Luminescence Spectra of Quantum Dots in Microcavities. II. Fermions.
Abstract
We discuss the luminescence spectra of coupled lightmatter systems realized with semiconductor heterostructures in microcavities in the presence of a continuous, incoherent pumping, when the matter field is Fermionic. The linear regime—which has been the main topic of investigation both experimentally and theoretically—converges to the case of coupling to a Bosonic material field, and has been amply discussed in the first part of this work. We address here the nonlinear regime, and argue that, counter to intuition, it is better observed at low pumping intensities. We support our discussion with particular cases representative of, and beyond, the experimental state of the art. We explore the transition from the quantum to the classical regime, by decomposing the total spectrum into individual transitions between the dressed states of the lightmatter coupling Hamiltonian, reducing the problem to the positions and broadenings of all possible transitions. As the system crosses to the classical limit, rich multiplet structures mapping the quantized energy levels melt and turn to cavity lasing and to an incoherent Mollow triplet in the direct exciton emission for very good structure. Less ideal figures of merit can still betray the quantum regime, with a proper balance of cavity versus electronic pumping.
pacs:
42.50.Ct, 78.67.Hc, 42.55.Sa, 32.70.JzI Introduction
In the first part of this work Laussy et al. (2008a), we have addressed the coupling of light and matter in the particular case where the material excitation follows Bose statistics, solving analytically this socalled linear model of two harmonic oscillators. We emphasized how a proper consideration of the incoherent pumping scheme was needed to describe the effective quantum state realized in the system, and how this bore consequences on the spectral lineshapes, in particular on the ability to resolve a Rabi doublet when the splitting to broadening ratio is small.
In this second part, we turn to the case where the material excitation follows Fermi statistics and explore the JaynesCummings model Jaynes and Cummings (1963). These two papers focus particularly on Quantum Dots (QDs) as the matter part of the system, which elementary excitation—the exciton—consists of an electron being promoted from the valence to the conduction band. The coupling of this electron with the vacancy it has created in the valence band (the “hole”), can be either a fermion or a boson, or possibly an interpolating case of the two Laussy et al. (2006). Strong Coupling (SC) regime requires efficient coupling of the dot with light, which can be enhanced from the dot point of view by increasing the dipole moment of the exciton (the coupling strength is proportional to where is the cavity electric field). Large QDs on the other hand would favor the overlapping of many excitons, whereas small QDs, by confining separately the electron and hole wavefunctions, fully exhibit Pauli blocking Laussy et al. (2006). The former case was studied in part I and we now turn to the latter. We use the same formalism and similar techniques, what allows for a comparison and a clear understanding of the specifics of both cases. It is known that in absence of nonlinearity or saturation of some sort, the quantum case is equivalent to the classical one Rudin and Reinecke (1999). In particular, the PL spectrum exhibits a Rabi doublet at resonance, which can be equally well accounted for by a purely classical model Zhu et al. (1990). There is therefore a strong incentive to evidence nonlinear deviations and attribute them to quantum effects Schneebeli et al. (2008); Steiner et al. (2008); Srinivasan and Painter (2007); Press et al. (2007); Kroner et al. (2008). In studying the coupling of light and matter, be it with atoms or semiconductors, spontaneous emission from a given initial state has been overly privileged as the case of study SanchezMondragon et al. (1983). Even when the emitter was modelled as a twolevel system, this configuration allowed to reduce it to the linear model by considering a single excitation as the initial state Carmichael et al. (1989); Andreani et al. (1999); Cui and Raymer (2006); Auffèves et al. (2008); Inoue et al. (2008). Most of the times that the excitation scheme was considered at the same level as the rest of the dynamics, this was for coherent pumping Mollow (1969); Savage (1989); Freedhoff and Quang (1994); Clemens and Rice (2000); Barchielli and Pero (2002); Florescu (2006); Bienert et al. (2007). There has been less considerations for the luminescence spectra under incoherent pumping Löffler et al. (1997); Clemens et al. (2004); Perea et al. (2004); Karlovich and Kilin (2007, 2008), that is the most adequate to describe semiconductors. In the atomic literature, Löffler et al. Löffler et al. (1997) have considered spectral shapes for the oneatom laser at resonance by numerical integration of the master equation and in this context have obtained some of the lineshapes of the best system that we study below. Karlovich et al. Karlovich and Kilin (2008) concentrated on strong coupling at resonance and low pump. In the wake of Part I, we rely on semianalytical results that put clearly apart the spectral and dynamical aspects of the problem. We shall discuss how our approach allows in general to track the transitions between the different regimes and in particular to identify the quantum to classical one.
One of the most important current task of the SC physics in semiconductors in our view is the quantitative description of the experiment with a theory that can provide statistical estimates to the data, in particular intervals of confidence for the fitting parameters. In this respect, there would be little need for fitting an experiment that would produce a clear observation of the JaynesCummings energy levels, which is a strong qualitative effect. But no such structures have been observed so far and the deviations to Rabi doublet have been understood as non fundamental features of the problem Hennessy et al. (2007). The most likely reason for this lack of crushing observations of the quantum regime in the PL lineshapes is that the best systems currently available are still beyond the range of parameters that allows the quantum features to neatly dominate. Instead, they are still at a stage where it is easy to overlook more feeble indications, as shall be seen in what follows for less ideal systems that are closer to the experimental situation of today. Another possible reason is that the models are not suitable and a QD cannot be described by a simple twolevel system. Then more involved theories should take over, with, e.g., full account for electron and hole band structures and correlations Feldtmann et al. (2006); Gies et al. (2007); Schneebeli et al. (2008). However, if a simpler theory is successful, notwithstanding the interest of its more elaborate and complete counterpart, it clearly facilitates the understanding and putting the system to useful applications (especially in a quantum information processing context). At present, there is more element of chance left in the research for quantum SC than is actually necessary. If a quantitative description of even a “negative experiment” (not reporting a triplet or quadruplet) could be provided, this would help tracking and probably even direct the progress towards the ultimate goal: a fully understood and controlled SC in the quantum regime.
The rest of this paper is organized as follow. In Section II, we spell out the model. In Section III, we detail the formalism and provide the expressions for all—and only those—correlation functions that enter the problem, making it as computationally efficient as possible for an exact treatment. We provide a decomposition of the final spectra in terms of transitions of the dressed states, which gives a clear physical picture of the problem. In Section IV, we give the analytical expressions for the position and broadening of the resonances of the system at vanishing pumping. Weighting these resonances by the selfconsistent dynamics of the system established by finite pumping and decay, gives the final spectral shape. We discuss in particular the notion of SC that varies from manifold to manifold, rather than holding for the entire system as a whole. In Section V, we consider three particular points representative of the experimental situation, plus one point beyond what is currently available. We first discuss their behavior in terms of population and statistical fluctuations as imposed from the pumping conditions. In Section VI, we give the backbone of the final spectra at nonvanishing excitations. This is the numerical counterpart of Section IV, in the presence of arbitrary pumping. In Section VII, we present spectral shapes for the three points in a variety of configuration and compare them to each other. In Section VIII, we investigate the situation at nonzero detuning, which is a case of particular importance in semiconductor physics. Finally, in Section IX, we provide an overview of the results and conclude.
Ii Model
The Hamiltonian that describes a Fermionic QD in strong coupling with a singlemode microcavity is the JaynesCummings Hamiltonian Jaynes and Cummings (1963) ( is taken as 1 along the paper):
(1) 
with the cavity mode annihilation operator tuned at energy (obeying Bose statistics) and the exciton annihilation operator at energy (obeying Fermi statistics, are Pauli matrices). The two modes are coupled with the interaction strength and close enough to resonance to allow for the rotating wave approximation Carmichael (2002). The detuning between the modes is defined as . The Liouvillian to describe the system in the framework of a quantum dissipative master equation, , has the same form as that in Part I of this work Laussy et al. (2008a):
(2a)  
(2b) 
where is the density matrix for the combined Fermiemitter/cavity system. The only change in both Eqs. (1) and (2) with respect to their counterpart in Ref. Laussy et al. (2008a) is the replacement of the Boson operator (as it was called in part I) to describe the matter field, by a Fermion operator: . This interchange has far reaching consequences, as will be seen in the following of this text.
We shall not focus on the difference between the spontaneous emission (SE) of an initial state in absence of any pumping, and the steady state (SS) established in presence of this pumping, as we did in part I for the Boson case. SS is the most relevant case for the experimental configuration that we have in mind. Rather than contrasting the SE/SS results, as was done in Part I, we shall therefore contrast the Boson/Fermion cases. For this reason and for concision, we shall not use the “SS” superscript and assume that which of the SE/SS case is assumed is clear from context or from the presence of the time variable .
In the Boson case, the quantum state of the system is not by itself an interesting quantity as most of its features are contained in its reduced density matrices, that are simply and in all cases thermal states with effective temperatures specified by the mean populations of the modes and Laussy et al. (2008a), defined by:
(3) 
For this reason, the higher order correlator
(4) 
that measures the fluctuations in the photon numbers, does not contain any new information. In the Fermion case however, becomes nontrivial, because the saturation of the dot provides a nonlinearity in the system that can produce various types of statistics, from the coherent Poisson distributions, encountered in lasers (where the nonlinearity is provided by the feedback and laser gain), to Fockstate statistics, with antibunching, exhibited by systems with a quantum state that has no classical counterpart. The fluctuations in particle numbers influence the spectral shape. The full statistics itself is most conveniently obtained from the master equation with elements for , photons and , exciton (, ). The distribution function of the photon number is simply .
Rather than to consider the equations of motion for the matrix elements directly, it is clearer and more efficient to consider only elements that are nonzero in the steady state. These are:
(5) 
and correspond to, respectively, the probability to have photons with () or without () exciton, and the coherence element between the states and , linked by the SC Hamiltonian. Both and are real. It is convenient to separate into its real and imaginary parts, as they play different roles in the dynamics. The equations for these quantities, derived from Schrödinger equation for the Liouvillian Eq. (2), read:
(6a)  
(6b)  
(6c)  
(6d) 
Note that in the steady state, Eqs. (6) are detailedbalance type of equations. The conditional photon statistics with and without the exciton are similar, and coupled through the imaginary part of the distribution (that is not a probability). At resonance, the real part of the coherence distribution, , gets decoupled and vanishes in the steady state. As a result, only Eqs. (6)–(6c) need to be solved. When vanishes, does not couple the two modes anymore, and their statistics become thermal like in the boson case. Through the offdiagonal elements , the photon density matrix can vary between Poissonian, thermal (superpoissonian) and subpoissonian distributions del Valle et al. (2007).
Iii Correlation functions and Spectra
The main quantity of interest of this paper is the luminescence spectrum of the system. In the Boson case, the symmetry allowed to focus exclusively on the cavityemission without loss of generality, as the direct exciton emission could be obtained from the cavity emission by interchanging parameters. Here, the exciton (Fermion) and photon (Boson) are intrinsically different, and no simple relationship links them. They must therefore be computed independently:
(7) 
We consider normalized spectra for convenient comparisons of the lineshapes. The normalizing factor is the population , as seen straightforwardly from .
As in Part I, we recourse to the quantum regression theorem to compute the two times average . We first identify the set of closing operators in the sense that, for any operator , the relation is satisfied for some to be specified. In the linear case, the set of with , is closed, what allows for an analytical solution. In the nonlinear case, four indices are required to label the closing operators, namely in with , and , . The links established between them by the Liouvillian dynamics are given by , with defined as:
(8a)  
(8b)  
(8c)  
(8d)  
(8e) 
and zero everywhere else.
We are interested in this text in with and on the one hand, to get the equation for that will provide the cavity emission spectrum, and with on the other hand, to get the equation for for the QD direct emission spectrum. Contrary to the Boson case, this procedure leads to an infinite set of coupled equations. The equations of motion for both and involve the same family of closing operators , namely with where the manifold of the boson case, and for :
(9) 
The links between the various correlators tracked through the indices , are shown in Fig. 1. To solve the differential equations of motion, the initial value of each correlator is also required, e.g., demands , etc. The initial values of (resp., ) can be conveniently computed within the same formalism, recurring to and with (resp., ). This allows to compute also the singletime dynamics , and their steady state, from the same tools used as for the twotime dynamics through the quantum regression theorem. The indices required for the singletime correlators form a set—that we call —that is disjoint from , required for the twotimes dynamics. The set has—beside the constant term —two more elements for the lower manifold (of the Boson case). This is because and invoke and for the cavity spectrum on the one hand, and and for the exciton emission on the other. At higher orders , all twotimes correlators otherwise depend on the same four singletime correlators . Independently of which spectrum one wishes to compute, these four elements , , and of are needed in all cases as they are linked to each other, as shown on Fig. 1.
On the figure, only the type of coupling—coherent, through , or incoherent, through the pumpings —has been represented. Weighting coefficients are given by Eqs. (8). Of particular relevance is the selfcoupling of each correlator to itself, not shown on the figure for clarity. Its coefficient, Eq. (8a), lets enter that do not otherwise couple any one correlator to any of the others. This makes it possible to describe decay by simply providing an imaginary part to the Energy in Eq. (1). The incoherent pumping, on the other hand, establishes a new set of connections between correlators. Note, however, that at the exception of , the pumping does not enlarge the sets , : the structure remains the same (also, technically, the computational complexity is identical), only with the correlators affecting each other differently. The addition of by the pumping terms bring the same additional physics in the Boson and Fermion cases: it imposes a selfconsistent steady state over a freely chosen initial condition. In the Boson case, the pumping had otherwise only a direct influence in renormalizing the selfcoupling of each correlator. In the Fermion case, it brings direct modifications to the JaynesCummings coherent dynamics. But its contribution to the selfcoupling is also important, and gives rise to an interesting fermionic opposition to the bosonic effects as seen in Eq. (8a) in the effective linewidth:
(10) 
For later convenience, we also define:
(11) 
In Eq. (10), it is seen that, whereas the incoherent cavity pumping narrows the linewidth, as a manifestation of its boson character, the incoherent exciton pumping broadens it. This opposite tendencies bear a capital importance for the lineshapes, as narrow lines favor the observation of a structure, whereas broadening hinders it. On the other hand, the cavity incoherent pumping always results in a thermal distribution of photons with large fluctuations of the particle numbers, that result in an inhomogeneous broadening, whereas the exciton pumping can grow a Poissonlike distribution with little fluctuations. Both types of pumping, however, ultimately bring decoherence to the dynamics and induce the transition into weak coupling (WC), with the lines composing the spectrum collapsing into one. Putting all these effects together, there is an optimum configuration of pumpings where particle fluctuations compensate for the broadening of the interesting lines, enhancing their resolution in the spectrum, as we shall see when we discuss the results below.
As there is no finite closure relation, some truncation is in order. We will adopt the scheme where a maximum of excitation(s) (photon plus excitons) is allowed at the th order, thereby truncating into manifolds of excitations, which is the most relevant picture. The exact result is recovered in the limit . As seen in Fig. 1, the number of twotime correlators from up to order is and the number of mean values from is . The problem is therefore computationally linear in the number of excitations, and as such is as simple as it could be for a quantum system. The general case consists in a linear system of coupled differential equations, whose matrix of coefficients [specified by Eqs. (8)] is, in the basis of , a square matrix that we denote . With these definitions, the quantum regression theorem becomes:
(12) 
where . Explicitly, for the lower manifolds, e.g., for :
(13) 
The ordering of the correlators is arbitrary. We fix it to that of Fig. 1, as seen in Eq. (13). With this convention, the indices of the two correlators of interests are:
(14) 
To solve Eq. (12), we introduce the matrix of normalized eigenvectors of , and the diagonal matrix of eigenvalues:
(15) 
The formal solution is then . Integration of and application of the WienerKhintchine formula yield for the th and th rows of the emission spectra of the cavity, , and of the direct exciton emission, , respectively. We find, to order :
(16) 
where and are the real and the imaginary part, respectively, of :
(17) 
and (when we refer to elements of a matrix or a vector by its indices, we enclose it with square brackets to distinguish from labelling indices). Further defining and as the real and imaginary parts, respectively, of
(18) 
we can write Eq. (16) in a less concise but more transparent way. To all orders, it reads:
(19) 
Equation (19) brings together all the important quantities that define the luminescence spectrum of a quantum dot in a microcavity. The lineshape is composed of a series of Lorentzian and Dispersive parts, whose positions and broadenings are specified by and , cf. Eq. (18), and which are weighted by the coefficients and , cf. Eq. (17). The former pertain to the structure of the spectral shape as inherited from the JaynesCummings energy levels. They are, as such, independent of the channel of detection (cavity or direct exciton emission). We devote Section IV to them. The latter reflect the quantum state that has been realized in the system under the interplay of pumping and decay. They determine which lines actually appear in the spectra, and with which intensity. Naturally, the channel of emission is a crucial element in this case. We devote Section V to this aspect of the problem.
Iv Spectral structure
In this Section, we discuss the series of coefficients and that in the luminescence spectrum, Eq. (19), determine the position and the broadening (HalfWidth at Half Maximum, HWHM) of the lines, respectively, be it the cavity or direct exciton emission. The case of vanishing pumping is fundamental, as it corresponds to the textbook JaynesCummings results with the spontaneous emission of an initial state. It serves as the skeleton for the general case with arbitrary pumping and supports the general physical picture. Finally, it admits analytical results. We therefore begin with the case where . The eigenvalues of the matrix of regression , are grouped into manifolds. There are two for the first manifold, given by:
(20) 
and four for each manifold of higher order , given by, for :
(21) 
( is defined as for and otherwise), in terms of the thmanifold (half) Rabi splitting:
(22) 
and of the thmanifold (half) broadening:
(23) 
For each manifold, we have defined the in order by increasing value of the line position .
According to Eq. (18), these provide the position of the line and its halfbroadening through their imaginary and real parts. is always real, so contributes in all cases to only. is (at resonance) either pure real, or pure imaginary, and similarly to the boson case, this is what defines SC. This corresponds to an oscillatory or damped field dynamics of the twotime correlators within manifold , which lead us to the formal definition: WC and SC of order are defined as the regime where the complex Rabi frequency at resonance, Eq. (22), is pure imaginary (WC) or real (SC). The criterion for th order SC is therefore:
(24) 
SC is achieve more easily for given system parameters ( and ), with an increasing photonfield intensity that enhances the effective coupling strength. The lower the SC order, the stronger the coupling. This corresponds to the th manifold (and all above) being in SC (aided by the cavity photons), while the manifolds below are in WC. First order is therefore the one where all manifolds are in SC. Equation (24) includes the SC in the standard boson case Laussy et al. (2008a), , as the first order SC of the fermion case, that is shown in green (thick) in Fig. 2. The same position of the peaks and the same (half) broadenings is also recovered (in the absence of pumping). Note that similarly to the boson case, the SC is defined by a comparison between the coupling strenght with the difference of the effective broadening and . The sum of these play no role in this regard.
The and are plotted in Fig. 2(a) and 2(b), respectively, as function of . Note that only depends on and , whereas also depends on (that is why we plot it for ).
The , Eq. (21), have a natural interpretation in terms of transitions between the manifolds of the socalled JaynesCummings ladder. The eigenenergies of the JaynesCummings Hamiltonian with decay granted as the imaginary part of the bare energies (), are given by with
(25) 
for the th manifold. The four possible transitions between consecutive manifolds and give rise, when , to the four peaks we found:
(26a)  
(26b) 
In the case , only the two peaks common with the linear regime arise, , given respectively by Eqs. (26a) and (26b) with . The fact that the correspond to shows that, although the positions of the lines are given by a difference, their broadenings are given by a sum (because of complex conjugation). Physically, the uncertainties in the initial and final states indeed add up in the uncertainty of the transition energy.
The ladder is shown (at resonance) in Fig. 2(c). Let us discuss it in connection with our definition of SC in this system, to arbitrary . When , each step of the ladder is constituted by the two eigenstates of the Fermion dressed by the cavity photons, resulting in a splitting of . This dependent splitting produces quadruplets of delta peaks with splitting of around , as opposed to the boson case where independently of the manifold, the peaks are all placed at around . In a more general situation with , there are three possibilities for a manifold :

Both manifold and are in SC. The two Rabi coefficients and are real. This is the case when
(27) The luminescence spectra corresponds to four splitted lines , coming from the four possible transitions [Eqs. (26), shown as and in Fig. 2(c)] between manifolds and . The emission from all the higher manifolds also produces four lines. They are grouped pairwise around [Fig. 2(a)] and all have the same broadening, contributed by only [the single straight line in Fig. 2(b)].

Manifold is in SC while manifold is in WC. In this case, is pure imaginary (contributing to line positions) and is real (contributing to broadenings). This is the case when
(28) This corresponds to two lines in the luminescence spectrum, coming from the two possible transitions [shown as in Fig. 2(c)] between the SC manifold and the WC manifold . Each of them is doubly degenerated. The two contributions at a given have two distinct broadenings around . [cf. Fig. 2(b)]. The final lineshapes of the two lines is the same. In this region, all the emission from the higher manifolds produce four lines and all from the lower produce only one (at ), being in WC.

Both manifold and are in WC. The two Rabi coefficients and are pure imaginary. This is the case when
(29) This corresponds to only one line at in the luminescence spectrum, coming from the transition from one manifold in WC to the other [shown as in Fig. 2(c)]. The line is fourtime degenerated, with four contributions with different broadenings , as seen in Fig. 2(b).
Figure 2 is the skeleton for the luminescence spectra—whether that of the cavity or of the direct exciton emission. It specifies at what energies can be the possible lines that constitutes the final lineshape, and what are their broadening. To compose the final result, we only require to know the weight of each of these lines.
In the SE case, the weights and include the integral of the singletime mean values over . Therefore, only those manifolds with a smaller number of excitations than the initial state can appear in the spectrum. Each of them, will be weighted by the specific dynamics of the system. The “spectral structure”—i.e., the and —depends only the system parameters ( and ). Therefore, in the SE case, the resulting emission spectrum is an exact mapping of the spectral structure of the Hamiltonian, Fig. 2.
In the SS case, the weighting of the lines also depends on which quantum state is realized, this time under the balance of pumping and decay. But the excitation scheme also changes the spectral structure of Fig. 2. When the pumping parameters are small, the changes will mainly be perturbations of the present picture and most concepts will still hold, such as the definition of SC, Eq. (24) for nonzero in (11). However, when the pump parameters are comparable to the decay parameters, the manifold picture in terms of Hamiltonian eigenenergies breaks. The underlying spectral structure must be computed numerically for each specific probing of the system with and . It can still be possible to identify the origin of the lines with the manifold transitions by plotting their position as a function of the pumps, starting from the analytic limit. SC of each manifold can be associated to the existence of peaks positioned at . We address this problem in next Sections.
V Population and Statistics
To know which features of the spectral structure dominate and which are negligible, one needs to know what is the quantum state of the system. In the boson case, it was enough to know the average photon () and exciton () numbers, and the offdiagonal element . In the most general case of the fermion system, a countably infinite number of parameters are required for the exact lineshape. The new order of complexity brought by the fermion system is illustrated for even the simplest observable. Instead of a closed relationship that provides, e.g., the populations in terms of the system parameters and pumping rates, only relations between observables can be obtained in the general case. For instance, for the populations:
(30) 
This expression is formally the same as for the coupling of two bosonic modes. The differences are in the effective dissipation parameter (instead of the bosonic one, ) and the constrain of the exciton population, . One solution of Eq. (30) is and , which corresponds to the case , where each mode reaches its thermal steady state independently (Bose/Fermi distributions, depending on the mode statistics). With coupling , we can only derive some analytical limits and bounds. For example, when , one sees from Eq. (30) that , with the condition for the cavity pump (since ). If only the dot is pumped, , and if both , then, also with the same temperature. As, in this case, must be strictly smaller than , the exciton population prevents an inversion of population, as is well known for a twolevel system.
When , we get the following bounds for the cavity populations in terms of the system and pumping parameters:
(31) 
When , the cavity is in thermal equilibrium with its bath, , and with the dot . In this case, the pump is limited by , and again . Again, the inversion of population cannot take place putting the system in contact with only one thermal bath. In all these situations where an analytic expression for the population is obtained, the detuning between cavity and dot does not affect the final steady state, although it determines, together with the coupling strength, the time that it takes to reach it. An interesting limiting case where inversion can happen, is that where and are negligible, then . When the pump is low and , grows with pumping, but when the dot starts to saturate and the cavity population starts to quench towards Benson and Yamamoto (1999). Here, all values of bring the system into a steady state as cannot diverge. However, if we allow some cavity pumping, given that does not saturate, is bounded. A rough guess of this boundary is, in the most general case:
(32) 
If Eq. (32) is not fulfilled, the system diverges, as more particles are injected at all times by the incoherent cavity pumping than are lost by decay. Numerical evidence suggests that the actual maximum value of depends on . To some given order , divergence typically arises much before condition (32) is reached, although it is difficult to know if a lower physical limit has been reached or if the order of truncation was not high enough.
The second order correlator can be expressed as a function of only:
(33) 
Obtaining the expression for the th order correlator and setting it to zero would provide an approximate (of order ) closed relation for . We shall not pursue this line of research that becomes very heavy.
As an overall representation of the typical systems that arise in real and desired experiments, we consider three configurations, shown in Fig. (3), scattered in order to give a rough representative picture of the overall possibilities, around parameters estimated in Ref. Khitrova et al. (2006). Point 1 corresponds to the best system of our selection, in the sense that its decay rates are very small (, ), and the quantum (Hamiltonian) dynamics dominates largely the system. It is a system still outside of the experimental reach. Point 3 on the other hand corresponds to a cavity with important dissipations, that, following our analysis below, precludes the observation of any neat structure attributable to the underlying Fermi statistics. According to numerical fitting of the experiment, real structures might even be suffering higher dissipation rates Laussy et al. (2008b). Point 2 represents other lead systems of the SC physics, that we will show can presents strong departure from the linear regime, in particular conditions that we will emphasize. The best semiconductor system from Fig. 3 is realized with microdisks, thanks to the exceedingly good cavity factors. We shall not enter into specific discussion of the advantages and inconvenient of the respective realizations and the accuracy of these estimations. From now on, we shall refer to this set of parameters as that of “reference points”, keeping in mind that points 1 and 2 in particular represent systems that we will refer to as a “good system” and a “more realistic system”, respectively.
In Fig. 4, the three observable of main interest for a physical understanding of the system that we have just discussed—, and —are obtained numerically for the three reference points. Electronic pumping is varied from, for all practical purposes, vanishing () to infinite () values. Various cavity pumpings are investigated and represented by the color code from nocavity pumping (dark blue) to high, near diverging, cavity pumping (red), through the color spectrum. We checked numerically that these results satisfy Eq. (30). The overall behavior is mainly known, for instance the characteristic increase till a maximum and subsequent decrease of with has been predicted in a system of QD coupled to a microsphere Benson and Yamamoto (1999). This phenomenon of socalled selfquenching is due to the excitation impairing the coherent coupling of the dot with the cavity: bringing in an exciton too early disrupts the interaction between the excitonphoton pair formed from the previous exciton. Therefore the pumping rate should not overcome significantly the coherent dynamics. Too high electronic pumping forces the QD to remain in its excited state and thereby prevents it from populating the cavity. In this case the cavity population returns to zero while the exciton population (or probability for the QD to be excited) is forced to one. The cavity pumping brings an interesting extension to this mechanism. First there is no quenching for the pumping of bosons that, on the contrary, have a natural tendency to accumulate and lead to a divergence. Therefore the limiting values for when or are not zero, as in the previously reported selfquenching scenario Benson and Yamamoto (1999). They also happen to be different:
(34a)  
(34b) 
and therefore satisfy . Eq. (34b) follows from the decoupled thermal values for the populations, , and corresponds to a passive cavity where the quenched dot does not contribute at all. In this case, the emission spectrum of the system is expected to converge to
(35) 
for the cavity, and for the dot. The other limit when , shows the deleterious effect of the dot on cavity population. The dot fully enters the dynamics contrary to the quenched case where it is subtracted from it.
Important application of SC for singleatom lasing are to be found in good cavities 1 and 2, where the coupling is strong enough Mu and Savage (1992); Ginzel et al. (1993); Jones et al. (1999); Karlovich and Kilin (2001); Kozlovskii and Oraevskii (1999). Lasing can occur when the pumping is also large enough to overcome the total losses, . Setting , , Eqs. (6) can be approximately reduced to one for the total probability Scully and Zubairy (2002); Benson and Yamamoto (1999):
(36) 
The parameters that characterize the laser are the gain and the self saturation . Far above threshold (), the statistics are Poissonian, , with a large intensity in the emission, , and half filling of the dot, . However, this analytic limit from the standard laser theory is not able to reproduce the selfquenching effect induced by the incoherent pump, nor the subpoissonian region () where quantum effects are prone to appear. The validity of the laser theory is restricted to the narrow region, , where is the boundary for the selfquenching. In the weak coupling regime, is the well known Purcell enhanced spontaneous decay rate of an exciton through the cavity mode. In the strongly coupled system, it can be similarly understood as the rate at which an exciton transforms into a photon. If the excitons are injected at a higher rate, there is no time for such a coherent exchange to take place and populate the cavity with photons. Fig. 4 shows that lasing can be achieved with system 1 in the corresponding region of pump. Here, we will solve the system exactly, covering this regime and all the other possible ones with the full quantum equations of motion.
The effect of cavity pumping depends strongly on the experimental situation. In the case of an exceedingly good system, has little effect as soon as the exciton pumping is important, . Cavity pumping becomes important again in a system like 2, where it enhances significantly the output power, with the price of superpoissonian statistics (). With a poorer system like point 3, some lasing effect can be found with the aid of the cavity pump: there is a nonlinear increase of and approaches 1 for . However, the weaker the coupling, the weaker this effect until it disappears completely for decay rates outside the range plotted in Fig. 3. In all cases, the selfquenching leads finally to a thermal mixture of photons () and WC at large pumping.
Vi Weights and Renormalization
To give a complete picture of the spectral structure, that we have obtained analytically in Section IV, we need to consider how this limiting case of vanishing pumpings evolves with finite pumping. Here again, we have to turn to numerical results.
Two cases of finite pumpings are shown in Fig. 5 for the finite pumping counterpart of Fig. 2(a), namely , (a), and , (b) and (c). We take as the reference energy for the remaining of this text. Panel (a) shows how the limiting case () is weighted and deviates rather lightly from the analytical result. The computation has been made to truncation order and we checked that it had converged with other truncation orders giving exactly the same result. In the figure, only whose weighting in the cavity emission (Lorentzian part) is nonzero are shown, although most of them are very small. If we plot only those with , only the usual vacuum Rabi doublet (in green in Fig. 2) would remain. In addition of the weight, also the degeneracy (number of peaks) at a given resonance should be taken into account to quantify the intensity of emission at a particular energy. This information is not apparent in the figures, where we only show in Blue or Red the cases of positive or negative, respectively, weighting. In some cases, many peaks superimpose with opposite signs, possibly cancelling each other. We plot negative values last so that a blue line corresponds to a region of only positive values, while a red line may come on top of a blue line. This figure gives nevertheless an insightful image of the underlying energy structure and how they contribute to the final spectrum as an addition of many emitting (or interfering) events. In (b) we show a case of higher pumping, with the same principal information to be found in the mapping of the eigenvalues. The characteristic branchcoupling of the JaynesCummings, still easily identifiable in Fig. 5(a), has vanished, and lines of external peaks directly collapse toward the center. A zoom of the central part, panel (c), shows the considerable complexity of the inner peaks, forming “bubbles” around the central line, due to intensityaided SC fighting against increasing dissipation that ultimately overtakes.
The origin of the lines can be better understood if we plot them as a function of pumping, as we commented in Section IV. In Fig. 6, the same weighted peak positions are shown (with the same color code) for Point 2 as electronic pumping is varied from to (). This last picture supports the idea that quantum effects (such as subpoissonian statistics, Fig. 4) are observed at small pumpings, with optimal range being roughly , where only the lowest manifolds are probed. This is the range of pumping where the JaynesCummings manifold structure is still close to that without pump. Further pumping pushes the lines to collapse, starting by the vacuum Rabi splitting which closes, evidencing the loss of the first order SC at . Here again we observe this phenomenon of bubbling, with a sequence of lines opening and collapsing, that makes it impossible to specify the exact pump at which the transition takes place. From this point, SC is lost manifold by manifold similarly as in the case where was increased. When , all lines have collapsed onto the center and will remain so at higher pumpings. The dot saturates and the cavity empties with thermal photons in a WC regime.
Vii Luminescence Spectra at Resonance
Now we have all the ingredients to present the final result: the spectral shapes of the system in a broad range of configurations and parameters. We cannot give a comprehensive picture as any set of parameters is by itself unique, but will instead illustrate the main trends, using specifically for that purposes the three reference points of Fig. 3. They give a good account of the general case and one can extrapolate from these particular cases how another configuration will behave. To get exact results for a given point, numerical computations must be undertaken.
From now on, we shall represent in Blue the spectra that correspond to cavity emission and in Violet those that correspond to direct exciton emission. The main conclusions, based on semianalytical results, are different for different points or family of points. Point 1, that represents a very good system, is the one that is best suited to explore quantum effects. Its spectral shape is unambiguously evidencing transitions in the JaynesCummings ladder, as shown in Fig. 7 with a clear “JaynesCummings fork” (a quadruplet). The outer peaks at are the conventional vacuum Rabi doublet, whereas the two inner peaks correspond to higher transitions in the ladder. Observation of a transition from outer to inner peaks with pumping such as shown in Fig. 7 would be a compelling evidence of a quantum exciton in SC with the cavity. Fig. 8 shows another multiplet structure of this kind for Point 1. The intensity of emission is presented in logscale and for a broader range of frequencies, so that small features can be revealed. Transitions from up to the third manifold can be explicitly identified. The decay from the second manifold, that manifests distinctly with peaks labelled 2 (although it also contributes to peaks labelled 0), is already weak but still might be identifiable in an experimental PL measurement. Higher transitions have decreasing strenght. This can be checked with the probability to have photons in the cavity, , computed from Eqs. (6). Whenever the mean number is low (as is the case here), this probability is maximum for the vacuum ( for all ), independently of the nature of the photon distribution (sub, super or Poissonian). Only when , in the best of cases (for a Poissonian distribution), does this trend start to invert and . This makes it impossible, even in the very good system of Point 1, to probe clearly and independently transitions between manifolds higher than , as their weak two outer peaks (approximately at ) are completely hidden by the broadening. A stronger manifestation of nonlinear emission is to be found in the pool of pairs of inner peaks from all highmanifold transitions (labelled 0 in Fig. 9), at approximately . Not only the inner peaks coming from different manifolds are close enough to sum up, but also they are more intense than their outer counterparts. This can be easily understood by looking at the probability, , of transition between eigenstates through the emission of a photon, , or an exciton, . This probability, , estimates the relative intensity of the peaks depending on the initial, , and final, , states of the transition and on the channel of emission, Laussy et al. (2006). A discussion in terms of the eigenstates of the Hamiltonian is still valid in the regime of Point 1 (very good system) at very low pump. At resonance, neglecting pumps and decays, the eigenstates for manifold , are . The outer peaks arise from transitions between eigenstates of different kind, , while the inner peaks arise from transitions between eigenstates of the same kind, . Their probability amplitudes in the cavity emission,
(37a)  
(37b) 
evidence the predominance of the inner peaks versus the outer ones, given that one expect the same weighting of both transitions from the dynamics of the system. The doublet formed by the inner peaks is therefore strong and clearly identifiable in an experiment. On the other hand, in the direct exciton emission, the counterparts of Eqs. (37) are manifoldindependent and equal for both the inner and outer peaks:
(38a)  
(38b) 
In this case, therefore, one can expect similar strength of transitions for both the inner and outer peaks with a richer multiplet structure for the direct exciton emission.
In Fig. 9, we give an overview of the PL spectra as is varied from very small to very large values. For point 1, as we already noted, the cavity pumping plays a relatively minor quantitative role. Therefore we only show two cases, of nocavity pumping (first row) and highcavity pumping (second row). As can be seen, there is no strong difference from one spectra with no cavity pumping to its counterpart with large cavity pumping. Third row shows the direct exciton emission that, with no cavity pumping, corresponds to the first row. Indeed, one can observe the richer multiplet structure up to in the direct exciton emission, whereas only inner peaks are neatly manifest in the cavity emission. This region corresponds to a quantum regime with a few quanta of excitations (and subpoissonian particle number distribution,