Berry phases and optical spectra of atoms placed in

Home

Search

Collections

Journals

About

Contact us

My IOPscience

Berry phases and optical spectra of atoms placed in an external adiabatically varying electric field

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1991 J. Phys. B: At. Mol. Opt. Phys. 24 2305 (http://iopscience.iop.org/0953-4075/24/9/010) View the table of contents for this issue, or go to the journal homepage for more

Download details: IP Address: 150.203.180.16 This content was downloaded on 10/09/2014 at 01:42

Please note that terms and conditions apply.

J. Phys. B: At. Mol. Opt. Phys. 24 (1991) 2305-2312. Printed in the U K

Berry phases and optical spectra of atoms placed in an external adiabatically varying electric field V I Matveev and M M Musakhanov Physics Depanment, Tashkent State University, 700095 Tashkent, USSR

Received 13 June 1990. in final form 9 November 1990

Abstract. The behaviour of an atom placed in an external adiabatically cone-rotating constant electric field is considered. Taking into account the Berry phase i n this problem is shown to be necessary. The radiation spectrum for this atom in such a field is obtained and photon polarizations are considered.The Berry phases areshown to beeadyobsewable during optical transitions i n atoms placed in an external varying electric field.

1. Introduction

The problem of the states and spectrum of an atom placed in an external constant electric field rotating in a plane is applicable, first of all, to the consideration of the interaction between circularly polarized electromagnetic ( E M ) radiation and atoms (Delone and Krainov 1984) as well as Stark broadened spectral lines of the atom colliding with charged particles (Lisitza 1970, 1977). To treat these problems in the presence of an external constant electric field one should consider a more general problem of the spectrum and states of an atom placed in an external cone-rotating constant electric field. Besides, this problem is of independent interest as an example in which the so-called Berry phases (Berry 1984) are to be taken into account. Berry phases are very important in quantum low-energy chromodynamics where the effective chiral action involves a Wess-Zumino-Witten term, which by its construction is regarded to be a magnetic Dirac monopole in the space of variables of the chiral pion field (Witten 1983). The occurrence of the Berry phase in our case is physically accounted for by the fact that while solving the problem of an atom state in a rotating field one should, as a rule, proceed with the reference frame rigidly connected with the field. In this reference frame rotating together with the field the effective magnetic interaction under the applicability of the adiabatic approximation may be interpreted (Moody et al 1986) as the field of the Dirac magnetic monopole, and the relevant phase factor of the wavefunction as a Berry phase. Previously we published a short report (Matveev and Musakhanov 1989) where we obtained Berry phases for an atom placed in an external adiabatically cone-rotating electric field. The radiation atomic spectrum with Berry phases taken into account was also obtained. Berry phases were shown to be easily observable during optical transitions in atoms placed in an external varying electric field. In the present paper we detail this problem, particularly we consider the polarization of emitted photons and offer more concrete recommendations for the experiment, 0953-4075/91/092305+08$03.50

0 1991 IOP Publishing Ltd

2305

2306

V I Matveev and M M Musakhanov

Note that at present there are a lot of approaches available to allow observation of Berry phases experimentally. As an example we will refer to a paper by Tucko (1987) who discusses experimental data of the observation of Berry phases for nuclear quadrupole resonance in a sample adiabatically rotating in an external constant magnetic field. A nuclear magnetic resonance ( N M R ) probe designed for magic-angle spinning N M R experiments was used. The NaCIO, crystal was oriented with a cleavage plane perpendicular to the rotation axis. Simple rotation is shown to induce frequency splitting in nuclear-quadrupole-resonance spectra. The splitting is interpreted both as a manifestation of Berry's phase, and as a result of a fictitious magnetic field associated with a rotating-frame transformation. Segert (1987) proposes to observe the nonAbelian Berry phase in atoms placed in external parallel electric and magnetic fields; the value of these fields being chosen so as to attain the level degeneracy needed for the non-Abelian Berry phase to occur by compensating the Stark effect due to the Zeeman effect. As distinct from the latter proposal, ours is a different, more simplified approach. We propose to observe the Abelian Berry phase in optical spectra of atoms placed in an external adiabatically cone-rotating constant electric field. This situation is readily realized by the addition of a circularly polarized monochromatic E M wave to the constant electric field along which the wave is propagated?. Therefore, only the presence of maser radiation makes our proposal distinct from a standard experiment on Stark splitting of atomic levels in a constant electric field.

2. Berry phases in problems of quantum mechanics

In many physically illustrative examples the division of subsystems into fast ( r ) and slow (F) cases is approved. Correspondingly, the Hamiltonian of the system has the form

H = Ho(F)+ h ( F , r ) .

(1)

The wavefunction of the fast subsystem 'remembers' a slow case parametrically, i.e.

h ( F , r)ln(F))= E,(F)In(F)).

(2)

Let a slow subsystem perform closed motions F( T)= F(O) for a long time T. At the instant of time f = O a fast subsystem is fixed to be in the ln(F(O)))state. The evolution of the fast subsystem is defined by the Schrodinger equation

(Here, and in the following, atomic units are used.) Usually the solution is taken in the form (Migdal 1977)

t As shown below, the line shift due to the Berry phase turns out to be not much less than the Stark splitting of lines in a constant electric field.

2307

Behaviour of afoms in an external electricfield

The zero approximation a‘,“’=a,(O) will be assigned. Then the assumption that d F / d f is small allows us to restrict ourselves to the first correction (Berry 1984)

a?’=S,, exp(iyn(t))

where V F= J/JR If In(F)) states are non-degenerate for any value of F they may apparently be chosen to be real valued. In this case y,( 1 ) = 0. More interesting is the case when there is a degeneracy at any value of F = F*. In this case y.(f) # 0, particularly, from ( 5 ) it follows that

x(T ) = fc A . ( F ) d F

(6)

where A, =i(n(F)IV,[n(F)), and the contour C is a closed trajectory ( F ( O ) =F ( T ) ) described by the variable F. If the contour C is close enough to the degeneracy point F* it is easy to show (Berry 1984) that the potential A.(F) is the ’field‘ of the magnetic Dirac monopole placed at the point F*. Note that in a more complicated case when there is degeneracy of the fast subsystem for all F and F ( 0 ) = F ( T ) , the corresponding gauge ‘field A,, may be non-Abelian (Wilczek and Zee 1984). For the case when F is rendered to be a dynamic variable, the gauge ‘field‘ A, may be directly interpreted as the effective magnetic tield taking the following example. In quantum mechanics one sets the task to determine the probability amplitude K.,(F(O), F ( T ) , T ) for the transition from the present initial ( I n ) , F ( 0 ) ) state into the present tinite (Im), F ( T ) ) state. Let the initial and finite states coincide. It is evident then that the relevant probability amplitude K , ( T ) is defined by the Feynman integral over the closed trajectories C:

K , ( T )=

Dc exp( i

loT

H o ( F )d f + i

joT

E.(F(

t ) ) d f +iy,(

T)).

(7)

Thus, as seen from ( 6 ) and (7),the effective Hamiltonian of motion along the variable F has the form dF H N ,= ~ Ho(F)+&n(F)+An(F)-. df

(8)

Let, for example,

H , = - P+: V(F) p F = -iV, (9) 2m where p F is momentum, and V(F) is the potential. Then the solution of the quantum problem (i.e. the calculation of the Feynman integral (7)) implies the solution of the Schrodinger equation of the form i-Pn(F, J Jf

f)= --(-iV,+A,(F))’+ m ;

[

V(F)+&.(F)]P.(F,

t)

(10)

that is pertinent to a particle having electric charge e = 1 placed in external electromagnetic ‘field’ A,,. In particular, the interpretation of the Berry phase that is analogous to formula (10) allows the incorporation of the ’field’ A, of the ‘Dirac monopole’ into the Hamiltonian of a two-atom molecule (Moode el a1 1985).

2308

V I Matveev and M M Musakhanoo

3. Atom in a n external adiabatically varying electric field Consider first of all an atom placed in an external gradually varying electric field F( t ) that is rather weak to be taken into account by perturbation theory; at the same time it is such that the Stark splitting is great as compared with the fine structure. Let F( t ) change only on orientation, i.e. ( F (t)i = F = constant, and F ( 0 ) = F( T ) . Following the meaning of the adiabatic approximation we solve first of all the problem of a hydrogen atom placed in a fixed external electric field. As is known (Bethe and Salpeter 1957) regular zero-approximation wavefunctions result from the hydrogen atom quantization in parabolic coordinates with the axis of the orbital momentum projection aligned with F. If the direction of F changes slowly with time (limitations to the rate of variation are given below) the regular zero-approximation wavefunctions are adiabatically adapted to it. Thus, the wavefunctions for the arbitrarily aligned field F( t ) = F( 8,'p) as described by spherical angles I9 = 6(t ) and 'p = p( 1 ) result from those for 19 = 'p = 0 when the rotation transformation ln(F(t)))=l&p.F)= Wt)lO,o,F)

(11)

~ ( t=exp(-ii,p) ) exp(-ii,B) exp(ii,p) (12) is applied. Here i, and f, are the operators of the angular momentum projection onto the y and z axes, respectively. Hereafter, following Moode et al (1985) we see that A. =i(n(F)IV,ln(F)) is a vector potential of the magnetic Dirac monopole whose charge g coincides with the angular momentum projection onto F (here, unlike Moody et al(1986), we have chosen for the rotation operator a very commonly used representation (Biedenham and Louk 1981)). Let us consider, for instance, a helium atom as the example of a non-hydrogen-like case. For the sake of simplicity we will assume that in the helium atom one electron is in the ground state whereas the other is in the excited state described by quantum numbers N,l, I,, where N is the principal quantum number, I is the orbital momentum and I, is its projection onto the z axis. Generally speaking, level splitting in such an isolated atom includes three parts (Bethe and Salpeter 1957) namely: 6, fine splitting related to the relativistic terms taken into account in H: A,, the energy difference between the states having the same principal quantum numbers, but different I ; and A,,,, the energy difference between the states having principal quantum numbers N and N + 1. These energies satisfy the inequality 6,,,v,, "L L l l F ,,,IC- p - " La LllC b n l l l C _I

..--:.:-..

as in the absence of rotation, while the two other lines are shifted to the right ( p = + 1 ) and to the left ( p = - l ) , respectively, by A E = 6. If the line strength in the absence of field rotation is assumed to be unity then in the presence of field rotation with the angle 8 the line strength will be equal to l d ~ o ( t 9 ) lwith z condition P,=,,,, ld,!,,(19)l2= 1. The transition from state 10, 1 , O ) behaves in such a manner. For the transitions from the states 10,0, + l ) to 10,0,0) we have p ' = *1 and p = * l , 0 therefore, according to (23) the lines will be shifted linearly to cos 9, for the rest the line behaviour is the same. The case considered is of a rather general character since it allows us to observe all the peculiarities resulting from the rotation effectt. Therefore, in our opinion, the experiment on the observation of the Berry phase in optical spectra differs little from that of the investigation of Stark splitting in a constant external electric field. A e m e n t i n n d ahnw il rnnstant mne-mtztino field riln be ob!;ljncd_ by scmmino _".._ ._ a the constant electric field Fl and the field F2, for instance, of circularly polarized maser radiation, whose beam is aligned with F, .The resulting field F = F, + F2is cone-rotating

.."._ .-....-.."-

-

t The conditions for the adiabatic (&4,5)+

approximation (14) ti) be applicable (for instance, for the Balmer series sin 19