Electric- and magnetic-dipole contributions to the reaction field theory

Physica 93A (1978) 316-326 (~) North-Holland Publishing Co.

E L E C T R I C - AND M A G N E T I C - D I P O L E C O N T R I B U T I O N S TO THE REACTION FIELD THEORY A.-S.F. O B A D A * and A.M.M. A B U - S I T T A

Maths. Department, Faculty of Science, AI-Azhar University, Cairo, Egypt

Received 2 January 1978

A theory for the interaction between electromagnetic radiation and a two-level atom in the electric- and magnetic-dipole approximation is presented. The Dyson-like equation satisfied by the propagators are generalized and a coupled set is obtained. This set is solved, and the susceptibilities are calculated. T h e dynamical equations for the fields are d i s c u s s e d when the field is s u p p o s e d to be an unpolarized and isotropic field. It is remarked that the Lamb-shift is enlarged, and the line shape is enhanced. Near r e s o n a n c e the fields are comprised of resonant and off resonant parts. The reaction field is identified.

I. Introduction

The problem of the interaction of an atom with the electro-magnetic field has been the core of a great deal of interest for a long t i m e ~ ) . The spontaneous emission is well understood using the perturbation theory methods either in the non-relativistic region 3) or relativisticallyS). It has been pointed o u t 6) that a dynamical theory of spontaneous emission which exhibits the radiation damping m e c h a n i s m as the consequence of a radiation reaction field is desirable. This is due to the d e v e l o p m e n t of intense fields and m e a s u r e m e n t techniques, and hence systematical investigation of for example, the optics of systems of atoms is needed without the inclusion of imperical damping factors or shifts, a priori. Because of interactable problems involved, any serious investigation resolves to the electric-dipole approximation. H o w e v e r , it is known, that the addition of a term representing the magnetic-dipole to the interaction makes the problem difficult7). This inclusion corrects, for example, the binding and van der Waals energies 8'9) and makes a p h e n o m e n o n like optical rotation * P r e s e n t address: Faculty of Science, King Abdul Aziz University, Jeddah, Saudi Arabia. 316

DIPOLAR CONTRIBUTIONS TO THE REACTION FIELD

317

explicable~°). Therefore, we would expect that this inclusion would contribute to Lamb-shift and effect the line shape, and consequently damping in the system. Thus the aim of this paper is to exhibit such effects. This is being tackled through the f r a m e work of a field reaction theory. This theory and its relations to current theories in the literature like the neo-classical t h e o r y " ) and the master equation~2), are discussed in some detail inl3). The outline of the perturbation theory and the decorrelation procedure, which is used in this study, is discussed in section 2. The Hamiltonian is written down for a two-level atom in interaction with the electromagnetic field. The field is c o m p o s e d of a probe (a c-number classical field) and a quantized field. A set of equations is obtained for the electric and magnetic dipole operators. These equations are taken up and developed in section 3. A set of coupled equations for a set of propagators is obtained. These propagators satisfy a generalized D y s o n - t y p e equation. The consequences of this set of equations is analyzed in section 4. The equations of motions for the different vectors in an isotropic unpolarized field are obtained. The reaction fields are identified. The a p p e a r a n c e of parts of the fields which m a y be interpreted to have an off-resonance behaviour is noted. Finally section 5 concludes with a discussion of these results.

2. The perturbation expansion The Hamiltonian of the system is written in the usual way H=no+Hint,

(2.1)

where H0 designates the Hamiltonian for the system of the unperturbed atom Hmat, and the electro-magnetic field Hrad; the contribution due to the classical probe can be ignored because of its c-number nature. The contributions of the electric and magnetic dipoles are taken to represent the interaction between the atom and the field. Thus the interaction Hamiltonian density is written in the form Hi.t(x, t) = - g ( x , t ) . [e(x, t) + E(x, t)] - m(x, t ) . [h(x, t) + H(x, t)],

(2.2)

where bt(x, t) = ~ ( t ) 6 ( x - Xo), m ( x , t) = m ( t ) 8 ( x - Xo), e(x, t) and h ( x , t) are the appropriate operators. All these operators are taken in the Dirac-interaction representation while E and H represent the classical probe. We specify now the atomic system to be an two-level atom with separation

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A.-S.F. O B A D A AND A.M.M. ABU-SITTA

e n e r g y hw~. For this s y s t e m Hmat is written in the f o r m (2.3a)

Hmat = lhtoso'z.

We a s s u m e that the dipole m o m e n t o p e r a t o r s can be written in the following simple f o r m l a ( t ) = tXo~trx(t)ut

and

re(t) = mosoy(t)u2

(2.3b)

with u, and u2 unit vectors. N o w , we look at the H e i s e n b e r g e q u a t i o n of m o t i o n of the operators. W e shall c o n s i d e r the e q u a t i o n of m o t i o n for the electric-dipole m o m e n t operator, as an example. This e q u a t i o n is

0._(;_) 0--7 -

[bt(t), H~,t(t)].

(2.4)

This equation is solved by s u c c e s s i v e iterations, which is well-known as the K u b o - f o r m a l i s m of the p e r t u r b a t i o n theory. In this formalism, a nest of c o m m u t a t o r s appears. In the following we shall c o n s i d e r the first few nests in s o m e detail. It is d o n e to establish a pattern which is carried o v e r in nests of higher order. T h e first of these nests is the c o m m u t a t o r [/.t(t), Him(t0] = - u t [ t t ( t ) , I ~ ( t O e ( t O + m ( t O h ( t O ] ,

(2.5)

where we have used u~. e = e and u2" h = h. W h e n the e x p r e s s i o n (2.3b) is i n t r o d u c e d , the a b b r e v i a t i o n Si = S ( t i ) and the properties of the o p e r a t o r s tr. are used, we arrive at [la(t), Hi,t(td] = -ul{(~0s)2(2 i sin w~(t - t,))tr:e~ +/x0sm0~(2 i cos to~(t - t0)o'zh~}.

(2.5a)

T h e s e c o n d iteration is proportional to the nest: [[/x(t),/xlel + mlht], ~2e2+ m2h2] = (/Xo,)2(2 i sin ~o~(t -- tO){iZo,O-:Xrx(t2)[el, e2]+ + mos~rztr~(t2)[el, h2]+} + (~o~mo~)(2 i cos to~(t - tl)){lXo~tr.
(2.5b)

T h e [ . . . . . ]+ is the a n t i c o m m u t a t o r of the two operators. Taking this result into consideration, we find that the nest in the third iteration b e c o m e s [[[/z (t),/zlel + m t h l ] , tz2e,_ + m2h2], ~3e3 + m~h3] ~- (/Z0s)2(2 i sin to~(t - t0)[el, e2]+(/I, Os)2(2 COS tOs(t 2 -- t3))Crze 3

+ (tzo~)2(2 i sin w~(t - t0)[e> ed+(/xosmoD(-2 sin to~(t2- t3))tr~h3

DIPOLAR CONTRIBUTIONS TO THE REACTION FIELD

319

+ (/~os)2(2 i sin tos(t - t0)[el, h2]+(mod~o~)(2 sin t o s ( t 2 - t3))trze3 + (/~os)2(2 i sin tos(t - t 0 ) [ e , hE]+(m0s)2(2 cos tos(t2 - t3))trzha + (/~o~mo~)(2 i cos tos(t - t 0 ) [ h , e2]+(~oD2(2 cos to~(t2 - t3))tr~e3 + (g.o~mo,)(2 i cos tos(t - t 0 ) [ h , e2]+(~o~moD(-2 sin oo~(tz- t3))tr~h3 + (/~osmoD(2 i cos tos(t - t0)[hb hz]+(mod~os)(2 sin

tos(t

2 --

t3))trze3

+ (p.o,moD(2 i cos to~(t - t 0 ) [ h , hz]+(mos)2(2 cos tos(t 2 - ta))trzh3.

(2.5c)

It is seen that the iteration pattern is established. W h e n we take the e x p e c t a t i o n value of the operators in the field state [ph), we get the following expression for the operators

× (a +(tz- t3)" E(t3) + ~¢+(t2- t3)" H(t3)) + I ~ ( t , - t2)(~+(t~ - t3)" E(t3) + y+(tl - t3)" H(t3))} +" • "]

+f

t,)[,,(t,) + f f

× { I ~ ( t ~ - t z ) ( a + ( t z - t3)" E ( t 3 ) + # + ( t z - t3)" H ( t 3 ) ) + J~(tl-

t 2 ) ( ( + ( t : - t3)E(t3)+ ~'+(t2- t3)H(t3))} +" • "1'

(2.6a)

J

and

f

f f .,..t.

x { J ~ ( a +E + ~+H) + I~(t~+E + y+H)} + . • "/

+f dt.~,,[H+ f f dtzdt3(J~(ff+E+,+H)+ ,y(a+E + CH,} + . . . ] in an abbreviated form. (2.6b) T h e definitions of the coefficients J~, J~, I~ and I3 are as follows: J+(t

-

t') = ( d x ' u i • F+(x, x'; t - t ' ) . ui6(x - x'),

(2.7a)

J

with i

[X+(x, x', t - t') = ~ O(t - t')(phl[e(x, t), e ( x ' , t)]+lph) = i O(t - t')(phl[h(x, t), h ( x ' , t)]+lph),

h

I~(t

t') = f d x ' G + ( x , x', t - t ' ) 6 ( x - x'), .I

(2.7b)

32O

A.-S.F. OBADA AND A.M.M. ABU-SITTA

where - t ' ) ( p h [ [ u ~ . e ( x , t), uz " h ( x ' , t')]+lph)

G + ( x , x'; t - t ' ) = ( ~ ) O ( t

and

I~(t

t') = f dx@+(x, x'; t - t ' ) 6 ( x - x'),

(2.7c)

with

(i)

.

G÷(x, x', t - t') = ~ 0(, - t')(ph[[u2- h ( x , t), u ~ . e ( x ' , t')]+lph). The propagators which appear here differ from those appearing elsewhere ~°) in that they contain the a n t i - c o m m u t a t o r s instead of the c o m m u t a t o r s which a p p e a r there. We develop eqs. (2.6) in the following section.

3. The iteration scheme The d e p e n d e n c e of the last expressions on t can be convoluted when we Fourier transform these equations, and take the expectation value in the matter state. Then we obtain p(~o) = - (Oss - p o o ) ( H ( ~ o ) E ( w ) + ~(~o)H(~o)),

(3. la)

m (~o) = - (P,~ - poo)(Z(~o)E(~o) + M ( w ) H ( ¢ o ) ) .

(3. lb)

The propagators H, ~ , Z and M have the following iteration formulae: / / ( ~ ) = ~ (0.,) + ( a ( ~ o ) J ; ( ~ ) + ~:(~o)I~(0.,)) • H*(e,) + ( ~ I ((~o) + ~J~-(oJ)) • Z+(¢o),

(3.2a)

~(¢o) = ~(oJ) + (a(oJ)J~(oJ) + ~l+(w)) • ~+(w) + ( a I ] + ~J~). M+(o.,), (3.2b) Z ( t o ) = ~(¢o) + ( y ( w ) J ~ ( w )

+ ~I[(to))Z+(to) + (¢J~((to) + yl~(to)) • H+(o)),

(3.2c) M ( o J ) = y(oo) + ( y J ~ + ~I T)M+(~o) + (~J~ + y I ~) . =-+.

(3.2d)

It a p p e a r s that they depend on a, ~, ~', y which are the Fourier t r a n s f o r m s of the electric-electric dipole, electric-magnetic, magnetic-electric and magnetic-magnetic dipole c o m m u t a t o r propagators in ~o-space. T h e y also depend on H +, ~+, Z +, M ÷. These have iteration formulae formally similar to (3.2) with the quantities a +, ~+, ~'+ and y+ appearing instead of a, ~:, ~ and % The a+ . . . . are the electric-electric dipole . . . . Fourier t r a n s f o r m s of the

DIPOLAR

CONTRIBUTIONS

TO THE REACTION

FIELD

321

a n t i c o m m u t a t o r p r o p a g a t o r s in the ~o-space. F o r the m o d e l i n t r o d u c e d here, we find that:

2(/Zos)2O)s a(o~)

2(mo~)2O)s

= h(oj2 _ ~o2),

v(a,)

= ~(co2 _ ~o2)

and ~(co) = 2 i(tzosm0s)~o = -~(~o), h(o~2 - ~o2)

(3.3a)

while 2(/.~os)2~

o:(o~)

= h(a,2 _ ~o2),

~,+(~o) =

2(m0s)2~o ~(a,s 2 _ 0~2)

and ~+(~o) = 2i(~os)(mo,)O~, h(o~]-

co 2)

-~[+(co).

(3.3b)

=

T h e p r o p a g a t o r s J+, I + on the other hand, h a v e the following f o r m s for a field state of n - p h o t o n s .

J~((o)= ~

1

J"

f f d3k(2nk+ l,u, • ( ~ e~(k,e~(k,), ul

e '''~_

× tck-Tg+ i8)

e-"'"

]

e-ik ••

"1

ck J,-~w~- i8)~ 8(r) dr

(3.4a)

and 1 eik • r

X (ck_-~w--+ iS)

ck ~ - ~ ~ iS)~ 8(r)dr"

(3.4b)

T h e generalized D y s o n - l i k e iteration equations for the p r o p a g a t o r s H +, ~ * , Z + and M + which are similar to eqs. (3.2) can be solved in t e r m s of a +, C , K+ and y+. T h e solution is written in the following f o r m H + = Ya+ + asr +

xy - ab '

(3.5a)

Z+ = x~ ++ ba + xy - ab '

(3.5b)

~ + = Y~:+ + a y +

xy - ab '

(3.5c)

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A.-S.F. O B A D A

AND A.M.M. ABU-SITTA

and m + -

xy + + b~ + xy - ab

(3.5d)

'

where y = 1 - - ( ' Y + J 2 + -f- ~ + 11), + x

= 1 - (ot+J~

+ ~ + I 2+) ,

(3.6a) (3.6b)

a = a+I~+UJL

(3.6c)

b = y+I~+

(3.6d)

and ~+J~.

These terms depend on ~o through the d e p e n d e n c e of the polarizabilities a +, s~+, sr+ and y+, and the d e p e n d e n c e of the terms J+ and IT on w. When expressions for the polarizabilities (i.e. eqs. 3.3) and that of 17(~o) and J j ( w ) are substituted in (3.5) and then in eqs. (3.2), we get the f r e q u e n c y dependence of the propagators //, E Z and M. This d e p e n d e n c e and especially the d e p e n d e n c e of the denominator is our primary interest. It is observed that all the propagators have the same denominator. In what follows we shall discuss the case when we have isotropic fields. This means that the n u m b e r nk depends on (k) only.

4. The dynamical equations In the case of isotropic un-polarized fields, nk depends on (k) alone, and the expressions in (3.4) can now be calculated easily. For such fields it is found that J~(o~)= J~-(o~)= J](w)6), while we find that 1~(~o) and l ~ ( w ) vanish, In this case, we can write the denominator which is c o m m o n to all propagators in the f o r m ( 1 - a+J~-)(l- a _ J ~ ) . Consequently we are able to express the propagator 11,... in the following forms. Al

A2

11 = 1 - a + J o + 1 - a _ J ~ "

_ --

B~

Z =

C1 l-a+J~

M

+

1 - a+J~

B~

(4.1a) (4. I b)

1 - a J~"

t-

C2

(4. lc)

l-a_Y~'

D1 D2 = 1 - a+J-~ + 1 - a J o

(4.1d)

D I P O L A R C O N T R I B U T I O N S TO T H E R E A C T I O N F I E L D

323

where +

a± =

_ ( a + ~ + - ~÷~÷)

AI = L { o t ( a + - 3~+)+ ~'+},

A2 --- L{ot (3 ~+ - a _ ) - ~ + } ,

B, = L{~(a+

B2 = L { ~ ( o t ÷ - a _ ) - a ~ ÷ } ,

- a +) + a ~ + } ,

C 1 ~--- L{~(a÷ - ~/+) + y~'+},

C2 = L{~(~/+ - a_) - y~+},

D, = L { y ( a + - a ÷) + ~ ÷ } ,

D2 = L{V(a + - a _ ) - ~'~:*}

(4.2a)

(4.2b)

with L = (a+ - a_) -l. W e n o w u s e the e x p r e s s i o n s (4.1) in (3.1). It is p l a u s i b l e to a s s u m e t h a t e a c h field is b r o k e n into t w o p i e c e s , w h e r e e v e r y p i e c e satisfies a d i f f e r e n t e q u a t i o n . B e a r i n g this in mind, w e n o w w r i t e p = p, + P2 a n d m = ml + m2, a n d c a s t e q u a t i o n s (3.1) in t h e f o l l o w i n g f o r m : (1 - a+Jg)pt = A ~ E + B~H,

(4.3a)

(1 - a-J~)p2 = A ~ E + B ; H ,

(4.3b)

a÷J~)mj = C~E + D~H,

(4.3c)

(1 - a-J~)m2 = C~E + D'2H,

(4.3d)

(1 -

w h e r e A ' = - (p~ - poo)A . . . . etc. T h e d e p e n d e n c e o f A, B . . . . . a ÷ a n d a - on to is v e r y c o m p l i c a t e d . T h u s to w r i t e eqs. (4.3) in d y n a m i c a l f o r m is n o t o b v i o u s . W e shall m a k e an a p p r o x i m a t i o n w h i c h is e s s e n t i a l l y as f o l l o w s . A f t e r i n t r o d u c i n g t h e e x p r e s s i o n s f o r a . . . . o f (3.3) in (4.2a), w e w r i t e to, = to in t h e n u m e r a t o r o f t h e r a d i c a n d o f (4.2a). T h i s is e q u i v a l e n t to a n e a r r e s o n a n c e a p p r o x i m a t i o n . W h e n this is p e r f o r m e d , w e find t h a t (4.3a) f o r e x a m p l e t a k e s the f o r m (tos2 _ to2_ 2to(tz2 + m2 )j~(to))p~ = - (ps, - poo){2/Xo2~to~E- 2 itxosmo~tosH}.

(4.4a)

W i t h i n t h e s a m e a p p r o x i m a t i o n (4.3b) is w r i t t e n as f o l l o w s toP2

=

, • - 2 i/x~sm0s H ~. 2 2 / ~,(~ o, + m0,)

-- (Pss -- po0)

(4.4b)

O n t h e o t h e r h a n d , t h e e q u a t i o n s f o r m~ a n d m2 a r e e x p r e s s e d in t h e f o l l o w i n g way: (tos2 _ to2_ 2to(ix2 + m~,)J-~(to))m. = - (Pss - poo)(2 i/zosmo~tosE + 2m o2~tosH),

(4.4c)

324

A.-S.F. O B A D A

t°m2=

--(Pss--

AND A.M.M. ABU-SITTA

"[2 i/x°sm °~s )

P o o J l 2 -2 X/J, Os t /r/0s

E

.

(4.4d)

It is apparent now, that the two fields p, and rn,, have the equations of motions p~(t) +/~p'~ + ,O2spt(t) = a~E + b~H,

(4.5a)

fh~(t) + Fro'fit) + 12~ml(t) = a l E + b]H.

(4.5b)

In obtaining these last equations we have used + J2~2 - to~2 - 2w~(/xo2~+ mo2D(Re Jo(ws))

(4.6a)

P = 2(/zg, + mgs)(Im J'~(Ws)).

(4.6b)

and

It is obvious that the governing equation of these fields is the driven damped oscillator equation. The driving force depends upon the electric and magnetic fields. Under certain conditions 6) the second term in (4.5) could be replaced by a term proportional to the 3rd derivative. With this in mind, the interpretation of reaction field in the sense of W e i s s k o p f and Wigner is apparent. The other two fields P2 and m2 satisfy the equations of motion p2(t) = d H ( t )

(4.5c)

rh2(t) = - d ' E ( t ) .

(4.5d)

and

These equations could be understood as off-resonance fields. It is noted here that the field P2 is produced by the magnetic field H, while m2 is given by the electric field E.

5. Discussion

Consideration of a magnetic dipole associated with a two-level atom, generalizes the Dyson-like equation for the dipole propagator. The representation of the magnetic dipole in terms of Pauli matrices changes the pattern. The susceptibilities now change, and depend on J+ and I +. For an isotropic unpolarized field the tensor J+ b e c o m e s isotropic, while I+ vanishes. This makes the problem s o m e w h a t simpler and consequently the fields are broken into two parts. The first of these parts which we m a y call

DIPOLAR CONTRIBUTIONS TO THE REACTION FIELD

325

resonant, at a frequency/2s. Near resonance this frequency is shifted from tos by the frequency shift A = (0z2s + rn2,) • Re J~(to,).

(5.1)

It contains the vacuum shift (Lamb shift) and the field dependent shift. This can be exhibited as follows. kc

(meC2"~ + 8_8

Re J~(co) = ~4 ( c ) f k dk

k 3 dk

0

(5.2) where we have made a cut-off at the electron rest energy htoc meC 2. The second term is proportional to the Lamb shift. The shift A is enlarged by a factor (1 + (m0J~os)2). It is apparent that it is due to the introduction of the magnetic-dipole interaction to the theory. For a distribution of the profile =

nk=n=constant =0

for a < t o / c < b ,

otherwise,

the second integral has the form 3,n. n ( ( ~to) ( l3n b 2 - ( t ° [ c ) 2 ~ + ( b (~olc) 2 - a }

2 - a2)}"

(5.2a)

For thermal photons, this integral becomes

31r

(x 2 - x~)(ex - 1)'

(5.2b)

0

with x0 = hco/¢3, /3 = kT, and hence, the conclusion that it is proportional to T 2 (for high temperature) and T 4 (for low temperature)~4). The damping f a c t o r / ~ of (4.6b) on the other hand does not depend on the profile of the photon distribution. In effect

/ •=

2 3 2 ~(toslc) (IZo., + m2s) " (2n(~/c) + 1)sgn(tos).

(5.3)

It is remarked that the inclusion of the magnetic dipole enhances the damping by the same factor {1 + (moJiZo,)Z}. For to~ > 0 and the distributions underlying (5.2) we find respectively /~ = (ire + Fm)(2n + 1) = (Fe + F r o )

for a < toslc < b, otherwise,

(5.4a)

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A.-S.F. OBADA AND A.M.M. ABU-SITTA

and = ( F e + Fro) coth(htos/2/3),

(5.4b)

w h e r e Fe a n d / ' m are the E i n s t e i n f a c t o r s for the electric- a n d m a g n e t i c dipole, respectively. T h e a l m o s t at r e s o n a n c e a p p r o x i m a t i o n w h i c h we h a v e used, r e s u l t e d in m a k i n g the s e c o n d part of the fields to satisfy the o f f - r e s o n a n t e q u a t i o n [(4.5b) or (4.5d)]. It is not feasible, in g e n e r a l , to write d o w n e q u a t i o n s of m o t i o n s similar to (4.5), u n l e s s the a p p r o x i m a t i o n is made. This is due to the c o m p l e x f o r m u l a e of the coefficients of (4.2). In c o n c l u s i o n , the i n c l u s i o n of the m a g n e t i c dipole m o m e n t , e n l a r g e s the f r e q u e n c y shift, e n h a n c e s the line b r e a d t h , or the r e a c t i o n field, a n d adds new f e a t u r e s to the fields. W e hope to c o n s i d e r the case of an e x t e n d e d s y s t e m of particles in a later investigation.

Acknowledgment W e wish to e x p r e s s o u r g r a t i t u d e to Prof. R.K. B u i l o u g h of T h e U n i v e r s i t y of M a n c h e s t e r , I n s t i t u t e of S c i e n c e a n d T e c h n o l o g y for a v a l u a b l e d i s c u s s i o n .

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