A more} precise ^statement is that successive terms are organized according ^to the smallness of dimensionless quantities of the form

In particular, ^{it is not} necessary ^{to assume} that the

magnetic ^{field is a} periodic function of time. A

systematic procedure ^{for the} derivation of an expan- sion of the above nature is provided by ^{a well-known}

variant of the WKB expansion [2, 3].

As was explained in section 2, ^{it is} sufficient to work out the solution of the spin-1/2 problem (2.17)

in the form (2.19). The intermediate steps ^{of the} calculation are simplified by using ^a « rotating

frame » defined from

Here C = (C+ , C_ ) ^{is the} two-component wavefunction of (2.17) ^{and T =} (F+ 9 F- ) ^{is a} rotating wavefunction. Using (3.2) ⁱⁿ (2.17) ^{leads to}

or, more explicitly,

The rotating wavefunction (3.2) ^{should not be}

confused with the rotated wavefunction defined by

the matrix go ^of (2.31). ^In view of the conventions of

figure ^{1 we note that}

and U(t) :A go ^{for all} intermediate times. In practice,

we will view (3.2) ^{as a} convenient change of variables and defer for the moment a discussion of its physical

content. Thus we seek a solution of (3.4) in the form

which is the rotating-frame analog ^of (2.19). ^The amplitudes ^{a and} /3 ^of (3.6) ^{are related to the} amplitudes a ^{and b of} (2.19) by

or

In particular, applying (3.7) ^{for t}

^{T and} using (3.5)

establishes that the amplitudes a (T) ^{and f3} (T)

differ from the rotated amplitudes ar (T ) ^and br (T ) by ^{an overall} sign. ^{Such a} simple relationship

between (a, f3) ^and (ap br) ^{does not hold for}

intermediate times.

We now come to the main point of this paper which is a systematic ^WKB solution of equation (3.4). With the notational abbreviations

the system ^of equations (3.4) is written as

where the dot stands for differentiation with respect

to the time variable t. On eliminating ^{r_ we arrive}

at a second-order differential equation ^for r + :

In order to derive a WKB solution of (3.11) ^we

first introduce ^a scaled time variable ^T

Et. Here

E is the inverse of some long-time scale whose

precise specification is unnecessary. The WKB ^ex-

pansion derived below as a formal series in the parameter E ^{will be} effectively ^an expansion ^{in the}

dimensionless parameters ^of (3.1). ^The resulting approximations will thus be valid for a wide range of

slowly varying magnetic fields which are not neces-

sarily periodic.

Denoting ^the derivative with respect ^{to T} by ^a prime ^the definitions (3.9) ^{are rewritten as}

and the first equation ⁱⁿ (3.11) ^as

with

The second equation ⁱⁿ (3.11) ^becomes

We follow the usual WKB procedure ^and represent

T+ ^{as an} exponential:

Substituting (3.16) ⁱⁿ (3.13) gives

or

which is a typical system of recursive WKB equations

that can be used to determine the unknown functions

Fo, F1... by elementary integrations. ^To appreciate

the relative significance ^of the various levels of

approximation ^we proceed in several steps.

The first two equations ⁱⁿ (3.18) yield

To handle the sign freedom in (3.19) efficiently ^we

retain the symbol ^F for the solution with the upper

sign and introduce the symbol

for the solution with the lower sign. ^{We thus find}

that

The coefficients of G’ would differ from those of F’ only by ^{an overall} sign ^{if we had} ignored ^{the total}

derivative.

The general solution for r+ ^{is a linear} superposi-

tion of the form

where > ^{and v are} arbitrary complex constants. The

corresponding solution for F- is obtained by ^insert- ing (3.22a) ⁱⁿ (3.15) :

The functions F and G contain additive integration

constants which are clearly redundant because of the arbitrariness of the constants > and v in (3.22). ^A

convenient choice of the integration ^{constants ob-}

tains by imposing the condition

Therefore, applying (3.22) ^{for t}

^{0, we find that}

or

We may then ^cast the general ^solution (3.22) ^{in the}

form (3.6) ^with

For future reference, these relations will also be written as

Once the rotating-frame amplitudes a ^and f3 ^have

been computed ^from (3.26) ^or (3.27) ^the fixed-frame

amplitudes a ^{and 6 are} given by (3.8).

For the moment, ^we approximate ^{F and G} by ^the

first two terms in the respective ^WKB expansions :

where the coefficients ^are calculated by ^{an elemen-} tary integration ^of (3.21) choosing ^the integration

constants in accordance with (3.23) :

The auxilliary parameter ^e is absent from the right-

hand sides of (3.29) ^{because we} have restored the

original time variable. Quantities for which no time argument ^is displayed ^{are defined at} time t whereas

Z1 (0 ), B (0), ^{etc., are their values at t}

^{0. A} slight exception ^to this rule is the definition (J 0 = (J (0) adopted throughout ^{the text} for notational conveni-

ence.

To complete the calculation of ^a and 8 ⁱⁿ (3.27)

we need the leading-order result for the functions M and N of (3.22b) :

so p

M(0)/N(0)

0(e2) ^and 1/N(0)

O(e).

Therefore, ^to obtain the leading ^or adiabatic ap-

proximation ^we may ^{set p}

0

1 /N (0 ) ⁱⁿ (3.27) :

The adiabatic approximation ^{for the} amplitudes

a and b in the fixed frame is calculated by inserting (3.31) ⁱⁿ (3.8) :

where the phase X ^{is the} only quantity ^that depends

on the history of the time evolution of the magnetic

field. The quantities 8 ^{and cp are the} spherical

variables defining ^{the field at} time t while 8 (0)

00

and cp (0 )

^{0. The} preceding ^result, together ^with

the discussion of section 2, provides ^a complete description ^{of the adiabatic} approximation ^{for arbit-}

rary spin.

As ^an illustration, ^one may apply (3.32) ^for

B

const, 0

0 0

^{const, and cp}

^{w t to recover}

the adiabatic approximation ^{of the} explicit ^solution (2.33) given ⁱⁿ (2.34). ^{One can further} apply (3.32)

for ^an arbitrary periodic field with period ^{T. After a} complete cycle ^{we set 9 = 0} (T)

00 ^and

cp

cp (T)

^{2 7T in} (3.32) ^{to obtain}

where the phase ^{y is} given by

and coincides with Berry’s ^result quoted earlier in

(2.39). Viewing ^the solution from a fixed frame whose z-axis coincides with Bo amounts to perform- ing the rotation (2.32) using ^as input (3.33). ^A simple calculation yields

which is the expected ^{result. A} faster way ^to arrive at this result is to use the rotating-frame amplitudes

« and 0 ^{for t}

T and include ^a minus sign ⁱⁿ

accordance with (3.5) :

Equation (3.35) is then rederived by employing ⁱⁿ (3.36) the adiabatic approximation ^{for a and} {3 given ⁱⁿ (3.31).

Having explained ⁱⁿ detail the strategy ^{in the}

context of the adiabatic approximation, ^{the deri-}

vation of nonadiabatic corrections is more or less

straightforward. Hence, the third WKB equation ⁱⁿ (3.18) ^is solved for F2 ^to yield ^two solutions which

we denote by F2 ^and G’ 2

We can also derive an updated version of the coefficients M and N by using ^the expression ^for F2 ⁱⁿ (3.30) :

The coefficients _p and 1 /N (0 ) ⁱⁿ (3.27) ^are given by

Therefore, p is still negligible ^{in this} approximation

and (3.27) may be written ^as

The calculation of the first nonadiabatic correction is completed by employing values for F and G including ^the correction (3.37). Imposing ^the

initial condition (3.23) ^and returning ^{to the} original

time variable we find that

Although the calculation of exp(G/s) ^is equally straightforward, ^an interesting subtlety arises be-

cause of the total derivatives present in the coef- ficients G1 ^and G’2. ^After integration, the function

G/ £ consists of a term - i X /2, ^which depends ^on

the history ^{of the} cycle, in addition to the contri- butions of the total derivatives which depend only ^on

the initial and final values of the magnetic ^{field. It}

would thus be natural to include the latter contri- butions as corrections to the coefficient of

exp (- i X /2 ) rather than as corrections to the phase

X. Such a task is accomplished by ^{a consistent} reexpansion ^{of those} portions ^{of the} exponential ^that

contain the contributions of the total derivatives.

Note that the total derivative in G2 is of order E ; it is thus negligible within the current approxi-

mation because the function ^a in (3.40) ^{does not} depend ^{on G} and the function f3 already ^{contains an}

overall power of ^{E .} However, the total derivative in the expression ^of G1 given ⁱⁿ (3.21) ^{must be included}

to yield

where X ^{is the same} phase ^{as in} (3.41). Again, ^we

have returned to the original ^time variable, setting

A

£.1 = 6 - i sin 0 cp , according ^to (3.12). ^{A simi-}

lar substitution must be made in (3.40), namely cA(0) = A(0), ^where A(O) is the initial value of A. Inserting (3.41) ^and (3.42) ⁱⁿ (3.40) gives

with

Therefore, ^an explicit calculation including ^the

first nonadiabatic correction is now possible for any

specific choice of the magnetic field. For example,

the fixed-frame amplitudes a ^{and b are} computed by

a simple substitution of (3.43) ⁱⁿ (3.8) :

which is an updated version of the adiabatic approximation (3.32).

To illustrate the preceding ^{result we} may again ^use the solvable case with B

const, 0

^{const and}

cp

wt. We thus set 0 0 ^{= 0,} B (0 ) ^{= B and A = i co sin 8}

A (0) ⁱⁿ (3.44) ^to obtain after some algebra

These expressions agree with those derived by pushing ^{the direct} expansion ^{of the exact solution} (2.33) beyond the adiabatic approximation (2.34).

In order to further appreciate the relative signifi-

cance of the calculated nonadiabatic correction, ^we

consider the general periodic ^case and view the system ^{from a} reference frame whose z-axis coincides with the initial direction of the magnetic field. The

amplitudes a, ^and br ^{after a} complete cycle ^are given by (3.36) ^{with « and} f3 calculated from (3.43). ^Note

that in the present ^case B(0) = B(T)p B and A (0)

A (T) =- ^{A = 6 - i sin 8cp.} Therefore,

with

The time argument ^is suppressed ^{in the} right-hand

sides of (3.46a) because the initial and final values of the field are the same and a specific ^{choice of} origin

on the contour C is obviously arbitrary ^{at this} point.

It is now clear that transitions between states

quantized along the initial direction of the field can occur after a complete cycle. ^For instance, ^{in the} spin-1/2 ^{case, the} probability for transition between states with azimuthal spin ^{± 1/2 is} given by

where ⁿ

B/B is the unit vector along ^the magnetic

field. A calculation of transition probabilities ^for general spin ^{s is also} possible using the results of section 2.

Our final task is to provide explicit expressions ^for

the next nonadiabatic correction. The two solutions of the fourth WKB equation ⁱⁿ (3.18) ^{are found to}

be

the total derivative in G3 ^{is a} complicated expression

which is omitted from (3.48) ^{because it does not}

contribute at this level of approximation. However,

the total derivative in F3 ^is important. ^{We will}

further need ^a slightly corrected version of the coefficients M and N calculated earlier in (3.30) ^and (3.38) :

With these new ingredients ^at hand, ^{we return to} (3.27) and extend the calculation of « and f3 ^to

include one more nonadiabatic correction. The

emerging strategy ^{for a} systematic incorporation ^of higher-order corrections can be simply ^summarized

as follows. The functions ^a and /3 ^are superpositions

of exponentials of the form exp (± i X /2 ) ^{where the} phase X is sensitive to the history ^{of the time}

evolution. On the other hand, ^the coefficients in front of those exponentials depend only ^on the initial and final values of the magnetic field and its derivatives. If n terms are retained in the WKB

expansion ^{of the} phase X, it is sufficient to keep

n - 1 terms in the expansion of the coefficients. In

particular, ^the leading ^{or adiabatic} approximation ^is

obtained by retaining ^{two terms in the} expansion ^of

the phase, i.e., by including ^the Berry phase, ^while calculating the coefficients to zeroth order.

Implicit ^{in the} preceding description ^{are two facts.}

First, the coefficients of exp (± i X /2 ) ^{are calculated} by ^a consistent, Taylor-like expansion in powers of

E which is continuously updated. ^Second, the contri- butions from the integration of total derivatives are

absorbed in the coefficients of exp (± i X /2 ) ^also by

a consistent expansion in powers of c. When we

apply the above procedure, using all the available information through equation (3.49), ^{we find that}

with

and

To be _sure, functions such as B, ^{A... for which no} time argument ^is displayed ⁱⁿ (3.50) ^{are defined at}

some arbitrary time t, ^whereas B (0 ), A (0),

^stand

for their values at t

0. As a check of consistency

one may verify the initial conditions a (t

0) ^{= 1}

and 13 (t

0 )

0, ^{and the} unitarity ^{relation « a +} iil3 ^{=1 to within terms of} order E 3. Although ^the parameter e ^{is no} longer present ⁱⁿ (3.50) ^the

relative orders of magnitude ^{are made} apparent by

the corresponding powers of 11B.

Equation (3.50) gives the final and most accurate WKB approximation derived in the present paper.

This result can be used for the calculation of various

quantities of interest along the lines of the simpler approximations discussed earlier in the text. For instance, ^the fixed-frame amplitudes a ^{and b can be} computed by ^a substitution of (3.50) ⁱⁿ (3.8) ^{to .}

obtain a more accurate version of (3.44). ^The resulting expressions ^{are somewhat} complicated ^and

will not be quoted explicitly. ^The phase

y

2 7r + y (T ) ^of (3.34) ^and (3.46) ^{must be} updated according ^to

Of _course, this relation cannot provide ^{a substitute}

for the detailed results of (3.50) because it does not

by ^{itself account} for transitions between spin ^states

which are now possible. However, y is the only quantity ^that depends ^{on the} history ^{of the} cycle ^and

is thus a natural generalization ^of Berry’s ^{result to}

include nonadiabatic corrections.

One can verify ^that (3.50) is consistent with the direct expansion ^{of the exact solution} (2.33) ⁱⁿ

powers of w /B. ^{The same solution can be used to} provide ^some insight ^{into the relative} significance ^of

the calculated nonadiabatic corrections. In figure ²

we plot ^{the real} part of the function a

a (t ) using ^as input ^{the exact solution} (2.33), its adiabatic approxi-

mation (2.34) ^and the WKB result derived from

(3.50). ^{It is seen that} (3.50) ^{is a} significant improve-

Fig. ^2.

Illustration in the solvable case (B

const, 8

const, cp

w t ) ^{for 6}

^{60° and} to IB

^{1/5. The}

dashed line represents the adiabatic approximation (3.32)

whereas the solid line corresponds ^to the WKB result

(3.50) ^{and is} graphically indistinguishable ^{from the exact}

result (2.33).

ment of the adiabatic approximation for the rela-

tively large ^ratio colb = 1/5. One should also note that the WKB expansion ^{of the exact} solution has ^a finite radius of convergence when it is viewed ^{as a} series in powers of w /B. ^In general, ^{the WKB series}

is expected ^{to be} asymptotic.

The eventual usefulness of the current calculation lies in its potential ^{to be} applicable ^{for a} wide class of

slowly varying magnetic fields for which exact sol- utions are not known. The derivation of accurate solutions for specific choices of the magnetic ^field

continues to attract some interest in the NMR literature [11]. Perhaps, ^we should reiterate here that the WKB calculation does ^not necessarily require that the field should be a periodic function of time.

4. Concluding ^remarks.

The WKB approach provides ^an illuminating perspective of the adiabatic limit and the inherent

Berry phase. It is thus useful to examine whether or not similar calculations can be carried out for other

physical systems. Although ^{the answer to the above} question ^is clearly affirmative, ^the actual calculations may strongly depend ^{on the} details of each particular

case. In the remainder of this paper ^we will briefly

discuss a simple extension of the methods developed

here to the study ^{of a class of} parametrically ^excited

harmonic oscillators. These systems ^{were used} by

Hannay ^to illustrate ^a classical analog ^{of the} quantum

adiabatic phase [12]. The actual connection was later

studied by Berry using the standard version of the

WKB method [13].

We consider a general harmonic Hamiltonian which is invariant under spatial rotations :

with

The _ai and ai* ^are the usual creation-annihilation operators ^{with i}

1, 2, ^{..., N where N is the} spatial

dimension which is left arbitrary. The coefficients

{l 0’ {1 ^{and 0 in} (4.1) ^{are some} functions of time.

The case studied in references [12, 13] ^{was such that}

f2 0 2>. 0 f2, while the space dimension ^{was set} equal

to one (N =1 ).

The system ^defined by (4.1a) ^is closely ^{related to}

the spin problem because the bilinear operators

(4.1b) close the algebra ^of SU(1,1). ^{A technical}

difference arises from the fact that the unitary representations ^{of this} noncompact algebra ^{are infi-}

nite dimensional. For instance, the Fock space associated with the harmonic oscillator may be

decomposed according ^to infinite dimensional ir- reducible representations ^which belong in the discre- te series of SU(1,1) ^{and are} characterized by ^a

Casimir invariant equal ^to k (k -1 ) ^with

k

(N/2 + l )/2 ; ^{here 1 =} 0, 1, 2,... ^{is the} angular

momentum. The one-dimensional case is obtained

by setting k

^{1/4 or} 3/4 for the subspace ^{with even}

and odd states, respectively [10].

In general, ^a restriction to a subspace with definite

angular ^momentum is achieved with ^a holomorphic

realization which is a simple analog ^of (2.6) :

The associated scalar product ^is given by

An important difference from (2.4) is that the

integration ⁱⁿ (4.3) ^{extends over} the finite disk

IZI -- 1.

Otherwise, ^the general formulation of the current

problem ^follows closely the work of section 2 and a

WKB expansion may be derived along the lines of section 3. In particular, ^{the WKB} calculation can be carried out entirely within the two-dimensional

spinor representation ^of SU(1,1), eventhough ^the

latter is not a unitary representation and, therefore,

does not correspond ^{to a} physical subspace. ^A

connection with the infinite dimensional physical subspaces ^of (4.1) ^{can be made} by ^{a relation} analogous ^to (2.23). Furthermore, ^the auxilliary spinor problem ^{can be used} to construct the solution for classical precession ^{of the} pseudospin ^variables Ko, ^{K and K*} by analogy with the classical spin precession ^of (2.27) ^and (2.28). ^{Such a} procedure

would lead to a direct link between Hannay’s angle

and Berry’s phase in view of the closing ^{remarks of}

section 2.

This paper will be concluded with the following

comment. The holomorphic realization (4.2) ^is appli-

cable to any rotationally ^invariant one-body ^Hamil-

tonian. A bosonized version of (4.2) in the form of a

Holstein-Primakoff realization ^was used earlier for the derivation of the 1 /N expansion ⁱⁿ quantum mechanics [14]. The results of that reference may be

reproduced by employing instead the realization

(4.2). ^More importantly, ^the holomorphic realization

yields ^an elegant formulation of both bound-state and scattering problems ^{within a} given subspace ^with

definite angular ^momentum. It is feasible that such a

formulation will lead to efficient methods for ap-

proximate, analytical ^or numerical, calculations of bound-state spectra ^{as well as} scattering phase ^shifts.

Acknowledgments.

I am grateful ^to Carl Bender for an earlier collabor- ation, in reference [4], which led to the present calculation. This work was supported ⁱⁿ part by ^the

U.S. Department ^of Energy.

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