PHYSICAL REVIEW D VOLUME 37, NUMBER 6 15 MARCH 1988

The Berry phase and the Hannay angle

Gautam Ghosh and Binayak Dutta-Roy Saha Institute of Nuclear Physics, 92, A.P.C. Road, Calcutta 700 009, India

(Received 28 May 1987)

The analysis provided by Berry of the extra phase acquired by a system undergoing adiabatic ex- cursion in a closed path in parameter space is extended in this paper to the case of nonstationary states. It is shown that, for certain integrable dynamical systems, wave packets exist which under continuation in parameter space behave in a manner that can be understood entirely in classical terms, thus providing a connection between the quantum adiabatic phase and the classical Hannay angle shifts.

If the Hamiltonian of a system depend on a set of pa- rameters which are made to change through a closed path ( C ) to return to their original values, and if the variation is slow enough to warrant the validity of the quantum adiabatic theorem'12 (whereby the system con- tinues to support a discrete nondegenerate spectrum throughout its history), then in the process the state vec- tor acquires a "geometric" phase3-6 y ( C ) depending on the particular contour C chosen in the parameter space, over and above the familiar dynamical phase associated with the time evolution of the state being transported. This phase, however, is nonzero only for such systems as are governed by Hamiltonians which are not invariant under time reversal. An important feature of the geo- metrical phase is its insensitivity to the rate of traversal of the circuit C provided, of course, the motion is adia- batic. In some simple situations such as a two-state sys- tem described by ' Hamiltonian characterized by three parameters, at a certain set of values of which the levels cross each other, the "geometric" phase turns out to be the solid angle in the parameter space subtended by a surface bounded by the contour C at the degenerate point. This observation has evoked the concept of a gauge potential in parameter space, holding forth, perhaps, the promise of a deeper insight into the nature of adiabaticity. Thus, for example, simon7 views the adiabatic limit as yielding a way of transporting a vector along a curve, i.e., a connection and the phase two-form as emerging naturally as an associated curvature. The universality of the adiabatic phase can be grasped from the variety of contexts in which it has surfaced such as the Born-Oppenheimer approximations8-12 in molecular physics, fractional statistic^,'^"^ anomalies in gauge fields,I5-l7 and the quantum Hall effect.'8319

Analogously, a classical integrable system (amenable to solution through reduction to action-angle variables) depending again on a set of parameters which are made to traverse a closed path sufficiently slowly and to return to their initial values, conserves, according to the classi- cal adiabatic theorem,20 the values of its action variables but suffers in the conjugate angle variable a change in addition to that originating dynamically. This extra change in angle21s22 can again be shown to be noninteg-

rable in that it can be cast into the form of an integral over a surface spanning the particular contour described in parameter space.

A connection between the extra phase (in the wave function in the quantal description) and the angle change at the classical level was established by Berry through semiclassical torus quantization,22 the angle change be- ing related to the derivative of the phase change with respect to the quantum number of the state being adia- batically transported, a result which constitutes in effect Bohr's correspondence principle for adiabatic phases. However, it is our present contention that it is instruc- tive to establish this correspondence in a different manner by extending Berry's analysis to nonstationary states representing localized wave packets following clas- sical trajectories. To illustrate the approach, we choose the context of two systems for which the quantal adia- batic phases and the corresponding classical angles have already received considerable attention.

The generalized harmonic oscillator. This system is governed by the Hamiltonian

which is obtainable from the usual oscillator Hamiltoni- an by a combined scaling and rotation in phase space to- gether with appropriate symmetrization. Through the introduction of annihilation (and corresponding creation) operator

with w = ( x z -y * satisfying the commutation rela- t tion [ a , a ] = 1, the Hamiltonian is thrown into the stan-

dard quadratic Hermitian form

and the evolution of the system under adiabatic trans- port around a closed circuit in parameter space [spanned in this case by R = ( x , y , z ) ] may be unraveled through the coupled Heisenberg equations of motion

1709 @ 1988 The American Physical Society

1710 BRIEF REPORTS 37 -

da ' aa 1 - + i f r [ a t , ~ l d t at

The terms representing the coupling between the creation and annihilation operators contributing as they do to second order in adiabaticity may be disregarded to obtain the leading behavior for adiabatic transport around a closed circuit C accomplished in time T, to wit,

X exp J t% [:Idt 1 , wherein the two exponents represent, respectively, the dynamical and the geometric phases, and the latter (A@) which is of interest in the present context can be transformed by writing

and through the application of Stokes's theorem to yield

which, it may be noted, is precisely the expression for the angular shift obtained classically by Hannay. To clinch the correspondence, however, it is illuminating to construct a wave packet (a coherent state23-25) which is an eigenstate of the annihilation operator:

which for an oscillator represents a minimum- uncertainty, nonspreading packet undergoing motion along the classical trajectory, as evidenced by the expec- tation value of the position operator a. Furthermore, from the form of H given by Eq. (3) it is evident that in

the classical limit (fi-0, / a / + co, g% / a / -finite) the action is given by fi I a / 2 and the corresponding an- gle variable by the phase of a , and it is for this reason that the additional phase of a worked out to be the Han- nay angle. Indeed for the coherent state of a generalized oscillator the geometric phase arising from adiabatic cy- cling can be understood entirely from classical argu- ments.

Spin in a magnetic field. The dynamics of a particle with spin angular momentum fiJ interacting with a mag- netic field is described by the Hamiltonian

where g is a constant involving the gyromagnetic ratio; and the energy levels of the system, Em ( B ) =figBm, with m taking on any of the ( 2 j + 1 ) integrally spaced values lying between - j and + j , depend on the parameters B and become degenerate at the point B=O. Adiabatic transport over a closed path in the parameter space (slow cycling of the magnetic field components), will in- duce a phase factor on the eigenfunctions of H (apart from that representing dynamical evolution), and this is given by3

where R ( c ) is the solid angle subtended by a surface bounded by the closed curve in parameter space (corre- sponding to the slowly effected cyclic change in the mag- netic field), at the origin where the levels become degen- erate. Once again to expose the semiclassical connec- tions it is appropriate to investigate the effect of adiabat- ic transport on a wave packet26'27 centered around a given value of the angular momentum d=W. Such a state is indeed available to us, having been constructed through the stratagem of a noncompact extension of the SU(2) and reads

I j ,m ) being the usual angular momentum states, a and B being complex members, which when parametrized as a = exp( i4) tane/2, B= exp(i$) / B , represent in the classical limit a spinning rigid body with its figure axis having an orientation (8 ,$ ) and an angular momentum +fig-d [where c= P 1 ( 1 + 1 a 2 ) ] . On adiabatic transport through a closed circuit carried out slowly in a time interval T, this nonstationary state I a , P ) will have been altered to

cz + j - m p j 1 a , P ) ' = ( c o ~ h c ) - ' / ~ 2 2 T

exp 1-imRic)-igm J B ( t idr 1 j , m ) ,= , m z - j g(j +m)! ( j -m)! o i

T i R ( c ) + i g J n T ~ ( t ) d t ] , pexp [--ifllc)-ig

The expectation values of the components of J in these angular momentum coherent states being given by

BRIEF REPORTS

on adopting the parametrization of a and /3 introduced above the classical limits of the expectation values of the angular momentum components in the transported state are given by

The change in the phase of f l reflects the alteration in the nodal angle, i.e., the angle between the nodal line (in- tersection of the body-fixed x' -y ' plane and the space- fixed x-y plane with the z ' axis oriented along J) and the space fixed x axis. Thus the geometric phase Wc), once again, entering as it does into the stationary components of the nonstationary state generates in the classical limit the Hannay angle. This situation is analogous to the one considered in the experimental ~ e r i f i c a t i o n ~ ~ - ~ ' of Berry's phase by Tomita and Chiao. There, a coherent beam of photons was sent through a helically wound single-mode optical fiber. The adiabatic invariant in that case was the helicity of the photon which clung to the instantaneous propagation vector ( K ) as it was trans- ported along the fiber. The degenerate point being the

origin in the K space, the spin state of the photon un- derwent a change in phase (in addition to the dynamical and material-dependent phases) that equaled the solid angle subtended by a surface spanning the contour in K space at the origin and this resulted in a rotation of the plane of polarization of the emergent photon beam by an amount that was precisely the classical Hannay angle (a calculation from the angle two-form is given in err^^).

Thus, in conclusion, the correspondence between the apparently innocuous Berry phase for stationary states adiabatically cycled in parameter space and the classical Hannay angle has been shown to be more directly amen- able to physical interpretation in the context of suitably constructed wave packets. That the wave-packet argu- ments reproduced the exact classical limit of course reflects the fact that the Hamiltonians considered were linear in the action variables so that the Hannay angles were independent of the action variables and the Berry phases came out to be linear in the quantum numbers as expected from the semiclassical theory of err^.'^ It would be instructive to extend the present analysis to more general Hamiltonians.

We thank Professor M. V. Berry for his helpful com- ments and suggestions for modification~ of the paper.

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