Delayed transition to coherent emission in nanolasers with extended gain media

Delayed transition to coherent emission in nanolasers with extended gain media

F. Lohof Institute for Theoretical Physics, Otto-Hahn-Allee 1, P.O. Box 330440, University of Bremen    R. Barzel Institute for Theoretical Physics, Otto-Hahn-Allee 1, P.O. Box 330440, University of Bremen    P. Gartner Centre International de Formation et de Recherche Avancées en Physique - National Institute of Materials Physics, P.O. Box MG-7, Bucharest-Măgurele, Romania    C. Gies Institute for Theoretical Physics, Otto-Hahn-Allee 1, P.O. Box 330440, University of Bremen
July 20, 2019

The realization of high- lasers is one of the prime applications of cavity-QED promising ultra-low thresholds, integrability and reduced power consumption in the field of green photonics. In such nanolasers spontaneous emission can play a central role even above the threshold. By going beyond rate-equation approaches, we revisit the definition of a laser threshold in terms of the input-output characteristics and the degree of coherence of the emission. We demonstrate that there are new regimes of cavity-QED lasing, realized e.g. in high- nanolasers with extended gain material, for which the two can differ significantly such that coherence is reached at much higher pump powers than required to observe the thresholdlike intensity jump. Against the common perception, such devices do not benefit from high- factors in terms of power reduction, as a significant amount of stimulated emission is required to quieten the spontaneous emission noise.

Valid PACS appear here
preprint: APS/123-QED

I Introduction

Lasers have been around for almost 60 years and most of the underlying physics is well established and understood. The threshold, in particular, is the central defining property of any laser device, as it separates the regime of spontaneous emission from that of phase-coherent operation. In conventional laser devices, the threshold can easily be identified from the input-output characteristics alone: Below threshold, a majority of photons spontaneously emitted from the gain material is lost, and only a fraction described by the factor ends up in the modes that can ultimately achieve lasing. Above threshold, stimulated emission into these modes completely overpowers the losses, which gives rise to an intensity jump typically over several orders of magnitude, which clearly marks the laser threshold (“intensity threshold”). This fundamental behavior can already be understood in terms of coupled rate equations for the excitation of the gain material and the photon number Yokoyama and Brorson (1989); Björk and Yamamoto (1991).

In recent times, the laser threshold has been revisited due to the uprise of nanolasers that operate in new cavity-enhanced lasing regimes Chow et al. (2014); Rice and Carmichael (1994); Blood (2013). Nature Photonics was amongst the first journals to issue a laser “checklist” to ensure a certain standard in identifying laser operation, which becomes even more important for small lasers noa (2017). Cavities with ultra-small mode volumes with dimensions of the light’s wavelength allow for tinkering with spontaneous emission itself. Quantum-electrodynamical effects make it possible to enhance the emission rate Wang et al. (2016); Strauf and Jahnke (2011), to suppress emission into nonlasing modes Stepanov et al. (2015); Gevaux et al. (2006); Takiguchi et al. (2013), or do both at the same time – thereby fundamentally changing the contributions from spontaneous and stimulated emission. When spontaneous emission is nearly completely directed into the laser mode (), the intensity jump even disappears and makes an identification of the threshold from the input-output curve alone impossible Prieto et al. (2015); Ota et al. (2017); Khajavikhan et al. (2012). In such thresholdless lasers, it has become common practice to investigate fluctuations in the photon number that are captured in the two-photon correlation function Chow et al. (2014); Strauf et al. (2006); Ulrich et al. (2007)


For coherent emission, takes on the value of 1, whereas for spontaneously emitted light below threshold it has a fingerprint . As we show, the increased role of spontaneous emission in nanolasers can substantially delay the formation of coherence to higher pump powers.

In the present work, we refine the definition of the laser threshold by going beyond the rate-equation approximation to take a combined look at the intensity threshold and the transition of the emission from thermal to coherent light reflected in (“coherence threshold”). While one might intuitively believe that both characteristics are interlinked and one necessitates the other, we show that there are operational regimes of nanolasers that give rise to a separation between the two thresholds, so that the emission becomes coherent at much higher excitations, after the intensity threshold has been crossed. An important result of this finding is that one commonly used criterion for lasing in cavity-enhanced nanolasers, i.e. the mean intracavity photon number , can be arbitrarily insufficient and truly only holds in the limit of the single-emitter laser Nomura et al. (2010); Strauf and Jahnke (2011); Gies et al. (2011). Furthermore, in those regimes an increase of the factor has no benefit in terms of reducing the threshold power, an aspect that has, to the best of our knowledge, not been recognized in the literature so far. The implications are both of fundamental relevance for understanding the laser threshold, and apply to present-day nanolaser devices. Despite the simplicity of our model, it captures all relevant effects of the laser dynamics and allows for the definition of analytical expressions for the above-mentioned threshold criteria.

Ii Characterizing the laser threshold beyond the rate-equation approximation

At the rate-equation level the laser threshold is defined in terms of the output intensity. This criterion we define as the intensity threshold. In conventional lasers it is well established that the threshold marking the transition to coherent emission is accompanied by a sharp increase of output intensity. This intensity jump is less pronounced or vanishes completely in high- lasers. Here we consider independent two-level emitters in resonant Jaynes-Cummings interaction with a single mode of an optical cavity Gartner and Halati (2016). This simplified laser model provides analytic access to quantities related to the threshold, yet it contains the relevant physics to render its implications applicable to current cavity-QED nanolaser devices. Starting from the system Hamiltonian we derive equations of motion for the upper-level population (the lower-level population is ) and the intracavity photon number that resemble standard laser rate equations Coldren et al. (2012) (see Appendix A): \cref@addtoresetequationparentequation




is the spontaneous emission rate in the cavity mode, is the light-matter interaction strength, is the rate of radiative losses, is the pump rate, and is the cavity decay rate. The factor is defined as the ratio of the spontaneous emission into the laser mode to the total spontaneous emission including losses, i.e.


In their stationary limit, the rate equations can be used to define the pump value of the intensity threshold. In Appendix A, we show that it obeys the equation

Figure 1: a) Output intensity and b) values according to Eqs. (2) and (6) are shown for different values of . The parameters , , and correspond to typical high- quantum dot (QD) nanolasers Ulrich et al. (2007). The intensity thresholds are marked as dashed lines. In this parameter regime, they nearly coincide with the threshold to coherent emission, which are indicated by the shaded regions for each value of .

The functionality of the threshold pump rate defined this way is illustrated in the top panel of Fig. 1. For typical quantum-dot (QD) nanolaser parameters taken from [16], output intensities are shown (solid lines) together with the respective values for marked by the vertical lines. Different values of are used to characterize different device efficiencies as explained below. As can be seen, the threshold pump rates determined from Eq. (5) lie exactly in the middle of the corresponding jump intervals in the output intensities.

Due to excitation-induced dephasing, the emission rate depends on the pump rate. With it, is implicitly excitation dependent. Nevertheless, since a constant factor is well established throughout the literature to characterize laser efficiency, we choose to use rather than the emission rate into nonlasing modes, which is more commonly used in theory. To do so, all given values of are evaluated at . While nonconstant factors have been investigated in different contexts Gericke et al. (2018); Lichtmannecker et al. (2017); Gregersen et al. (2012), this approximate treatment fully serves the purpose in this work.

For approaching unity, the intensity jump goes to zero, reflecting the concept of thresholdless lasing often referred to in the literature Prieto et al. (2015); Ota et al. (2017); Khajavikhan et al. (2012). However, it is well established that coherent emission can take place at finite excitation even in the limiting case Gies et al. (2007). With no distinct threshold nonlinearity in the output intensity of high- lasers, one has to rely on statistical properties of the emission in order to identify lasing Strauf et al. (2006); Chow et al. (2014); Ulrich et al. (2007). In a classical picture, the output intensity fluctuates around its mean value, which allows distinguishing a thermal from a coherent light source. The laser model we employ is fully quantum mechanical. In this picture, a coherent (thermal) state of the light field is characterized by a Poissonian (Bose-Einstein) photon distribution function . This quantity can be accessed both in theory by using master-equation approaches Gartner (2011); Jagsch et al. (2018); Leymann et al. (2013); Carmele and Knorr (2011), and in experiment by using photon-number-resolved detection schemes Schlottmann et al. (2018); Calkins et al. (2013). In practice, it is much easier to use the second-order photon autocorrelation function to characterize the emission, as it can be readily measured using Hanbury Brown and Twiss setups Hanbury Brown and Twiss (1956). It is defined in Eq. (1), with , and the transition to its coherent value of 1 has become an established method to characterize nanolasers that operate close to the thresholdless regime with Strauf et al. (2006); Chow et al. (2014); Ulrich et al. (2007).

We access the two-photon correlation function beyond the rate-equation limit by extending Eqs. (2) to contain higher-order correlations and truncate the arising hierarchy of quantum-mechanical expectation values up to and including two-photon correlations Gies et al. (2007). This approach, the details of which are given in Appendix B, allows us to arrive at an analytic expression for the photon autocorrelation function that only contains the photon number , the carrier occupation , and fixed system parameters


where we define . By using the stationary values for and provided by the rate equations and taking advantage of the relationship between them


which follows from Eq. (2b) in steady state, Eq. (6) can be further simplified to obtain a closed analytic expression for in terms of the photon number only:


This expression is a key result of the paper, it provides access to the statistical properties of the emission even when values for the photon number obtained from rate equations are inserted. In Appendix B we show that the results obtained from Eq. (8) agree extremely well with full solution from the extended quadruplet equations so that, in fact, it can be used to augment results of rate-equation calculations with the corresponding values.

In Fig. 1 the photon correlation function as obtained from Eq. (6) is shown together with output intensities for different factors. At low excitations exhibits a value of 2. As it is customary in the literature, we model the excitation as an incoherent process, described by a Lindblad term (see Eq. (A2)). It represents physically the scenario where intermediate scattering processes from higher excited ones to the final states before recombination dephase any phase information that may have been imprinted by the exciting laser. As a result, the emitted photons exhibit no phase coherence leading to thermal emission with . For all values of , the transition of to 1 (marked by the shaded regions) indicating coherent emission closely follows the intensity threshold (marked by the vertical lines). This behavior corresponds to the common conception of the intensity threshold being interlinked with the transition to coherent emission. Only for the intensity threshold vanishes and solely indicates the existence of a transition.

We define the coherence threshold as the point, where indicates the onset of coherent emission. Here we choose with the aim to identify the beginning of an interval in which stays practically constant around unity. Of course, the choice of 1.1 for as the onset of coherence is to a certain extent arbitrary, however, a lower value would only push the coherence and the intensity thresholds even further from one another. It is important to point out that a unique definition of is only possible in the limit . For large values of that are relevant in nanolasers, approaches 1 gradually over a wide range of excitation powers.

Figure 2: For a nanolaser with an extended gain material, a) output intensity and b) values are shown for different . The large carrier number in the gain media is reflected in , whereas the other parameters are , . For all values of , the intensity threshold (dashed vertical lines) is clearly separated from the transition to coherent emission (indicated by the gray area). Furthermore, the coherence thresholds indicate that increasing has no benefit in terms of reducing the pump power to reach coherent emission.

Iii New Laser Regimes

We now turn to a different class of nanolasers and show that they elude the common conception of the laser threshold. Until now, high- nanolasers were mostly realized in high- cavities with discrete emitters, such as quantum dots, as gain material. The previous section illustrated that for these systems, the formation of coherence is linked to the intensity threshold as one typically expects. A fundamentally different behavior is encountered in a very different, yet highly relevant regime of cavity-QED nanolasers that operate with extended gain materials instead of individual emitters. Atomically thin layers of semiconducting transition-metal dichalcogenides (TMDs), such as WSe or MoS, have recently moved into the focus of optoelectronic device applications Tian et al. (2016) including nanolasers Li et al. (2017); Salehzadeh et al. (2015); Ye et al. (2015); Wu et al. (2015). These new materials combine advantages in ease of fabrication over conventional III/V semiconductors with a plethora of possibilities to engineer their electronic and optical properties. The possibility of population inversion Chernikov et al. (2015) and room-temperature lasing with monolayer TMD flakes on top of photonic crystal Wu et al. (2015) or nanobeam Li et al. (2017) cavities is being vividly discussed in the literature. Also recently, room-temperature lasing from a two-dimensional GaN quantum well (QW) embedded in a nanobeam cavity has been demonstrated Jagsch et al. (2018).

Typical carrier densities in extended gain media (in the order of in two-dimensional TMD materials) strongly exceed carrier numbers in discrete gain media, such as quantum dots. This results in very different operational regimes with severe implications regarding the threshold behavior. While more complex semiconductor laser models can be employed Jagsch et al. (2018); Hofmann and Hess (1999); Chow and Crawford (2015), for the present purpose it suffices to bear with the formalism used up to this point by mapping the number of electron-hole pairs in the two-dimensional gain media to our discrete-emitter model. This notion is formally justified, since the creation of cavity photons results from the recombination of an electron-hole pair both in the case of independent emitters each containing a single excitation, or excitations in an extended system. Assuming an active region of and a typical carrier density of yields an estimate for the maximum number of excitons of . The parameters we use in the following correspond to that of the nanobeam-quantum-well laser investigated by Jagsch et al. Jagsch et al. (2018). In addition to much larger , reported factors for both the nanobeam and TMD nanolasers are significantly lower and are typically around 2500 Jagsch et al. (2018); Li et al. (2017); Salehzadeh et al. (2015); Wu et al. (2015); Ye et al. (2015).

Input-output curves and photon-autocorrelation functions corresponding to the parameters of Ref. [26], but also applicable to recent work on TMD gain media, are shown in Fig. 2. As for the few-emitter case (Fig. 1), the intensity threshold, marked by the intensity jump in the input-output curves, moves to lower pump rates with increasing factor, suggesting that the threshold current can be strongly reduced by maximizing the factor. The corresponding autocorrelation functions are shown in the lower panel and reveal a very different picture: The transition to coherent emission is strongly offset to higher pump rates from the intensity jump, so that, in fact, the coherence and intensity thresholds are no longer aligned. Coherent emission is reached at a pump rate of about irrespective of the value of , implying that for devices with extended gain media, a reduction of the threshold current is no longer possible by reducing radiative losses (i.e., increasing ).

This behavior, which is a key insight of this work, can be attributed to the very distinct parameter regimes: Room-temperature lasing from the extended gain media in Refs. [26; 33] takes place in a regime, where cavity losses are the dominant quantity in determining the laser threshold. Despite the relatively low factors, lasing is still possible due to the large number of carriers producing gain. For few-emitter nanolasers, as the systems summarized in Ref. [8], a much larger cavity is required to attain lasing, which sets an upper bound for . In this low- regime, radiative losses dominate over the cavity losses, so that the threshold pump rate is strongly dependent on . These results are summarized in Fig. 3, which compares the pump rate , at which coherent emission is reached, for the few-emitter (dashed lines) and extended gain media (solid lines) as a function of the cavity loss rate. The cavity loss rates for both operational regimes we consider are indicated by vertical lines.

Figure 3: Pump rate at the coherence threshold as a function of cavity loss rate . Parameters (, ) correspond to those in Figs. 1 (dashed) and 2 (solid). Vertical lines indicate respective values for . One observes the different dependence on for the two cases as well as the transition between the regimes for varying . For small number of emitters there is an upper bound for beyond which lasing can not be achieved and thus the dashed lines terminate above a certain value.

More insight is revealed by condensing the information in Fig. 2 into a depiction of as a function of the mean photon number. Surprisingly, for the parameters of the extended gain media (solid lines), Fig. 4 shows that the transition to coherent emission is solely determined by the mean intracavity photon number completely irrespective of the value of . Coherence is reached at , which greatly exceeds the criterion that has frequently been used to indicate the threshold Björk et al. (1994); Chow et al. (2014); Kreinberg et al. (2017). To emphasize the contrasting behavior to the few-emitter nanolaser, the results of Fig. 1 are shown as dashed lines. Here, coherence is reached at significantly lower mean photon numbers ( for ), and one observes a clear reduction in the photon number necessary to reach coherence by increasing . The consequences of this finding are highly relevant for the design of future nanolasers that aim to exploit the unique properties of two-dimensional TMD and QW gain materials: Here, a reduction in energy consumption requires improving the cavity instead of minimizing radiative losses.

Figure 4: Two-photon correlation function as a function of photon number. For high- quantum-dot nanolasers (dashed lines) an increasing factor leads to fewer photons required for coherent emission. For a high number of emitters and lower cavity (solid lines) coherent emission happens at much larger photon numbers and independent of .

The only available measurements performed in the parameter regime we associate with nanolasers with extended gain media are published in Ref. [26]. Upon close inspection, the results shown there indicate that the coherence threshold is also offset to the intensity threshold thereby confirming our prediction. At the same time, the results of our discrete-emitter model are in quantitative agreement with the semiconductor laser model used in the reference, which indeed supports the sufficiency of our approach. For the parameter regime of few-QD nanolasers, recent results have revealed mean photon numbers of at the threshold without an offset between the intensity and coherence thresholds Kreinberg et al. (2017), which is exactly what is observed in Figs. 1 and 4 for a similar value of .

The origin of the delayed formation of coherence and the insensitivity of the coherence threshold to the factor can be understood from the contributions to the photon emission as expressed in Eq. (2b) assuming that coherence arises when stimulated emission dominates. Both stimulated emission and spontaneous emission scale with the amount of gain material . On the other hand, in the lasing regime the carrier population is close to and the inversion is then approximately given by and, therefore, decreases with the number of emitters like . Corrections to this approximation are provided by the right-hand side of Eq. (19) in the Appendix, which is again of the order . This is the reason why establishing coherence requires large photon numbers when is large, and much lower values in the few-emitter case, as demonstrated in Fig. 4. Technically, this behavior is also present in conventional lasers (such as gas and edge-emitting lasers) but is rendered irrelevant and goes unnoticed as the transition to coherent emission occurs sharply on a very narrow interval in such devices due to low factors.

More insight is obtained from the analytical expression for the photon autocorrelation function. From Eq. (8) it is clear that as becomes large, the first terms both in the numerator and the denominator become dominant and approaches unity. Of course, the precise onset of coherence depends also on all parameters, but the dependence on appears through and , which contain the sum of the loss rates . Operating at higher cavity losses, the extended active medium system becomes insensitive to the rate of radiative loss. Two important conclusions are drawn from this observations. On the one hand, the benefits of high- nanolasers in terms of at threshold and low threshold pump powers due to increasing rely on being close to the true few-emitter limit of nanolasing. On the other hand, in systems with many emitters or a large number of charge carriers in extended media, the influence of cavity losses on dominates that of radiative losses in device regimes relevant to high- room-temperature lasing with 2D-gain media. Increasing in those regimes becomes less important than aiming for improved cavity design with higher quality factors .

Iv Conclusion

The prospect of strongly reducing the excitation power required to reach the threshold by minimizing radiative losses (i.e. ) is a key motivation in the development of nanolasers. By going beyond rate equations, we demonstrate that while this is indeed possible in operational regimes found in high- nanolasers with few emitters, novel devices based on extended gain media behave differently: There, coherence is established irrespective of the value of . Having defined closed expressions for the laser intensity and coherence thresholds (the first indicating the intensity jump, the second the transition to coherent emission), we are able to identify that for the latter, the pump rate at which coherent emission is achieved is rather determined by the cavity losses and can only be improved by increasing rather than improving . Our results stress the importance of statistical properties of the light emission, given by , and we provide an analytic expression that can be used in conjunction with rate-equation theories to access to very good approximation. In the light of strong interest in novel ways to design nanolasers by means of new gain media and device geometries, our findings point to possible limitations of future nanolasers based on quantum-well and transition-metal dichalcogenide gain materials. More insight will be obtained once results on the statistical properties of such devices become available.

The authors gratefully acknowledge funding by the DFG (Deutsche Forschungsgemeinschaft) and the DFG graduate school “Quantum Mechanical Material Modelling QM”. P.G. would like to thank the University of Bremen for their hospitality and the DFG for funding the visit and acknowledges financial support from CNCS-UEFISCDI Grant No. PN-III-P4-ID-PCE-2016-0221.

Appendix A Derivation of the rate equations

The system consists of identical two-level emitters in resonance with a cavity mode. In the rotating frame the Hamiltonian reads ( throughout)


with operators for the photon mode, for the upper (conduction-band) emitter states and for the lower (valence-band) ones. We assume a single carrier per emitter, so that the language is equivalent to the pseudospin formalism by and .

The equation of motion (EOM) for the expectation value of a given operator contains, beside the coherent, von Neumann part, the dissipation expressed in the Lindblad form


where is the rate associated to the scattering process defined by the operator . Three such processes are considered: the radiative loss in each emitter with denoted by and , the cavity losses with the rate and operator and the pumping simulated as an upscattering process, , with the rate .

We apply Eq. (10) to calculate the photon number and the upper level population . Note that the latter does not depend on the emitter index . One obtains


Both are expressed in terms of the real part of the photon-assisted polarization . The EOM for brings in higher-order expectation values, such as . Such terms are treated in a mean-field approximation, which amounts to the truncation at the doublet level in the cluster expansion formalism Kira and Koch (2011); Florian et al. (2013)


The evolution of also generates terms involving polarization operators from pairs of different emitters. They are discarded on reasons discussed in the next section. As a result one obtains


where . This also shows that is, in fact, a real quantity. The steady-state solution is identified by setting the time derivatives to zero. By eliminating in the resulting equations one obtains for the populations \cref@addtoresetequationparentequation


with the rate of spontaneous emission given by


The equations above are balance conditions. Their left-hand side contains the net photon generation rate, made up of spontaneous and stimulated emission, as well as absorption. In Eq. (15a) this is balanced against the rate of the emitter excitation, taking into account both the pumping and the radiative losses, while in Eq. (15b) the photons generated by all emitters compensate the cavity losses.

Combining these equation one obtains


It is obvious that is the steady-state upper-level population in the absence of the Jaynes-Cummings interaction ().

Eq. (15b) can be rewritten as


where is the population for which the gain (stimulated emission minus absorption) exactly compensates the cavity losses. As seen from the above equations, one has i.e. both and are upper bounds for the true steady-state solution. Multiplying Eqs. (17) and (18) one obtains a quadratic equation for


whose lower solution is the physical one. In turn, the photon population is given by


This solves the problem of steady-state populations for the carriers and for the photons at the rate-equation level.

In the limit the rhs of Eq. (19) vanishes and, as the pump increases, the solution changes abruptly from to , always following the smaller. The case corresponds to pumping spent exclusively for exciting the carrier subsystem. This takes place until reaches the value with sufficient inversion to compensate the photon losses. From then on massive photon generation takes place, as seen in Eq. (20), which shows the photon density changing sharply from to . This is the lasing regime (hence the index ), in which the photon number becomes macroscopic, i.e. scales like . It becomes clear that in the limit one obtains a sharp crossover from a “normal” regime to lasing. The threshold is defined by the pump value , which obeys the Eq. (5), expressing the crossing condition .

For finite values of , the transition is no longer abrupt. The rhs of Eq. (19) being nonzero, a smooth change (anticrossing) from to takes place. The lasing transition becomes gradual, without a well-defined threshold. Now a transition interval around still separates two contrasting behaviors: Above it the photon number grows with system size , while below it their number stays finite.

Appendix B Beyond rate equations: accessing

The second-order autocorrelation function requires the cluster expansion at least up to the quadruplet level. This introduces a set of new expectation values besides the populations and . The following notations are used below:


With a single carrier in each emitter, these averages are not independent since, obviously , but we keep the notation for convenience. Also, is the average photon number , and is the same as of the previous section. The index is spurious here, since all emitters are identical.

Figure 5: Photon autocorrelation function obtained by using full quadruplet (solid) and rate-equation (circles) populations. Both the few- (left) and many-emitter (right) cases are represented.

Quadruplet averages generated by equations of motion for doublet quantities are not factorized any more. As an example, consider the case of the one-photon-assisted polarization , for a generic emitter


In contrast to the rate equation, where one approximates , and , here we write new EOM for them. Having an additional factor in its definition, the evolution of generates the two-photon-assisted polarization . The same holds for , which is the central quantity of the two-photon autocorrelator . The equation for reads


where . Quantities such as and are cluster expanded in terms of correlators as Kira and Koch (2011); Florian et al. (2013)


and similarly for . As a sixtuplet correlator, is neglected, and using , , one obtains the expression of in its truncation at the quadruplet level as


The summations appearing in both Eq. (23) and (24) describe correlations of polarizations and photon-assisted polarization, respectively, between pairs of emitters. Bearing in mind that quantities on different sites are less correlated than same-site ones, we adopt the procedure of factorizing into averages on separate sites. For example,


This expresses the sum in Eq. (23) in terms of on-site polarizations. But these are anomalous averages, not driven by the system excitation, and as such are vanishing quantities. Therefore this sum is taken as zero. This is not the case of the two sums in Eq. (24), where by the same procedure we get and , respectively. Since the equations obeyed by show that this is a real quantity, the two expressions are equal up to the prefactor, and a partial cancelation takes place, leaving the net result as . Let us emphasize that such terms, which express correlations between two emitters mediated by the photon field are usually neglected when one is not interested in superradiance effects Leymann et al. (2015). In the present approach they are included at a mean-field level.

The other EOM do not generate higher-order averages. As a result of the procedure described above, one is left with a closed system of EOM for and . In the steady state the time derivative vanishes and the system becomes algebraic. From this, one obtains an expression of and thus of in terms of and only, by eliminating the other unknowns. The result is expressed in Eq. (6) of the main text. Further simplification arises by eliminating the upper-level population in favor of the photon number and inserting the rate-equation value of the latter, as described in the derivation of Eq. (8) of the main text. The approximation is very accurate, as shown in Fig. 5, which compares it to the result of using full quadruplet-computed populations in Eq. (6).


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