Path-summation approach to the dynamics of radiative phenomena

Physica A 166 (1990) North-Holland

361-386

PATH-SUMMATION RADIATIVE

APPROACH

TO THE DYNAMICS

OF

PHENOMENA

Petr CHVOSTA’ Service de Chimie Physique II, Facultk des Sciences, Brussels, Belgium

Received

27 June

C. P. 231, Universitk Libre de Bruxelles,

1989

We investigate a new form of the path-summation method, which is related to the Feynman-Vernon influence-functional approach, within a fairly standard model describing the two-level atom interacting with the (quantum) electromagnetic field. Our method, being essentially a nonperturbative one, is developed with perspective of application to dynamical problems, where the parametrs of the (weak) atom-field interaction are combined with the parameters describing the (strong) pulse of radiation in the initial condition and/or in situations where one desires to calculate the time evolution of the field quantities. However, in this paper, after displaying the path-summation method in its general form, we test its possibilities just in the simplest case of populational-difference time evolution. The populational difference is formally expressed as a series in the level-splitting parameter. In the terms of this series, field averaging is explicitly performed and the physical analysis of the path contributions leads to a hierarchical class of nonperturbative partial-summation approximations based on the damping of the field correlations. We propose a trial form of the “correlation-damping” function, which enables the explicit discussion of the resulting time evolution. Finally, we compare the dynamics so obtained with the usual approach using the perturbative (in the atom-field interaction) expansion of the memory operator.

1. Introductory Starting ents,

the

remarks.

with Einstein’s investigation

The definition 1917 paper, of the

of the model which introduced

interaction

of the

the A and B coeffici-

two-level

atom

with

the

electromagnetic field has attracted the attention of physicists. Since then, several times a new dimension was added to this seemingly simple problem, but a lot of questions still remain open. In recent years, it is the dynamical description of the related phenomena above all, which is actively investigated. Thorough physical discussion of the related difficulties can be found in the recent papers of F. Henin and J. Jeener, [l, 21 which, in fact, have served as our guide and provided us with the motivation for the present study. I On leave

from

0378-4371/90/$03.50

Institute

0

of Physics

1990 - Elsevier

of Charles

Science

University,

Publishers

121 16 Prague,

B.V. (North-Holland)

Czechoslovakia.

362

P. Chvosta I Dynamics of radiative phenomena

Let us first approach the problem in question in its simplest possible form. The salient features of the two-level-atom-plus-field system are believed to be described by the Hamiltonian [l, 3,4]

(1) d escribes the isolated two-level Here the operator M=$(le)(el-jg)(g() atom with the stationary states le) (the excited state) and 1g) (the ground state), hA is the energy difference between these two states. The atom-field interaction is that of the dipole-moment versus electric-field type. D = 1(14(kA+ Id(4 comes from the atom dipole-moment operator and ; HF= : qB;Bi, i=l

R = : i=l

(h,B;

+ ATB,) .

(2)

The field operator R originates from the intensity of the electric field with the magnitude of the dipole-moment matrix element being already included in the coupling constants Ai, i = 1, . . . , M. Starting with an initial conditions, ~(t = O)%ffp,, one pursues the time evolution of the (normalized) mean value of some physical observable which is represented by the Hermitian operator Q. To put it more exactly, one examines the real function w(t)

~

(Q

I expW=Qld (Qb,)

Here .9=(llh)[H, *] is the Liouville superoperator [2,5] (we shall use the term “operator” since the distinction between operators and superoperators will be given by the symbols used, e.g. 2 versus L). In (3), (Xl V) = Tr,+,(X+Y) denotes the Hilbert-Schmidt scalar product in the atom-plus-field Liouville space. Usually, the initial condition is assumed to be of the type p0 = le) (el @ rr. For example, one takes “the atom in a bare excited state and the field in its own equilibrium state at t = 0”. However, as stressed in ref. [l], this initial state cannot be prepared experimentally nor it corresponds to any physically plausible past for t < 0. That is why the authors of [l, 21 explore the time evolution of a realistic and stable initial condition, which can be described as “the atom-plus-field system in the true ground state, IgA+F), and a quasicoherent pulse of radiation created in the field”. In our approach, it is essential that this initial condition can be formally obtained by acting with some exponential field operators on the bare basis kets. Actually, the variational analysis of the

P. Chvosta I Dynamics of radiative phenomena

true ground

state

leads

Ih+d -[t&W+

to the ansatz

363

[6,7]

+ SF->lKld + l4Plva41

+ P&C

+ 5~+)lKld - le))@lvac)l~

F, = exp

(4)

(5)

parameters chosen to minimize 5, &, i = 1, . . . , M, are some variational I/-/ ]g,,,). Thus the ground state is given with high the expectation value (g,,, precision by displaced oscillator functions and contains an admixture of both bare states I g), le). Th e creation of the incident pulse can then be done by where

acting

with the Glauber

coherent-state

operators

[l, 2,8]:

Id = exp( a ij {ais’ - ai*&,*I) llh+,>( h+A).

(6)

Here (Yis the characteristic magnitude of the one-mode coherent-state parameof the (field) operators X ters (Ye,. . . , aM and {X, Y} is the anticommutator and Y. The coherent-state parameters are chosen in such a way that the corresponding classical pulse would have been quasiresonant with the atom and it would have moved towards the atom. Consequently, as for the state of the atom, the time evolution stemming from (6) should be close to that given by the usual Maxwell-Bloch equations [3,4] (in the latter case, however, the results are obtained within has performed the transition

the semiclassical to the interaction

approximation). Now, after one picture in the manner of ref. [l],

the combination ah begins to appear in the solution. Obviously, provided the magnitude of the (quasiresonant) pulse (Yis not small, the traditional perturbative approach

can give misleading

conclusions.

As for the operator Q in (3), the calculations are usually directed towards the quantities describing the atom itself, such as the populational relaxation and/or the dephasing. This corresponds to Q = QA @ 7,. Nevertheless, in a real experiment, one can never observe the state of the atom per se. Instead of that, one measures the changes in the field caused by the atom-field interaction. From this point of view, the approaches using the reduced information provided by the density matrix a(t) = Tr,[p(t)], such as the generalized master equation (GME) mains of heuristic

method with the trace-making projector [5,9,10-121, reinterest. In fact, these approaches orginate from Brown-

particle type of problems. A physically more satisfactory description of the atom-field system is given by subdynamics theory (for recent developments see refs. [13-16]), where the subspaces and the projectors are connected with the

364

P. Chvosta

I Dynamics

of radiative phenomena

spectral properties of the whole Hamiltonian. Dealing with the field-observable Q = I, 8 Q,, we can again use an exponential operator and rewrite the function (3) as w(c) = qt, x = 0)

qt,

x) =

d &

)

(L+FIexp(xL@0,) exp(-i~%d (I*@ QFIPO) .

(7)

Let us now introduce the parameters of the above problem. Clearly, the function w(t) is only dependent upon (1) the level splitting A, (2) the coupling constants h,i=l,..., M, and the individual-mode frequencies oi, i = M, (3) the parameters pi, &, . . . describing the initial condition, and which specify the operator Q in eq. (3). In the infinite-volume limit, the number of the field modes, M, tends to infinity and the distribution of the coupling constants together with the density of the field modes can be described by the single function

Y(W)= jpx

L$, Ih,12S(o - q)

)

(8)

which is often referred to as the strength function [17,18]. In nonrelativistic quantum mechanics, the strength function possesses two essential features. First, as A, 0~a, one obtains (after integration over the angle coordinates of the wave vector) y( w) m w for small enough w. Second, it is well known that the relativistic approach brings on the present nonrelativistic level the notion of cutoff frequency. This means that the function (8) is effectively damped for w > 6.1,where hw, is of the order rnec2. The precise form of this cutoff is not important, and we shall use the specific form of the strength function according to the formula y(w) = y

zc exp ( -

F

c)

The parameter y measures the overall strength of the coupling and W, gives the relevant interaction bandwidth. On the whole, in the above infinite-volume limit sense, the parameters of the Hamiltonian are A, y and w,. However, it is an important feature of the nonperturbative approaches that “all frequency scales are coupled together” [19,20] and the parameters of the model can appear in quite unexpected combinations, in different regimes. Technically, this is due to the fact that one deals with a complex system of field excitations, which is usually referred to as “the cloud of photons”. The description of such

P. Chvosta ! Dynamics of radiative phenomena

365

a system can only be possible if one “keeps the coupling parameters hi in the This is precisely an essential feature of our approach and it exponent”. contrasts our method to the perturbative ones. On the whole, apart from the fact that the atom-field interaction is weak, it is highly desirable to have an nonperturbative scheme for dynamical calculations. Of course, such a scheme should reproduce the Wigner-Weisskopf (or the generalized master equation) results in the traditional situations. On the other hand, it should not be limited by the requirement of the abovementioned simple initial condition and/or by the reduction of information. In this article, we propose such a scheme and we wish to analyze its possibilities. The plan of the paper is as follows. In the second section we present the path-summation method in its general form. To proceed further, one must calculate the matrix elements of some exponential-type field operators and solve the problem of the A summation. These two points are dealt with in the third section in some more detail. However, to test our method in its “pure form” as well as to keep this paper to a reasonable length, we confine ourselves to the simplest case of the dynamics of the populational difference and the calculations in the above more interesting situations are postponed to a forthcoming paper. It should be clear that all essential steps used in the present study of the populational difference can be carried out in the above more interesting situations, too. Up to the fourth section, our conclusions are exact. In the fourth section, and hierarchical class of nonperturbative approximations is introduced (independent clusters approximations - ICA). Further, we study the way in which the Born approximation (BA) is contained within the path-summation scheme. Eventually, the fifth section presents numerical demonstrations and the discussion.

2. General form of the path-summation

method

Let us introduce the basis 11) = lie) (el), 12) = llg)( gl), 13) = lie){ gl), 14) = IIg) (el) in the Liouville space of the atom (notice the ordering of the basis), which is relevant in the matrix expressions below. Using the scalar product in the Liouville space of the atom, the full Liouville operator can be expressed in the form of a 4 x 4 matrix whose elements are operators in the reservoir space:

366

P. Chvosta / Dynamics of radiative phenomena

0

z~F=;

0

-(:R)+& i +(R*)

-(*R)

-(*R)

+(R*)

+(R*)

-(*RI

0

0

0

0

’

(10)

Here the symbol (* R) means the field operator

acting as (* R)jX) = IX/?). In the form (lo), the Liouville operator is a convenient starting point for the usual timer-dependent perturbation theory. The motivation of our next step is based on the Feynmann-Vernon pathintegral theory [19,21] (f or recent developments within the quantum-coherence problem see refs. [22-261). Imagine that A = 0; then, of course, the Hamiltonian (1) ceases to have any physical meaning. However, using e.g. the polaron transformation method [27], it is then exactly solvable. Now, our A is neither zero nor a small parameter. Nevertheless, according to the Feynman-Vernon influence-functional scheme, we are free to express the evolution as a sum (Gfunctional integral) of all possible ways (=paths) of the initial-state-finalstate transitions, each of them being weighted by the influence functional. The path contributions reflect the transitions between four possible “channels” of evolution with the probability (per second) A/2. Within a specific channel, the dynamics is influenced solely by the variables of the field. Our approach does not use the influence functional, explicitly. Instead of that, it is based solely on the algebraic properties of the field operators and it possibly offers a deeper insight into the technical possibilities. Nevertheless, the underlying idea of path summation is the same and it will clearly emerge after we switch to the new basis in the Liouville space of the atom, Ii) = $]Il) + 12) + 13) + 1411 > 12) = f 1% = 12) =

[ll) + 12)-

13) - 14)l 1

tm - I4 - 13) + t[ll) - 12)+ 13) -

1411 >

(11’

l4)l )

in which the Liouville operator

looks like

(12)

367

P. Chvosta I Dynamics of radiative phenomena

Here the diagonal term, T,,, corresponds to ZA, + $A 8 ZF in (10). We have introduced the assignments 9 = 1 [R, * 1, %! = f {R, * }, with R as in (2) and with {X, Y} being the anticommutator of the (field) operators X and Y. The operator Ju acts, in fact, in the Liouville space of the atom and one can show JlJ13=&. Employing the usual repeated-convolution expansion of an exponential operator with two non-commuting operators in the exponent (see e.g. refs. [3, 12]), the resolvent exp(-iG’)dzf%(t) can be expressed as cc

%(t) =

c %‘“‘(t) n=O

= z.

(-id)”

/ dt,, . - * f dt, exp[-i(t 0

0 x

- t2n-1 ).Zn] . . . Ju exp(-it&)

exp[-i(t,,

At

- fZn)&,] .

(13)

Under a closer examination of the series (13), it turns out that its first term is diagonal, all even terms Qc2”)(t), n = 1,2, . . . , have their non-zero matrix elements just in the upper-diagonal and lower-diagonal 2 x 2 blocks and all odd terms OU(2n-1)(t), n = 1,2, . . . , have the non-zero terms just in the upper-right and lower-left 2 X 2 blocks. To proceed further, let us introduce the field operators which will play a fundamental role in the following: 92*(t) = exp[-it(&

* S)]

B,(t) = exp[-it(&

+ %!)I,

,

do(t) = w+w + SQ-(91> So(t) = @B+(t)

+ B-(t)]

(14)

.

Thus e.g. the first term of the series (13) reads

Q”‘(t) = exp(-it%,) = diag[d+(t),

d_(t), B+(t), LX_(t)] ,

(15)

and the operator J! merely combines SII* and S3, in the higher terms of (13). Consequently, it is possible to eliminate Ju completely and express the matrix elements of the terms a”“‘(t) as e.g. %cy’(t) = (-id)*”

[ dt,, . . . [ dt, a+(t - t2n) Bo(t2, - t,,_,) 0

0 x

Eventually,

~o(fz,-,

-

t,,-2).

* . 93o(t2

we use the above expressions

-

4)

d+(fl).

for the resolvent operator

(16)

in the

368

P. Chvosta I Dynamics of radiative phenomena

definition (3). To this end, we expand the initial condition and the operator Q in the new basis (11) (this can be done quite generally): (Q] = Cf=, (F]@(ai] and 1po) = Cp=1 Ii”) C3(Gi). The object function, w(t), then acquires the form of a linear combination of the matrix elements of the field operators,

C (cl,l’ij(‘)l’j) w(t>= iTi= 4

.

(17)

Here, the field-operators Qi, are the matrix elements of “u(t) in the new basis, i.e. the series of terms like (16). In eq. (17), both scalar products are in the Liouville space of the field. The natural question to ask is whether any advantage has been acquired by expressing the function w(t) in the form (17). In the forthcoming sections, we would like to show that the answer is affirmative. Presently, two general points should be stressed. First, any further development of the whole scheme rests on the calculation of the matrix elements of some exponential field operators between (Gil and I;,), i, j= 1,. . . ,4. Surprisingly enough, this can be done for quite a broad class of initial conditions and/or the operators Q. Second, one is left with the problem of A summation. These two points are analyzed in the next section.

3. Exact expressions for the populational

difference

Starting with this section, we shall concentrate our efforts on the dynamics of the populational difference resulting from the initial condition “atom in the bare excited state and the field in the thermal equilibrium (chaotic) state”. Thus the function (3) acquires the form w(t) = 2(M 63 7,j exp(-it2)

lie) (e] 8 nr)

(18)

with M as in eq. (1) and nF = exp(-pH,)lTr,[exp(--p&)1, p = l/k,T. One has that w(0) = 1 and lim,,, w(~)~Z~W, is supposed to be a negative real number reflecting the fact that the dynamics should result in an equilibrium value. Repeating step-by-step the procedure from the preceding section, the resulting path-summation form of the function w(t) reads

w(t) = p(t) - f-(t) + s(t)

2

(19)

369

P. Chvosta I Dynamics of radiative phenomena

where

At>

parameter

= ( IF1 ~&I/

~~1 and

r(t)

Mt))

is a series in odd (even) powers

of the

A: I

r(t) = 2

r,(t) = 2

(-iA)2nP’

s(t) = "g, s,(t) = n!l

0

(-id)*”

are matrix

(20)

i dt,, * * * f dt, s”,(t, t2,,,

elements

..

0

0

The integrands

. . . 1 dt, r”,(t, tzn_,,

1 dt,,_,

n=l

??=I

12

(21)

. . , t,)

0

of the field operators: (n- 1) times

~(7F~~+~o~o~~~~o~o~~~~o~o[~+ -&]I&

. . . ,t1)=

r”,(f, &,,

(22) (ns”,k

&,

. f . , f,)=

1) times

~(~,~~+sB,~o~o~~~~o~~~~~~o~o[~+

- %_]17fF)) (23)

with

the

time

arguments

t - tZn_, , t2n_1 - t2n_2, . . . , t, in (22)

t 2n -t 2n-1,. The formulae . . 5 t, in (23). all our subsequent discussion. 3.1.

Algebraical

In this sub-section,

procedure

(19),

(20)-(23)

and

t - t,, ,

will form the basis of

for the field averaging

our concern

is to calculate

a typical

Notice that all matrix elements which occur in p(t), type. In a sense, this is also true in the more general

matrix

element

r(t) and s(t) are of this situations mentioned in

the first section. In keeping with the terminology, which was established in the functionalintegral approach to the spin-boson problem 119,231, the operator $21Pi(b,)in (24) can be thought as if it represents the ith “blip”, i = 1, . . . , n, pi = tl being its sign and bi = t2i - &_, its length. Thus there are it time-ordered blips in the matrix element (24). Similarly, think of the operator &Jai) as of the

370

P. Chvosta

I Dynamics

of radiative

phenomena

ith “sojourn,” i = 1, . . . , n, where (Y~= ?l is its sign and a, = t2i_1 - t2r_2 its length. First, it is quite easy to prove that the matrix element (24) can be written as a product of M matrix elements, each for one individual mode of the field. The calculation of an one-mode matrix element gives an exponential (see below). Hence, in the final product, the summation over the modes occurs in the exponent which is afterwards included in the functions fi(t) and &(t) (see below). Accordingly, consider first a fixed one-mode matrix element, Zn,i. Hence we drop the mode-index i and we designate u = Aiw, r = wt. The following procedures involve just five one-mode field operators and the essential point is that they form the Lie algebra [28]. Table I gives their definitions as well as their mutual commutators. This Lie algebra and/or its subalgebras can be treated by the parameter-differentiation method. The Wilcox’s article [28] presents the modus operandi at great length and we restrict ourselves to the list of identities which will be used in the following. (1) Let (Ybe an arbitrary complex number. Then the following factorization takes place: exp[a(g

? s)] = exp(&)

exp[ + F sinh (Y? “21(1 - cash CX)], (25)

exp[cY(%’? &!)I = exp(cY%) exp[+ C%sinh (Y+ p(l - cash CX)], (2) Let (Y,p be two complex numbers. The left-commutation exp(@) can be carried out using the formulae exp(cu9 + pa) = exp(@)

of the operator

exp( $?) expl( CYcash y - /3 sinh y)%+

(-a

sinh y + p cash

y)‘%] ,

exp(a& + p9) exp(y9) = exp(-$?) exp[(a cash y - /3 sinh y)a

Table I The commutators

9

8

are given. 9

0

9

w

4

s

*}

-9

0

0

0

u’9,

B, *)

-&

0

0

-u29,

0

*I

-%

0

l2.9,

0

0

-3

- u’ca,

0

0

0

in the form

9e[l3+/3, *] &=$lu{B++B, pp-

tu{s+-

57,-~u[B++B, 4-&u[B+-B,*]

+ (--a sinh y + p cash y)Y] .

(row,

column)

3

011

371

P. Chvosta I Dynamics of radiative phenomena

let

exp(&!

p, y

S be

+

=

exp( @

73 +

density

matrix

(7,( exp( p!%! + v~)]T~)

numbers.

6021)

exp(&

(4) Let p, v are arbitrary the canonical

complex

+

complex from

exp[u2(&

numbers (24).

- Pr )] .

(27)

and 7rF the one-mode

version

of

Then

= exp[ iu’( p2 - v’) coth( I#+)]

.

(28)

Let us now return to the one-mode matrix element of the type (24). Its evaluation proceeds through the following four steps, which are closely connected to the four above identities. (1) Using (25), one can factorize all the exponential operators, representing blips and sojourns, i.e. one can write them as a product of the two exponentials. The first exponential will be of the form exp(cYg), and the second will have only the operators &, 9 (for &,) or only the operators s, 021(for CB3,,) in the exponent. of the form (2) As ( I,/ exp(c-Y% = ( f,i, one wants to get all operators exp(&‘) to the left. This is done using the formulae (26). Pushing exp(c&‘) to the left will merely change the time arguments of the functions sinh and cash. (3) Similarly, operators of the to do it. Loosely commuted over all previous

as both (7,].? and ( 7F]% equal to zero, one commutes form exp(@ + I,!&) to the left. The formula (27) shows speaking, for a given blip, all the previous sojourns have it (towards ( 7F]). This gives the “correlation” of this blip --(bms)in the final expression soujourns and results in the term e,

below. (4) Eventually, &)I TV) where

one

has only

certain

element

of the

6 and E are some sums of the functions

form

all how to be with (29)

(7,] exp(@

sinh and cash.

+

At this

point, one makes use of the averaging prescription (28), and two temperaturedependent factors appear in the final formula (29). The first of them, e”:), expresses for each given blip (the summation over i in the exponent) the self-interaction of this blip. This factor does not depend on the signs of the blips. The second, e”pmb), describes blip-blip correlations for all possible pairs of blips. For a given pair of the blips (index i and j in the summation in exponent), this correlation depends on the relative sign p,p,. A given blip is correlated only with the previous ones. Leaving out all further details, we now write the final result in a fairly compact form (the product of one-mode matrix elements has already been

372

P. Chvosta I Dynamics of radiativephenomena

performed),

e,(4

nr

&I,.

= fi

..

) t,,;

t,

3

a!,)

. .

)

a!,;

p,

)

.

.

.

PiPjft”‘)

r=l

e”Pmb) q-“’

p,) = p

)

exp(i z$

‘c’

p(t2i - t2,-,) exp($

1=l

.

j=1

Pfajfl”“)

.

(29)

fl”” =flCt*i-

+fitt2i-l -

t*j-l)

-fi(t2i -

t2j-2)

-f*(&-I

t2j-2)

-

t2j_l)

7

(30) f2

(i’j)

=f*(It21

-

t*,)

+f2(t2ikl

-

t2j-*)

-f*(t*i

-

t2jm*)

-f*(t*i-I

-

t,j)

.

(31) The coupling parameters enter only through the functions and which appear as a result of the above-mentioned mode summation in the exponent, fi(t)

fi(t) e g

($1’

f2(t),

sin(w,t) , (32)

f2(t)

=

5

(!$)2,1

i=l

cos(w;t)] coth($&w;)

-

,

.

In leaving this subject, one should note that the operators B3,, L$* are combined in a special way in the relevant matrix elements (23), (22). Specifically, they appear in the combinations zZ,, and W,. Therefore, it is possible to combine appropriate terms of the form (29) in such a way that the four-terms appear as the arguments of the functions sin and cos. combinations Similarly, the four-term combinations j, appear as the arguments of the functions sinh and cash. On the whole, all terms ri(t) in (20) and s,(t) in (21) are real and they have the form similar to that exemplified below. As an example of the whole procedure, we give the explicit forms of the functions p(t), r,(t), s,(t) and r2(t) from (19) and (20,21). One has p(t) = f

t,j’,

i

j,

2

f

f*j),

i

>

exp[-f2Wl and I

r,(t) = A

dt,

p(t -

f

f2

t,)

sin(

f

(1l.l))

(33)

,

I

0

dt, 0

0

p(t -

t2) p(tl)

cos(

f

(1272))

sinh(

f

r31))

,

(34)

P. Chvosta

r2(t) = -A

I Dynamics

of radiative phenomena

dt, p(t -

f3)

~(4)

[sin( fi2,“) cos( fi”“)

cosh( fy”‘)

+ cos( fi2”‘) sin( fi”“)

sinh( fp”‘)]

x

313

~os(f(l~*~))

.

(35)

3.2. Problem of the A summation The importance of infinite A summation in the formula (19) is evident from the observation that the asymptotic value of the function w(t) should be A dependent in an nontrivial way. In the Born approximation (BA), for example, one has that wEA = -tanh( ifid@). The capability of the BA to induce this reasonable asymptotic value stems from the fact that one performs the perturbation truncation in the GME memory operator and not in the resolvent %(t) as in usual time-dependent perturbation theory. Truncation of the memory operator to a finite order (in coupling parameter y) gives certain infinite-order contributions (in y ) in the direct perturbative expansion of w(t). This suggests that in order to sum-up certain “relevant” contributions in the A expansion of the function w(t), it should be sufficient to take a finite number of terms in the A expansion of the memory operator. However, as A is not small parameter, the natural question arises: what is the physical meaning of the A truncation of the memory operator? We postpone the study of the answer to the next section. Presently, we derive an exact expression for the Laplace transformation of the function w(t) in the form of a fraction which has the A series in both the denumerator and the nominator. With this strategy in mind, we expand the memory operator [5] X(t) = ~o.ZSJ exp[-itgrT]

~lZ~o

(36)

with

in the A series similar to (13). Formal solution of the GME then leads to the function w(z) expressed through the Laplace transform of the matrix elements of X(t) in the basis (11). We leave out the details of this time-consuming calculation and write down the final result:

42) = P(Z) -P(Z)

42)

- 42)

p(z) _ +)

.

(38)

374

P. Chvosta I Dynamics of radiative phenomena

Here

the functions

m(t)

m

m(t) =

and n(t)

are series

in the level-splitting

parameter:

x

c m,(t) = c k=l

(-iA)2k-’

k=l

/dfzk-C--j&

Gk(t,t2k_1,.

0

. . ,f,),

0

(39)

n(r) = ,c, n,(t)

= k;, (-iA>‘”

[ dt,, . . . j dt, n”,(t, t,,, . . . , t,) . 0

The integrands

are matrix

elements

of the field-operators: (k-l)

&&,

&,-,,

. . ’ 9 tl> = ;

(40)

0

times

~sdo%o~* * do voe,[a+

&l~+~o~oe,*

(k-l)

times 3

&&,

&,

. woLdo.. . ceodo[%+

...

~-ll~FF)~ (42)

The t 2k

time -t

*km,,

arguments are t - tzk_,, tzk_, - tzkp2, . . . , t, in (42) t, in (41). We have introduced the field operators . . ’ >

Ye,(t) = exp[-it(ZF

Taking

X(t)

up to (and

+ P,L%!)] ,

including)

q)(t)

the term

= $[%+(t) + K(t)]

m Ak corresponds

and

t -

.

t,,,

(43)

to taking

the

series m(t) up to (and including) m,(t) and the series n(t) up to (and including) n,(t). The next term of the memory operator, Xck+‘)(t), then brings out the term nk+i (t) to the formula (38). Eq. (38) is precisely of the anticipated form and it suggests a possible way of constructing

the infinite

A summation

approximations.

Indeed,

one can take

e.g. just the terms m,(z) and n,(z) of the series m(z) and n(z), which already induces contributions of all orders in A in the direct expansion (19). There is still one complication in this direction - the operators V. have the projector P1 in the exponent and one can not apply the rules from the subsection 3.1. This makes a direct calculation of the matrix elements (41), (42) impossible. Fortunately, on comparing the two exact expressions (19) and (38), it is evident that a connection should exist between the terms ri, sj in eqs. (20), (21) is based on the repeated and mk, n1 in eqs. (39), (40). This connection application of the following theorem.

375

P. Chvosta I Dynamics of radiative phenomena

Theorem. Let % be an operator in the reservoir Liouville space, IX), 1I’) two elements of this space and 9, the projector from eq. (37). Then the Laplace transform of the function

u(t) = (X]exp(-itP’,

C!?)]Y)

(44)

can be expressed by means of the Laplace transforms a(t) = (I,] exp(-it%)]7rr),

b(t) = (7,l exp(-it%)]

of the functions Y) , (45)

c(t) = (Xl exp(-it%)]7rr)

d(t) = (Xl exp(-it%)]

,

Y)

as

dz> = d(z) -

* zu(z)

[Zb(Z) - b(t = O)] .

(46)

In the proof, the above functions are expanded into their Taylor series. We have e.g. a(t) = Cr=, (-it)“a,ln! with the moments a, = ( I,]%~]~~), n = O,l,.... Using the definition of the projection operator, P1, the moments a,, b,, c,, d,, u, are then shown to obey certain recursive equalities. Eventually, the Laplace transforms of the Taylor series are used and the terms corresponding to the same powers of l/z are compared. The typical usage of this theorem will be quite clear from the example: let us take~=~~+~,(XI=(7,1~=((1,1~andIY)=~l~~)=~;!~~).Onedesires to calculate the function h(t) = (7,/!3? exp[-itP1(.&

+ S?)]~?]G-~).

(47)

The above theorem together with ( lF](ZF + ?A!)]~,) = 0 then yields h(x) = --z + 1 /p(z). Thus the matrix element including the projector CP1in the exponent has been expressed through the matrix element without PI. We now again drop out a considerable deal of mathematics in the spirit of the above example and write down the resulting connection. One gets m,(z) = rl(z),

n,(z)

= s,(z)

and i-l C

m,(z) = r,(z) - ‘=’

i-l m,(z)

C

‘i-,Cz)

P(Z)

ni(z)

si_j(z)

n,(z) = Si(Z) - j=’ ’

P(Z)

’

(48) for i = 2, . . . . These formulae, together with the eq. (38) constitute the starting point for the discussion of the independent clusters approximations in the next

376

P. Chvosta I Dynamics of radiative phenomena

section. Note that the series m(z), n(z) with the terms according to (48) can be summed to give

m(z)= P(Z)

r(z)

p(z)

+

s(z)

3

s(z) P(Z)+ s(z) ’

n(z)= P(Z)

(49)

and on substituting into eq. (38) one recovers w(z) = p(z) - r(z) + s(z). This confirms the consistency of the whole derivation. One can raise the objective that the above procedure rests on the GME, i.e. that it is of the reduction type, which has been criticized in the first section. This is not the case. in fact, the memory operator has played an subsidiary role and the whole procedure can be viewed on the level of connection between the matrix elements Y;,, S; and rii,, il. As such, it can be fruitfully adapted to the more general situations. One then deals with t_he matrix elements ( 6il . . . 1;;i) and with the set of the projectors LPi,j= Iv?~)(C?,[,i, j = 1, . . ,4.

4. Approximative

treatments

Due to the mixture of the “bosons” and “fermions” degrees of freedom in the Hamiltonian (l), the exact diagonalization is inaccessible and one has to admit of an approximation. To get some orientation, let us consider the typical orders of the parameters of the present atom-field problem [3,4]: A = 10” s-‘, y = lOI s-l and w, = 102” s-l. It will be useful to introduce the combinations y”=ylw,,~=A/~,and~=k,Tlhw,.Thusonehasd”~10~5andy”~10~7.As for the temperature, the asymptotic value in the BA distinctly delineates a low-temperature region and a high-temperature one: for ?= 10m6 (i.e. T = lo3 K) one obtains d”l?= 10 and wrA) = -l+ 2exp(-d”/?)= -1. For ?1O-4 (i.e. T= IO5 K) one has d”/?= lo-‘e and WY*) = -i/(2?) = -0.05. It is only in the intermediate region lop6 < ? < 10e4 that the asymptotic value significantly depends on f. 4.1. Independent

clusters approximations

(ZCAs)

In the preceding section, we were able to express the exact solution, w(t), through the two functions of central importance, f,(t), &(t). Using the strength function (8)) one can carry out the integration stemming from the modes summation in (32). The result is

fr (t) = 7 arctan( tw,) ,

f2(t) = -7

ln(vm

“(‘rr:tii’)‘z)

.

(50)

P. Chvosta I Dynamics of radiative phenomena

377

Here r(z) denotes the Euler gamma function of a complex argument. As ? 6 1 in the whole interesting region, we can adopt the corresponding approximative expression for the gamma function [29]. This being done, one has

f*(f) = rrTTy”&‘tw, + 7 In dm (

1 - exp(-2n 2&“,

ho,) 1.

(51)

A number of features should be mentioned in connection with the functions fi (t), f2(t) - see also refs. [19,30-321 where the same function appear in the context of a quantum-coherent effect. (1) The most rapid changes are those which take place on the time scale acquires its asymptotic value, t, = w-lc . Within the time scale, the functionf,(t) TV/~. It then follows from the disposition of the time arguments in (30) that the blip-sojourn correlation function fy’” vanishes whenever the time interval between the non-neighbouring (i > i) blip-sojourn pair is greater than t,. As for the neighbouring blip-sojourn correlation, the factor f(li’i) consists of three terms, and it acquires its asymptotic value TIT/~ whenever the length of both the ith blip and the ith sojourn is greater than t,. (2) The function f*(t) is an increasing function of the time, and one can clearly identify three time scales. For t > t,, the square root in (51) can be approximated by to,. Next, for t > t, =h/2-rrk,T the second term in (51) becomes constant and the whole function f2(t) is, the fact, a linear function of time. Therefore, once the time interval between any two blips in the matrix elements (22), (23) is longer than t,, one has that f$‘” = 0 (cf. again the disposition of the time arguments in (31)) and these two blips are not correlated any more. Eventually, after_ t, the function f*(t) is still very small and it is only on the scale tp = 1 /nyT that it becomes significantly different from zero. As p(t) = exp[-f,(t)], this time scale is also relevant for the decreasing of the function p(t). For f= lo-“, t, = lo-l5 and tp = lo-‘. For f= 10m4, t, = 10-l’ and t = 10-9. These observations su[gest that one should look at the various matrix elements (22), (23) as if they describe a system of correlated blips and sojourns. Once the time interval between two groups of subsequent blips and soujourns is long enough, these two groups are not correlated. Formally, such “splitting” leads to the possibliltiy to write the Laplace transforms of the higher terms ri, si in (20), (21) as a product of Laplace transforms of lower terms. This is the essence of the following construction. Suppose, first, that we decided to ignore all interblip correlations as well as all blip-sojourn ones: fr’” = 0, i > j, fy’j’ = 0, i 2 j. It is then very easy to show that all matrix elements Fi, $ are equal to zero (cf. also examples of their explicit forms (33)-(35)). T o b e more definite, let us call this approximation

378

P. Chvosta I Dynamics of radiative phenomena

the independent clusters approximation of the zeroth order, ICA”‘. In this approximation, r(‘)(t) = 0, s(‘)(t) = 0 and W(O)(~)= p(t) (the upper index denotes the ICA”‘). The recursive formulae (48) then give m, = 0 and Al;= 0, i = 1,2, . . . . Next let us admit just of a blip-preceding-sojourn correlation: ffXi) # 0 and fji.j) = 0, i > j, fy%j) = 0 as above. Each term in (21) contains at least one factor sinh(ft’j)) and so s(‘)(t) = 0. In r(t), the only surviving term is r,(t), which represents just the correlation of one blip and its preceding sojourn. We have to take it exactly, i.e. according to the eq. (33). Therefore, in the ICA”‘, ~(l)(t) =p(t) - rl(t). Notice that the ICA”’ is effectively induced by the assumptions mi +1 = 0 and IZ;= 0, i = 1,2, . . . In the next step, we admit also of the correlation of the neighbouring blips: ff+lXi) #O and the other correlation as above. Presently, all ri, si will have some surviving terms. Namely, rl(t) and sl(t) have to be taken exactly, i.e. according to (33) and (34), and the first term to be influenced is r2(t) - only the second part of eq. (35) survives with cos(f, (*,I)) = 1 in it. Let us call this approximative form r?‘(t). Now, it is a main point of the whole construction that this approximative form is such that r?)(z) = TV s,(z)lp(z). The first relation from (48) then implies m2 = 0. Similarly, on approximative forms of higher terms r,, si, i > 1 to (48), these latter relations give mi = 0, n, = 0, i > 1. Eqs. (49) then yield

,(d”!

r(*)(z)= P(Z)

(z)

s(*)(z) = P(Z),,;“‘i

,

(z)

1

3

(52)

I

and w(*)(z) as given below. One expanding these fractions into power series in A, one can clearly identify the surviving terms in all ri(z), si(z). They are all built just with the lowest-order cluster functions p(z), rl(z) and sl(z). Putting together the considerations of the several above paragraphs, an hierarchical family of approximations clearly emerges. The formal rules leading to the final expression in the ICA’“‘, ,@) (t) are the following. (1) For n = 2k, put mi(t) =0 and ni(t) =O, i= k+ 1,. . . . For 12=2k+ 1, put mi(t) =O, (2) Using the recursion formulae (48) and express n,+,(t)=O,i=k+l,.... the nonzero terms m,(z), nj(z) through r;(z), sj(z). (3) Substitute the results in eq. (38). For it even, the resulting function, W@)(Z), has a polynomial in A of the degree n - 1 in the denominator and that of the degree IZin the numerator. For it odd, the degrees are n and n - 1, respectively. We now give several lowest-order examples of the ICAs. Besides W(O)(~)= p(t) and ~(l)(t) = p(t) - rl(t), we have obtained

[P(Z) - r1(z>l

w(*)(z) =P(Z)

[p(z)

_

s,(z),

7

(53)

P. Chvosta I Dynamics of radiative phenomena

w(3)(z)

=

PC4

[P(Z)

-

r&)1 - TI(Z)b(z) - h(Z)1 7 b(z) - s1(z)l

P(Z)b(z) - r&)1 - TI(Z)[P(Z)- SI(Z)l w(4)(z)= dz) p(z) [p(z) - sz(z)]- s,(z) [p(z) - s1(z)] .

379

(54) (55)

Althogether, guided by these rules, we have reduced the calculation of the populational difference to the numerical integrations (r”,, 5. + ri(t), s,(t)) and to the computation of the direct (r,(t), sj(t)+ ri(z), sj(z)) and the inverse (w’“‘(z)+ w’“‘(t)) Laplace transformation. 4.2. Recovering the BA results The BA results can be obtained by solving the GME with the projector (37) and with the second-order (in coupling parameters Ai) memory operator. This procedure is very well known [lo, 111 and we merely quote the final result,

)pyZ)

=

[z +.Mz)l - ;m Here f,(t) =fo(t)

cos(At),

g,(t)

t q,(z)/zl -f,(z)1 - [z + h(z)1 - Ml(z)-L(z)1 . (56)

= go(t)

to(t) = (7,122 exp(-i&)%]rr)

sin(At) and = 2 ~Ai(*cos(~,t)coth(~~/3wi), i=l

go(t)

=

i( 7,]C%!exp(-itZr)+,)

= ,g, [hi]’ sin(+).

(57)

(58)

Thus the function wCBA)( t ) consists of two terms. The first of them starts from unity, and reflects the damping (to zero) of the initial value. The second starts from zero, and builds up the asymptotic value WY*) = -tanh( ihA@). The time scales for both parts are the same: performing a transition to the master equation (the Markovian approximation [ 10-121) one obtains

dBA)(t) = exp( - $)

+

wp*)[l

-exp(&)j

,

BA

(59) t BA=

2 tanh[d”/(2?)] v(A>

’

With the above set of parameters t,, = llrry?= lo-’ for ?= 1O-4 and t,, = 2/1r5ryd1: lo-’ for ? = 10-4 In the high-temperature region, both the relaxation

380

P. Chvosta

I Dynamics

of radiative phenomena

time and the asymptotic value do not depend on the level-splitting parameter, A. In figs. 1-3, we have contrasted the function (59) against the output of the ICAs (see the next subsection). We now wish to address the question: in what sense is the BA result (56) generated within the path-summation scheme? In search for the answer, it will be useful to cast the exact expression (38) into the shape

w(z)=

[l/p(z)]

1 [m(z) /P(Z)1 - [n(z) /p2(2)] - [l/p(z)] - [n(z) lp2(z)] .

(60)

Now, one would like to show that the square-bracketed functions in the BA formula (56) are the lowest-order (in coupling) limits of their counterparts in the exact expression (60). This can be actually done and we sketch the main points of the proof. As for l/p(z), the connection is quite straightforward: using the example from sub-section 3.2 one has that l/p(z) = z + h(z) and it then follows from (47) that the lowest perturbative order of h(t) is just &(t). Next, one should notice that all square-bracketed functions in (56) are of the second order in the coupling parameters A;. On the other side, e.g. the function g,(z)/z can be written as a series in the odd powers of A and as such it can be compared to the w(f) 0.75 0.50 0.25 0 - 0.25 - 0.50 - 0.75 -1.00

Fig. 1. The time and the temperature dependence of the populational difference in the ICA”’ (the against the BA results (the three thin three thick curves, computed from eq. (64)) contrasted curves, computed from eq. (59)). Each pair of curves (thick curve: ICA”‘; thin-curve: BA) namely (from left to right) ?= 10m4, T= 10m5 and corresponds to a different temperature, ? = 3 X lo-“. Other parameters: o, = ~O*‘S~~, y = 10” s-i and A = 10” s-l. Notice that the timeaxis is logarithmic one (the decadic logarithm was used).

381

P. Chvosta I Dynamics of radiative phenomena

WItI 0.75 0.50 0.25 0

- 0.25 - 0.50 - 0.75 - 1.00 -11

-10

-9

-a

Fig. 2. The time and the temperature dependence of the populational difference in the ICA”’ (the three thick curves, computed from eq. (65)) contrasted against the BA results (the thin curves, computed from eq. (59)). See fig. 1 for explanation and parameters. For ?= 10m4, the ICA”’ curve coincides with the BA one in the resolution of the figure.

w(t) 0.75 -. 0.50 -0.25 -0 -0.25

..

- 0.50 -. - 0.75 -- 1.00 -11

-10

-9

-a

-7

log(t)

Fig. 3. The time and the temperature dependence of the population difference in the ICA”’ (the three thick curves, computed from eq. (66)) contrasted against the BA results (the thin curves, computed from the eq. (59)). See fig. 1 for explanation and parameters. For ? = 10e4, the ICA”’ curve coincides with the BA one in the resolution of the figure.

382

P. Chvosta I Dynamics of radiative phenomena

series formed by m(z)/p( z ) in (60). The first term r1 (z) /p(z). In seeking for the lowest-order perturbative to substitute l/p(z) by z (because of the same reasons the argument

of the sin function

in (33).

Therefore,

of this latter series is contributions, we have as above) and take just we have

f

T’(Z) /p(z)%

ZY~“(Z)

)

~P”(t)=A~dt,[f,(r-r,)+f,(r,)-f(t)l. (61)

Using the expression (32) for the f,(t) together with the definition of the strength function and interchanging the integration over the time with that over the frequency, the function ryA( z ) is shown to be equal to g, r(z) /z*, where g,,i(t) = Atg,(t). Generally, in order to reproduce the term =d(2n+‘) in the A expansion of g,(z)/z one begins with r,(z)lp(z), substitutes l/p(z) by z and takes just the peripheral field operators in (41) (all inner ones have to be substituted by exp(-it&) with their original time arguments). In other words, one has to keep the correlation between the last (in time) blip and the first (in time) sojourn. Obviously, whenever the isolated-blip function p(t) is essentially damped, the above picture soujourn “gas”. The proof done along similar lines.

cannot reproduce the real situation in the blipof the transition r(~)/p*(z)~f~(z) -f,(z) can be

Altogether, the answer to the above question reads: the BA-result cannot be obtained (in the sense of the small-coupling limit) once we invoke the ICA’“‘. This means that the BA takes into account the correlations within arbitrarily large clusters of blips and sojourns. However, within the BA, one approximates all cluster-contributions perturbatively. Contrary to this, in the ICA’“‘, one describes exactly (in the coupling) the correlations in the clusters of the orders

up to n and one approximates

the higher-order complementary tive expansion 4.3.

them

(in a nonperturbative

manner)

in

clusters. From this point of view, the ICAs are seen to be to the family of approximations stemming from the perturbaof the memory

operator.

Explicit expressions for populational

difference

To push further the analysis of the ICAs we now utilize once more the observation that the exact solution depends on the coupling and on the temperature solely through the functions (50). The essential features of these functions were listed above. In fact, instead of assuming some specific form of the strength function (8), one can start with the explicit forms off,(t) andf,(t). We now take them so as they reflect: (1) distinct separation of the time scale t, from the other ones, (2) qualitatively different behaviour of f2(t) in the two regions t E (0, tT) (where it is close to zero) and t 2 t, (where it linearly increases, with the slope given by w,, = 1 ltp = ry?):

383

P. Chvosta I Dynamics of radiative phenomena

T

f1(t> =

o(t)

)

(62)

f*(t) = w,(t - tT) @(t - CT).

Here O(X) = 0 for x < 0 and O(X) = 1 for x 2 0. With these trial forms we were able to realize the whole program delineated at the end of section 4.1. Quite generally, the multiple integrals ri(t) and si(t), have the form of some combinations of the factors i=l,..., (w,t)” exp(- w,t). A ccordingly, the object functions w’“‘(z) are rational functions in both A and z and it is easy to Laplace-invert them. For example d” sin( jk/2) rl(t) = 7 2T %I2

x (opt + @(t - tT){l - w,(t - tr) - exp[-w,(t

- t,)]}) .

(63)

This formula points to several important features which are common to all multiple integrals in (20), (21). All functions rj(t) converge to a nonzero limit and they approach it as (opt)’ exp(- o,t). The limit does depend upon 7, indicating this dependence also in the true asymptotic value w, (renormalization of A). All functions s,(t) converge to zero as (opt)’ exp(-w,t). Further, M ,., rj(t) = SCziP1)and si(t) = c?(*~),where 6 dzfA/T. Accordingly, w(*)(z) is a rational function in 6 and this is also the case for w,(n)dZflim,,, w@)(t). In this argument, we have used the relation lim,,, f(t) = lim,,, zf(z). Eventually, all functions r,(t), s,(t) display qualitatively different behaviour in the initial time interval of the order t, (e.g. rl(t) is linear and s,(t) is equal to zero) in comparison with their nonexponential asymptotic behaviour (for the general analysis of these two points see refs. [33,34]). We have evaluated the explicit forms of the populational difference exactly as they follow from (62). For instance, the second part of w(r)(t) is, of course, identical to (63). However, we give here the forms which result from neglecting the factors of the order coptT = y”/2 (these effects are “invisible” with regard to the above set of parameters and in the scale of the following figures). Then our final results assume the surprisingly transparent form

w(O)(t) = exp(w(l)(t) = @ w(*)(t) = w(*) = m

wy

w,t) ,

(64)

+ (1 - ~2’) exp(-ww,t) ,

+(1-

w’2’)exp[-~Pf(l+

6

2 + [8/(27F)12 .

w(l) m = -612 )

(65)

-$)I,

(66)

384

These

P. Chvosta

functions

I Dynamics

are exemplified

of radiative phenomena

in figs. 1-3 and they are contrasted

against

the

BA curves (59). From the comparison of both aproximative following conclusions emerge. (1) In the ICAs, the (“proper”)

devices, the asymptotic

value

of 6. If the

w?*)

temperature

= -tanh(6/2) is too

low

reproduce this asymptotic the relaxation time, t,,.

is approached (i.e. value

S too

with

high),

properly.

rational

the

functions

lowest-order

(2) A similar

ICA’“’

statement

do not

is valid for

(3)

In the high-temperature region, already the lowest-order ICAs reproduce the BA curves along the whole time-axis. These observations clearly indicate the range of validity for the ICAs and we naturally concede that, with the relevant set of parameters used above and with regard to the particular case of the populational difference, the ITAs are succesful only in so far as they are able to reproduce well-known BA results. However, the path-summation scheme enables the analysis of the BA “from outside” and, at the same time, its “complementarity” (in the sense of the section 4.2) offers new possibilities. This is probably the most important point: once a reasonable agreement with the BA has been gained within its range of validity (the weak coupling), it then follows from the given construction that the ICAs can be trusted for stronger coupling and/or higher temperatures.

5. Summary In this paper we have considered the most commonly used model for the atom versus electromagnetic field interaction and our main emphasis has been on a nonperturbative, time-resolved description. We have traced the nonperturbative analysis in spite of the fact that the atom-field interaction is weak. The reasons for this have been twofold. First, it is the crucial role of the present model in physics which deserves the exploration of new approaches. Second, in the present form, the path-summation device is prepared for fruitful applications within experimentally accessible situations such as the (nonquasiclassical) study of laser-pulse interaction with the atom and/or temporal changes

of the field-energy

flows in the vicinity

of the atom.

Summarising what has been done, first, we have found the exact expression for the populational difference, which has the form of a series in the levelsplitting parameter - this is given by the set of formulae (19)-(23) - and we know how to evaluate all the matrix elements in (22), (23). From this angle, the time evolution can be viewed as a succession of correlated events and one can explicitly investigate the relations between these events. Second, we have constructed the family of the ICAs. We claim that our procedure is not only a nonperturbative one, but it also does not require the level-splitting to be small. This should be understood in the following sense.

385

P. Chvosta I Dynamics of radiative phenomena

The arguments

resulting

in the ICA’“’ are not connected

with the magnitude

of

A. It is only on the face of this procedure subsequently, truncated the A expansion

that we assume A to be small, and, of the GME memory operator. Of

course,

this truncation

effectively,

sequence of certain These

relations,

we did it. However, relations

in turn,

within

the correlated

are not dependent

was merely

a con-

system of blips and sojourns.

on the A at all. On the other hand,

in the final formula, the approximate forms of the cluster contributions are combined with the parameter A in such a manner that the relative error of a of A. given ICA’“’ 1s smaller for a smaller magnitude Eventually, we have chosen the trial forms for the correlation functions and calculated the populational difference in the lowest order ICAs in quite explicit forms. The initial zero-derivative region of the interblip correlation function, i.e. the initial nonexponential region of the blip self-interaction function p(t), was proved to be of fundamental importance. Without it, one simply has that w(t) = p(t) and the evolution does not depend on A at all. Differently speaking, the short-time behaviour of the interblip correlation function influences the asymptotic properties of the populational difference. This evidences the importance of the initial nonexponential period of evolution (Zeno effect) which was recently analyzed in refs. [33,34]. One may wonder what is the physics underlying the family of ICAs. We sketch it very broadly: during the time evolution, the electron accomplishes a succession of jumps between the two superpositions of the states ]g), le). It has to do so because of the non-zero value of the transition probability, which is proportional to A. While remaining in the definite “channel”, the electron is not influenced by the field. However, it influences the field oscillators pushing them to the (displaced) equilibria. After the electron jumps to the opposite “channel”, these equilibrium positions are changed and the field oscillators tend to relax to these new equilibria. are long enough, the field oscillators the preceding

episode

and they start

Now, provided the individual episodes “forget” the details of their evolution in the evolution

in the new one from

the

equilibria congruent with the preceding channel. Increasing the accuracy of the description, we treat the time evolution in several episodes exactly and we make use of the “memory-lost” assumption just after the electron acomplishes the series jumping events. Finally, our study is yet another example for the complexity turbative approach to the time-evolution problems in quantum

of any nonperstatistics.

Acknowledgements The author would like to express his gratitude to Professor Ilya Prigogine stimulating interest in this work and many important comments regarding

for the

P. Chvosta I Dynamics of radiative phenomena

386

atom-field problem. Many thanks are also Professor Fernand May& for their helpful sions. Financial support from the “Instituts Chimie, fond& par E. Solvay” is gratefully

due to Dr. Tomio Petrosky and advices and enlightening discusInternationaux de Physique et de acknowledged.

References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] (171 [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

J. Jeener and F. Henin, Phys. Rev. A 34 (1986) 4897. F. Henin, and J. Jeener, J. Stat. Phys. 48 (1987) 1321. W.H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973). L. Allen and J.H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975). H. Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechanics, Springer Tracts in Modern Physics, vol. 95 (Springer, Berlin, 1982). M. Wagner, J. Phys. A: Math. Gen. 18 (1985) 1915. H.B. Shore and L.M. Sander, Phys. Rev. B 7 (1973) 4537. B.R. Mollow, Phys. Rev. A 12 (1975) 1919. P. Grigolini and G. Pastori Parravicini, Adv. Chem. Phys. 62 (1985) 1. H. Spohn, Rev. Mod. Phys. 53 (1980) 569. H. Spohn and J.L. Lebowitz, Adv. Chem. Phys. 38 (1978) 109. E.B. Davies, Quantum Theory of Open Systems (Academic Press, London, 1976). T. Petrosky and I. Prigogine, Physica A 147 (1988) 439. T. Petrosky, and I. Prigogine, Physica A 147 (1988) 461. C. George, F. MaynC and I. Prigogine, Adv. Chem. Phys. 61 (1985) 223. C. George and F. Mayne, J. Stat. Phys. 48 (1987) 1343. P. Ullersma, Physica 32 (1966) 27. H. Spohn and R. Diimcke, J. Stat Phys. 41 (1985) 389. A.J. Leggett, S. Chakravarty, A.T. Dorsey, P.A. Fisher, A. Garg and W. Zwerger, Rev. Mod. Phys. 59 (1987) 1. U. Weiss, H. Grabert, P. Hanggi and P. Riseborough, Phys. Rev. B 35 (1987) 9535. R.P. Feynman and F.L. Vernon, Ann. Phys. 24 (1963) 118. A.O. Caldeira and A.J. Leggett, Ann. Phys. 149 (1983) 374. S. Chakravarty and A.J. Leggett, Phys. Rev. Lett. 52 (1984) 5. A. Garg, Phys. Rev. B 32 (1985) 4746. U. Weiss and M. Wollensak, Phys. Rev. B37 (1988) 2729. A.T. Dorsey, M.P.A. Fisher and M. Wartak, Phys. Rev. A 33 (1987) 1117. M. Wagner, Unitary Transformation in Solid State Physics, Modern Problems in Condensed Matter Sciences, vol. 15 (North-Holland, Amsterdam, 1986). R.M. Wilcox, J. Math. Phys. 8 (1967) 962. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York, 1980). W. Zwerger, Phys. Rev. B 35 (1987) 4737. C. Aslangul, N. Pottier and D. Saint-James, J. Phys. (Paris) 47 (1986) 1657. C. Aslangul, N. Pottier and D. Saint-James, J. Phys. (Paris) 47 (1986) 1671. T.Y. Petrosky and H. Hasegawa, Physica A 160 (1989) 351. T.Y. Petrosky and W. Saphir, Physica A, submitted.