Abstract
We report on nonconventional lasing in a photoniccrystal nanocavity that operates with only four solidstate quantumdot emitters. In a comparison between microscopic theory and experiment, we demonstrate that irrespective of emitter detuning, lasing with \({g}^{\mathrm{(2)}}=1\) is facilitated by means of emission from denselying multiexciton states. In the spontaneousemission regime we find signatures for radiative coupling between the quantum dots. The realization of different multiexciton states at different excitation powers and the presence of electronic interemitter correlations are reflected in a pumprate dependence of the βfactor.
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Introduction
Selfassembled semiconductor quantum dots (QDs) exhibit a discrete density of electronic states due to quantization effects owed to their nanometersized dimension^{1}. In analogy to real atoms, QDs are often termed artificial atoms and are wellestablished in numerous research fields, such as solidstate cavity quantum electrodynamics (QED), photonic quantum technologies, and semiconductor lasers. The incorporation of QDs into high quality factor (Q) and low mode volume (V) optical microcavities^{2}, like photonic crystals (PhCs)^{3}, enables the exploration of the miniaturization limit of lasing, where the optical gain medium consists of a few, or even a single quantum emitter coupled to a single cavity mode^{4,5,6,7}. In this regime, the unique electronic structure of each solidstate emitter matters and differences to natural fewatom lasers become apparent. Highly excited individual QDs possess a rich emission spectrum consisting of a multitude of closelying (meV) discrete neutral and charged multiexciton states^{1} that have been demonstrated to couple to the cavity mode even if they are not in perfect resonance^{4, 8, 9}. Recently, another effect has been explored in atomic^{10} and solidstate lasers with ensembles of QDs^{11}, in which few discrete emitters can exchange photons via a highQ cavity mode, thereby establishing electronic interemitter correlations that are connected to superradiance. Analyzing the properties of fewQD nanocavity systems beyond the limitations of conventional laser models for independently acting atomic emitters can provide new insight into the fascinating working principles of cavityQED driven nanolasers.
Here, we study a PhC nanocavity laser with an energytunable mode, pumped predominantly via the discrete multiexciton states of only four QD emitters by combining confocal and photon correlation spectroscopy with a quantumoptical theory. The microscopic approach accounts for the semiconductor properties of the solidstate emitters by allowing multiexcitonic states of each emitter to couple to the cavity mode, and to form electronic correlations between different emitters. We identify nonresonant coupling as the underlying mechanism for lasing over a wide range of spectral detunings up to \({\rm{\Delta }} \sim 17\,\) meV between emitter and cavity mode. Evidence for lasing is based on measurements and calculations of the autocorrelation function \({g}^{\mathrm{(2)}}(\tau =\mathrm{0)}=(\langle {n}^{2}\rangle \langle n\rangle )/{\langle n\rangle }^{2}\), with n being the number operator for photons in the laser mode. Furthermore, we find signatures of electronic correlations that are established between the emitters via the common light field of the PhC cavity mode. These correlations are not related to stimulated emission, which similarly establishes phase correlations in the presence of a sizable photon population in the mode^{10}. Our theory accounts for these effects and predicts a strongly reduced spontaneous emission rate and superthermal photon bunching with \({g}^{\mathrm{(2)}}(\tau =\mathrm{0)} > 2\). Due to pumprate dependent realizations of different multiexciton states that contribute to the emission and the presence of interemitter correlations, the system cannot be described by a singlevalued βfactor, which is widely used as key characteristic of nanolaser devices^{4, 6, 12}. Instead, β itself becomes pumprate dependent. With the combination of experiment and microscopic theory, we are able to provide insight into fewemitter solidstate lasing that will aid the design of future highly efficient, low threshold nanolasers.
The investigated samples are freestanding GaAs membranes, loaded with a single layer of InGaAs QDs at its centre (areal density ~20 μm^{−2}) and embedded into twodimensional PhCs with L3 line defect nanocavities^{13} (Further details on sample growth and fabrication are given in the Supplementary Information). The inset of Fig. 1(a) shows a representative scanning electron microscope image of this structure. A typical microphotoluminescence (μPL) spectrum of a QDcavity system subjected to resonant excitation via a higherorder cavity mode^{14} is shown in the upper panel of Fig. 1(a). The spectrum shows the fundamental (CM) cavitymode emission at \({E}_{{\rm{CM}}}=1257.1\) meV with \(Q=\mathrm{12,000}\) and several emission lines stemming from few (N = 4) QDs located in the PhC nanocavity, the strongest one at \({E}_{1X}=1263.1\,\) meV is labelled with 1X. Cavityresonant excitation via a higher order mode guarantees that predominantly QDs inside the defect region are excited quasiresonantly^{14}.
In order to test the system for lasing, we first recorded μPL spectra as a function of excitation power density P, as shown on a semilogarithmic scale in the lower panel of Fig. 1(a). The corresponding inputouput curves of CM (squares) and 1X (circles) are shown on a doublelogarithmic scale in Fig. 1(b). The dominant emission line, labelled 1X, stems from a single QD, confirmed by secondorder photon correlation measurements \({g}_{1X}^{\mathrm{(2)}}(\tau =\mathrm{0)}\), where we obtain a reduced multiphoton emission probability of \({g}_{1X}^{\mathrm{(2)}}\mathrm{(0)}=0.25\pm 0.16\) ^{6, 15} (shown in the Supplementary Material). The weaker emission lines visible at lowest pump power are most likely either stemming from charged excitons of the same QD or from single exciton transitions of other QDs located inside the cavity region. The inputoutput curves are fitted by a power law \(I=A\cdot {P}^{m}\), where I and P denote the intensity and pump power density, respectively. We observe an exponent \({m}^{{\rm{1X}}}=0.91\pm 0.03\) for the 1Xemission, indicating excitonic character^{16}, before it saturates at \({P}_{{\rm{sat}}}^{{\rm{1X}}}=0.14\pm 0.1\) kW/cm^{2}. The CMemission is mainly determined via nonresonant cavity feeding^{17, 18} from the 1Xemission despite the large detuning of \({{\rm{\Delta }}}_{0} \sim 7\) meV and, thus, exhibits a similar slope \({m}_{1}^{{\rm{CM}}}=1.06\pm 0.02\) below \({P}_{{\rm{sat}}}^{{\rm{1X}}}\). This is further confirmed by secondorder photon correlation measurements between 1X and CM, \({g}_{1XCM}^{\mathrm{(2)}}(\tau )\) (i.e. crosscorrelation) shown in the Supplementary Material. For \({P}_{{\rm{sat}}}^{{\rm{1X}}} < P < {P}_{{\rm{sat}}}^{{\rm{CM}}}\) we observe a slight superlinear increase of CM (\({m}_{2}^{{\rm{CM}}}=1.29\pm 0.07\), highlighted in orange in Fig. 1(b)), which starts at \({P}_{{\rm{sat}}}^{{\rm{1X}}}\) of the QD when multiexciton states become increasingly populated with significant probability. Most remarkably, we observe for \(P > {P}_{{\rm{sat}}}^{{\rm{CM}}}=4.7\pm 0.4\) kW/cm^{2} a complete saturation of CM, strongly suggesting the suppression of nonsaturable background contributions, such as 0D–2D transitions between QD and wetting layer states^{19}. This is in contrast to QDlasers with shallow confinement^{4} and reflects the finite gain provided by the multiexciton states in the fewQD nanolaser presented here. The moderate kink in the inputouput curve of CM combined with the saturation behaviour leads to a slightly sshaped curve, which we attribute to ultralow threshold lasing^{4, 20}, a conclusion supported by the measurements presented below. We note that an unpronounced kink in the inputoutput characteristic as observed in our study makes it usually difficult to unambiguously claim lasing for the device. Therefore, we suggest that further advanced spectroscopy techniques, such as first and second order correlation measurements should be applied in order to test the device under study for coherent light emission.
To support our experimental findings and our assignment of lasing, we evaluate a theoretical model that accounts for the key physical elements, namely the multiexciton states of the QD emitters, and their lightmatter interaction with photons in the cavity mode. Strong excitation is typically required to drive a laser across the threshold. In this regime, multiple carriers accumulate in the QD so that a multitude of manyparticle states arises due to the Coulomb configuration interaction. An illustration is given in Fig. 1(c). In terms of the number n of excitations (eh pairs), for the n = 1 to n = 0 transition only wellseparated emission lines exist, whereas for higher manifolds (gray shaded regions) denselying sets of transitions form, which are observed in the experiment as a broadband background (visible in Fig. 1(a)). Some of these transitions can be in resonance with the mode even though the excitontogroundstate transition is detuned by several meV^{8, 9}. While singlephoton sources use the excitontoground state transition, these higher lying transitions are the ones that drive the emission into the laser mode. Which ones are realized depends on the excitation level and can vary as a function of the pump power.
For the numerical evaluation, we select transitions from different manifolds that are in resonance with the mode and account for their interplay by calculating the dynamics of the density matrix of the coupled fouremitterphoton system. We numerically solve the von NeumannLindblad equation \(\frac{\partial }{\partial \,t\,}\rho =i[{H}_{{\rm{JC}}},\rho ]+ {\mathcal L} \rho \) with the JaynesCummings interaction Hamiltonian \({H}_{{\rm{JC}}}=g{\sum }_{\alpha ,i}[{b}^{\dagger }{D}_{\alpha ,i}^{l}+b{({D}_{\alpha ,i}^{l})}^{\dagger }]\). \({H}_{{\rm{JC}}}\) describes the nonperturbative lightmatter interaction between all dipoleallowed transitions and manyparticle configurations \({i}_{\alpha }\rangle \mathrm{,\ }{l}_{\alpha }\rangle \) of QD α, which are represented by \({D}_{\alpha ,i}^{l}\), and the quantised field of the PhC nanocavity (\({b}^{\dagger }\) denotes the creation operator for photons in the laser mode). Excitation and relaxation processes are accounted for via Lindblad terms \( {\mathcal L} \). Further details and information on the numerical implementation are given in the Supplementary Material. The density operator \(\rho \) yields the outputintensity and \({g}^{\mathrm{(2)}}\mathrm{(0)}\). Moreover, it accounts for all correlations between electronic and photonic degrees of freedom, enabling the study of radiative coupling effects in our system. The red curve in Fig. 1(b) shows the calculated inputoutput curve of the fewQD nanolaser, which is in excellent agreement with the experiment. In particular, it reflects the varied photonemission contributions from different multiexcitonic emission channels that are in resonance with the mode at different pump rates. As our laser model explicitly accounts for four QD emitters, both the slope and the saturation of the emission are well reproduced.
To better understand the consequences of lasing via multiexciton states, we vary the spectral cavitymode position, and thereby the QDcavitymode detuning \({\rm{\Delta }}\), by local inert gas deposition onto the PhC^{21, 22}. In Fig. 2(a), PL spectra are shown of the very same PhC nanocavity for detunings \({{\rm{\Delta }}}_{0}\simeq 7\,\) meV, \({{\rm{\Delta }}}_{1}\simeq 11\) meV and \({{\rm{\Delta }}}_{2}\simeq 17\,\) meV in blue, black and red, respectively. The photon autocorrelation function \({g}^{\mathrm{(2)}}\mathrm{(0)}\) has become the central tool for identifying the threshold in nanolasers, taking on the thermal (coherent) value of \({g}^{\mathrm{(2)}}\mathrm{(0)}=2\) (1) below (above) the threshold^{12, 20, 23}. Unambiguous evidence for lasing at all detunings \({\rm{\Delta }}\) is provided by measuring the autocorrelation function of the \(CM\)emission as a function of P. Zerotimedelay values are obtained from measured \({g}_{CM}^{\mathrm{(2)}}(\tau )\), shown in Fig. 2(b) for \({{\rm{\Delta }}}_{0}\), by fitting \({g}^{\mathrm{(2)}}(\tau )=1+A\cdot \exp (2\tau /{t}_{0})\) (fits shown as solid lines)^{15}. The resulting \({g}_{CM}^{\mathrm{(2)}}\mathrm{(0)}\) is shown in Fig. 2(c) (blue), together with the results for \({{\rm{\Delta }}}_{1}\) (black) and \({{\rm{\Delta }}}_{2}\) (red). For all \({\rm{\Delta }}\), with increasing P we observe a clear transition from \({g}_{CM}^{\mathrm{(2)}}\mathrm{(0)} > 1\) to coherent lasing with \({g}_{CM}^{\mathrm{(2)}}\mathrm{(0)}=1\), demonstrating a surprising robustness of our fewQD nanolaser with respect to spectral cavityemitter detunings up to 17 meV due to efficient nonresonant coupling.
When compared to previous experiments, the pronounced bunching with \({g}_{CM}^{\mathrm{(2)}}\mathrm{(0)}\)values up to 2.7–exceeding the classical limit of 2 for thermal light^{15}–strongly distinguishes the results in Fig. 2(c) from previous experimental and theoretical studies^{4, 8, 20}. Recent theoretical work^{11, 24, 25} have identified \({g}^{\mathrm{(2)}} > 2\) as a fingerprint for a new regime of spontaneous emission with radiatively enhanced correlations between distant emitters mediated by the nanocavity. Our theoretical model accounts for the interplay of multiexcitonic emission channels between four QDs as well as their lightmatter interaction with cavity photons and reproduces both, the superthermal bunching, and the transition to lasing as shown by the solid line in Fig. 2(c) very well. The insight from microscopic theory allows us to attribute the enhanced two and multiplephoton emission probability to two effects: (i) Competition between different resonant transitions in each QD enables simultaneous emission of photons into the mode, and (ii) strong radiative coupling between different emitters results in radiative coupling (subradiant regime). The latter effect has been reported recently under pulsed excitation in ref. 11 in accordance with earlier predictions^{24, 26}.
Finally, we demonstrate that nanolasers containing only few, discrete quantum emitters as gain medium are not well characterised by the conventional spontaneous emission coupling factor β. Typically, β is obtained from rate equations for twolevel systems^{27} that quantifies the fraction of the total spontaneous emission into the laser mode, \(\beta ={\gamma }_{{\rm{l}}}/({\gamma }_{{\rm{l}}}+{\gamma }_{{\rm{nl}}})\), \({\gamma }_{{\rm{l}}}\) and \({\gamma }_{{\rm{nl}}}\) denoting the emission rates into lasing and nonlasing modes, respectively. According to the observation (i), in our fewQD nanolaser multiexciton transitions from different emitters tune in and out of resonance with the cavity mode as pumping is varied, which is beyond the twolevel approximation. Furthermore, the rateequation approach assumes that all emitters act individually, which is in contrast to our observation (ii) of superthermal emission as a collective effect. Therefore, we account for the varying coupling efficiency into the lasing mode for each of the multiexciton emission channels by introducing a pumprate dependent factor \(\beta (P)=\frac{{\rm{\Gamma }}(P)}{{\rm{\Gamma }}(P)+{{\rm{\Gamma }}}_{{\rm{nl}}}(P)}\), where the spontaneous emission rate into the cavity mode \({\rm{\Gamma }}(P)\) and into nonlasing modes \({{\rm{\Gamma }}}_{{\rm{nl}}}(P)\) contain contributions from all bright multiexciton configurations that are realized at a certain pump rate in an averaged form. To assess the impact of (i), we plot in Fig. 3 \(\beta (P)\) without radiative coupling effects (black curve), where the asymptotic values at low (\(\beta (P) > \mathrm{90 \% }\)) and high (\(\beta (P)\approx \mathrm{50 \% }\)) excitation reflect the conventional βfactor associated with the transitions that dominate at low and high excitation. For intermediate pump powers, we observe a transition as the system switches between multiexcitonic emission channels. We note that often a kink in the inputoutput curve is used to quantify a constant β, which for fewQD nanolasers may actually result from transitions between multiexciton states of various emitters tuning in and out of resonance at different excitation powers.
To assess the impact of (ii), we quantify the role of radiative coupling effects from their contribution to the spontaneous emission rate \({\rm{\Gamma }}(P)\) defined as^{24}
Here, the operator \({D}_{\alpha ,i}^{l}\) describes an allowed (bright) dipole transition between multiexciton states in QD α, with the initial state \({i}_{\alpha }\rangle \) and the corresponding recombination rate \({R}_{i}(P)\). The quantummechanical average is taken with respect to the steadystate density operator. The expression in the brackets contains two contributions: The first summation accounts for the spontaneousemission contributions from all QDs α individually. The second sum is the contribution of dipolecorrelated transitions in different emitters α and β that arises due to radiative coupling. While this second contribution has been neglected for the black curve in Fig. 3, the red curve shows \(\beta (P)\) including radiative coupling. In the lowexcitation regime, where the superthermal bunching is observed in \({g}_{CM}^{\mathrm{(2)}}\mathrm{(0)}\), interemitter coupling leads to a strong inhibition of the spontaneous emission rate and of the \(\beta (P)\)factor by nearly a factor of 2. This reduced photon output is caused by the buildup of dipolecorrelations between different emitters, which has been discussed in the context of superradiant lasing in an atomic system^{10} and for QDmicropillar lasers under pulsed excitation^{11}. At \(P\mathop{ > }\limits_{ \tilde {}}20{P}_{sat}^{QD}\) the spontaneous emission becomes enhanced as the second term in Eq. (1) changes sign and the radiative coupling changes from sub to superradiance^{24, 26}. The enhancement in the highexcitation regime is smaller in comparison to the suppression at low excitation due to stimulated emission, which overrules the effect. Further details on the theoretical description can be found in the Supplementary Material.
In summary, we present new insight into the extraordinary operational regime of a few (~4) QD PhC nanolaser that operates solely by nonresonant coupling of QD multiexciton transitions to the cavity mode. In a systematic study we demonstrate that this mechanism gives the laser emission a surprising stability against spectral detuning of the mode up to 17 meV. On the basis of a microscopic theory and \({g}^{\mathrm{(2)}}\mathrm{(0)}\) measurements exhibiting superthermal values of up to 2.7, we find strong evidence that the spontaneousemission regime is subject to radiative coupling between emitters in the form of subradiance. In the presence of such effects, fewemitter cavityQED lasers are not well characterized by the conventional singlevalued βfactor. A factor \(\beta (P)\) is suggested that is pumprate dependent and strongly determined by radiative coupling especially in the lowexcitation regime, shining new light on the practice to determine β from the shape of the inputoutput curve alone.
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Acknowledgements
We thank F.P. Laussy and E. del Valle for fruitful discussions and acknowledge financial support from the DFG (SFB 631, GI1121/11, JA619/103, JA619/131), the German Excellence Initiative (NIM), and the BMBF (Q.com).
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M.K. and J.J.F. conceived and designed the experiments. M.Bi. grew the quantum dot sample. T.R. fabricated the photonic crystal nanocavities. S.L. and M.B. performed the optical measurements. M.F., F.J. and C.G. performed the theoretical calculations. S.L. and M.K. analysed the data. All authors discussed the results and reviewed the manuscript. S.L., M.K., M.F. and C.G. wrote the paper with contributions from all other authors. M.K., J.J.F., C.G. and F.J. inspired and supervised the project.
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Lichtmannecker, S., Florian, M., Reichert, T. et al. A fewemitter solidstate multiexciton laser. Sci Rep 7, 7420 (2017). https://doi.org/10.1038/s41598017070979
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DOI: https://doi.org/10.1038/s41598017070979
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