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Comment on"Nonlinear photoluminescence spectra from a quantum dot-cavity system revisited"by Yao et al. and"Photoluminescence from Microcavities Strongly Coupled to Single Quantum Dots"by Ridolfo et al

We refute the criticisms of our work on strong-coupling in the presence of an incoherent pumping.

Both groups focus on our boson model only, and in fact only on our Letter on the topic [1]. Many of their statements are already addressed in our full boson text [2] and more crucially in its fermion counterpart [3]. Throughout, we made clear that the boson model is adequate either in the limit of small pumping, or in the limit of bosonic 0D system. The interest of the boson case is in its analytical solutions, that allow to explain transparently fundamental features of pumping, such as the effect of the effective quantum state on the spectral shapes. We will show that the analytical solutions for the fermion models from [4] and [5] are in fact valid only in trivial cases (namely, the boson limit or the uncoupled limit).
In the following, we shall clarify, on the one hand, that our approach does not suffer from any inconsistency or pathology, and on the other hand, that in the particular cases where the reservoirs of excitations are thermal baths, our model also applies (by enforcing the ad-hoc constrain in the equation) and show the errors made by the approximations of Refs. [4,5].

I. VALIDITY OF OUR MASTER EQUATION
Our model describes the linear regime (of vanishing excitations) analytically and the nonlinear regime semi-analytically. Its master equation reads: where a is the cavity mode and σ the exciton in the QD. The cavity is always a bosonic mode. In Ref. [1] and [2], σ is also a Bose operator, that we note b for clarity, while in Ref. [3], σ is a fermion operator, describing a two-level system [34]. This choice of σ as a Bose operator addresses two important cases: i) the limit of vanishing pumpings (even if the QD is indeed a fermion emitter) and ii) the case where the excitons follow bose statistics. The latter case could be realized in large QDs, that recover the physics of quantum wells where excitons are known to behave as good bosons [6].
The main reproach of [4,5] is that in Eq. (1), the effective pumping rates P a,σ can vary independently of the effective decay rates γ a,σ [35]. They observe that this can lead to some divergences beyond some critical values of pumping, as we have ourselves discussed before, and they conclude that the model is flawed. They propose instead to use thermal reservoirs, that do not exhibit such divergences, at all values of pumping.
A thermal reservoir for a bosonic mode a (with frequency ω a and H a = ω a a † a) at temperature T leads to the effective rate of excitation: withn T given by the reservoir Bose-Einstein distribution. It vanishes at T = 0. In thermal equilibrium, the system is loosing excitations at a larger rate of: The parameter κ a is the spontaneous emission (SE) rate at T = 0. The steady state thermal equilibrium reads At very high temperatures, as the effective income of particles approaches the outcome, P a ≈ γ a , the number of particles remains finite, since P a < γ a . As long as γ a = 0, any combination of parameters γ a , P a corresponds to a given thermal bath (with κ a = γ a − P a and T > 0). The linewidth of the optical spectrum of emission is: and is independent of temperature (i.e., of the population of the mode), since it is always equal to the spontaneous emission decay rate κ a . It is clear from the above results, that a bosonic thermal bath cannot provide gain and does not exhibit any line-narrowing in its luminescence spectrum. A thermal bath is a medium of loss as P a < γ a by definition.
A thermal reservoir is, however, a particular case. In out-of-equilibrium conditions, especially under externally applied pumping, one can expect deviations from the thermal paradigm.
In contrast to the thermal case, a gain medium can be derived with bosonic baths out-of-equilibrium. This is discussed in textbooks, e.g., in Chapter 7 of Gardiner and Zoller's text [7]. A linear gain can be obtained with an "inverted" harmonic oscillator maintained at a negative temperature −T . The effect in the master equation is that the effective parameters are now given by a relation opposite to Eqs. (2)(3): In this way, G a is the gain or input of particles into the mode at zero temperature. Obviously, given that now P a > γ a , there is no stationary solution to the master or rate equations. Since the absence of this stationary solution is the source of confusion in Refs [4,5], we quote here Gardiner and Zoller's comment, p • 216: If γ < κ [that is γ a < P a with our notation], there is no stationary situation, and the amplifier gives a signal that increases without limit. Essentially, the power being fed into the cavity cannot escape fast enough. (Of course the idea of an inverted medium which maintains its inversion independent of power output is not exactly valid, and depletion effects will then need to be considered. The system is then essentially a laser).
Obviously, an ever-growing population will always be stopped by some external physical effect (the sample will have burnt, the reservoir will be depleted, etc. . . ). There is however nothing pathologic in this behaviour. In particular, this does not invalidate the results for values below the critical pumping rates. In the case where γ c > P c (c = a, b), there is a physical solution to the dynamical master equation, starting at t = 0: at all (finite) times, there is a valid master equation, with positive trace, normed to unity, etc. . . However, indeed, the system diverges with time. There is no unphysical behaviour or flaw of some sort. Not all dynamical systems have a steady state, some because they are oscillatory, others because they increase without bounds.
In our work [1,2,3], we have naturally considered the configurations which admit a steady state. We have even given analytical solutions for their domain of convergence, supported by a clear physical picture [36].
In a microcavity QED system (a QD in a microcavity), which is a complicated solid state system, open to many sources of excitations [8,9], one should also include gain effects in the most general case. Granted all together, the microscopic coefficients are likely terms of the form: that is, including loss media and gain media. Net losses in the case of a cavity mode comes, among other reasons, from the fact that the photons can escape the cavity through the imperfect mirrors. Net gain could come from surrounding off-resonance or weakly coupled QDs, high energy QD levels or the wetting layer. A gain medium can be obtained by the very configuration realized with self-assembled QDs in a microcavity. Gardiner and Zoller, p • 140 (ibid) study a bath of uncorrelated twolevel emitters that are kept in average in the excited state. Quoting them again: Note that there is no restriction on N + a and N − a [that is, in their notations, the number of emitters in the excited and ground states, respectively, that provide the pump and decay terms for the bosonic mode]-this means that N + a > N − a is permissible. Physically such a population inversion could only be maintained by some kind of pumping, as indeed happens in a laser. and as indeed could happen in an incoherently pumped microcavity. The previous discussion on gain-media that applies for the cavity mode can also be extended for bosonic emitters (QDs, in our case). In this case, for example, the electron-hole pairs that form excitons inside the QD decay from the wetting layer (at higher energy) by, e.g., emitting phonons with the corresponding energy difference. Such phonons will not be reabsorbed to bring back the electron-hole pair to the wetting layer, leading to a net source of particles. Net losses take place through the spontaneous decay into leaky modes.
In general, there is thus no reason to restrict the master equation (1) designed to account effectively for the largest possible amount of physical effects, to a given type of reservoirs (namely, thermal baths). Our study aims at the greatest level of applicability and generality and provides the tools for the understanding of any case. Therefore, better than trying to apply at the theoretical level any criteria (other than convergence) to choose γ a , P a , we prefer to consider them independent. So far, the dynamics of lines (1b-1c) found its most important domain of applicability with atom lasers and polariton lasers [10,11,12,13,14,15,16,17], that is, systems where a condensate (or coherent state) is formed by scattering of bosons into the final state from another state rather than by emission. In both cases, scattering or emission, the process is stimulated. In this case the income and outcome of particles is a complicated function of the distribution of excitons (or polaritons) in the higher k-states. See, e.g., Ref [11]. In a pulsed experiment, it is typically time dependent (see, e.g., Fig. 2 of Ref. [18]). Line narrowing is a natural feature of this dynamics in such systems.
It remains, of course, possible that a particular experiment corresponds to a thermal bath of excitation. In this case, a fitting analysis with our model should indicate this constrain in correlations between the γ and P coefficients. As a result of this analysis, one will then understand that the given system refers to the particular cases of Eqs. (2) and (3).
The above discussion concerns the bosonic mode. We now turn to the fermion mode.
The thermal equilibrium case, at temperature T , gives the counterpart of the boson case (given above): where κ σ is the Einstein A-coefficient and P σ is the Einstein B-coefficient. The steady state is the Fermi-Dirac distribution: The maximum occupation for the emitter is 1/2, at infinite temperature. It is, therefore, not possible to invert the two-level system with a thermal bath, (where, again, γ σ > P σ ), which is a well known result. In the fermion case, even a gain-medium does not lead to a divergence thanks to the intrinsic saturation of a two-level system. The emission spectrum is also a Lorentzian, with effective linewidth: which broadens from the decay rate at zero temperature, κ σ when the temperature increases. This elementary discussion illustrates the notorious fact that thermal reservoirs are unable to achieve population inversion of a two-level system [19]. Therefore, the choices of excitation reservoirs of Yao et al. [5], forbids lasing in such systems, which is already contradicted by experiments [20].
The issue of achieving gain with a master equation for a fermion emitter has been extensively discussed in the lasing literature, in particular in the one-atom laser case [21,22,23,24,25,26,27,28,29,30,31,32], and it is typically described theoretically also by considering an effective negative-temperature thermal bath. A popular notation to consider gain and dissipation is [25]: With Γ σ > 0 (the linewidth broadening) and 0 ≤ s ≤ 1, including both the situations with net losses (s < 1/2) and gain (s > 1/2). In this case: This parameterization is not linked to any particular experimental realization but is designed to separate the physical effects that lead to line broadening, Γ σ , from those that change the population, s. Apart from that, it is equivalent to consider directly the effect of varying the effective decay and pumping parameters, γ σ and P σ , as we have done in Refs [3] and [33].
The Jaynes-Cummings model couples the fermionic and bosonic modes. It has been extensively studied in the case of thermal cavity bath and some gain-medium for the emitter (which is also the case of Ref. [4]). We studied it in its most general form, using the master equation (1). Again, if a particular constrain arises from a given realization of the reservoirs of excitations, such as (2-3), (6), (7), (8) or some other case, this would appear in our unconstrained case from correlations following these trends. We expect, as we previously discussed, that a successful statistical analysis would indeed inform about underlying microscopic details of the excitation scheme.

II. VALIDITY OF THE PROPOSED SUBSTITUTES TO OUR WORK
In the case of Ridolfo et al. [4], only the photonic mode was excited thermally while the excitonic pump was still allowing an unconstrained pumping and decay. Yao et al. [5], on the other hand, require both modes to be excited by thermal baths.
Thermal baths in the linear (boson) model reduce to results identical to the spontaneous emission of an initial state that is a mixture of excitons and photons in the ratio of population n a (t = 0)/n b (t = 0) = P a /P b . The thermal character of the bath merely prevents renormalization of the linewidths and of the Rabi frequencies.
The ratio P a /P b still determines the possibility to resolve the line-splitting. This fundamental consequence of the effective quantum state is independent of any choice of the reservoirs. It is a general result that we have amply discussed before [1,2,3] and that is "rediscovered" by Ridolfo et al.
Ridolfo et al. [4] otherwise have mistakes in their formulas, that certainly bias their analysis. For instance, their parameters n a and C should read, in their notations: (cf. their Eq. (7) & (8)) Also, their spectra are normalized to √ 2πn a , again apparently as an error since they compare them directly to ours which are normalized to unity. These mistakes do not seem to be a misprint, given that the authors state in their paper: Although at low pump intensities, our approach and that of Ref. [Laussy et al.] essentially represent models of a linear Boselike dynamics of two coupled harmonic oscillators, nontrivial differences can be appreciated.
There should be no difference in the limit of vanishing pump. In fact, once corrected as above, their formulas and lineshapes do converge to our results at low pumps.
In the non-vanishing case, of course, the thermal reservoir gives a different result than unconstrained parameters of our Eq. (1), even in the linear regime (that is, when n σ 1, although n a is not compulsorily also vanishing). The illustration of this fact was attempted in Fig. (1a) of Ref. [4], although here also the plot is wrong. With non-vanishing cavity pumping, the linear regime can be maintained only if the modes are almost uncoupled. This is shown in our corrected version (Fig. 1).  Fig. 1(a) of Ridolfo et al. [4]: ∆ = −3.6g, γσ = 1.48g, Pa = 0.49g, Pσ = 0.0078g. Like in their figure-but with the correct formulas (13)-we compare: in solid-blue, the case where γa = 1.96g and in dashed-purple κa = 1.96g (γa = κa + Pa, thermal bath). Both cases are indeed in the linear regime (nσ 1) since the fermion model [3] gives the same results. The system is however almost decoupled due to the large detuning. The emission is thus basically that of the bare cavity (Lorentzian).
We now turn to the approach of Yao et al. [5]. They miss a factor Γ c in the second term of the denominator of their Eq. (4), although in their case this is a misprint only since their figures match the formulas that follow from their approximations. These approximations, however, are incorrect.
As we discussed above, it is straightforward in our approach to constrain the coefficients, following a given choice of the reservoirs of excitations. If we choose the thermal baths advocated by Yao et al., we are able with our approach [3] to apply the full fermion model with no truncation in the number of excitations, for the parameters of these authors. We find that their approximation is a poor one, as seen in Fig. 2, where we superimpose the exact (numerical) result, in solid black, to their approximate (analytical) formula, in dashed red. As could be expected, the agreement is good only at very low pumping (linear regime) and very high pumping (uncoupled regime). It is incorrect in the most relevant region of the transition where the doublet collapses (as has been reported before, e.g., Fig. 13 of our Ref. [3]).
Their approximation is also basically flawed in that it allows an inversion of population for the two-level system, although it is excited by thermal reservoirs, as seen in the magnified version of their Fig. 3(a), that we reproduce in dashed lines in our Fig. 3, along with the converged solutions from our model [3] (with their parameters and choices of reservoirs). Beyond the poor quantitative agreement when pumping is non-vanishing, at the point indicated by the arrow and above, the QD is inverted, which indicates a pathology of their approximation.
Their implication that cavity pumping is determinant to achieve lasing is in contradiction with well-known and established facts of the one-atom laser theory. See for instance our text Ref. [3] where lasing is achieved without cavity pumping, thanks to the gain-medium that our general model allows. On the other hand, thermal reservoirs of Yao et al. forbids lasing, regardless of the magnitude of cavity pumping (note that with their parameters, very high cavity populations are already achieved, but they have thermal statistics, with second-order correlator g (2) that increases rapidly towards 2 with pumping).
Finally, we want to stress that their Fig. 2(b), that supposedly represents our model, does not make any meaningful comparison, since, fitting some data with their model [that we have just shown is wrong, but even if it was correct], they proceed to plot our boson model with their fitting parameters. It is obvious that, the two formulas being different, the best-fitting parameters for one of them will yield poor agreement on the same data for the other. Beside, they should have used our fermion model, since they consider a supposedly two-level emitter in a nonlinear regime. Fitting them independently, on the one hand, and comparing them on statistical grounds on the other hand, rather than settling for some aesthetic of the agreement, is the correct course of action. Fitting with the nonlinear fermion model is not a trivial task. With E. Cancellieri and A. Gonzalez-Tudela, we have recently obtained results in this direction, to be published shortly.
In conclusion, we have shown that our work [1,2,3] is correct and that the critics addressed against it [4,5] are unsubstantiated on the one hand, and the proposed substitutes are incorrect on the other hand. These authors do not derive any master equation. They settle for thermal reservoirs, which derivation is a standard textbook material. This is a particular case of our work that they apply incorrectly or beyond its limits of validity. We have already provided the valid limit for the nonlinear regime [3]. There remain many open questions in the field. Some can be settled by statistical analysis of experimental data with our model, Eq. (1), which correlations between the fitting parameters can teach about underlying microscopic mechanisms (such as the nature of the bath of excitations, among other). To this intent, we invite experimentalists to make their raw data available to everybody [37].