Transition between cooperative emission regimes in giant perovskite nanocrystals

MainLighting and display applications are almost exclusively making use of photoluminescence (PL), where large numbers of emitters radiate light essentially as independent entities (Fig. 1a). Cooperative effects, however, would allow a much higher photon flux and spectrally more brilliant emission. Most prominently, laser devices1 deliver coherent emission by placing the optically active material within a resonator in which the light is amplified in an avalanche-like process as it circulates in it. The stimulated emission process that is harnessed in lasers can also be used without a resonator to build sources in which the light output is spectrally much broader and less coherent. In such a waveguiding configuration, a few initial, spontaneously emitted photons become amplified through stimulated emission during their propagation within the excited material. In this amplified spontaneous emission (ASE) process, the photon flux increases exponentially before saturation is reached for a larger propagation length. The effective coupling between the emitters in ASE is induced by propagating photons, whereas the quantum mechanical phases of the emitters do not need to be coherent (Fig. 1b). The ASE emission spectrum typically deviates substantially from the PL spectrum, as the spectral region with the highest net gain is nonlinearly amplified through the stimulated emission process. This yields a substantially narrower emission peak that appears above a threshold on the red side of the PL due to reabsorption.Fig. 1: Photon emission processes.Full size imagea–c, Illustrations of standard PL (spontaneous emission) (a), ASE (b) and SF (c). The excited and ground states are described as \(\mid{\rm{e}}\rangle\) and \(\mid{\rm{g}}\rangle\), respectively. The green wavy arrow represents a photon. The white circles represent the emitters with the quantum mechanical phase of the excited electric dipoles as red arrows. τD indicates the time that the emitters need to spontaneously synchronize. d, Schematic of the emission regimes. The solid curves correspond to the borders between different emission regimes, and the dashed curves indicate their qualitative shift when the emitter density is decreased.Alternatively, superfluorescence2,3 (SF) can occur when the emitters have synchronized quantum mechanical phases, effectively behaving as one giant emitter. Even if the emitting dipoles are initially excited incoherently, they can spontaneously develop macroscopic coherence through interactions via the common electromagnetic field and collectively emit a burst of correlated photons (Fig. 1c). In contrast to ASE, a sub-wavelength-sized region is theoretically large enough to host this effect, but sufficiently low homogeneous and inhomogeneous broadening and high oscillator strength are necessary. With excitonic SF, the emission spectrum remains very similar to the PL spectrum, whereas its several key characteristics arise in the temporal emission dynamics that scale with the number of coupled excitons N, such as a nonlinearly growing emission peak after a build-up time τD that is followed by accelerated decay with lifetime τR and Burnham–Chiao ringing4,5 (Supplementary Note 1).ASE and SF are based on different physical mechanisms, namely, stimulated emission and coherent dipole–dipole coupling between excited emitters, respectively, and transitions between these regimes may occur6. Figure 1d illustrates the dominant emission regimes, depending on the excitation density and temperature. SF emission can be observed when the condition for the dephasing time (\({T}_{2} > \sqrt{{\tau }_{\text{D}}{\tau }_{\text{R}}}\)) is satisfied7,8. For materials in which T2 at low temperatures is on the order of the single-emitter radiative decay time τ0, the condition is already met for two coupled emitters. With increasing temperature, phonon-induced dephasing reduces T2 and, therefore, involves increasingly higher excitation strength to maintain SF, consistent with the N dependencies of τR and τD. Once the SF requirement of slow decoherence cannot be fulfilled any more, the emission occurs as either ASE or standard PL. The threshold from PL to ASE is given when the optical gain accumulated by propagation through the excited material exceeds the optical losses. In the time domain, the transition from PL to ASE is characterized by an abrupt change in the emission lifetime with increasing excitation density, as the stimulated emission rates depend on the photon density. At a given pump fluence, decreasing the density of emitters reduces both number and density of coherently coupled excitons for the SF regime and the modal gain in the ASE, thereby shifting the boundaries (Fig. 1d, dashed lines).Although ASE does not require coherence among emitters and has been demonstrated in many material classes, SF has only been observed in a few, select systems such as hydrogen fluoride gases9, O2− centres in KCl crystals10, CuCl quantum dots11, InGaAs/GaAs multiple quantum wells12,13 and nitrogen-vacancy centres in diamond14. However, the transition between the two regimes has only been observed in a singular system6, and only via temperature-induced dephasing. Recently, lead halide perovskites have attracted enormous attention due to their exceptional optical properties15,16,17,18,19,20 and technological relevance21,22. Their low inhomogeneous and homogeneous broadening as well as their high oscillator strength enabled the observation of SF signatures in ordered perovskite nanocrystal (NC) superlattices23,24,25, with superradiant emission ceasing around 100 K (refs. 25,26), in films even up to more elevated temperatures27,28. In experiments that included photonic resonators or distributed Bragg reflectors, transitions from SF to cavity-enhanced SF24 and cooperative exciton–polariton condensation29 have been observed, respectively. In different configurations, ASE has been realized in thin films30,31,32, but it has remained an open question whether the very same perovskite material can support both regimes and how transitions may be experimentally induced and observed.Here we report transitions between the emission regimes of SF, ASE and PL in films of giant CsPbBr3 perovskite NCs. The different regimes have been accessed by systematically altering the exciton dephasing time by the sample temperature, the number density of excitons by the used excitation fluence, the density of NCs by the dilution of NC solutions during film preparation and the photon propagation distance through the pumped stripe length. These results provide comprehensive insights into the different emission mechanisms and transitions between them, which are important for the fundamental understanding of cooperative emission processes and the development of compact, ultrabright light sources.SF in ensembles of giant NCs at 6 KWe synthesized CsPbBr3 NCs via the ligand-assisted reprecipitation (LARP) technique and prepared drop-cast films on Si/SiO2 substrates (Methods). The typical particle size is much larger than the exciton Bohr diameter of ~7 nm (ref. 21; Fig. 2a). These ‘giant’ NCs support excitons in the weak-confinement regime with characteristic fine structure and multiexciton states33. The excitons in an individual NC are within the Dicke regime2 because its volume V λ3). The build-up time τD shortens as a function of excitation fluence (Fig. 2d, bottom). The above-discussed excitation-fluence-dependent emission dynamics are typical signatures of SF.SF-to-PL transition in giant NCsSince faster phonon-induced dephasing at increasing temperatures may render it increasingly more difficult to build up a collective SF state, we investigate the anticipated transition from SF to standard PL or ASE6,34. The time-integrated emission spectra (Fig. 3a–c and Supplementary Figs. 2 and 3) do not exhibit any sign of an (abrupt) transition between regimes. Except for 300 K, the spectra are fit well by a single Lorentzian peak. The emission peak shifts towards higher energy, and its width increases with increasing temperature (Supplementary Figs. 4 and 5). The peak energy shift is consistent with the temperature dependence of the bandgap energy of halide perovskites35,36, and spectral broadening arises from coupling to thermally excited phonons, indicating decreasing coherence.Fig. 3: Temperature dependence of emission dynamics of the low-density thin film of giant CsPbBr3 NCs.Full size imagea–c, Emission spectra for the several excitation fluences at 45 K (a), 125 K (b) and 300 K (c). The inset indicates a change in the main emission regime in the emission regime diagram under the change in excitation fluence. d–f, Spectrally integrated emission intensity time traces for the several excitation fluences at 45 K (d), 125 K (e) and 300 K (f). g, 1/e emission decay time as a function of excitation fluence at different temperatures. h, 1/e emission decay time as a function of temperature for several excitation fluences. The green- and blue-shaded areas indicate the estimated exciton dephasing time T2. For the estimation, we have taken the reported T2 value of smaller CsPbBr3 NCs at 4 K from ref. 15 as a reference. The temperature dependencies were estimated based on the evolution of the emission linewidth of NCs38 (green trace, labelled as Zhu et al.) and on four-wave-mixing measurements with CsPbBr2Cl perovskite NCs16 (blue trace, labelled as Becker et al.).In contrast to the time-integrated emission spectra, the time-resolved emission (Fig. 3d–f and Supplementary Fig. 6) shows a pronounced transition in the emission dynamics with temperature. Below 45 K, the emission is consistent with SF, inferred from the accelerated decay with increasing excitation fluence and the observation of Burnham–Chiao ringing. For temperatures between 45 K and 150 K, a rather abrupt acceleration of decay occurs above a threshold at increasingly higher fluences. In particular, the approximately constant 1/e decay time from PL at a lower fluence suddenly drops to the same SF-accelerated decay time as that at low temperature.Quantitative analyses of emission dynamics also indicate the change in emission regime (Fig. 3g and Supplementary Fig. 7). The black curves in Supplementary Fig. 7 represent the fitting results based on a model function, which comprises a constant decay for the PL regime and a threshold above which the SF-accelerated decay regime shows its characteristic fluence dependence (Methods). The peak intensities Ipeak follow a power law with exponents α = 1.4–1.8, except for 300 K. Remarkably, they do not exhibit a threshold-like change, as would be expected from the transition between PL (α = 1) and SF (α = 2). Such a superlinear growth of Ipeak in the PL regime could be a consequence of gradual filling of below-bandgap trap states37 with increasing excitation fluences, given the non-unity PL QY at high temperature (Extended Data Fig. 1b). In addition, the experimental data of the pulse build-up time, below 100 K, shows the typical trend expected for SF: the emission occurs earlier with increasing fluence. Above 100 K, no fluence dependence is observed for the build-up time.Figure 3h shows the temperature dependence of the decay time for different fluences. We attribute the observed behaviour to the decrease in dephasing time T2 with temperature; therefore, the SF condition time \({T}_{2} > \sqrt{{\tau }_{\text{D}}{\tau }_{\text{R}}}\) requires a larger number of excited emitters N to be fulfilled, because τR ∝ τ0/N. In these perovskites, T2 is comparable with τ0 at low temperatures15, and therefore, SF starts already from the lowest excitation fluence, whereas at elevated temperatures, an increasingly higher excitation fluence is required for a transition from PL to SF (Fig. 3a,b, insets). Hence, although at low temperature, SF-accelerated decay occurs without a pronounced threshold because the low dephasing does not prevent collective emission even for weak excitation, at intermediate temperatures, a threshold develops because a certain N is needed to realize a cooperative state with sufficiently fast decay that is faster than T2. The trend for the transition between PL and SF is qualitatively similar to the expectation from the T2 time (Fig. 3h, shaded region), whereas quantitatively, there can be differences depending on how the T2 estimate is obtained16,38 (Fig. 3h). At room temperature (Fig. 3c), even at very high excitation fluence (>300 µJ cm−2), no SF signatures were observed.SF-to-ASE transition in dense NC ensemblesNext, we study high-density thin films of NCs obtained by drop casting polymer-free solutions of NC (Methods and Extended Data Fig. 2). The layer thickness and effective refractive index is sufficiently high to allow the guiding of light in the NC layer. Time-integrated PL spectra of these films (Fig. 4a–c and Supplementary Fig. 8) show that below 45 K, the single-peak structure persists with increasing excitation fluence, whereas above 45 K, first a shoulder and then a clearly defined new peak appears above a certain threshold excitation fluence. This emerging peak exhibits strong superlinear growth above the threshold (Supplementary Fig. 9). It is redshifted by up to 10 meV until ~86 K, but then at higher temperatures, by up to 35–45 meV (Supplementary Fig. 10), larger than any trion or biexciton binding energy33. It remains narrow at around 10 meV over the whole temperature range (Supplementary Fig. 11). A substantial redshift and narrowing is typical for ASE because due to the small Stokes shift, the peak centre of the excitonic emission is reabsorbed, and optical gain is the highest at wavelengths red-detuned from the PL maximum. Similar energy redshifts and narrowing in the ASE regime can be re-enacted with numerical finite-difference time-domain simulations (Extended Data Fig. 3), showing that narrowing and redshift of tens of millielectronvolts is a robust characteristic of ASE, independent of temperature.Fig. 4: Temperature dependence of emission dynamics of the high-density thin film of giant CsPbBr3 NCs.Full size imagea–c, Emission spectra for several excitation fluences at 6 K (a), 130 K (b) and 300 K (c). The inset indicates the sampled regimes in the emission regime diagram with the change in excitation fluence. d–f, Spectrally integrated emission intensity time traces for different excitation fluences at 6 K (d), 130 K (e) and 300 K (f).The temporal dynamics (Fig. 4d–f and Supplementary Figs. 12 and 13) at 6 K are qualitatively similar to the low-density sample, with a gradual emission acceleration under increasing fluence, consistent with SF. Comparable acceleration at comparable excitation fluence may suggest that the coupling of excitons within individual NCs would contribute more strongly to SF acceleration than the coupling between various NCs that occurs on top of it, which, however, would need to be confirmed through single-NC experiments. For a quantitative analysis, we use the same fit model as for the low-density sample, as it can capture the threshold behaviour reasonably well. Starting at ~44 K, the development of a threshold for accelerated decay is observed for both low- and high-density samples.However, two important differences arise above the threshold, indicating that the emission from the high-density sample evolves towards the ASE, whereas SF may still coexist at intermediate temperatures. First, as discussed above, there is a concomitant change in the spectrum in which a redshifted, narrow peak in the time-integrated spectra appears suddenly above the threshold. Below 90 K, its energy shift remains within the red-peak’s linewidth, which is an indication for a mixture or crossover of SF and ASE above the threshold. Above this temperature, its redshift exceeds its linewidth, becoming consistent with the behaviour expected from the ASE simulations (Extended Data Fig. 3). Besides, streak camera measurements (Extended Data Fig. 4) indicate that the accelerated emission in the time traces corresponds to this redshifted emission peak in the PL spectra. Second, the subsequent growth of the time-resolved peak intensity Ipeak occurs with a strongly superlinear power-law exponent α > 2 (Supplementary Fig. 13), whereas the exponent in the regime below the threshold is only slightly superlinear (α = 1.0–1.6), similar to the low-density sample. This concurs with the temperature at which the redshifted shoulder in the time-integrated spectra appears. Presumably, the temperature-induced dephasing increasingly impedes the establishment of SF, favouring ASE at higher temperatures. This would be in line with the evolution of the thresholds (Extended Data Fig. 5), showing that although the threshold for the low-density sample increases by 20 times between 45 K and 130 K, in the high-density sample, it is only 4 times in the same temperature range. This difference is probably attributable to the strong temperature dependence of SF as it is sensitive to emitter decoherence, whereas ASE is not. Although for InGaAs quantum wells, a transition from ASE to SF has been reported at cryogenic temperatures39, we have not observed such transition in our measurements.The different characteristics of dense and diluted films at intermediate temperatures can be understood by considering that SF and ASE are competing processes. In the high-density film, the excitation density is much higher than in the low-density film, and scattering losses in the densely packed film are lower. Therefore, in the low-density film, the net modal gain within the pump spot and the number of PL seed photons is insufficient for a stimulated emission avalanche, and hence, the excitons have time to synchronize spontaneously via coherent interactions and decay cooperatively via SF. Although at higher temperatures at which dephasing is faster, a transition to ASE could nevertheless be possible at an even higher excitation fluence; we did not observe ASE from the low-density film within the excitation fluence range of our experiments. In comparison, in the high-density sample, the higher exciton density increases the optical gain, effectively favouring ASE already at lower temperatures and excitation fluences. As a result, the excitons decay via ASE before forming SF states, which is consistent with vanishing ringing that is observed above ~86 K.Experiments and simulations of ASE with variable excitation stripe lengthsInterestingly, with a much smaller excitation spot (Methods), no redshifted peak was observed, even up to much higher fluences at room temperature (Extended Data Fig. 6). In contrast to SF, which needs only (sub)wavelength-sized volumes, a substantial pumped gain length is required for ASE. To assess the effect of propagation length on the emission dynamics, we performed variable stripe length measurements40,41 (Methods and Extended Data Fig. 7). As shown in Fig. 5a–c (insets), the redshifted emission peak appears at weaker excitation fluence for longer excitation stripe length L. In terms of the temporal emission dynamics, fast decay components are more pronounced with larger L (Fig. 5a–c), in line with a higher overall gain.Fig. 5: Emission dynamics under variable stripe length excitation at 300 K.Full size imagea–c, Spectrally integrated time-resolved emission intensity traces detected from the edge of the sample for several excitation fluences with excitation stripe lengths of 400 µm (a), 600 µm (b) and 800 µm (c). The insets show the corresponding time-integrated emission spectra. d–f, Numerically simulated emission intensity time traces for different excitation stripe lengths of 400 µm (d), 600 µm (e) and 800 µm (f). The total number density of the emitters in the system was set as Ntotal = 1018 cm−3. Nfrac indicates the fraction of the excited emitters to Ntotal and does not scale linearly with the excitation fluence due to nonlinear processes such as absorption saturation. The simulation model and parameters are discussed in Methods.What is unexpected, however, is that some of the emission time traces for larger L and higher excitation fluence exhibit very pronounced ringing, resembling the Burnham–Chiao ringing. This contrasts with the assumption of dephasing being too fast at room temperature to allow SF. Instead, ASE should be the dominant process, as concluded before. Ringing in ASE has been previously observed in host–guest systems due to the presence of intermediate states42. Moreover, CsPbBr3 has been reported to exhibit delayed optical gain due to slow carrier cooling at high excitation fluence43. To probe whether the latter could lead to ringing, we performed numerical simulations using a three-level scheme to account for hot carriers (Methods and Supplementary Note 2).The simulations (Fig. 5d–f) reproduce not only the emergence of a short ASE pulse for sufficient stripe length and excitation but also the ringing for large L and strong excitation. This can be understood by the hot-carrier relaxation continuously filling the emissive band edge state until its population exceeds the ASE threshold and is depleted in a burst of ASE. Repeating this until the population does not reach the ASE threshold any more effectively produces a ringing behaviour. Photon propagation appears to be essential, as for short L or low-density films, no ringing is found (Extended Data Fig. 8).ConclusionsWe present characteristics and transitions between the different collective emission regimes in halide perovskites that pivotally depend on density, temperature and excitation area. At cryogenic temperature, we demonstrated that SF occurs in giant perovskite NCs that are much larger than the exciton Bohr diameter, whereas a transition from typical SF signatures to normal PL (for low-density films) or ASE (for high-density films) was observed at elevated temperatures. Surprisingly, with large excitation beams, a combination of carrier relaxation and propagation effects can mimic SF-like pulse ringing even at room temperature in the ASE regime, suggesting small excitation beams to circumvent similar experimental signatures. Hence, the combined analysis of temporal emission data (decay acceleration, peak growth and ringing) allows extracting information about the coupling in the system, but they always have to be interpreted very carefully together with spectral features and pump geometry (for instance, redshift and excited area size) to allow faithful attribution to the specific emission regime. Our results provide essential understanding and guidelines for the interpretation of peculiar emission dynamics in perovskites, constituting a unique platform to study cooperative phenomena. Furthermore, these can be exploited for the design of ultrabright perovskite light emitting devices such as quantum light sources and lasers/amplifiers.MethodsSynthesis and sample preparationCsPbBr3 NCs were synthesized using the LARP technique following the procedure of ref. 33 with slight modifications. The hybrid solution was prepared by mixing solutions of 90 µl of PbBr2 (0.67 M in DMF), 465 µl of CsBr (0.043 M in DMF-DMSO in a 1:1 ratio) and 300 µl of OGB (1 M in DMF), where DMF, DMSO and OGB denote dimethyl formamide, dimethyl sulfoxide and oleylguanidinium bromide, respectively. Then, 75 µl of the hybrid solution was injected into a 4-ml vial filled with 2.5 ml of mesitylene under vigorous stirring to initiate the fast nucleation of CsPbBr3 NCs. The green solution with bright green luminescence could be observed in 15–20 s. For the purification of CsPbBr3 NCs, 0.25 ml of ethyl acetate was added to 0.5 ml of crude solution followed by centrifugation for 4 min at 8 krpm (4,250g). The supernatant was discarded, and the precipitate was retained. For preparing low-density NC thin films, the precipitate was redispersed in 0.4 ml of 1% solution of poly(1-vinylpyrrolidone)-graft-(1-triacontene) in toluene and 30 µl of the resulting solution of CsPbBr3 NCs was drop cast on Si/SiO2 substrates and dried under a vacuum. For preparing high-density NC thin films, the precipitate was redispersed in 0.4 ml of toluene and 30 µl of the resulting solution of CsPbBr3 NCs was drop cast on Si/SiO2 substrates and dried under a vacuum.Optical spectroscopyTime-integrated and time-resolved PL measurements were conducted by mounting the sample in a helium exchange-gas cryostat that operates in a temperature range from 6 K to 300 K. As the excitation source, we used a frequency-doubled regenerative amplifier running at 400 nm with a repetition rate of 1 kHz, delivering pulses with a duration of about 150–250 fs. To prevent parasitic excitation light, short-pass filters (cut-off wavelength, 442 nm) were used. For both excitation and detection, we used the same focusing lens with a 100-mm focal length, resulting in an excitation spot size of about 200 \({\mathrm{\upmu}}{\mathrm{m}}\) in diameter. The recorded PL was spectrally filtered by means of a long-pass filter (cut-off wavelength, 480 nm). For the time-resolved measurements, the emission was dispersed by a 150 lines mm−1 grating in a 0.3-m-long monochromator and detected with a streak camera with a nominal time resolution of 2 ps and instrument response function full-width at half-maximum (FWHM) of 4 ps. The time-integrated PL spectra were recorded by a 0.5-m-long spectrograph equipped with a 300 lines mm−1 grating and a nitrogen-cooled charge-coupled device camera.For our experimental conditions, we calculate that an excitation fluence of 1 µJ cm−2 leads to the creation of ~44 excitons per NC, which is a fraction of the Mott density and would correspond to 0.15 excitons per NC for 10-nm-sized NCs. In our measurements, at cryogenic temperatures, we excite with fluences in the range of 0.2–14 µJ cm−2 and increase up to 640 µJ cm−2 at room temperature, which is then far above the Mott density43.With these giant CsPbBr3 NCs, up to ten times less excitation fluence compared with formerly reported superlattices of 10-nm NCs23,24,25,44 is needed to achieve the same SF emission acceleration. Presumably, this is due to the higher absorption cross-section at 400 nm, which is supposed to scale linearly with the NC volume45, as well as low (in)homogeneous broadening in the giant NCs.For the measurements that use excitation with a much smaller excitation beam (Extended Data Fig. 6), the laser was focused with a ×100 objective lens (Mitutoyo) to a Gaussian spot diameter with FWHM of 3.5 µm.For the variable stripe length measurements (Extended Data Fig. 7), the samples were cleaved to allow the light to be collected from the cleaved edge. The stripe-shaped beam spot was obtained by focusing the excitation laser pulse with a cylindrical lens to FWHM of 40 µm × 2,000 µm, with both dimensions well approximated by Gaussians, as confirmed by imaging the excitation spot on a camera. The stripe length was controlled by partially blocking the light with a razor blade on a micrometre stage.SF fit modelWe conducted curve fitting on the rising signal and the initial decay of the emission time traces with an empirical model function to extract the peak intensity and the time at peak. The model function is a convoluted function of an exponential decay and a Gaussian, that is,$$I\left(t\right)=A\times \exp \left[\frac{{w}^{2}}{4{\tau }^{2}}-\frac{\left(t-{t}_{{\rm{c}}}\right)}{\tau }\right]\times \frac{1}{2}\left(\mathrm{erf}\left[\frac{1}{\sqrt{2}}\left(\frac{t-{t}_{{\rm{c}}}}{w}-\frac{w}{\tau }\right)\right]+1\right).$$ (1) The extracted peak time values are plotted as relative delay time (bottom panels in Supplementary Figs. 7 and 13) by setting the peak time at the weakest excitation fluence as zero for each dataset.The 1/e decay time values were extracted directly from the emission time traces. Further analysis of the obtained decay times was performed with curve fitting with the following model function, which comprises a constant decay for the PL regime and the fluence dependent SF regime above threshold, that is,$$\tau (F\,)=\left\{\begin{array}{l}{\tau }_{\mathrm{Flat}}\left(F\le {F}_{\mathrm{th}}\right)\\ \frac{{\tau }_{0}}{\zeta \times F+1}+{y}_{0}\left(F > {F}_{\mathrm{th}}\right)\end{array}\right.$$ (2) where Fth is the excitation fluence threshold of emission lifetime shortening. Below the threshold, we use a constant emission lifetime \({\tau }_{{\rm{Flat}}}\), as is expected for PL in a simplified picture without traps or other fluence-dependent quenching. Although, above the threshold, we assumed that the number of coherently coupled dipoles is proportional to the excitation fluence F with the proportionality constant ζ. τ0 is fixed to values obtained from the time-resolved PL measurements of NCs at weak excitation fluences (Supplementary Fig. 1). y0 is inserted to account for effects such as the finite time resolution.Computational simulation of ASEAs the excitation energy is much larger than the bandgap energy of the NCs, it has been observed that photoexcited carriers may take several picoseconds to relax into the band edge state43,46. Therefore, excitation energy well above the bandgap causes an effective delay in the pumping process from the ground state to the band edge state, that is, the emitting state. Moreover, it is known that the competition between stimulated emission and a delayed pumping process can lead to the ringing of ASE emission pulses42. We simulated the ASE dynamics under variable stripe length excitation. First, the propagation of electromagnetic waves in media is described by the following Maxwell’s equations:$$\nabla \times {\bf{E}}=-{\mu }_{0}\frac{\partial \bf{H}}{\partial t},\nabla \times {\bf{H}}={\varepsilon }_{0}\frac{\partial {\bf{E}}}{\partial t}+\frac{\partial {\bf{P}}}{\partial t}.$$ (3) Here E(t, x) and H(t, x) are the electric and magnetic fields with the electric and magnetic constants ε0 and μ0, respectively. The polarization P(t, x) represents the sum of classical oscillator dipoles in the medium. For simplicity, we only considered the propagation of a transverse-electromagnetic wave in the x direction.To replicate the delayed pumping and emission process of the perovskite material, we used a three-level system as the medium. Carriers are excited from the ground state to the high-energy state and then relax to the band edge state non-radiatively. A photon is emitted when a carrier relaxes from the band edge state to the ground state. This system is described using the following pair of rate equations1:$$\frac{{\rm{d}}{N}_{{\rm{h}}}}{{\rm{d}}{t}}=-\frac{1}{{\tau }_{{\rm{h}}}}{N}_{{\rm{h}}},$$ (4) $$\frac{{\rm{d}}{N}_{{\rm{e}}}}{{\rm{d}}t}=\frac{1}{{\tau }_{{\rm{h}}}}{N}_{{\rm{h}}}-\frac{1}{{\tau }_{{\rm{e}}}}{N}_{{\rm{e}}}+\frac{1}{\hslash \omega }E\frac{{\rm{d}}P}{{\rm{d}}t},$$ (5) $$\frac{{\rm{d}}{N}_{{\rm{g}}}}{{\rm{d}}{t}}=\frac{1}{{\tau }_{{\rm{e}}}}{N}_{{\rm{e}}}-\frac{1}{\hslash \omega }E \frac{{\rm{d}}{P}}{{\rm{d}}{t}}.$$ (6) N(t, x) is the number density of carriers at a given time t and position x. The subscripts h, e and g correspond to the high-energy state, emitting band edge state and ground state, respectively. τh is the time constant of the hot-carrier relaxation process from the high-energy state to the band edge state, and τe is the spontaneous emission lifetime from the band edge state to the ground state. ħω corresponds to the energy separation between the band edge state and the ground state.According to the Lorentz model, the interaction between the electric field and polarization in the media can be described using the following equation:$$\frac{{{\rm{d}}}^{2}P}{{\rm{d}}{t}^{2}}+\Delta \omega \frac{{\rm{d}}P}{{\rm{d}}t}+{\omega }^{2}P=\kappa \Delta {NE}.$$ (7) Δω is the linewidth of the transition between the band edge state and ground state. Here κ = e2/m, where e is the elementary charge and m is the reduced mass of the electron. ΔN is the population difference and defined by ΔN(t, x) = Ng(t, x) – Ne(t, x).Since ASE is initiated by spontaneous emission, we introduced an artificial source of the electric field to simulate spontaneous emission numerically47. The artificial source is described as follows:$${E}_{\mathrm{source}}\left({t,x}\right)=\sqrt{{N}_{{\mathrm{e}}}\left({t,x}\right){\hslash }\omega \beta \eta } \times \sin \left({\omega t+{\phi }_{\mathrm{random}}}\right).$$ (8) Here η is the wave impedance. β is a phenomenological parameter, which represents the fraction of photons that is spontaneously emitted in the ASE mode, and β is set as 0.01 in our simulation. At each given position x, the artificial source has an individual random phase ϕrandom.Figure 5d–f shows the calculated emission intensity time traces. The total number density of the emitters in the system was set as Ntotal = 1018 cm−3, and Nfrac indicates the fraction of the excited emitters to Ntotal. The calculation qualitatively reproduced the experimental results. First, when the system reaches population inversion, the time traces show a fast emission burst because of ASE. The emission burst peak intensity increases with the excitation stripe length because the emitted light is increasingly amplified for a longer excitation stripe length. Besides, the ringing behaviour of the emission pulse is reproduced well in the calculation. It shows that pulse ringing in the time domain is not necessarily a unique signature of SF, but can also occur with ASE due to a combination of relaxation and retardation. On the basis of the similarities of the experimental and calculation results, we conclude that the observed emission dynamics at 300 K can be explained only with ASE, also because for small excitation spots, no acceleration or ringing can be observed at all. The ASE behaviour is expected when dephasing is much faster than SF timescales, as is anticipated at room temperature.The set of rate equations was solved numerically using Euler’s method. The excitation from the ground state to the high-energy state was modelled as an impulsive response, and the spatial profile was homogeneous all over the stripe-shaped excitation beam spot. The temporal and spatial calculation step sizes were set as Δt = 1 × 10−16 s and Δx = 3 × 10−8 m. The parameters were set as τe = 6 ns and τh = 5 ps. The value of τe was obtained from the emission lifetime measurements at 300 K under a weak excitation fluence (Supplementary Fig. 1h). As for τh, we took a value from the typical timescale of the hot-carrier relaxation in CsPbBr3 (ref. 43). Even though it is known that hot-carrier relaxation times in lead halide perovskites depend on the excitation fluence, we use a constant value in our calculation for simplicity.For simulation of the transmission and amplified spectra (Extended Data Fig. 3), the commercial finite-difference time-domain software Lumerical Inc. FDTD (Version 25R1) was used. The material dispersion is modelled using multioscillator Lorentz models by fitting the spectroscopically obtained spectra as discussed in the figure caption, using positive and negative imaginary permittivities for absorption and gain, respectively.