Open quantum system approach to single-molecule spectroscopy

Open quantum system approach to single-molecule spectroscopy

Adrián A. Budini Consejo Nacional de Investigaciones Científicas y Técnicas, Centro Atómico Bariloche, Av. E. Bustillo Km 9.5, (8400) Bariloche, Argentina, and Consortium of the Americas for Interdisciplinary Science and Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA
July 5, 2019
Abstract

In this paper, single-molecule spectroscopy experiments based on continuous laser excitation are characterized through an open quantum system approach. The evolution of the fluorophore system follows from an effective Hamiltonian microscopic dynamic where its characteristic parameters, i.e., its electric dipole, transition frequency, and Rabi frequency, as well as the quantization of the background electromagnetic field and their mutual interaction, are defined in an extended Hilbert space associated to the different configurational states of the local nano-environment. After tracing out the electromagnetic field and the configurational states, the fluorophore density matrix is written in terms of a Lindblad rate equation. Observables associated to the scattered laser field, like optical spectrum, intensity-intensity correlation, and photon-counting statistics, are obtained from a quantum-electrodynamic calculation also based on the effective microscopic dynamic. In contrast with stochastic models, this approach allows to describe in a unified way both the full quantum nature of the scattered laser field as well as the classical nature of the environment fluctuations. By analyzing different processes such as spectral diffusion, lifetime fluctuations, and light assisted processes, we exemplify the power of the present approach.

pacs:
42.50.Ct, 42.50.Ar, 03.65.Yz

I Introduction

In spite that in recent years many different experimental techniques have been established, single molecule spectroscopy (SMS) barkaiChem (); michel (); jung (), that is, the study of single nano-objects is predominantly realized by purely optical methods, i.e., by measuring the far-electromagnetic field scattered by the system when it is subjected to laser radiation.

In contrast to standard quantum optical systems breuerbook (); carmichaelbook (); milburn (); scully (); loudon (); yariv (), where the reservoir is only defined by the background (free) quantized electromagnetic field, in SMS the supporting nano-environment is highly structured. Its definition and specific properties depend on each experimental setup. In fact, the local environment felt by the fluorophore may involve cryogenic single-molecules tamarat (); donley (); kulzer (), molecules at room temperature, either immobilized trautman (); MoernerOrrit () or diffusing in solution Elson (), the solid state matrix supporting a nanocrystal nirmal (); PMichler (); brokmann (); verberk (); kuno (); potatova (); cichosLaser (); cichos (); vanOrden (); bianco (), or single bio-molecules Dickson (); xie (); WEMoerner (); caoYang (); Lu (); SWeiss (); valle (). Even more, the proper definition of the background electromagnetic field may be modified because the local surrounding of the system may develop fluctuations in its dielectric properties cichos (); valle ().

Each specific environment leads to different underlying physical or chemical processes that in turn modify the emission properties of the fluorophore. In most of the situations, it can be model as a two-level optical transition. The central theoretical problem of SMS is to relate the scattered laser field statistics with the underlying nanoscopic environment dynamic. Due to its complexity, its dynamical influence is usually taken into account by adding random elements to the characteristic parameters of the system barkaiChem (); michel (); jung (). For example, spectral diffusion processes are taken into account by adding classical noise fluctuations to the transition frequency of the system, while lifetime fluctuations may be associated to transitions between different conformational (chemical or physical) states of the environment. Stochastic Bloch equations zheng (); he (); FLHBrown (); BarkaiExactMandel (), stochastic decay rates wang (); brown (), modulated reaction models caoYang (); xieSchenter () or stochastic reaction coordinates mukamel () are some of the associated theoretical models. On the other hand, radiation patterns whose statistics depend on the external laser power Dickson (); WEMoerner (); hoch (); barbara (); talon (); moerner (); wild (); cichosLaser () are modeled by introducing extra states coupled incoherently to the upper level of the system molski ().

The previous approaches are well-accepted and standard theoretical tools for modeling SMS experiments. They provide a solid basis for describing different experimental observables, such as those associated to the photon counting statistics. Nevertheless, as the influence of both the background electromagnetic field and the reservoir fluctuations are represented in a unified way by a classical noise, in general it is not clear how to calculate arbitrary observables associated to the scattered laser field. This limitation is evident when considering, for example, the optical or absorption spectrum, which is defined in terms of the scattered field correlations.

The previous drawback or limitation could be surpassed if one is able to describe SMS experiments on the basis of a full open quantum system approach. By an open quantum systems approach we mean: (i) the possibility of describing both the fluorophore system and the reservoir fluctuations through a density matrix operator, whose evolution, i.e., its master equation, can be written and adapted to each specific situation. (ii) Describing the quantum nature of the scattered electromagnetic field through operators and providing a closed solution for their evolution, in such a way that a simple procedure for calculating arbitrary field correlations is established. Then, the calculation of observables like the optical spectrum follows straightforwardly. (iii) Characterizing the photon counting process through a Mandel formula and establishing a manageable analytical tool for the explicit calculation of the photon counting probabilities. The main goal of this paper is to demonstrate that it is possible to build up such kind of powerful and general approach, which not only recovers the predictions of standard approaches, but also allows to characterize, in different situations, arbitrary observables associated to the scattered quantum electromagnetic field.

The scarce application of an open quantum system approach to SMS has a clear origin. Due to the complexity of the underlying nano-environment, a microscopic description, from where to deduce the system density matrix evolution, is lacking. We overcome this difficulty by noting that the noise fluctuations induced by the reservoir barkaiChem (); michel (); jung () can be associated to a coarse grained representation of its complex structure vanKampen (), allowing us to write microscopic interactions that take into account its leading dynamical effects and at the same time are analytically manageable. From the effective microscopic dynamic the system and reservoir fluctuations result described in terms of a Lindblad rate equation. The theoretical validity of these equations for describing non-Markovian open quantum system dynamics was established in Refs. rate (); breuer (). The possibility of establishing a quantum-electrodynamic treatment of SMS experiments also relies on the results of Ref. QRT (), where the quantum regression hypothesis was analyzed for Lindblad rate master equations.

We remark that recent author contributions anticipated the possibility of establishing the powerful formalism developed in this paper. In Refs. rapid (), it was demonstrated that an anomalous fluorescence blinking phenomenon can be dynamically induced by a complex environment whose action can be described by a direct sum of Markovian sub-reservoirs. In Ref. mandel (), for the same situation, observables associated to the photon counting process and a mapping with triplet blinking models barkaiChem (); molski () were characterized. In Ref. luzAssisted (), we shown that fluorescence blinking patterns whose statistics depend on the external laser power can be model through an underlying tripartite interaction. Both situations lead to quantum master equations that correspond to particular cases of the present general formalism. Since in both cases the calculations relied on specific generalizations of a quantum jump approach plenio (), there is not a recipe for calculating the scattered field correlations. The present quantum-electrodynamic treatment also fills up this gap.

The paper is outlined as follows. In Sec. II, the effective Hamiltonian microscopic dynamic that defines the full approach is introduced and the evolution of the fluorophore density matrix is obtained. In Sec. III, observables associated to the scattered laser field are derived from the full microscopic approach. In Sec. IV, the photon counting statistics is characterized through a Mandel formula and a generating function approach cook (); mukamelPRA (). In Sec. V, the formalism is exemplified by analyzing different kind of environment fluctuations. Processes like spectral diffusion, lifetime fluctuations, and light assisted processes are explicitly characterized through the scattered field observables. In Sec. VI we give the conclusions.

Ii Effective Microscopic description and density matrix evolution

The total Hilbert space associated to a SMS experiment is defined by the external product where each contribution denotes respectively the Hilbert space of: the fluorophore system, the background electromagnetic field, and the rest of the degrees of freedom that define the local nano-reservoir felt by the system. In general, it is impossible to know the total microscopic dynamic of the reservoir . In order to bypass this task, following an argument presented by van Kampen vanKampen (), we notice that the complexity of the reservoir may admits a simpler general description. Since the nano-reservoir can only be indirectly observed through the fluorophore system, its Hilbert space structure can not be resolved beyond the experimental resolution. Therefore, it is split as where each subspace is defined by the set of all quantum states that lead to the same system dynamic vanKampen (). As the reservoir may be characterized by inordinately dense as well as by discrete manifolds of energy levels, some may be of finite dimension.

Clearly, the subspaces define the maximal information about the reservoir Hilbert space structure that can be achieved from a SMS experiment. Then, our approach consists in representing the reservoir through a set of coarse grained “configurational macrostates”, , each one being associated to each subspace, and writing the total microscopic dynamic in the effective Hilbert space The configurational Hilbert space is expanded by the (unknown) configurational macrostates. The approach is closed after defining both, the field quantization and its interaction with the system in the total effective Hilbert space, and the self-dynamics of the configurational states.

ii.1 Electromagnetic field quantization

The classical Maxwell equations landau ()

(1a)
(1b)
provide the basis for the quantization of the electromagnetic field in a material media loudon (); yariv (). The macroscopic field vectors are related by the constitutive relations and where and  are the macroscopic material constants and is the light velocity in free space. For an absorptionless dielectric, the Maxwell equations lead to the classical Hamiltonian field
(2)

where is the volume of integration.

The canonical field quantization follows from the expansion of the electric and magnetic field in normal modes and by associating to each component a creation and annihilation photon operator. Here, the domain of quantization is defined by the local surrounding of the system and not by the full volume of the supporting media. In order to take into account the possible dependence of the local field on the configurational bath states, the macroscopic material constants associated to the volume are written as operators in

(3)

Then, depending on the environment configurational state the local field is quantized in a media characterized by the (real) constants Consistently, as the system have nanoscopic dimensions we assume that these constants, in the local surrounding of the fluorophore do not have any spatial dependence. After standard calculations carmichaelbook (), both the electric and magnetic field operators can be written in terms of positive and negative frequency contributions, each one related to the other by an Hermitian conjugation operation. They read

(4a)
(4b)
Here, we assumed that inside the volume the normal modes can be well approximated by plane waves. As usual, the creation and annihilation operators satisfy the commutation relation where and denote the wave vector and polarization of the quantized mode respectively. In agreement with the transversability of the electromagnetic field, the polarization of the magnetic field reads where denotes the position vector. The dispersion relation associated to each configurational state is
(5)

From now on, we take As is well known landau (), this assumption is always valid in an optical regime.

ii.2 System and field Hamiltonians

The fluorophore system is defined by a two-level optical transition with natural frequency The upper and lower states are denoted as and respectively. We take into account that, depending on the configurational state the natural frequency may be shifted a quantity These parameters define the spectral shift of the system associated to each state of the reservoir barkaiChem (). Therefore, the system and field Hamiltonians are written as

(6)

where follows from Eqs. (4) and (2). is the z-Pauli matrix written in the base The frequency operator of the system is defined by

(7)

while for the field it reads

(8)

The frequencies are defined by the dispersion relations Eq. (5).

ii.3 Dipole-field interaction

The natural decay of the fluorophore is induced by the coupling of its electric dipole with the quantized electric field. Their interaction can be written as

(9)

where is the (electric) dipole operator associated to the optical transition and is the electric field operator at the position of the system. To take into account the different configurational states of the reservoir, the standard definition of the dipole operator carmichaelbook () is generalized as

(10)

where are vectors (assumed real) with units of electric dipole. and are respectively the raising and lowering operators acting on the states The diagonal contributions define the dipole associated to each configurational state. The nondiagonal contributions, with take into account the possibility of coupling different configurational states (those of finite dimension) through system transitions.

The dipole-field interaction Eq. (9), in a rotating wave approximation carmichaelbook (); loudon (); milburn (); scully (), from Eqs. (4) and (10) reads

(11)

where the interaction parameters are defined by

(12)

ii.4 Environment configurational fluctuations

The previous definitions effectively take into account the influence of the different configurational states of the environment on the fluorophore and the background electromagnetic field as well as into their mutual interaction. Now, in agreement with the van Kampen argument vanKampen (), and consistently with real specific situations barkaiChem (); michel (); jung (), where the properties of the nano-environment fluctuates in time, the configurational states (associated to dense bath manifolds) must be endowed with a mechanism able to induce transitions between them. We remark that the results developed in Refs. rapid (); mandel (); luzAssisted () do not take into account neither rely on this extra dynamical effect, which in the context of an open quantum system approach can only be recover with the present treatment.

In order to maintain a full microscopic description of the effective dynamics, here the (incoherent) transitions are introduced through the Hamiltonian

(13)

is the free Hamiltonian of define an environment () responsible for the transitions while defines their mutual interaction. We assume that the states are the eigenvalues of with eigenvalues and is defined by a continuous set of arbitrary bosonic normal modes

(14)

and are the  creation and annihilation operators of respectively. The Hamiltonian reads

(15)

Then, the transitions are assisted by the creation or destruction of bosonic excitations in each of the modes of frequency

ii.5 System density matrix evolution

The previous analysis allow to define the unitary dynamic, of the density matrix associated to the Hilbert space The total Hamiltonian read

(16)

where each contribution follows from Eqs. (6), (11) and (13). describe the statistical dynamical behavior of the system, the electromagnetic field, and the configurational states. The joint dynamic of the fluorophore and the configurational states is encoded in the density matrix which follows after tracing out the degrees of freedom of the background electromagnetic field  and the reservoir i.e., The system density matrix can always be written as

(17)

where the base was used for taking the trace over The auxiliary states define the system dynamic “given” that the reservoir is in the configurational state The dynamic of the configurational states follows from the populations

(18)

which provide the probability that the reservoir is in the configurational state at time Therefore, these objects encode the statistical properties of the noise fluctuations introduced in standard stochastic approaches barkaiChem (); michel (); jung (). The expression Eq. (18) follows straightforwardly from the definition where

Both, the electromagnetic field  and the bath are assumed to be Markovian reservoirs. Then, their influence can be described through a standard Born-Markov approximation. After some algebra carmichaelbook (), the evolution of each state can be written as a Lindblad rate equation rate (); breuer ()

where denotes an anticonmutation operation. The Hamiltonian reads

(20)

and the remaining system operators are defined by

(21)

The rates associated to the reservoir  are given by

Here, is the density of states of and are the transition frequencies of The rates associated to the field read ()

(22)

Here, is the density of states of the quantized electromagnetic field associated to each configurational state. By writing

(23)

where is the solid angle differential and the contribution follows from the dispersion relation Eq. (5), after a standard integration carmichaelbook () it follows

(24)

For simplicity, it was assumed that the dipole vector [defined by Eq. (10)] can be written as where is the modulus of the dipole vectors, and the unit vector does not depend on the coefficient and While this simplification forbid us to analyze the angular dependence of the scattered radiation, the general case can be easily worked out from the present treatment.

The previous expressions rely on the condition i.e., the transition frequencies of are much smaller than the optical frequency To simplify the analysis, in Eq. (II.5) we have discarded any shift Hamiltonian contribution induced by the microscopic dynamic. Furthermore, since the fluorophore is an optical transition, we have assumed that the average number of thermal excitations of the electromagnetic field at the characteristic frequency are much smaller than one, i.e., where being the Boltzmann constant and the temperature associated to the electromagnetic field This inequality allows to discard in Eq. (II.5) the contributions that leads to thermal excitations in

When the fluorophore is subjected to a resonant external laser field of frequency the system Hamiltonian becomes with

(25)

As before, the operators and are the raising and lowering operators acting on the states The system-laser detuning is given by

(26)

The Rabi frequency reads

(27)

where each measure the system-laser coupling for each configurational state.

Figure 1: Scheme associated to the fluorophore evolution Eq. (II.5), see text.

The Lindblad rate equation Eq. (II.5) is the central result of this section. It effectively describes the action of the nanoscopic fluctuating environment over the fluorophore. The scheme of Figure 1 symbolically represents all processes associated to this equation. The first line of Eq. (II.5) describe the self-system dynamic for each configurational state The decay rate {}, the natural transition frequency {}, as well as the Rabi frequency {} may depend on the bath state. The second line describe transitions between the configurational states (with rates {}), whose dynamic does not depend on the state of the fluorophore (property symbolically represented in Fig. 1 by the surrounding ellipse). Finally, the third line describes configurational transitions that are attempted by a transition between the upper and lower states of the system (rates {}). In Sec. IV, a detailed analysis of each contribution and its associated “kinetic environment evolution” [i.e., the dynamics of Eq. (18)] is presented.

As neither the initial conditions or the dynamics [Eq. (II.5)] introduce any coherence between the bath macrostates, their density matrix can be written as From the relation it follows that the dynamics of operators  acting on is classical and dictated by the probabilities On the other hand, the system dynamic arise after “tracing out” [Eq. (17)] all internal transitions between the configurational states of the reservoir (see Fig. 1). Therefore, the evolution of the system density matrix must be highly non-Markovian. In fact, taking into account the results presented in Ref. rate (), it follows

(28)

defines the system unitary dynamic. The equation that defines the superoperator can be explicitly written in a Laplace domain in terms of the propagator associated to Eq. (II.5) (see Eqs. (58) and (62) in Ref. rate ()).

Iii Scattered field observables

In SMS experiments, the fluorophore dynamic is indirectly probed by subjecting the system to laser radiation [Eq. (25)] and measuring the scattered electromagnetic field. Therefore, while the master Eq. (II.5) completely characterizes the system dynamic, one is mainly interested in observables associated to the scattered laser field. In general, these observables can be written as a function of the electric field. For example, the flux of energy per unit area per unit time (module of the Pointyng vector) reads landau ()

(29)

where the second equality follows from the relation between and [see Eq. (4)].

The time evolution of the electric field operator follows from a Heisenberg evolution with respect to the total Hamiltonian Eq. (16), i.e., Taking into account Eq. (4a), the time dependence of can be obtained from the evolution of the operators We get

(30)

where and the coefficients are defined by Eqs. (5) and (12) respectively. Furthermore, the operator is defined by

(31)

Eq. (30) relies on a set of approximations consistent with the effective representation of the reservoir. Since the dielectric constant of each configurational state is well defined, any non-diagonal non-linear coupling between the field modes is discarded. Terms arising from [Eq. (13)] are also disregarded because they only introduce a small modification to the natural (optical) frequency of each mode.

The dynamics of can be written as the addition of two contributions, each one associated to the homogeneous and inhomogeneous terms in Eq. (30). Then, the electric field is written as carmichaelbook (); breuerbook ()

(32)

The contribution defines the free evolution of the field. In terms of positive and negative frequency contributions, it is defined by

(33)

The scattered field contribution , associated to the inhomogeneous term in Eq. (30), after some algebra carmichaelbook () can be written as where

(34)

with and As before [Eq. (24)], for simplicity we assumed that

The previous expression allows to obtain the electric field in terms of operators defined in the system and configurational Hilbert spaces. Consistently with the effective representation of the reservoir, it is rewritten as

(35)

where each contribution reads

(36)

Since the background electromagnetic field [Eq. (4a)] does not involve any coherence between the configurational states, the operator appearing in Eq. (35) must be read as follows. It labels all contributions to that at time are attempted by the configurational transition (). Then, the “conditional operator”  defines the electric field restricted to the condition that at time the reservoir is in the configurational state and change to the state Similarly, the diagonal contributions define the electric field “given” that at time the reservoir is in the configurational state Consistently, the conditional operator gives the evolution of the system operator under the same condition.

Any scattered field observable must be written in terms of the conditional system operators Below, we characterize the field correlations.

iii.1 Correlations

When the scattered field is measured with photoelectric detectors, the usual observables can be written in terms of two time (normal order) correlations breuerbook (); carmichaelbook (); milburn (); scully (); loudon ()

(37)

as well as

(38)

The overbar denotes quantum average with respect to the total initial density matrix. The symbol takes into account the right interpretation of Eq. (34) and denotes a summation over all internal configurational paths defined through the conditional contributions Eq. (36). Then, the correlations are explicitly written as

(39)

and

(40)

where This definition guarantees that both correlations can be related to observables defined in terms of energy (photon) fluxes [see Eq. (29)]. It also takes into account that before the transition the dielectric constant of the bath is

By using Eq. (36) and the rate expressions Eq. (24), the first order correlation can be written as

(41)

where for brevity we define and

(42)

Eq. (41) can be considered as a natural generalization of the expression corresponding to the Markovian case Markov (). Similarly, the correlation of the (conditional) raising and lowering operators can also be obtained from the system density matrix evolution [Eq. (II.5)] after invoking to a quantum regression theorem. For Lindblad rate equations it reads QRT ()

(43)

where and are arbitrary system operators. Each contribution [indexed by defines a conditional average of the involved operators: at time the configurational bath state is  while at time it is denotes the generator of the Lindblad rate equation,

(44)

From Eqs. (43) and Eq. (41) it follows

(45)

where By using the same calculations steps, the second order correlation reads

The expressions Eqs. (45) and (III.1) are the central results of this section. They allow to characterize observables such as the spectrum of the radiated field as well as the intensity-intensity correlation.

iii.2 Spectrum

The spectral intensity radiation (in units of energy ) per unit of solid angle carmichaelbook (); breuerbook () is defined by the dimensionless expression

(47)

where is the frequency of the laser excitation, [Eq. (25)]. As usually the spectrum can be split in a coherent and incoherent contributions

(48)

consists in a Dirac delta term

(49)

that measures the scattered radiation emitted at the frequency of the laser excitation. From Eq. (45), it follows

(50)

The incoherent contribution can be written as

(51)

where is the Laplace transform of the function

In a Markovian limit, i.e., when the configurational space is unidimensional, from Eq. (51) it is possible to write the spectrum as an addition of three Lorentzian functions (Mollow triplet) whose widths and heights depend on the natural decay of the system and the Rabi frequency breuerbook (); carmichaelbook (); milburn (); scully (). In the general non-Markovian case, the spectrum also has a strong dependence on the parameters that define the environment fluctuations (see next sections). In spite of these dissimilarities, in both cases the spectrum is mainly related to the dynamic behavior of the system coherences [see Eq. (50)].

iii.3 Intensity-Intensity correlation

The normalized intensity-intensity correlation milburn () reads

(52)

where is the intensity operator. From Eqs. (45) and (III.1) it follows

(53)

where

(54)

and the normalization constant reads

(55)

As in the Markovian case carmichaelbook (), Eq. (53) corresponds to the probability density of detecting one photon in the stationary regime () and a second one in the interval The factor takes into account all possible emission paths that leave the system in the ground state and the reservoir in the configurational state The sum over the index takes into account all photon emissions that happen in the interval and leave the reservoir in the state The normalization factor  defines the average stationary intensity emitted by the fluorophore.

Iv Photon counting statistics: a generating function approach

In most of the SMS experiments the scattered laser radiation is measured with photon detectors. Then, the photon counting statistics is also an usual observable. As in standard fluorescent systems, the probability of detecting photons up to time follows from a Mandel formula carmichaelbook (); breuerbook (). Here, it is generalized as

(56)

where the normalized intensity operator has units of photon flux. denotes the distance between the fluorophore and the detector. denotes both an usual (normal) time ordering and a summation over all internal configurational paths, whose definition follows straightforwardly from Eqs. (39) and (40).

While Eq. (56) allows to characterize the probabilities a simpler technique to calculate these objects is provided by a generating function approach cook (); mukamelPRA (). This very well known technique was also used in the context of SMS when dealing with stochastic Bloch equations zheng (); FLHBrown (); he (); BarkaiExactMandel () and related approximations. In contrast, here we formulate the generating function approach cook (); mukamelPRA () on the basis of one of the central results of this contribution, i.e., Eq. (II.5). Added to its broad generality, the present formulation avoid the use of any stochastic calculus.

By writing the system density matrix as

(57)

where each state corresponds to the system state conditioned to photon detection events plenio (); carmichaelbook (), the probability of counting -photons up to time reads

(58)

A “generating operator” cook () is defined by

(59)

where is an extra real parameter. This operator also encodes the system dynamic, The conditional states can be decomposed into the contributions associated to each configurational state of the reservoir, leading to the expression

(60)

Each matrix defines the state of the system under the condition that, at time photon detection events happened and the configurational state of the environment is Consistently, each contribution defines the (conditional) generating operator “given” that the reservoir is in the configurational state Its evolution, from Eq. (II.5), can be written as

where is defined by Eq. (42). Notice that the parameter is introduced in all terms (third line) associated to a photon detection event, i.e., those proportional to plenio (); carmichaelbook ().

In the context of SMS zheng (); FLHBrown (); he (); BarkaiExactMandel (), the matrix elements of in an interaction representation with respect to are usually denoted as

(62a)
(62b)
(62c)
(62d)
[ while their evolution is called “generalized optical Bloch equation.” From Eq. (60), it is possible to write each matrix element as a sum over the parameter In Appendix A, we provide the evolution of each component [Eq. (88)].
After getting the matrix elements of the generating operator, Eq. (62), the photon counting process can be characterized in a standard way vanKampen (). From the definition of the generating operator, Eq. (59), the photon counting probabilities Eq. (58) follows as
(63)

The function also allows to calculate all factorial moments

(64)

in terms of its derivatives

(65)

Furthermore, the first two moments of the photon counting process, (), read

(66a)
(66b)
Both moments encode important information about the scattered radiation. The line shape of the fluorophore system is defined by barkaiChem ()
(67)

while the Mandel factor is defined as

(68)

As is well known breuerbook (); carmichaelbook (); milburn (); scully (), it allows to determining the sub- or super-Poissonian character of the photon counting process. In Appendix B we show the consistency between the generating function approach, the Mandel formula Eq. (56), and the results obtained in the previous section. In Appendix C, it is shown that the stationary Mandel factor

(69)

can be obtained in an exact analytical way after solving the evolution Eq. (IV) in the Laplace domain.

V Examples

In this section, different processes covered by Eq. (II.5) are analyzed. The examples are classified in accordance with the evolution of the environment fluctuations, i.e., the evolution of the configurational populations Eq. (18). In each case, observables such as the spectrum Eq. (47), intensity-intensity correlation, Eq. (53), line shape, Eq. (67), and Mandel factor, Eq. (68), can be calculated in an exact analytical way. In fact, in a Laplace domain, both the evolution of the auxiliary density states, Eq. (II.5), and the evolution of the auxiliary generating operators, Eq. (IV), become algebraic linear equa