Non-Poissonian intermittent fluorescence from complex structured environments

Adrián A. BudiniInstituto de Biocomputación y Física de Sistemas Complejos, Universidad de Zaragoza, Corona de Aragón 42,

(50009) Zaragoza, Spain�Received 1 February 2006; published 14 June 2006�

We characterize the photon-counting statistics of single-chromophore systems submitted to continuous laserradiation and embedded in a complex structured environment able to induce strong non-Markovian effects inthe decay dynamics. In the weak-laser-excitation regime, the photon-emission process can be describedthrough a delay function. We demonstrate that the dynamics, by itself, is able to develop a non-Poissonianintermittent fluorescence phenomenon where the bright and dark intervals are characterized by probabilitydistributions that may depart from an exponential one.

DOI: 10.1103/PhysRevA.73.061802 PACS number�s�: 42.50.Ar, 42.50.Lc, 05.40.�a, 33.80.�b

While the fluorescence signal of single-atomic systems�1–6� can be well described in a Markov approximation,due to the complex structure of the supporting nano-environment, different nanoscopic systems such as singlemolecules �7–16� and nanocrystal quantum dots �17–19� sub-ject to continuous laser radiation produce a scattered radia-tion pattern that cannot be understood or modeled in terms ofstandard Markovian decay dynamics. In fact, while in theshort-time regime the scattered radiation is characterized byantibunching �12,14�, indicating single-photon emission andatomiclike behavior, at longer times a photon-bunching be-havior is found, which in turn may be associated with anon-Poissonian intermittence effect, i.e., the radiation ran-domly switches between bright and dark intensity stateswhose durations are described by probability distributionsthat departs from an exponential one �7–9,17–19�. This char-acteristic implies strong departure from Markovian dynam-ics.

There exist different physical effective processes andminimal theoretical models that may reproduce such anemission pattern, like environment conformal fluctuations�9–11�, jumps to extra dark states where the system is unableto radiate �12�, and spectral diffusion processes �13–16�,where the nanoenvironment leads to random changes in thetransition frequency of a single molecule, which in conse-quence comes in and out of resonance with the external laserfield. This process can be modeled with generalized opticalBloch equations �14–16�.

The aim of this work is to present an alternative physicalmodeling that also leads to the previous emission pattern. Wecharacterize the photon statistics of a single-chromophoresystem whose radiative decay is produced by a complexstructured environment able to induce strong non-Markovianeffects in the system decay dynamics �20�. In the weak-laser-excitation regime, by using a generating operator formalism�6,14�, the photon statistics can be approximated by a re-newal process �2–4�. Non-Poissonian intermittent fluores-cence arises as an environment-induced effect without invok-ing extra effective physical processes. This result establishesan alternative physical scenario for modeling and under-standing single-chromophore spectrography in nanoscopicenvironments �7�.

We consider a two-level single-chromophore system withHamiltonian

H�t� =��A

2�z +

��

2��†e−i�At + �e+i�At� . �1�

The first term, proportional to the z Pauli matrix, defines theenergy levels �with transition frequency �A�, and the secondone represents the interaction with a resonant external laserfield, with Rabi frequency �, where �† and � are, respec-tively, the raising and lowering operators acting on the states�± � of the system.

First, we consider the case in which the chromophore ra-diative decay is induced by a Markovian environment. In thissituation, the direct photon-counting statistics of the scat-tered radiation can be described through the system densitymatrix written as �R�t�=�n=0

� �R�n��t�. Each term represents the

reduced system density operator in the subspace containingexactly n=0,1 ,2 ,¯ photons in the scattered field. The prob-ability of counting n photons up to time t reads

Pn�t� = Tr��R�n��t�� , �2�

where Tr�¯� is a trace operation in the system Hilbert space.For unit detector efficiency, the set �R

�n��t� evolves as �2,3�

d�R�n��t�dt

= LH�R�n��t� + ��L + �RLD��R

�n��t�

+ �R�1 − n,0�J�R�n−1��t� , �3�

where LH�·�= �−i /���H�t� , · � is the system Liouville super-operator. The superoperators

LD�·� = −1

2��†�, · +, J�·� = � · �†, �4�

where �¯+ means an anticonmutation operation, correspondto a dissipative environment, which produces radiative tran-sitions between both levels with rate �R. The contributionL�·�= ���z · ,�z�+ ��z , ·�z�� /2, corresponds to an extra dis-persive reservoir which destroys coherences with rate �,without affecting populations.

By using Eqs. �2� and �3�, it is possible to completelycharacterize the photon-counting statistics. As is well known,the emission pattern can be described through a delay orwaiting-time distribution �1–5�.

Now we assume that the radiative decay is produced by a

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complex structured environment whose influence can be de-scribed in a generalized Born-Markov approximation �20�.Thus, its dynamical influence can be written in terms of adirect sum of Markovian subreservoirs, each one character-ized by a different rate �R and participating with weightPR. The state of the system follows from the average��t�=�RPR�R�t���R�t��, where now each �R�t� correspondsto the dynamics induced by each subreservoir. Similarly, thephoton-counting statistics reads

Pn�t� = Tr���R�n��t��� Tr���n��t�� . �5�

This averaging procedure is a direct consequence of describ-ing the full environment in terms of a direct sum of nonin-teracting subreservoirs. The average �¯� over the set��R , PR can be managed after introducing the generating op-erator �6,14� G�t ,s��n=0

� sn��n��t�, which also encodes thesystem dynamics

��t� = �n=0

�

��n��t� = �G�t,s��s=1. �6�

Under the conditions of weak laser intensity �� / ��R��2�1and � / ��R��1, from Eq. �3�, it follows that �21�

dG�t,s�dt

= LHG�t,s� + �0

t

d� K�t − ���LD + sJ�G��,s� ,

�7�

where K�u�= ��R�u+�R�−1���u+�R�−1�−1 defines the memorykernel in a Laplace domain, u being the conjugate variable.Therefore, ��t� is characterized by a non-Markovian evolu-tion.

In order to characterize the set of probabilities Pn�t�, wenotice that Eq. �7� implies the evolution

d��n��t�dt

= LH��n��t� + �0

t

d� K�t − ��

�LD��n���� + �1 − n,0�J��n−1����� , �8�

with ��0��0�=��0�, and ��n��0�=0, for n�1. From here, theprobability of counting n photons up to time t, Eq. �5�, canbe written as

Pn�t� = �0

t

dtn�0

tn

dtn−1 ¯ �0

t2

dt1P0A�t�tn�

wA�tn�tn−1� ¯ wA�t2�t1�wA�t1�0� , �9�

where we have introduced the survival probability

P0A�t��� = Tr�T�t����̄0� , �10�

and the delay �2,5� or waiting-time distribution

wA�t��� = Tr J��

t

dt�K�t − t��T�t�����̄0� . �11�

The propagator T�t ��� is defined through the evolution

dT�t���dt

= LHT�t��� + ��

t

dt�K�t − t��LDT�t���� , �12�

with T�� ���= I. These expressions follow after integratingthe infinite set of coupled evolutions Eq. �8� as��0��t�=T�t �0���0�, and for n�1 as ��n��t�=�0

t d� T�t ����0�dt�K��− t��J��n−1��t��. Furthermore, in deriv-

ing Eq. �9� we used the property J�·�=Tr�J�·��̄0, with�̄0�−��−�, and Tr�LD�·�=−Tr�J�·�, expressions valid forany input operators; for simplicity, we have assumed the ini-tial condition ��0�= �̄0.

Equation �9� completely characterizes the statistics of theradiation scattered by the non-Markovian chromophore sys-tem. As in the case of Markovian evolutions, the structure ofEq. �9� indicates that the full photon-counting statistics canbe intended as a renewal process �2–5�. In fact, noting thatwA�t ���=−dP0

A�t ��� /dt, it is possible to interpret the inte-grand in Eq. �9� as an n-joint �exclusive� probability densityfor having n emissions, each one at times �tii=1

n with prob-ability density wA�ti � ti−1�. Correspondingly, P0

A�t ��� is theprobability of not having any emission in the interval �t ,��given an emission at time � �22�.

By writing Eq. �12� in an interaction representation withrespect to �� /2��A�z, it is possible to demonstrate thatwA�t ��� and P0

A�t ��� depend only on the time difference�t−��. In a Laplace domain we get

wA�u� =K�u�/2

u + K�u�/2 �2h�u�u2 + uK�u� + �2h�u�� . �13�

Here, u represents the conjugate Laplace variable to t−�, andwe defined h�u�= �u+K�u� /2� / �u+ ��R� /2�. The survivalprobability reads P0

A�u�= �1−wA�u�� /u.Up to now, we left the properties of the complex environ-

ment unspecified. Its characteristics, i.e., the set ��R , PR, areintroduced through the kernel K�u�, and in turn, from Eq.�13�, determine the radiation properties. Therefore, our for-malism allows us to establish a kind of “environment spec-troscopy,” where the bath properties are linked with thephoton-counting statistics. Here, we assume the form

K�u� =�

1 + ��/u�1−� , �14�

where 0���1. After taking �= ��R� and�= ���R

2�− ��R�2� / ��R�, this kernel can be associated with anexponential dependence on index R of both the rates �R andweights PR �20�. These functional forms completely charac-terize the environment in the framework of the generalizedBorn-Markov approximation. Below, we demonstrate thatthis kernel shape leads to an anomalous intermittent effect.

The radiation properties corresponding to the kernel �14�can be clearly shown through the delay function. In Fig. 1 weplot wA�t ���, Eq. �13�, along with its memoryless limit�=0. As ���, the two distributions are indistinguishable inthe short-time regime. Equation �13� with K�u�=� reduces tothe delay function in a Markovian environment �4�. Thus,consecutive photon emissions inside the short-time regimeare indistinguishable from that of a Markovian quantum

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emitter radiating with intensity 1/�M, where �M=2/�+� /�2 is the average �Markovian� waiting time �2,4�,i.e., �M = ���

�dt twA�t �����=0. The antibunching characteristicof these emission processes is manifested by wA�� ���=0.

In the asymptotic time regime �t−����M, the twodelay functions differ strongly. From Eqs. �13� and �14�,for u�1 we get the series expansion wA�u��1−2�1−�u� /�−�Mu+¯, which implies the asymptotic be-havior �23� wA�t ����2��1−� / ���t−��1+��, and an exponen-tial one for �=0. In correspondence with wA�t ���, in theshort-time regime the survival probability can be fitted by anexponential decay, while a power-law behavior arises in theasymptotic regime,

P0A�t��� � exp −

�t − ���M

�, P0A�t��� �

2�1−�

��t − ��� . �15�

These features are clearly seen in the inset of Fig. 1.As it is well known, the existence of two drastically dif-

ferent time scales in the survival probability may lead tointermittent fluorescence �2,5�. In Fig. 2 we plot a realizationof the intensity corresponding to the renewal process definedby the objects of Fig. 1. The intervals between consecutiveemissions �t−�� were determined from the survival probabil-ity as P0

A�t ���=r, where r is a random number in �0,1�. Fur-thermore, for getting the intensity we have introduced anadequate time flag averaging, which leads to a noisy signalsimilar to that of an experimental realization �19�. Consis-tently, we observe periods of brightness, with an averageintensity 1/�M, and periods of darkness, which correspond to

successive emission distributed in the power-law regime. Wenotice that these “rare events” occur with a small probability.In fact, when the survival probability attains the power-lawregime, allowing large intervals between successive emis-sions, we have P0

A�t ����1 �see inset of Fig. 1�. Thereforemost of the photon emissions correspond to events in theexponential regime, which produce the periods of brightness.

The intermittence phenomenon can be characterizedthrough the statistical distribution of the durations of thebright �WB�T�� and dark �WD�T�� states. In order to definethese objects �2,5�, we introduce a time delay T0��M suchthat successive emissions separated with a bigger time definethe beginning of a dark state. Consistently, a bright statemaintains its status while the time between emissions isshorter than T0.

The bright distribution is determined by the relationWB�T�dT= pndn, where pn is the probability of having nemissions separated by a time interval shorter than T0, con-ditioned to the beginning of a bright state. Thus, n�T� countsthe average photon emissions from the beginning of a brightstate up to time T. Introducing the constant p= P0

A�T0+� ���,we can approximate pn��p / �1− p���1− p�n. The first factortakes into account that pn is a conditional probability, i.e., abright state begins after the ending of a dark period.Using that p�1, and expressing n�T��T /TM withTM =�0

T0dt twA�t �0���M +O�pT0�, we get an exponentialbright distribution

FIG. 1. Waiting-time distribution wA�t ��� �full line� and its cor-responding Markovian limit �dotted line�. The parameters are�=1/2, � /�=0.01, and � /�=5. The inset shows the correspond-ing survival probabilities P0

A�t ���.

FIG. 2. Stochastic realization of the emission intensity. The av-erage intensity of the bright intervals is 1 / ��M���0.18.

FIG. 3. Probability distributions WB�T� and WD�T� for the brightand dark periods, along with their theoretical fitting �indistinguish-able�, Eqs. �16� and �17�, respectively. We take T0 /�M =50.

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WB�T� �1

TBexp −

T

TB�, TB =

TM

p. �16�

The dark distribution WD�T� is proportional to the asymptoticbehavior of the delay function �2,5�

WD�T� �wA�T�0�

p�

1

T T0

T��

, T � T0, �17�

where the factor 1 / p takes in account the ending of a brightstate. Furthermore, we have used that T0��M, which impliesthat P0

A�T0+� ��� falls in the asymptotic power-law regime,i.e., p�2�1−� /�T0

�.In Fig. 3 we plot the bright and dark distributions ob-

tained from successive intensity realizations and the previous�indistinguishable� theoretical fitting. While the bright distri-bution is exponential, as in the standard Poissonian intermit-tent fluorescence, the dark distribution is a power law, indi-cating strong deviations with respect to Markoviandynamics.

We studied the fluorescence signal of a single two-levelchromophore system driven by a resonant external laser fieldand embedded in a complex structured environment. In theweak-laser-intensity regime, the photon-counting statisticscan be specified through a delay function Eq. �13�. Due to

the presence of strong non-Markovian effects, the dynamicsis able to develop quantum and classical properties like an-tibunching and non-Poissonian intermittent fluorescence. Weremark that the latter effect arises as a pure dynamical effect,without the introduction of extra �effective� physical pro-cesses. We studied a case that leads to exponential andpower-law distributions for bright and dark periods, respec-tively.

The present treatment relies on the applicability of a gen-eralized Born-Markov approximation, which allows one todescribe the environment action through a set of subreser-voirs, each one being able to induce by itself a Markoviansystem dynamics �20�. This modeling arises naturally fromglassy environments, where the underlying disorder inducesa shell structure of localized modes, each set having a differ-ent interaction strength with the system �24�. Our resultsdemonstrate that interaction with this kind of environment isan alternative underlying physical mechanism that may pro-duce the radiation statistical pattern found in different nano-size quantum emitters embedded in condensed phaseenvironments.

This work was supported by Secretaría de Estado de Uni-versidades e Investigación, MCEyC, Spain.

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ph/0601140.�21� In deriving Eq. �7�, we assumed that the rate �, while small

when compared to ��R�, is able to destroy any asymptoticmemory contribution in the coherence components, which im-plies the condition �exp�−��R /2+��t���exp�−��R�t /2�. Forthe kernel Eq. �14� this approximation implies the constraint�����. Furthermore, in order to avoid writing the finalevolution in components, we assumed K�u± i�A���K�u��u=�

=�, which is always valid for optical frequency �A.�22� Equation �9� admits this interpretation only when P0

A�t ��� de-cay in a monotonic way, guaranteeing the positivity of wA�t ���.Consistently, this condition is satisfied for weak laser excita-tion, where Eq. �7� is valid. Outside this regime, the scatteredradiation cannot be described through a renewal process.

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