EI~qEVIER

12 December 1994

Physics Letters A 195 (1994) 296-300

PHYSICS LETTERS A

Adiabatic approximation and Berry's phase in the Heisenberg picture

Yves Brihaye a, Piotr Kosifiski b

Department of Mathematical Physics, University of Mons, Av. Maistrian, 7000 Mons, Belgium b Department of Theoretical Physics, University ofL6dL Ul. Pomorska 149/153, 90-236 L6dL Poland

Received 21 September 1994; revised manuscript received 5 October 1994; accepted for publication 10 October 1994 Communicated by J.P. Vigier

Abstract

The adiabatic approximation and Berry's phase are discussed within the framework of the Heisenberg picture.

In order to describe the Berry phenomenon [ 1 ] one usually refers to the Schr6dinger picture. This is mainly because the Schr~Sdinger picture provides a natural framework for the adiabatic approximation [2]; moreover, in the geometric setting [ 3 ], Berry's phase and its generalization [4] are defined in terms of holonomies in Hermitian line bundles, which again calls for considering the families of states rather than observables.

On the other hand, the Heisenberg picture seems to be more convenient if some general features of quantum theory are discussed. For example, the quantum counterpart of canonical equations of clas- sical theory is provided by the Heisenberg equations. Therefore, a natural question arises: How does the adiabatic approximation and the Berry phase emerge in Heisenberg picture? This is the problem the pres- ent paper is addressed to. Similar ideas are contained in the paper by Pereshogin and Promin [ 5 ].

Let us start with the following observation [ 6 ]. Let the initial state be a superposition of energy eigenstates

I~Uo)= ~ a ~ l n ) . (1) n

At the time T, after the parameters of the Hamilto- nian have been cycled, the final state takes the form

I~ur)= ~, exp[i(Tdn+y~)]anln) , (2) n

where 7~ is Berry's phase while

T

~)dn--m--- f d t E , ( t ) (3) 0

is the dynamical one. Now, for an observable A which does not commute with H ( T ) = H ( 0 ) one gets

( A ) r = ~ la, 12(nlA[n) n

+ ~. R e ( a m a , ( m l A I n ) ) m ~ n

X C O S [ (~dn "~- ~)n) - - (~)dm "~" ~/m) ]

- ~, I m ( a m a n ( m l A [ n ) )

×sin[ (Tdn +Yn) -- (Ydm +Ym) ] • (4)

Therefore, the oscillatory terms reveal Berry's phase.

0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0375-9601 ( 94 )00814-0

Y. Brihaye, P. Kosihski / Physics Letters A 195 (1994) 296-300 297

Most experiments concerning Berry's phase [ 7 ] em- ploy this idea•

The main lesson to be drawn from the above is that Berry's phase may be analysed by studying the time behaviour of the quantum expectation values• How- ever, this can be done in the Heisenberg picture as well. One should only keep in mind that the time be- haviour described by Eqs. (2) and (4) refers to the adiabatic approximation; so, our first task is to for- mulate the adiabatic approximation within the Hei- senberg formalism. This can be done provided we find the evolution operator in the adiabatic approxima- tion or, equivalently, its generator - the adiabatic Hamiltonian. The way to construct it is known [ 8 ]. However, for completeness we would like to present here a simple derivation. To this end let us recall first the essentials of the adiabatic approximation [ 2 ]. Let H( t ) be a time-dependent Hamiltonian with a purely discrete spectrum and let

H ( t ) = ~. E , ( t ) I I , ( t ) (5) n

be its spectral decomposition• If we admit no level crossing we can write

I I , ( t ) = U ( t ) H ~ ( O ) U + ( t ) ; (6)

U(t) is uniquely defined by the conditions

( J ( t ) = K ( t ) U ( t ) , U(0) = 1 ,

K ( t ) - ~ [ In ( t ) l l~ ( t ) . (7) n

K(t) has an important property,

H ~ ( t ) K ( t ) I I , ( t ) = 0 , (8)

for all n, which serves for cancelling non-oscillatory terms in the adiabatic approximation [ 2 ]. The adi- abatic solution to the Schr/~dinger equation

• d}~, ( t ) ) 1 - - = H ( t ) l ~ , ( t ) ) (9)

dt

reads

t

[ ~(t)).,d = ~ exp(--i ~ dz E,,(r)) 0

x u(t)H.(o) I ~,(o) >. (lo)

Now, we define an adiabatic Hamiltonian Had (t) by

• dl ~u(t) >ad 1 dt = H a d ( t ) l ~ ( t ) ) a d . (11)

Using I I 2 ( t ) = I l n ( t ) as well as Eq. (6) one easily finds

H a d ( t ) = H ( t ) + i K ( t )

= ~ [ E , ( t ) + i H n ( t ) ] I l n ( t ) . (12) n

Now, the adiabatic Heisenberg picture is defined as the Heisenberg picture which reproduces the time be- haviour of observable expectation values given by the adiabatic approximation in the Schr6dinger picture. It is then obvious that any observable A(t) (the hat denotes an observable in the Heisenberg picture while the subscript "ad" has been omitted) should obey the following equation,

d,4 1 [~ , /~ad] . j t_ a.4 ~ad=Ha d (13) dt - i ~- ' "

In particular, it follows from Eqs. (6) and (8) that

/ t ( t ) = ~ E . ( t ) H . ( O ) , (14) n

H. (0)R( t)H. (0) = 0 . ( 15 )

In particular, if the energy spectrum is time- independent,

i [B(t), grad(t)] + ~ (t) =0. (16) i

Let us consider the simplest example: a spin ½ parti- cle in a magnetic field. The Hamiltonian reads

H ( t ) = - g s . B ( t ) = - ½hgBa.n(t) ,

B=--Bn, / 3 = 0 , (17)

where we have assumed for simplicity that the mag- nitude of B is constant.

We have

H_+ = ½(l_+o.n) ,

K=H+ .17+ + [I_ .17_ = ½i(~iXn).o', (18)

and

1 /~an = -g.~.B- ~5 (BXB) .~ . (19)

The Heisenberg equations of motion read

298 Y. Brihaye, P. Kosihski /Physics Letters A 195 (1994) 296-300

s = - gn+ ~(Bxn) x~. (20)

We shall discuss them in some detail later on. The above formulae are easily generalized to any spin s. A straightforward calculation of Haa (or K) is rather cumbersome because the projectors are degree-2s polynomials in the initial Hamiltonian H ( t ) . How- ever, a derivation can be drastically simplified due to the following observation. Assume that the energies E, are time-independent. Let/~,/~' be two different solutions to Eqs. (15), (16). Then A~_-__/~-/~' com- mutes with ~ ( t ) and, by virtue ofEq. (14), with all /-/,(0). But then it follows from Eq. (15) that

M~= E/Tm(0)aRr/.(0) m , n

= ~ H , ( O ) A R H , ( O ) = 0 . (21) n

We can now immediately conclude that Eq. ( 1 9) is valid for any spin s. Indeed, if we choose B to be the direction of, say, the third axis, then, by Eq. (19), K is a linear combination of sx and sy, which is off-di- agonal in the basis diagonalizing sz; therefore, Eq. ( 1 5) is verified. On the other hand, it is easy to check that Had, given by Eq. ( 1 9), obeys also Eq. ( 1 6).

As a second example we consider the harmonic os- cillator with time-dependent frequency (obviously, this system does not exhibit Berry's phenomenon)

/~= ½ (p :+ 09ZqZ) =09[a + (o9)a(og) + ½ ] ,

l a(09) = ~ = ~ (p-i09q),

1 a+(09)= ~ (p+iogq). (22)

Let us find K. Define U(09) by

a(og) = U(09)a(ogo) U + (o9) ,

a + (o9) = U(09)a + (090) U + (o9) . (23)

Then

IIn(og) = U(09)1-I~(09o) U + (to) , (24)

and

~b a(09) = - ~-0~o9 a + (o9) = [f/U +, a(og) ] ,

6~ d + ( o g ) = - ~ - ~ a ( 0 9 ) = [ f / U + , a + ( o g ) ] . (25)

These conditions together with the off-diagonality condition (8) fix K= f/U + uniquely. We obtain

th /4ad= ½ (/52 +o92~ z) - ~ ( /~+ ~/5). (26)

Now, we can ask the question: What is the classical meaning of Had? For integrable systems the classical adiabatic approximation is obtained via the averag- ing principle [ 9,10 ]. We shall show that the classical adiabatic Hamiltonian is exactly the classical coun- terpart of (26). The generating function for the ca- nonical transformation to the action-angle variables, (q, p) ~ (~o, I), reads

S( q, I, t) =qx/½109-- ~o92q2

+ I arcsin ( ~ q) (27)

and gives

q = ~ s i n ~ 0 , p= 2 ~ c o s ~ 0 ,

H=ogI+ °)I sin ~ocos ~0. (28) co

If we apply the averaging principle to the canonical equations obtained from/7, we arrive at the follow- ing ones,

tb

(o z p= ~-~p-09 q, (29)

which correspond to the Hamiltonian

69 Ha d = ½ (p2+ o92q2) _ _f_~pq. (30)

The above correspondence between the classical and quantum adiabatic Hamiltonians is general. We will give here only very sketchy arguments in favour of this statement. To this end we use an elegant picture of semiclassical approximation to integrable systems given in Ref. [ I 1 ]. The semiclassical space of states is spanned by the common eigenveetors of the set of commuting action operators I,, k= 1 .... , N,

I k l n ) = h n k l n ) , n = - ( n l , . . . , n u ) . (31)

Y. Brihaye, P. Kosihski / Physics Letters A 195 (1994) 296-300 299

The angle variables (Pk are defined as follows,

Uk=e -i-k, U ~ l n ) = [ n ( k ) ) ,

n ( k ) = (nl ..... n k - 1 .... , F / N ) , (32)

In the classical limit, n i ~ , h--,O, hni fixed, we can view Uk as a unitary operator and Eq. (32) makes sense as a definition of ~Ok.

The Hamiltonian is a function of the action vari- ables. Assume that the spectrum of the Hamiltonian is nondegenerate and time-independent. The nonde- generacy implies that, apart from the action opera- tors Ik, there are no other independent operators commuting with the Hamiltonian. On the classical level this means that any single-valued function whose Poisson bracket with H vanishes is a function of ac- tion variables only [ 12 ].

In the semiclassical limit every quantum dynami- cal variable can be expanded in powers of Uk and U~ with the coefficients depending on Ik. The ~av- eraging gives the constant term in the classical expan- sion; on the other hand, the diagonal matrix elements give the U-independent term in the quantum expan- sion. Therefore, the conditions

1 OH _ [H(t) , Had(t) ] + ~ (t) =0 1

I I . ( t )K( t ) I I . ( t ) =0

imply on the classical level

OH {n( t ) ,Had( t ) }+ --~ (t) = 0 ,

1 ~v

(2i /) N f k_I~t d~ok Had

1 I f i d~okH. (33) - ( 2 / / ) N k = I

Again, this set of conditions gives a unique solution for//an. Indeed, the difference between two solutions commutes with H, so it depends on the/ ' s only and, by the second condition, must vanish.

To solve Eqs. (33) let us write the equations of motion for any dynamical variable F

b'={r,/4}~,p+ -~ ~,p

(O~) (34) F={F'I~tp'I'r -~ ~,i"

But { , },,p={ , )q.p, I t=H+OS/Ot; if we put F = H and take into account that the time-independence of the spectrum of the quantum Hamiltonian implies that the classical Hamiltonian does not depend on time if expressed in terms of action variables, we ob- tain ( (OH/Ot)q,p=_ OH/Ot)

OH/ Ot- { H, OS/ Ot} = 0 . (35)

Now, we can subtract from OS/Ot its ~average be- cause it depends on the/ ' s only and commutes with H. Therefore, the solution to conditions (33) reads

Haa+H+ ( ~ d~ok , (36) k=l Ot Ot

which obviously exactly reproduces the averaging principle.

Let us now go back to the problem of Berry's phase. Assume that the parameters of the Hamiltonian tra- verse a closed path in parameter space. It follows from the above considerations that the solution to the Hei- senberg equations in the adiabatic approximation can be written as

T

0

× U~ ( T)A,~.U.( T) , (37)

where

Am,, -- 1-/m (0)A (0)/-/n ( 0 ) ,

U.( T)=-17.(O)U( T)17.(O) . (38)

The appearance of the operators U, (T) in the above formula is just the Berry phenomenon. In the non- degenerate case U, (T) reduces to the pure phase fac- tor; in general, it is a unitary operator acting in the nth eigenspace of the initial Hamiltonian [4].

Let us further relate Eqs. (37), (38) to Eq. (4). Assuming no degeneracy we put U~(T)=exp(iy~) and insert it into right-hand side of (37). We can immediately check that the expectation value (g/olA(T) It.go), with Ig/o) given by Eq. (1), coin- cides with the right-hand side of Eq. (4) (note that I I , (O)= ln) ( nl ).

Using the above results we can now derive in a

300 Y. Brihaye, P. Kosihski / Physics Letters A 195 (1994) 296-300

simple way the relat ion between Berry's phase and Hannay ' s angle. According to Ref. [ 11 ] we choose the initial state as a coherent one,

o~ n l o>=exp - i l

ot=x/Q e -i~(°) . (39)

In the classical l imit I ~ we obtain the classical equat ions of mot ion for averages (q/ol/i(t)I~Uo) - A ( t ) . However, we have shown that the classical l imit for our adiabat ic quan tum dynamics is classical adi- abatic dynamics. In the classical case we have

A (T) = ~ An(I)e i~(r)n , n

T

~0(T)=~0(0) + J dtog(l, t ) + A~0, (40) 0

where A~o is Hannay ' s angle. But the same result should be obtained for A (t) from Eq. (37) in the l imit I o oo. Taking into account that the relevant sums are strongly peaked a round n = ]otl 2 we obta in the fol- lowing [ 11 ],

A~o= --07,~On. (41)

Again let us consider a spin ½ case as an example. It is easy to solve Eq. ( 20 ). Assuming n (T) = n (0) --- no, we get

g(T) = { g ( 0 ) - [g(0)'nolno} cosfl

+ [no ×~(0) ] sin f l+ [no's(0) ]'no,

f l = - g B T + ~ (no × n ) - d n (42) 1 +no-n

C

Here C is a closed curve in paramete r space start ing and terminat ing at no. The final direct ion o f the spin is obta ined from the ini t ia l one by a ro ta t ion by an angle p around no. There are two contr ibut ions to fl:

one resulting from the energy difference between the up and down states and one related to the difference in Berry's phases.

Finally, let us note that the problem of how the geometric phase emerges in the Heisenberg picture was considered, from a somewhat different poin t of view, in Ref. [ 13 ].

This paper was ini t ia ted by a quest ion posed by Kacper Zalewski in a discussion with P .K.P .K. grate- fully acknowledges the k ind hospi ta l i ty extended to h im during his stay in Mons. We are grateful to an unknown referee for bringing Refs. [ 8 ] and [ 13 ] to our at tention. This paper is suppor ted by grant KBN 202179101 o f the Commit tee for Scientific Research.

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