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VOLUME 93, NUMBER 5

PHYSICAL REVIEW LETTERS

Absorption and Emission in the Non-Poissonian Case

Gerardo Aquino,' Luigi Palatella,2 and Paolo Grigolini1'2'3

1Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427, USA

2Dipartimento di Fisica dell'Universita di Pisa and INFM, via Buonarroti 2, 56126 Pisa, Italy

3Istituto dei Processi Chimico Fisici del CNR Area della Ricerca di Pisa, Via G. Moruzzi 156124 Pisa, Italy

(Received 20 February 2004; published 28 July 2004)

This Letter addresses the challenging problems posed to the Kubo-Anderson (KA) theory by the

discovery of intermittent resonant fluorescence with a nonexponential distribution of waiting times. We

show how to extend the KA theory from aged to aging systems, aging for a very extended time period or

even forever, being a crucial consequence of non-Poisson statistics.DOI: 10.1103/PhysRevLett.93.050601

In the last few years, as a consequence of an increas-

ingly faster technological advance, it has become clear

that the conditions of ordinary statistical mechanics as-

sumed by the line shape theory of Kubo and Anderson

(KA) [1], are violated by some of the new materials. For

instance, the experimental research work of Neuhauser

et al. [2] has established that the fluorescence emission of

single nanocrystals exhibits interesting intermittent be-

havior, namely, a sequence of "light on" and "light off"

states, departing from Poisson statistics. In fact, the

waiting time distribution in both states is nonexponential,

and it shows a universal power law behavior [3]. In this

Letter, for simplicity, we assign to both states the same

waiting time distribution

Tv-l

c(t) = (v - 1) +T)(1)

with 1 < v < oo. The parameter T > 0 is introduced for

the purpose of making i(t) finite at t 0 so as to ensure

its normalization. We shall focus on the case when v < 3.

In accordance with Brokmann et al. [4], the experimental

condition v < 2 implies that the observed waiting time

distribution depends on the time at which observation

begins. Let us assume that the probability of the first

jump from the "on" ("off") to the "off" ("on") state

is given by Eq. (1), if the observation begins at t = 0. If

the observation begins at a later time ta > 0, the distri-

bution of the sojourn times, before the first jump, turns

out to be different from Eq. (1): it is ta dependent and, for

this reason, is denoted by f(t, ta). This is the property

responsible for the breakdown of the ordinary KA theory:

it is the aging effect on which we focus our attention in

this Letter. It is worth noticing that when v > 2, this

aging effect is still present, in a less dramatic form, given

the fact that a stationary condition exists, even if the

regression to it takes a virtually infinite time if v < 3 [5].

The authors of Ref. [6] showed how to derive the ab-

sorption line shape in the case v > 2, when the stationary

condition applies, and evaluated the form that the spec-

trum would have, immediately after switching on the

radiation field, when the nonstationary condition v < 2PACS numbers: 05.40.Fb, 78.47.+p, 87.15.Ya

holds true. Here we illustrate a way to evaluate the time

evolution of the absorption spectrum, so as to take into

account the aging effects of Brokmann et al., with v < 2,

as well as those of Ref. [5], with v > 2. We use the

following stochastic equation:d

-- (t) = i[wo + e(t)]p(t).

dt(2)

The quantity (t) is a complex number, corresponding to

the operator le)(gl of the more rigorous quantum me-

chanical treatment [7], le) and Ig) being the excited and

the ground state, respectively, wo is the energy difference

between the excited and the ground state, and e(t) denotes

the energy fluctuations caused by the cooperative envi-

ronment of this system. In the presence of the coherent

excitation, Eq. (2) becomesd

dpt(t) = i[woo + ((t)]u(t)+ kexp(iwt),

dt(3)

where w denotes the radiation field frequency. It is con-

venient to adopt the rotating-wave approximation. Let us

express Eq. (3) by means of the transformation /(t)=

exp(iwot) t(t). After some algebra, we get a simple equa-

tion of motion for /(t). For simplicity we denote / (t)

with the symbol (t) again, thereby making the resulting

equation read:d

d (t) = i[6 + ((t)]p(t) + k,

dt(4)

where 6 wo - w. The reader can easily establish the

connection between this picture and the stochastic Bloch

equation of Ref. [7] by setting t = v + iu. Note that the

three components of the Bloch vector in Ref. [7],

(u, v, w), are related to the rotating-wave representation

of the density matrix p, v, and u being the imaginary and

the real part of e-iotpge, and w being defined by w

(Pee - pgg)/2. Note that the equivalence with the picture

of Ref. [7] is established by assuming the radiative life-

time of the excited state to be infinitely large and the Rabi

frequency 1 k vanishingly small.0031-9007/ 04/93(5)/050601(4)$22.50 2004 The American Physical Society

week ending

30 JULY 2004050601-1

050601-1

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Aquino, Gerardo; Palatella, Luigi & Grigolini, Paolo. Absorption and Emission in the Non-Poissonian Case, article, July 28, 2004; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc67641/m1/1/: accessed March 1, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.