Absorption and Emission in the Non-Poissonian Case Page: 1
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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.
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) , are violated by some of the new materials. For
instance, the experimental research work of Neuhauser
et al.  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 . In this
Letter, for simplicity, we assign to both states the same
waiting time distribution
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. , 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 .
The authors of Ref.  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 < 2
PACS 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. , with v > 2. We use the
following stochastic equation:
-- (t) = i[wo + e(t)]p(t).
The quantity (t) is a complex number, corresponding to
the operator le)(gl of the more rigorous quantum me-
chanical treatment , 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) becomes
dpt(t) = i[woo + ((t)]u(t)+ kexp(iwt),
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
d (t) = i[6 + ((t)]p(t) + k,
where 6 wo - w. The reader can easily establish the
connection between this picture and the stochastic Bloch
equation of Ref.  by setting t = v + iu. Note that the
three components of the Bloch vector in Ref. ,
(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.  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
30 JULY 2004
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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 September 23, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.