## Abstract

The emission properties of a single quantum dot in a microcavity are studied on the basis of a semiconductor model. As a function of the pump rate of the system we investigate the onset of stimulated emission, the possibility to realize stimulated emission in the strong-coupling regime, as well as the excitation-dependent changes of the photon statistics and the emission spectrum. The role of possible excited charged and multi-exciton states, the different sources of dephasing for various quantum-dot transitions, and the influence of background emission into the cavity mode are analyzed in detail. In the strong coupling regime, the emission spectrum can contain a line at the cavity resonance in addition to the vacuum doublet caused by off-resonant transitions of the same quantum dot. If strong coupling persists in the regime of stimulated emission, the emission spectrum near the cavity resonance additionally grows due to broadened contributions from higher rungs of the Jaynes-Cummings ladder.

© 2011 OSA

## 1. Introduction

Studies of the light-matter interaction for quantum dots (QDs) in optical microcavities with three-dimensional photon confinement currently attract considerable interest. These systems allow to access quantum-optical effects in a solid-state environment and provide potential applications in quantum information technologies and novel light emitting devices. For individual QDs in optical microcavities, deterministic sources of single [1,2] and entangled photons [3–5] have been demonstrated. Stimulated emission has been investigated for a small number of QD emitters [6–10] and recently laser emission was observed in microcavities where the main emission contribution originates from a single QD (SQD) [11–14]. Characteristic for these systems is a transition from single-photon emission to stimulated emission, reflected in the corresponding changes of the second-order photon correlation function [13].

The realization of the strong coupling regime with individual quantum dots in microcavities [15, 16] has been another focus. The strong coupling of the lowest interband excitation of a single QD to the empty cavity mode manifests itself by two lines in the emission spectrum. Their energies correspond to the dressed states of vacuum-field Rabi oscillations and exhibit a characteristic anticrossing versus detuning. More intricate is the behavior in the strong-pumping limit. Typically the transition from strong to weak coupling with increasing pump rate appears *before* the onset of stimulated emission. Then the double-peak structure of the hybridized states merges into a single emission line due to excitation-induced dephasing as well as loss of oscillator strength for the QD transition. In this regime the behavior of the single-emitter is analogous to the polariton normal-mode coupling of spatially extended excitonic states [17]. In novel experiments [14], the aim was to maintain strong coupling of a single emitter to the cavity mode up to the onset of stimulated emission. There is no reason that lasing in the weak coupling regime would be fundamentally different from lasing in strong coupling regime. Technically achieving the latter is a much greater challenge, however, and we particularly focus on this matter in a later section of this manuscript.

When the cavity photon number increases *in the presence of strong coupling*, clusters of lines appear close to the degenerate cavity and exciton resonances. Their appearance has been reported in the theoretical work [18] for a two-level system coupled to a single mode of a microcavity and was further explored in great depth in [19]; for first experimental results with QDs see [20]. The dominating role of these lines in between the vacuum-field Rabi doublet and the disappearance of the vacuum doublet itself in the strong-pumping limit is not necessarily an indication for the transition to the weak-coupling regime. The behavior originates from higher rungs of the JC ladder and is accompanied by additional peaks further away from the cavity and exciton resonances.

Single-QD laser emission represents a considerable challenge regarding the interplay of material parameters and dissipative effects. On the one hand, a large QD light-matter coupling strength *g* (determined by the QD oscillator strength and spatial and spectral coupling of the QD to the cavity field) is required for the photon production rate to exceed the cavity losses. The maximum photon output does not only depend on the recombination rate, but also relies on fast feeding of carriers into the laser levels, a limit being set by saturation due to Pauli blocking. The fast carrier dynamics needed to drive the coupled QD-cavity system into stimulated emission also introduces dephasing that broadens the laser transition line. If the broadening becomes comparable to the light-matter coupling strength, strong coupling is no longer feasible.

When describing the situation where a single semiconductor QD dominates the emission, special attention should be paid to its multi-level structure and the possibility to excite more than one electron-hole pair. Furthermore, the excitation is typically performed off-resonantly into continuum states followed by relaxation into the localized QD states. As a consequence no output quenching is expected, as it appears e.g. when a two-level system is incoherently excited [19, 21, 22]. Instead, the photon production saturates when the valence-band states become depleted.

The aim of this paper is to present a quantitative analysis of the “single-QD laser” based on a model that accounts for the semiconductor properties of the QD emitter. We explicitly consider the QD multi-level structure and the excitation of multiple charge carriers. Starting from the single-particle states we determine the multi-exciton states due to the Coulomb interaction and their emission contributions into the cavity mode. Furthermore, we account for the connection between carrier scattering processes providing carriers in the states coupled to the optical transitions and the excitation-induced dephasing [23]. Another important aspect is the interplay of several electronic configurations to the emission process. For situations and parameters corresponding to present experiments we explore possibilities and signatures of stimulated emission in the strong coupling regime. This includes accounting for spectrally and spatially off-resonant emitters, which are assumed to make up a considerable fraction of the photon emission [13, 14]. Such background contributions counteract saturation and quenching effects that are typically associated with the emission from a single emitter.

## 2. Single-QD microcavity system

In this paper we consider self-assembled QDs with discrete electron and hole single-particle states. Their number and wave function symmetry (that determines optical interband transitions) in general depend on the particular QD properties like geometry, material composition, and strain. To model the embedded QD-microcavity system, we determine the density matrix *ρ*(*t*) in a state space defined by the confined QD states and the Fock states of the cavity mode. The electronic part is naturally limited due to the finite confinement potential for electrons and holes, which leads to a finite number of localized states to be occupied by carriers; for the cavity photons we consider a sufficiently large maximum photon number. The von-Neumann equation for the QD-photon density matrix *ρ*(*t*) has the form

*h̄*= 1). The non-perturbative light-matter interaction between the QD-interband transitions and the quantized field of the microcavity mode is described by the Jaynes-Cummings (JC) Hamiltonian

*a*

^{†},

*a*are fermionic creation and annihilation operators for electrons, and the index

*i*(

*j*) refers to valence (conduction)-band states, as the elementary processes in the JC Hamiltonian comply with the rotating wave approximation.

*b*

^{†}and

*b*are the bosonic operators for photons in the laser mode, and

*g*is the light-matter coupling strength for the respective electronic levels and the cavity mode.

_{i,j}Unlike in atomic models, we start from the single-particle states and explicitly consider the Coulomb Hamiltonian *H*
_{Coul} between the QD carriers in the von-Neumann equation. Using the full Coulomb Hamiltonian, the numerical solution is then equivalent to a configuration interaction (CI) calculation since it describes the time-evolution of the system in terms of interacting states, which are also eigenstates of the diagonalized CI Hamiltonian. For details of the considered electronic states and the Coulomb interaction, see Appendix A.

In contrast to isolated atoms, self-assembled QDs are embedded systems and are, therefore, strongly influenced by their environment. Delocalized states exist in close energetic proximity to the localized electron and hole QD states. Frequently the self-organized QD growth involves a wetting layer (WL) in which electrons and hole are delocalized in two dimensions with a quasi-continuum of states energetically above the bound states. Additionally, the surrounding barrier material provides states for three-dimensional carrier motion. For device applications it is preferable to excite carriers in the delocalized states by means of electrical injection, although many current experiments utilize optical pumping of carriers into the WL or barrier states.

Carrier scattering is an important part of the QD laser physics. As previous investigations have shown, carriers are captured rapidly into the QD states on a ps time scale [24, 25]. Scattering between the localized and delocalized states, as well as within the QD, is facilitated by the Coulomb interaction [26, 27] and by LO-phonons [28]. These processes supply carriers for the laser transition and are the dominant source of dephasing. The dissipative nature of these processes originates from the coupling of the discrete QD states to a continuum of states, which is provided by the lattice vibrations in case of carrier-phonon interaction. Efficient coupling to delocalized states is provided by Coulomb scattering. Examples for the latter are carrier relaxation within the QD, where the excess energy is transferred to either a WL carrier, or to a QD carrier that is scattered into the WL (Auger relaxation). We describe the carrier scattering between the QD levels with the Lindblad terms

*γ*are intraband scattering rates between the levels

_{ij}*j*and

*i*. In a similar fashion, carrier scattering can lead to the capture of electrons and holes from the delocalized states into the QD states and vice versa. The corresponding Lindblad terms

*i*and ${\gamma}_{i}^{\text{out}}$ from a QD state into delocalized states. Note that the scattering rates depend on the carrier population in the delocalized states. While the Lindblad terms originate from a Born-Markov treatment of the interaction processes, more advanced many-body methods can be used to calculate capture and relaxation rates [27, 28]. However, this is numerically challenging and requires the knowledge of the QD confinement wave functions to determine the interaction matrix elements.

The system is also coupled to its environment via other photon modes. Carrier recombination due to spontaneous emission into a continuum of non-lasing modes is modeled via the Lindblad term [29]

*j*and a hole in state

*i*at rate ${\gamma}_{ij}^{\text{nl}}$. Cavity-photon losses at the rate

*κ*are given by [29]

In this paper we assume that the pump process generates carriers in the delocalized states, where, in case of optical pumping, carriers are generated pairwise. The successive carrier scattering is expected to efficiently destroy initial electron-hole correlations, so that the subsequent carrier capture process into the QD occurs independently for electrons and holes. In case of injection pumping, initial electron-hole correlations are absent in the first place. The above-discussed Lindblad term (4) describes precisely this independent capture for electrons and holes. In contrast, a different carrier generation process is frequently used in the literature in a form derived from the reverse process of Eq. (5). This case describes pairwise and, therefore, fully correlated electron-hole capture, which represents the opposite limiting case. Note that the independent capture leads to the possibility of charged (multi-)exciton states, since the number of confined electrons and holes is not necessarily the same.

In a previous publication [23] we have studied a situation where in the regime of strong pumping the accumulation of carriers in the WL leads to excitation-induced effects. With increasing carrier density in the WL, the dephasing rate grows due to a larger number of possible scattering channels, and at the same time the Coulomb interaction is reduced due to screening. As a result, excitation-density dependent shifts and broadenings of the QD lines are observed, e.g. in Ref. [30]. In other experiments with short-pulse laser excitation [31, 32], on the other hand, no line shifts were detected. In this case, the WL carrier density created by the pump rapidly disappears either due to capture into localized states or via other processes, like carrier diffusion out of the excitation spot. Line shifts are also absent in the spectra measured in recent experiments on SQD lasing [14]. To describe this situation we assume in this paper that the Coulomb interaction strength and the scattering rates are determined only by the system properties itself, and remain fixed independent of the excitation density. We emphasize that experimental evidence exists for both situations.

## 3. Configuration space in a single- and two-spin description

The selection of the QD single-particle states determines the possible system configurations. For a QD with two confined shells for electrons and hole, which we consider throughout this paper, the lowest bright configuration (when successively filling the dot) is the exciton that contains an electron-hole pair of either spin direction. The highest excited state corresponds to the filled dot, which consists of four electron-hole pairs. In total, 256 configurations are possible, amongst them the ground state biexciton and trions, hot biexcitons and trions, and configurations carrying more charges. The basis of the interacting carrier-photon system additionally contains the Fock states of two orthogonal photon polarizations. The numerical time integration of the von-Neumann equation in such large basis poses a challenge.

Motivated by the intention to keep the number of possible configurations small enough in order to facilitate a transparent discussion of the interplay of configurations, as well as to reduce the numerical effort, the limitation to a single spin subsystem can be useful. The possible system configurations are different in this picture: Each configuration is independent of the presence of carriers with the opposite spin. As a consequence, the exciton is still the first bright excitation in the system, but effectively summarizes other configurations, like the ground state trions and biexciton, as they contain additional carriers with opposite spin. Higher excited configurations always contain *p*-shell carriers and may be called ‘hot’ configurations. The highest multi-exciton state is the *sp*-biexciton, which can be expected to behave similarly to the filled-dot configuration in the two-spin picture.

Under suitable circumstances the overall behavior of the SQD laser may be well captured without explicitly accounting for carriers of both spin directions. It is for example the case when the energetic shifts introduced by additional carriers of the second spin direction (responsible for the ‘ground-state’ biexciton binding energy) are small in comparison to the Coulomb exchange energy (responsible for the energetic separation between ‘hot’ configurations with additional *p*-shell carriers), which is often the case. The line broadening due to dephasing can mask the small splittings especially at elevated excitation conditions. Furthermore, at high excitation, the system dynamics is mainly determined by the highest multi-exciton configuration, in which case the influence of the other configurations plays a smaller role. Still, the larger number of possible configurations in a two-spin description can modify the system dynamics. Also a detailed study of the emission characteristics of e.g. the charged exciton line [33] would require a calculations beyond the one-spin restriction.

In Fig. 1 we show the input/output curve of a single-QD laser, where results for the full calculation with both spin subsystems and 256 electronic configurations are compared to a single-spin calculation and 16 electronic configurations. Two resonance conditions are considered: Either the *s*-exciton (top panel) or the highest multi-exciton state (bottom panel) is tuned to the cavity mode. The first is the preferred configuration in the weak excitation limit, while the latter dominates the emission at high excitation. We conclude that, for the considered situation, the single-spin model is able to capture the relevant physics with quantitative agreement. Deviations between both models result from the higher number of possible configurations in the two-spin model, as well as from different energetic positions of the contributing configurations due to the Coulomb interaction. For example, the smaller mean photon number in the top panel of Fig. 1 is the result of a larger detuning of the *sp*-biexciton for the used parameters, while in the two-spin case the additional Coulomb interaction reduces the energetic separation between the cavity mode and the configuration corresponding to the filled QD. A detailed discussion of the underlying physics is given in the remainder of this manuscript, whereas more information about the two-spin calculation will be found in Ref. [34].

Finally, we would like to point out that each spin sub-system couples only to one circular light polarization. Assuming optical excitation with circularly polarized light, only carriers in one spin-subsystem are excited and the considered (spin conserving) carrier relaxation and optical recombination processes are typically faster than spin-flip processes. In this case, the single-spin model provides an apt description.

## 4. Laser emission from a single QD

The aim of many recent experiments is to obtain stimulated emission driven by a single QD emitter, although to date contributions from background emission cannot be ruled out completely. In this section we analyze the ideal situation in which only the single QD emitter contributes. The light-matter coupling strength in current experiments [14] is known from the vacuum-field Rabi splitting to be of the order of 100*μ*eV, corresponding to a Jaynes-Cummings coupling between the electron and hole ground states (*s*-states) of *g* = 0.15/ps. The typical cavity decay rate of *κ* = 0.1/ps corresponds to a cavity-*Q* = 20,000 in the red spectral range of the InGaAs QD emission. Solely driven by a single QD emitter, we are unable to reach the regime of stimulated emission with these parameters. Determining the requirements for stimulated emission in the multilevel-QD system is not straightforward, and a discussion of emission rates in terms of QD parameters is given in Appendix B.

In all of the following calculations we consider a QD with two levels in each band indexed by *s* and *p*, as well as one spin sub-system as discussed in Section 3. The system is pumped by injecting electrons from the continuum into the conduction-band *p*-state and ejecting them out of the valence-band *p*-state (hole capture). These two processes are modeled by the Lindblad terms of Eq. (4) with the capture rates
${\gamma}_{{c}_{p}}^{in}$ for the former and
${\gamma}_{{v}_{p}}^{out}$ for the latter. Carrier relaxation inside the QD is described by Eq. (3), in which only the *p*-to-*s* scattering is considered in both bands, with the corresponding rates
${\gamma}_{r}^{e,h}$. This is equivalent to contact with a low-temperature thermal bath.

For the analysis of a genuine single-QD laser, in this section we use *g* = 0.3/ps, which we expect to be achievable now or in the near future. Independent calculations along the lines of [27, 28] for continuous wave WL excitation suggest electron and hole relaxation rates of the order of 1/ps, depending on excitation conditions and QD parameters. As discussed in Appendix B, the relaxation rates influence the population dynamics and contribute to dephasing in a way that the former increases and the latter decreases the emission rate. (Note that each optical transition between configurations is dephased by specific capture, relaxation or recombination processes, as we discuss in the context of the emission spectra further below.) For the used *g* and *κ* values, we find that *p*-to-*s* intraband relaxation rates
${\gamma}_{r}^{e,h}$ around 0.5/ps for electrons and holes maximize the emission rate and use this value in the following. Radiative losses into non-lasing modes are typically strongly suppressed in microcavity devices, and we consider *γ*
^{nl} = 0.01/ps. This set of parameters we label in the following as Set A. Using a constant carrier capture rate, we evolve Eqs. (1)–(4) for the density matrix elements of the initially empty system in time until steady state is reached.

**Mean photon number.** In the left column of Fig. 2 we show input/output curves for a SQD laser with the cavity being in resonance with the 1*X _{s}* exciton, the
$1{X}_{s}^{\pm}$ charged exciton, and the 2

*X*biexciton transition, respectively. The contributions to the mean photon number from the 1

_{sp}*X*, $1{X}_{s}^{\pm}$ and 2

_{s}*X*configurations are shown separately as dashed, dotted and dash-dotted lines, respectively. At low excitation powers they exhibit different slopes, the one of the biexciton being twice that of the exciton, and the one of the charged exciton states in between. For the considered confinement situation these are the only bright configurations allowing for a recombination at the

_{sp}*s*shell.

At low excitation the exciton is the most likely bright configuration, and the highest photon emission is obtained if the cavity is in resonance with it. The capture of an additional carrier in the *p* shell becomes more favorable for capture rates above 10^{−3}/ps and then leads to a preference of the two charged exciton configurations. As a consequence, the mean photon number is higher in this regime if the cavity is tuned to the charged exciton transitions. At high excitation carriers get captured into the higher QD *p*-states much faster than the optical recombination process, so that the *sp* biexciton configuration is always dominant. Thus, tuning the biexciton transition into resonance with the cavity maximizes the photon output. Only in this case stimulated emission with 〈*n*〉 > 1 can be reached for the chosen parameter set. At the same time, the realization of the other bright configurations becomes strongly suppressed, and this is one reason why quenching is observed in the cases of the cavity at the neutral or charged exciton resonances— dephasing introduced by the capture processes is the other.

In comparison to conventional lasers, no kink is observed in the input/output curve, because with a single emitter it is hardly possible to obtain stimulated emission at all in typical parameter regimes. To do so, losses into non-lasing modes, which are responsible for the reduced emission below the threshold and the resulting kink, must be kept at a minimum. Furthermore, the single QD saturates quickly at the necessary high excitation, which masks any slight kink otherwise visible. In contrast, for a QD ensemble, stimulated emission is typically reached closer towards transparency and saturation is shifted away from the threshold. Then a kink is visible, although it is also very small in QD-based microcavity lasers due to *β*-factors close to unity, see e.g. the review article [35] and references therein. In this cases, the threshold is no longer sharp and not well suited to define the onset of stimulated emission.

**Photon statistics.** The emission from a SQD at weak pump rates exhibits a characteristic photon antibunching [1]. Placed inside a microresonator that provides a long-lived mode, the emission properties can change severely. Photons stored in the cavity can be reabsorbed or enhance the emission. The antibunching feature is then no longer a property solely of the emitter, but of the coupled emitter-cavity system. Its persistence at low excitation depends strongly on the cavity lifetime as well as on scattering and dephasing rates. For example, if the cavity-*Q* is too high, the photon antibunching is suppressed [23]. Looking at the low-excitation regime of the autocorrelation function shown in Fig. 3, different values are obtained depending which QD configuration is in resonance with the cavity mode. Note that a value of 1 in the second-order correlation function at low pump rates is not necessarily a signature of coherent emission. While the autocorrelation function is of great significance in quantum-optical experiments, one should keep in mind that it is related only to the first and second moment of the photon statistics and not the photon statistics *p _{n}* itself. If one calculates

*g*

^{(2)}(0) = (〈

*n*

^{2}〉 – 〈

*n*〉)/〈

*n*〉

^{2}in terms of

*p*using 〈

_{n}*n*〉 = Σ

^{j}

_{n}*n*

^{j}*p*and considers only the leading terms for a photon statistics (thereby assuming that

_{n}*p*rapidly decays in

_{n}*n*), one arrives at ${g}^{(2)}\hspace{0.17em}(0)\hspace{0.17em}\approx \hspace{0.17em}2{p}_{2}/{p}_{1}^{2}$. Since in the discussed case of weak pump rates ${p}_{1}^{2}$ and

*p*

_{2}are very small numbers, the contributions of rather unlikely events are amplified in

*g*

^{(2)}(0) due the probability of having zero photons being practically 1.

By obtaining the density matrix of the system, we have direct access to the photon statistics and can check for the signature of coherent light with a Poissonian probability distribution, or thermal light with an exponential dependence on the photon number *n*. As an example we consider the case where the biexciton is tuned to the cavity mode. For weak pumping (left in Fig. 4) the photon statistics *p _{n}* deviates from a single-photon Fock state albeit the probability

*p*for n>1 is below that of either ideal thermal or coherent light with the same mean photon number.

_{n}At capture rates between 0.1 and 10/ps, *g*
^{(2)} exhibits bunching when the cavity is resonant with either the exciton or the charged exciton transition, while it does not if the biexciton is tuned to resonance. Looking at Fig. 2, we infer that bunching appears when more than one configuration contribute with comparable amounts to the total photon production. In other words, the bunching is a reflection of competing emission channels which are realized in the density matrix with comparable likelihoods.

For strong excitation, the second-order photon correlation function in Fig. 3 and the photon statistics in Fig. 4 indicate coherent emission if the cavity resonance is tuned to the biexciton emission. Only in this case the mean photon number exceeds unity thus enabling stimulated emission. When the cavity is in resonance with the neutral or charged exciton, quenching of the resonant contributions, in combination with the inefficient coupling of the detuned biexcitonic emission, prevent stimulated emission and lead to a nearly thermal photon emission for strong pumping, see Fig. 3.

**Spectra and linewidths.** The contribution of separate configurations is also reflected in the cavity emission spectrum *S*, which is given by the Fourier transform *ℱ* of the first-order two-time photon correlation function *g*
^{(1)} (*t,τ*) = 〈*b*
^{†}(*t*)*b*(*t* +*τ*)〉 with respect to the delay time *τ*

*τ*with modified initial conditions obtained from the steady state solution of the

*t*-time evolution

*ρ*(

*t*) [36]. In the right column of Fig. 2 spectra are shown for several capture rates and cavity tuning situations. At low excitation (black lines), peaks at three energetic positions are visible: The

_{ss}*s*-exciton resonance, the charged exciton contributions split off by the Coulomb exchange energy, and the

*sp*-biexciton separated by twice the exchange energy from the

*s*exciton. The increased photonic density of states at the cavity mode leads to a significant enhancement of the resonant emission. The broadening of the spectral lines reflects the possible scattering processes due to carrier capture, relaxation and recombination, which differ for various excitation configurations. Deduced from the contribution of the Lindblad terms to the equation of motion for the relevant polarization, a simple rule can be derived: A given optical transition is dephased by those processes changing either its initial or its final configuration. For example the 1

*X*excitonic transition is dephased by the carrier capture processes, which act upon the initial and final configurations, but is not influenced by the relaxation as long as one neglects

_{s}*s*-to-

*p*scattering processes, which are unlikely at low temperatures. For the biexciton transition it is the other way round, only the

*p*-to-

*s*relaxation, but not the capture rates contribute. For the charged exciton transition both do. Hence, the neutral and the charged exciton contributions to the photon output are subject to quenching by means of excitation-induced dephasing (see Fig. 2), while the biexcitonic contribution is not.

We first discuss the underlying physics of the spectral peak of the 1*X _{s}* → 0

*X*transition. At low capture rates the finite cavity lifetime is the main source of dephasing ( ${\gamma}_{\text{in}}^{e,h}\hspace{0.17em}\ll \hspace{0.17em}\kappa \hspace{0.17em}=\hspace{0.17em}0.1/\text{ps}$). As a consequence, if the inverse cavity lifetime is smaller than the light-matter coupling strength, and the

*s*-exciton transition is in close spectral vicinity to the cavity mode, both hybridize and exhibit the double-peak structure of the vacuum Rabi splitting— the well-known signature for strong coupling that is seen in the upper right panel of Fig. 2. With increasing capture of carriers into the QD

*p*-states, the capture process itself starts to contribute to the dephasing, resulting in broadening and finally the transition into weak coupling.

The weak-excitation spectra of the other two transitions differ significantly. Since the fast carrier relaxation causes strong line broadening exceeding the light-matter coupling strength ( ${\gamma}_{\text{r}}^{e,h}\hspace{0.17em}>\hspace{0.17em}g$) independently of the carrier capture rates, no signatures of strong coupling are found in the lower two panels of Fig. 2.

Another interesting feature is observed in the spectra of the *s* exciton being on resonance: Already at low pump rates an inner peak emerges within the Rabi doublet of the *s* exciton. It originates from the detuned resonances of the charged exciton and biexciton. These are too far detuned to hybridize with the cavity mode. As a consequence, each possesses *two* spectral contributions, one at the *s*-shell transition energy that is renormalized by the Coulomb interaction, and one at the bare cavity energy. In the spectra the bare cavity peak of the detuned configurations can be observed to emerge and gain oscillator strength as the charged exciton states and the biexciton state gain probability (see the upper right panel in Fig. 2). This feature has also been discussed in [37, 38] and must be taken into consideration in the interpretation of experiments.

## 5. Single QD with background contributions

In the following we discuss the influence of cavity-photon generation due to residual emitters. In microresonators like photonic crystals, VCSEL micropillars, or microdisks the selected high-*Q* mode is typically extended over a micrometer range in the emitter plane. Often additional emitters are present in this area that are also subjected to the pump process. While these emitters typically exhibit off-resonant emission with respect to the cavity mode, it is now widely accepted that even transitions detuned from the cavity mode by tens of meV can still contribute to cavity feeding.

In the literature a frequently used method to model contributions from additional photon-emitting sources is a “cavity-feeding” Lindblad term inverse to that describing cavity losses [14, 19, 39–41]

*κ*the two corresponding Lindblad terms simulate the contact of the photon subsystem with a reservoir of a temperature determined by the ratio Γ/

*κ*=

*e*

^{−βh̄ω}. This treatment of the residual emission can only produce thermal, incoherent photons. Moreover, the situation with Γ >

*κ*leads to a pathological, exponentially increasing “negative-temperature” photon statistics.

To solve these problems one has to consider that the “background” photon emission does not come from a thermal reservoir but from an active medium, able to produce coherent light and showing saturation effects due to Pauli blocking and photon reabsorption. As a simple way to include these features we resort to the random injection model of Scully and Lamb [42–44] by adding a corresponding feeding term to the von-Neumann evolution. This amounts to replacing Eq. (8) by

*S*. We use a value of $\Gamma \hspace{0.17em}=\hspace{0.17em}\alpha {\gamma}_{\text{in}}^{e,h}$ proportional to the capture rates and keep the saturation coefficient constant. Note that Eq. (8) is the

*S*= 0 limit of Eq. (9).

In the presence of background emission, stimulated emission has experimentally been observed at the exciton resonance [12–14], and this situation we focus on in the following. In Fig. 5 we show the mean photon number for the same parameters as used in Fig. 2, now with additional photon injection into the cavity. Instead of saturating, lasing with *g*
^{(2)} = 1 indicating coherent light emission is reached, see Fig. 6. The photon production does not saturate but keeps increasing with increasing capture rates. Regarding the carrier system, the capture rates appear together with population factors that ensure that the Pauli principle is obeyed and each state is filled with one fermion only.

At the highest capture rate one can see that the photon production is mainly due to residual emitters. Thus, the value for coherent light emission in the autocorrelation function can be attributed to the cavity feeding only. For comparison, the gray curve added to Figs. 5 and 6 has been calculated with the frequently used inverse loss term Eq. (8) with the same rate instead. Here the photon statistics reveals that a purely thermal component is added to the photon field and *g*
^{(2)} takes on a value of 2 instead. Note that a stable solution cannot be obtained over the whole excitation range, as Γ < *κ* must be fulfilled. Clearly, both methods to account for background emission channels are not arbitrarily interchangeable, as they describe different physical situations in the way we have discussed above. Their impact on the photon statistics does not depend on parameters, but lies in the origin of the terms. If background emission accounts for a significant part of the photon production, no coherent light emission can be expected from the inverse cavity loss term.

While it is possible to achieve lasing at the exciton resonance with the help of cavity feeding for the chosen parameters, the strong coupling signatures in the emission spectra, shown in the right panel in Fig. 5, are not maintained and disappear already at a mean photon number of about 0.05.

On the one hand, experimental evidence clearly points to the significance of residual emitters in current investigations on a SQD emitter in a microcavity. On the other hand, the modeling of photon injection via a reservoir coupling adds arbitrariness to the theoretical description. Accounting for additional emitters by a cavity feeding term omits their electronic degrees of freedom and leads to unsaturated photon output. Also the mutual effects of the emitters on each other are neglected. If estimates of number and properties of additional emitters were known, the development of a theory that accounts for their electronic degrees of freedom would be worthwhile. Note that the inclusion of additional emitters in the density matrix of the system complicates the numerical effort considerably.

## 6. Lasing in the strong coupling regime

For Parameter Set A used in the previous sections, the vacuum Rabi splitting disappears with increasing pump at a mean photon number of around 0.1 and before the photon statistics becomes Poissonian. Even in the presence of background emitters, strong coupling and lasing are not obtained at the same time. As we have discussed in the context of the cavity emission spectra in Section 4, each transition between configurations is dephased according to how different process act upon its initial and final state. The relation between the strength of this dephasing and the light-matter coupling determines whether strong coupling is observed for this transition. By altering the ratio, in the following we discuss two scenarios where strong coupling persists in the presence of stimulated emission.

**Strong coupling and lasing at the exciton transition.** Since the carrier capture rate determines the broadening of the 1*X _{s}* exciton transition, which eventually destroys the strong coupling, as well as all the necessary supply of carriers for the laser transition, an optimal balance between the two counteracting effects is required. With an interband relaxation rate of 2/ps and a light-matter coupling strength

*g*= 1.8/ps, strong coupling in the presence of stimulated emission can be obtained. We refer to this as Parameter Set B.

In the input/output curve shown in Fig. 7, a mean photon number of unity is reached at carrier capture rates of about 0.14/ps. The low-excitation spectrum exhibits signatures of strong coupling with additional peaks visible at ±2.9 and ±0.5meV, corresponding to the
$g(\sqrt{2}\pm 1)$ transitions between first and second rung of the Jaynes-Cummings ladder. The pronounced peak in the middle is the bare-cavity contribution from the charged and biexciton lines, discussed in the end of Section 4. The spectrum in the middle corresponds to the point where the mean photon number reaches unity. Signatures of strong coupling persist while lasing sets in, as one can infer from the Poissonian photon statistics shown for the same capture rate in the right panel of Fig. 7. The corresponding value of the autocorrelation function is *g*
^{(2)}(0) = 0.9 (not shown). Due to the increased probability of having several photons in the regime of stimulated emission, peaks from higher transitions between Jaynes-Cummings rungs appear, e.g. at 3.7meV, corresponding to the
$g(\sqrt{3}\hspace{0.17em}+\hspace{0.17em}\sqrt{2})$ transition. The inner lines at
$g(\sqrt{n+1}\hspace{0.17em}-\hspace{0.17em}\sqrt{n})$ cannot be resolved and appear as one broadened peak. At high excitation, the far-detuned biexciton transition takes over the emission completely and strong coupling is lost, see the uppermost spectrum.

**Strong coupling and lasing at the biexciton transition.** We now take the cavity to be in resonance with the *sp*-biexciton, which, of all configurations, maximizes the photon production rate at high excitation. In Section 4 we have identified the intraband carrier relaxation processes to act as the dominant dephasing channel for this transition. In a QD in which these scattering rates are strongly reduced, signatures of strong coupling are expected to be seen. To demonstrate this, we consider a third Parameter Set C, where the relaxation rates are reduced to
${\gamma}_{\text{r}}^{e,h}\hspace{0.17em}=\hspace{0.17em}0.02/\text{ps}$. While reducing the dephasing, also the efficiency of carrier scattering into the QD *s*-states after capture into the *p*-states is significantly lowered. In order to still obtain stimulated emission, an improved cavity *Q* of 100, 000 (corresponding to *κ* = 0.02/ps) is required, together with additional feeding of photons from background emitters parameterized with *α* = 0.001 and *S* = 3, cf. Section 5. The used light-matter coupling strength is *g* = 0.3/ps like in Parameter Set A.

The input/output curve in the left panel of Fig. 8 exhibits a similar shape as the case studied in bottom left panel of Fig. 2 up to a capture rate of 1/ps. Then the additional injection of photons into the cavity counteracts saturation and the mean photon number exceeds unity. Due to the decreased relaxation rates, the vacuum Rabi splitting is apparent at low excitation rates in the spectrum (middle panel), and more peaks appear on both sides as excitation increases. Strong coupling persist even at high excitation, where the photon statistics is Poissonian (right panel), reflecting the strong coupling in the presence of lasing at the biexciton transition.

## 7. Conclusion

We have analyzed the properties of an isolated single QD coupled to a high-*Q* cavity mode, the choice of parameters being close to what is possible to realize experimentally to date. The presence of carriers in excited states modifies the emission energy at the *s*-shell due to the Coulomb interaction, and we have discussed cases of particular interest. Choosing the highest multi-exciton configuration (in our particular QD the *sp*-biexciton) to be in resonance with the cavity mode maximizes the photon output from the device. The main reason for this lies in the fact that it is the dominating configuration at high excitation densities. Capture and relaxation processes are typically faster than the radiative interband recombination, so that the dot fills up before emission takes place. At the same time, higher multi-exciton configurations are subject to strong dephasing due to fast carrier relaxation, which prohibits strong coupling to be observed at these resonances. Nevertheless, stimulated and coherent laser emission can be achieved in the weak coupling regime, characterized by Poissonian photon statistics, *g*
^{(2)} ≈ 1, and a mean photon number exceeding unity.

The *s*-exciton transition, on the other hand, is not dephased by the relaxation but the capture processes. Thus, if resonant with the cavity mode, strong coupling becomes possible in the emission spectrum at low excitation densities for typical values of the light-matter coupling strength, but disappears as capture rates increase with pumping. The attainable photon output is significantly reduced due to quenching of the transition. At high excitation, only the off-resonant highest multi-exciton state contributes. Then the emission characteristics depends strongly on the detuning. Excitation-induced screening and dephasing can enhance the overlap in the presence of a high WL carrier density [23].

The observation of strong coupling in the presence of stimulated emission requires a sufficient splitting-to-linewidth ratio in order to resolve the peaks of the dressed-state resonances. To overcome the dephasing at the exciton transition, which is introduced by the carrier capture processes, an increased light-matter coupling strength is required to counteract both quenching of the photon output and broadening of the spectral line. Strong coupling and lasing at the highest multi-exciton transition, on the other hand, requires the dephasing due to intraband carrier relaxation to be smaller than the light-matter coupling strength. We have discussed strong coupling and lasing for both resonance conditions to demonstrate the interplay of scattering, dephasing and recombination that determines whether strong coupling can be maintained or not. Nevertheless, a clean-cut distinction of both regimes based on the spectrum alone is difficult to make: We have identified three possible contributions to the inner peak, which could serve as an indicator for the transition into the weak coupling regime: a) the bare cavity resonance of other, off-resonant transitions, b) the merged inner peaks broadened by dephasing of the higher Jaynes-Cummings transitions, and c) the actual peak of the resonant transition when strong coupling is lost.

Finally, we have addressed the reservoir-treatment of background emitters that contribute towards the production of cavity photons. We have demonstrated that the commonly used inverse cavity loss term can only produce thermal photons and is prone to delivering artifacts for ill-chosen parameters. Instead, we suggest an alternative method based on the random injection model that overcomes these weaknesses. Still, some freedom remains in these phenomenological models, and a more appropriate description is desirable.

## Appendix A: Electronic states and solution of the von-Neumann equation

For the calculations presented in this paper, we consider two confined states for electrons and holes which are only spin degenerate. The lowest confined QD conduction-band state and the highest confined valence-band state provide the laser transition. The energetic separation of the *p* and *s* shell from the WL continuum are, for the electrons, 40 and 80meV, respectively, and 15 and 30meV for the holes.

The Coulomb interaction between the QD carriers is considered explicitly in *H*
_{Coul} in Eq. (1) within our subspace of QD states. For confinement potentials of typical self-assembled QDs it has been discussed in [45] that the dominant contribution consists of direct and exchange terms. As a result the Hamiltonian can be expressed in occupation number operators

*λ*′ to the two middle states in the matrix elements

*W*of the screened Coulomb potential [46]. Given the limited number of one-particle states and their symmetry, the only non-diagonal Coulomb term appearing in the problem connects the

*s*-exciton and the

*p*-exciton, and has the value of 2–3meV. This is one order of magnitude smaller than the energetic separation between the states involved (tens of meV’s – see above) so that the mixing of these configurations can be safely neglected. Throughout the paper (except in Section 3) we also restrict ourselves to the simplified case of equal envelopes for the electron and hole wave functions [47], in which the Coulomb matrix elements do not depend on the band indices. In this situation the second line of the above equation can be rewritten as ${D}_{sp}\hspace{0.17em}({n}_{s}^{e}\hspace{0.17em}-\hspace{0.17em}{n}_{s}^{h})({n}_{p}^{e}\hspace{0.17em}-\hspace{0.17em}{n}_{p}^{h})$. This shows that an exciton, having ${n}_{i}^{e}\hspace{0.17em}=\hspace{0.17em}{n}_{i}^{h}$, does not contribute to the direct interaction, being neutral not only globally but also locally. The direct integrals appear in the excitonic binding energies (first line of the equation), but not in the exciton-exciton interaction. It is the exchange interaction (third line) which is responsible for the latter.

## Appendix B: Quantum-dot emission rates

An estimate for the recombination rate of the QD laser transition can be obtained by calculating the corresponding off-diagonal density-matrix elements due to the Jaynes-Cummings interaction in the adiabatic limit (we are considering properties of the stationary state) and inserting them into the equation for the diagonal matrix elements. This leads to an inverse spontaneous recombination time of

*γ*. The value of

*γ*is determined for each transition (1

*X*→ 0

_{s}*X*, $1{X}_{s}^{\pm}\hspace{0.17em}\to \hspace{0.17em}0{X}^{\pm}$, 2

*X*→ 1

_{sp}*X*) according to the discussion of the cavity emission spectra in Section 4.

_{p}Considering for the sake of argumentation only the exciton transition, assumed to be resonant with the cavity mode, the spontaneous recombination rate *R _{X}* follows by multiplying 1/

*τ*with the occupation probability

_{sp}*f*of the considered exciton configuration:

_{X}To obtain a recombination rate that equals the cavity decay rate for 〈*n*〉 = 1, electrons and holes need to be generated in the QD ground state with a rate of (at least) 0.1/ps (the assumed value for *κ*). This adds up to a dephasing rate = 0.2/ps (neglecting the small
${\gamma}_{ij}^{\text{nl}}$), and with *g* = 0.15/ps one finds 1/*τ _{sp}* = 0.3/ps. Only if the occupation probability of the corresponding electronic configuration exceeds the value of 1/3, photon emission balances the cavity losses. At this point the population dynamics comes into play. For a carrier generation rate slower than that of the spontaneous emission, the carriers in the QD-ground state recombine faster than they get refilled by the stationary pump process. Hence the QD-exciton population remains small. If the pump rate becomes faster than the spontaneous recombination rate, the QD-exciton population starts to increase, but so does

*γ*. This pump-induced dephasing, in turn, reduces the emission rate according to Eq. (11). If carriers start to populate also higher QD states, the probability of the excitonic configuration is reduced for the benefit of excited configurations. Note that adding confined carriers to a QD exciton leads to the formation of a higher multi-exciton state. Consequently the probability of the exciton configuration

*f*, and with it the exciton recombination rate

_{X}decreases*R*.

_{X}In view of obtaining a high photon output, one should tune the cavity to the highest excited QD configuration whose occupancy becomes the dominant one for strong pumping. On the other hand, in order to avoid quenching, this configuration should not be dephased by the pump itself. In our calculations these requirements are met by tuning the cavity to the biexciton transition. In a two-level model, irrespective of the pumping mechanism, quenching always takes place.

We would like to point out, that in our numerical calculations the approximation (11) is not used and all diagonal and off-diagonal density matrix elements are evaluated according to Eqs. (1)–(4).

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