Room temperature collective blinking and photon bunching from CsPbBr_3 quantum dot superlattice

Domain: biology
Published: 2026-03-27
Editorial outcome: major_revision

Abstract
Development of quantum systems supporting collective many-body states is crucial for advancement of quantum technologies, and perovskite quantum dots (QDs) have emerged as promising quantum light sources. However, photon bunching, a key signature of collective states, has been observed in perovskites only at cryogenic temperatures. Here, we report collective blinking and photon bunching in perovskite QD superlattices at room temperature. Sub-wavelength-sized CsPbBr superlattices exhibit distinct two-level blinking, and demonstrate photon bunching with a degree of up to 3.9. Time-resolved photoluminescence and super-resolution imaging reveal long lifetime components, and emission spatially confined to regions tens of nanometers in size, observations consistent with long-range exciton migration to a localized energy trap within the superlattice. Power-dependent degree of bunching and analysis of the bunching dynamics point to biexciton–exciton cascade emission as the origin of photon bunching. These findings establish perovskite QD superlattices as a promising platform for room-temperature collective optical phenomena.

Introduction
Quantum technologies, such as computing, communications or sensing, have the potential to tackle some of the most challenging problems facing science and industry today, and pave the way for future innovations. To realize such technologies, there is a growing demand for developing quantum light sources and finding quantum systems capable of supporting collective many-body states. Such states, which are critical for understanding fundamental physics and for developing the new technologies, are typically realized at cryogenic temperatures. This restriction could be lifted by exploring new classes of materials, and quantum light sources have been demonstrated, e.g., using single organic molecules[1], single semiconductor quantum dots (QDs)[2] or single nitrogen-vacancy centers in diamonds[3]. Among the new materials, metal halide perovskite QDs are promising candidates due to their potential for non-classical light emission, such as single photons or entangled photon pairs. These non-classical states of light have been demonstrated in single perovskite QDs as photon antibunching[4-9], as well as photon bunching[10-13]. The latter phenomenon, which arises, among others, from cascade-like emission of biexcitons, further enhances the potential of perovskite QDs as quantum light sources, as the cascade emission is a key mechanism for generating entangled photon pairs[14-22]. Therefore, it is crucial to explore alternative systems that exhibit the correlated emission of entangled photons, as well as other collective phenomena at room temperature, with the vision to integrate perovskite QD systems into practical quantum technologies.

Recently, superlattices formed by cesium lead bromide perovskite (CsPbBr₃) QDs have emerged as a highly promising platform for investigating collective phenomena, arising from the interactions of individual quantum emitters. A typical CsPbBr₃ QD superlattice is fabricated via self-assembly of individual QDs into a periodic structure. The highly uniform stacking present in the superlattice creates an environment that enables coherent interactions between individual QDs. This gives rise to various types of collective behavior[23,24], such as the typically cryogenic phenomenon of superfluorescence[25]. Superfluorescence is preceded by the formation of a collective quantum state which results in shorter lifetime, red-shifted spectrum and photon bunching of the emission from the QD superlattice[25].

Cubic-shaped CsPbBr₃ QD superlattices can also act as optical cavities, with cooperative interactions between excitons and cavity modes, enabling lasing and cavity-enhanced superfluorescence[26-30]. In addition to collective optical behavior, collective charge dynamics have also been observed in CsPbBr QD superlattices, leading to exciton delocalization and efficient exciton diffusion within the superlattice[29,31-33]. Beyond cubic shapes, superlattices with alternative shapes and building blocks have also been reported, and these exhibit unique characteristics as well. For example, didodecyldimethylammonium bromide (DDAB) treated CsPbBr QDs can be used to form rhombic superlattices[34]. Low-dimensional lead halide perovskite superlattices[35] and anisotropic nanocrystal superlattices[36] exhibit good potential for light emitting diodes and show efficient carrier dynamics[37]. Superfluorescence has also been observed in some of these structures, including all-perovskite multicomponent superlattices[38], pyramid-shaped QD superlattices[39] and superlattices from perovskite nanocubes[40]. Although superfluorescence from perovskite QD superlattices is generally limited to cryogenic temperatures due to thermal decoherence[41], time-resolved spectroscopic features consistent with superfluorescence have been reported in perovskite thin films at higher[42] or even room temperature[43]. These results point to the great unexplored potential of perovskite hierarchical super-structures for collective optical phenomena at ambient temperatures.

Here, we investigate the photophysical properties of well-ordered QD superlattices at room temperature using single-particle. We focus on sub-wavelength size (100 ~ 500 nm) superlattices self-assembled from CsPbBr QDs. We discovered collective blinking behavior and photon bunching at room temperature from individual superlattices, and propose exciton migration, biexciton formation and biexciton-exciton cascade emission as the origin of the observed phenomena.

Results
Fabrication and characterization of CsPbBr quantum dot superlattices

CsPbBr₃ QDs with edge size of approximately 10 nm were synthesized using a modification of a previously reported room-temperature method[44]. TEM images of the synthesized QDs are shown in Supplementary Fig. 1. CsPbBr₃ QD superlattices were prepared on glass substrates by a self-assembly method, as shown in Fig. 1a. The QD toluene solution was spread onto the glass substrate and placed in a toluene-rich atmosphere, allowing the toluene to evaporate slowly and the QDs to gradually self-assemble into superlattices on the microscope cover glass. SEM images (Fig. 1b,c and Supplementary Fig. 2) reveal self-assembled cubic-shaped particles with edge lengths ranging from 100 nm to 500 nm. This size range is on the order of the wavelength of light and is significantly smaller than the several micrometer-sized perovskite superlattices reported previously[25-31]. About 50% of the samples exhibited irregular aggregates formed by stacking of the cubic-shaped particles. Some of these are also shown in the Supplementary Fig. 2. However, SEM alone cannot provide sufficient resolution to determine the precise internal structure of these cubic-shaped particles. This leaves two possibilities: these particles could be either superlattices assembled from individual QDs or, less likely, they might be big single crystals formed by the fusion of multiple QDs. To distinguish the two cases, we studied individual particles sufficiently dispersed on a substrate (as shown in Supplementary Fig. 3) using a fluorescence microscope. We note that the large majority of the particles show no signs of photobleaching for the period of the experiments (> 10 min.). We compared the PL spectra of an individual particle and of a random ensemble of isolated QDs. As shown in Fig. 1d, the spectrum of the QD ensemble exhibits a width (FWHM) of approximately 25 nm with a peak centered at 515 nm. In contrast, the superlattice particle shows a narrower FWHM of 15 nm, while the peak position remains at 513 nm. This reduction in FWHM and consistent peak position suggest that the particles are indeed superlattices in which individual QDs retain their quantum-confined character. A large fused single crystal would undergo significant spectral red-shift compared to the QDs because of the diminished quantum confinement effect[45-47], but unlike the red shifted spectrum associated with superfluorescence the bandwidth of the fused crystal would be broader. Since QDs with similar sizes are more likely to assemble into superlattices the spectrum is narrower due to the decreased size dispersion.

Collective blinking and photon bunching at room temperature

The PL characteristics of individual sub-wavelength size superlattices were measured at room temperature. Fig. 2a presents the time trace of PL intensity for a single superlattice and the corresponding PL intensity distribution. We examined such PL intensity traces of 202 superlattices and found that 193 of them (more than 95 %) exhibit unusual PL fluctuations, including two-level blinking behavior similar to single QDs, with the PL intensity switching between a strongly emitting ON-state and a grey state. To discriminate between the grey state and a possible OFF state we used also the PL lifetime and spectra, as elaborated later. Compared to the blinking of a single constituent CsPbBr QD (Supplementary Fig. 4), the superlattices exhibit unique features in their blinking behavior. Specifically, the intensity of the ON-state is more than 20 times higher than that of the grey state, which itself is still weakly emissive. Also, under the same excitation conditions, the PL intensity of the ON-state of the superlattices is typically over 100 times larger than that of a single QD. These results suggest that the superlattice undergoes collective behavior, where the excitons within the superlattice transition synchronously between ON-states and grey states. For a reference, random QD ensembles of comparable sizes do not show any blinking due to averaging over many randomly blinking individual QDs[48] (Supplementary Fig. 5a).

We further carried out photon coincidence measurements using a 375 nm ps pulsed laser and the Hanbury-Brown Twiss detection scheme. As demonstrated in the second-order correlation function g(2)(t) (Fig. 2b) for the blinking superlattice, the photons of the ON-state are significantly bunched. The presence of bunching does not depend on the repetition rate within the range of 1 x 106 ~ 1 x 107 Hz. An example of photon bunching obtained with the repetition rate of 1 MHz is shown in Supplementary Fig. 6. For a statistical ensemble of 49 superlattices, the degree of bunching 𝑔(2)(0)/𝑔(2)(𝜏) can reach up to 3.9 (Supplementary Fig. 8) and is anticorrelated with the superlattice PL intensity obtained under the same excitation intensity (Fig. 2c and Supplementary Fig. 8). Interestingly, while many superlattices display clear two-level blinking, some particles show blinking with more than two levels. For example, the superlattice illustrated in Supplementary Fig. 9a exhibits four-level blinking, characterized by three ON-states and one grey state. Photon coincidence measurements (Supplementary Fig. 9b) reveal that the total PL from such multi-level blinking particle is not bunched but the PL from the weakest ON-state remains bunched. This behavior suggests that such particles could be non-uniform aggregates (such as those in the bottom row of Supplementary Fig. 2) where each unit acts as an individual emitting center, and where only PL originating from specific uniform cubic region exhibits bunching.

Another important observation is that the blinking and photon bunching are independent phenomena. High degree of photon bunching can be observed even for superlattices that do not show clear blinking, such as the example in Supplementary Fig. 10. On the other hand, no bunching is observed for random QD ensembles (Supplementary Fig. 5b), and single QDs show photon anti-bunching, as expected for a single photon emitter (Supplementary Fig. 11). We verified throughout the experiments that the phenomena we are observing are not artefacts originating from laser scattering or substrate emission. We note that the observations of anti-bunching from a single CsPbBr QD and of bunching from a superlattice of such QDs are not necessarily contradictory because of the superlattice size and high local concentration of excitons. Such transition from antibunching to bunching were observed for other systems as well[49,50].

Fig. 2d shows the PL lifetimes of the ON-state (blue) and the grey state (black) for an individual superlattice. Both states display two-component PL lifetimes, with 17 ns and 45 ns components for the ON-state, and 4.6 ns and 22 ns components for the grey state. The longer components of the ON-state lifetimes measured on other individual superlattices are distributed between 28 and 69 ns, as shown in a histogram in Supplementary Fig. 12. The ON-state lifetime is significantly longer than that of the random ensemble of isolated single QDs (Supplementary Fig. 13) which have components of 12 and 2.4 ns, suggesting that the superlattices possess unique exciton dynamics. The grey state lifetime is comparable to that of the random QD ensemble, and this fact also helped with classifying the particular PL intensity as the grey state (as opposed to an OFF state). Lifetime analysis obtained with different thresholding of the ON- and grey states (Supplementary Fig. 14) confirms the existence of the grey state and clearly distinguishes it from a potential OFF state.

PL blinking mechanism

The phenomenon of blinking is typically observed in individual nano-sized quantum emitters. Apart from that, collective blinking has also been reported in micro-sized objects and clusters of individual emitters. Such collective blinking behavior arises from various mechanisms. For instance, correlated blinking observed in bilayer 2D semiconductor heterostructures is associated with intermittent and random interlayer carrier transfer[51]. In perovskite microcrystals, spatially correlated blinking has been attributed to photon recycling caused by the reabsorption of emitted light and the waveguiding effect[52]. Various trap models have been proposed to explain collective blinking in semiconductor quantum wires[53] and perovskite sub-micrometer crystals[54,55]. Additional mechanisms include charge carrier[47,56,57] or exciton migration [58-62], which are linked to collective blinking observed in perovskite microcrystals and films, J-aggregates, multichromophoric dendrimers or CdSe semiconducting nanoplatelets.

To understand the mechanism of the observed collective PL blinking in the superlattices, we conducted super-resolution localization analysis. Fig. 3a presents the time trace of PL intensity for a single superlattice over a span of 600 seconds. In this specific case, the superlattice mainly remains in the grey state and switches to the ON state for short durations of time. Different PL intensity levels are represented by distinct colors, with the highest intensity marked in red and the lowest one in blue. Fig. 3b maps the localization positions of PL corresponding to these intensity levels. High PL intensity levels are confined within a region of approximately 30 nm in diameter. More examples for different types of blinking (shown in Supplementary Fig. 15) confirm the spatially confined nature of the emitting center within 20 – 30 nm. Such area is significantly smaller than the edge length of the superlattices (100 ~ 500 nm). We note that the super-resolution analysis provides only the size of the emitting area, not its position within the superlattice. The observation of such small emitting area suggests that even though absorption is supposedly occurring with the same probability at any location of the superlattice, the emission of the excitons is concentrated within a small localized region. Most excitons generated within the superlattice are funneled to an energy minimum (defect) located in this confined region, which acts as the primary luminescent center during the ON state.

More insights into the blinking dynamics were obtained through power spectral density (PSD) analysis of the PL time trace (Fig. 3c). More examples of blinking traces and PSDs are shown in Supplementary Fig. 16. The frequency f dependent PSDs are fit[63] using a stretched Lorentzian function

PSD(𝑓) = A / (1+(𝑓/𝑓0 )𝛽) (1)

which consists of a constant A in the low frequency range and describes a power law at high frequencies.  is a characteristic time scale of PL fluctuations and is related to the characteristic frequency f of the cross-over from saturation to power law as

2𝜋𝑓 = 1/𝜏 (2)

The majority of the blinking traces examined show saturation at the lower frequencies and the value of  above 1.3. Such stretched Lorentzian behavior suggests that a well-defined type of phototactivable quencher controls the collective blinking behavior. For  above 1.5 (such as the one in Fig. 3c) a single two-level photoactivated quencher with a specific switching rate is responsible for the blinking. Such quencher is associated with the spatially confined emission center where most excitons are funneled to and where they collectively transit between the emitting ON- and the grey states, leading to the observed collective blinking behavior. Deviation of  values toward 1 suggests participation of additional fast two-level quenching processes. About 20% of the blinking traces showed such power-law type behavior without the low-frequency saturation, indicating that multiple two-level photoactivated quenchers with a distribution of switching rates contribute to the collective blinking.

Further, the fluorescence lifetime intensity distribution (FLID) of an individual superlattice was analyzed to understand the nature of the quencher (Fig. 3d). The distribution reveals distinct averaged lifetimes associated with the ON and grey states, suggesting a type-A blinking, in which the ON state corresponds to a neutral excitonic state and the grey state arises from charged excitons (positive or negative trions)[64,65]. Example of blinking traces obtained for a series of excitation powers from 0.05 W/cm2 to 5.0 W/cm2 (Supplementary Fig. 17) show a trend of increase of grey states fraction with increasing power, providing a support for the proposed mechanism.

Model of the collective phenomena: Long-range exciton migration and biexciton-exciton cascade emission

We propose that the collective PL phenomena observed in the superlattices, including collective blinking and photon bunching, can be attributed to long-distance exciton migration, biexciton formation and biexciton-exciton cascade emission. When multiple excitons are generated within a perovskite QD, biexcitons can form depending on the QD size and halide composition[66]. The presence of biexcitons is typically confirmed by the saturation of emission intensity with increasing number of excitons[67], and by a spectrum which is red-shifted compared to single exciton emission due to the biexciton binding energy. The Coulomb binding energy can be affected by excitation energy[68], size of the QD[69,70] and degree of charge-carrier confinement[71], and in CsPbBr₃ QDs ranges from 5 to 100 meV[66-77]. Relaxation via biexciton-exciton cascade emission, where sequential one-exciton recombination events occur, results in photon bunching[78-82]. The cascade emission is a competing process with biexciton Auger recombination but it has been reported that the cascade emission in QDs has relatively high quantum efficiency[75]. The correlated photon pairs have been shown to exhibit quantum entanglement[14-22], making this process a promising candidate for quantum applications.

To confirm the presence of biexciton-exciton cascade emission in the superlattices, we investigated the excitation power dependence of the photon bunching degree and analyzed the bunching dynamics. We further supported the model by excitation power dependence of the PL intensity and by analysis of the PL spectra. Biexciton cascade emission is generally characterized by a distinct power dependence of the degree of photon bunching. Such dependence has been first observed on GaAs quantum dots[78], and later confirmed on InGaAs QDs[81], as well as perovskites[10,50]. The decrease of bunching degree with increasing excitation power is caused by the fact that at high excitation intensities (high number of biexcitons) the individual photon emission (from biexciton to exciton) surpasses the correlated photon pair emission from the biexciton-exciton cascade. The Fig. 4a shows the effect of excitation power on the degree of photon bunching for an example of two superlattices. With increasing excitation power, the degree of bunching decreases. A qualitative model proposed before[78,81] to explain the phenomenon is introduced in the Supplementary Information, together with the equation used to fit the data in Fig. 4a. Further, an example of the correlation function g(2)(t) taken at the lowest excitation intensities (Supplementary Fig. 18) confirms the bunching degree decrease with increasing power. We note that the trend of the decrease of the degree of bunching distinguishes biexciton-exciton cascade emission from other possible origins of photon bunching, such as correlated multiexciton emission,[83] superfluorescence[25] and super-radiance based lasing[84]. In these phenomena, an opposite trend of increase of photon bunching degree with increasing power has been observed. These findings support the interpretation that the observed photon bunching is a consequence of the biexciton-exciton cascade emission, and that potential contributions from higher multiexciton states can be neglected.

The biexciton formation is a result of accumulation of excitons in the confined location of the superlattice, as schematically summarized in Fig. 5. When a superlattice absorbs light, single excitons are generated at random positions throughout the structure. These excitons can either directly emit light through spontaneous emission, or migrate (funnel) to a spatially confined emitting site that works as an energy trap. Due to the trapping nature of this site the local exciton concentration increases, enhancing the likelihood of biexciton formation. It has been reported[75] that in perovskite QDs the exciton-exciton interaction is attractive, facilitating the formation and stabilization of biexcitons.

The excitation power dependence of PL intensity in Fig. 4b supports this hypothesis. The initial slope of the dependence is linear for the superlattice, the random QD ensemble, and for individual QDs, indicating the absorption due to single excitons in all cases. The single excitons created randomly in different parts of the superlattice then migrate and accumulate in the energy trap where they form the biexcitons. The trap is associated with the photoactive quencher that causes the collective blinking of excitons and biexcitons localized in the trap. We estimate (as detailed in the Supplementary Information) that at the excitation power of 1 W/cm2 on average N = 0.0075 excitons are generated in a single QD. This estimate, together with the random nature of the absorption across the superlattice, supports the initial linearity of the power dependence of the superlattice and confirms that mutiexcitons are not directly generated in this intensity regime. We note that all the experiments (except the power dependences) are carried out with the 1 W/cm2 excitation intensity. For higher excitation intensities the superlattice PL intensity saturates and this saturation is not observed in the random ensemble. Saturation of biexciton emission has been reported before[67]. However, in our case the saturation is a consequence of the energy funneling rather than a specific feature of the biexciton emission. In the Fig. 4b the single QDs show onset of PL saturation around the 100 W/cm2 intensities, corresponding to N = 0.75. For a typical superlattice containing 1000 individual QDs, the total number N under 1 W/cm2 excitation is 7.5. Assuming a realistic value of PLQY of 80%, this number would drop to N ~ 6, as 20% of the excitons would decay non-radiatively before reaching the emission center. Further, assuming that the size of the emission center of 20 – 30 nm would correspond to ~ 10 QDs would result in N = 0.6 per QD, which is on the order of the onset of saturation. On the other hand, such funneling is not present in the random QD ensemble where individual QDs emit independently and the linearity is maintained.

The PL lifetime data are consistent with this model. Both the ON state and the grey state exhibit lifetime components comparable to the PL lifetime of random ensemble of QDs, corresponding to excitons outside the spatially confined emitting site. We note that the short ns component (the 2.4 ns component in the QD ensemble) is negligible in the ON-state lifetime, pointing to reduced non-radiative relaxation. For the ON-state, an additional longer lifetime component is present, which corresponds to the long-distance migration of the excitons. If the exciton, instead of emitting photon at the location of absorption transfers its energy (by Förster mechanism) to a neighboring QD and this step is repeated many times before reaching the emitting site, the lifetime of emission gets longer, resulting in the tens of ns lifetime component. On the other hand, biexcitons generally show very short PL lifetime (tens to hundreds of ps) which mostly reflects the Auger recombination process[67, 71, 77]. We have not resolved such short components in our experiments. However, in our photon correlation experiments we are concerned with the mechanism of cascade-like biexciton-exciton emission. Analysis of the photon bunching dynamics can, in principle, provide information on the cascade-like process[85]. Specifically, fitting of the central 𝑔(2)(𝜏) peak and comparison with the fitting results of the side peaks enables extraction of a fast component which corresponds to the time delay between the emission of the first (biexciton) and second (exciton) photons in the biexciton-exciton cascade, and can be interpreted as the biexciton-exciton emission lifetime. We carried out such analysis on the 𝑔(2)(𝜏) of several superlattices, and present an example in the Fig. 4c. The central peak is well fit with a two-exponential function with lifetimes of 1.3 ns and 27 ns, respectively. The side peaks show single exponential decay with a lifetime of 27 ns. We can therefore attribute the 1.3 ns component to the biexciton-exciton delay (or biexciton lifetime). We note that the 1.3 ns value is very likely resolution-limited, and that the actual delay is on sub-ns scales. This is consistent with the observation on CsPbBr QDs that the radiative rate of the biexciton-exciton transition is more than two orders faster than that of the exciton transition[10].

Another experimental observation indirectly supporting this model is the dependence of the photon bunching degree on the PL intensity at the same excitation power (Fig. 2c) for different individual superlattices. The dependence shows an exponential-like decrease of the bunching degree with the PL intensity. Assuming that the PL intensity is directly related to the superlattice size, this result suggests that the larger the superlattice the smaller the degree of photon bunching. This might be a consequence of higher absolute number of excitons created by absorption in the larger superlattice which, when funneled from the whole superlattice to the low-energy trap site, leads to larger exciton density in the confined emitting space. Such situation is effectively identical to an increase of excitation power in a smaller superlattice, where we observed exponential decrease of the degree of bunching and explained that using the biexciton cascade-like emission model described by the Eq. (3). The same model can thus also explain the exponential-like decrease observed in Fig. 2c. A schematic figure comparing both phenomena is presented as Supplementary Fig. 19.

The Fig. 4d displays the PL spectrum of a single superlattice. The shape of the spectrum is identical for both the ON state and the grey state in the blinking traces (Supplementary Fig. 20). This fact also helped with classifying the particular PL intensity as the grey state (as opposed to an OFF state). Further, time evolution of the PL spectrum (Supplementary Fig. 21) does not show any signs of spectral changes (spectral diffusion), similar to other observations on CsPbBr QDs[48]. The PL spectrum in Fig. 4d can be deconvolved into two Gaussian bands: One centered at 512 nm with an FWHM of 15 nm (purple), and another one at 517 nm with an FWHM of 29 nm (green). The red curve is the measured spectrum and the dashed black curve is the sum of the two Gaussian peaks, which matches the measured spectrum very well. The spectra of all superlattices are characterized by the presence of the low-energy Gaussian band. We note that neither the random QD ensemble (Fig. 1d) nor of the single QDs show the low-energy shoulder. Examples of the single QD spectra in the Supplementary Fig. 22 are characterized by mostly Lorentzian lineshapes. One plausible explanation of the unique superlattice spectral lineshape might be to attribute the red-shifted peak at 517 nm to biexciton-to-exciton transitions[71,76,77], and the peak at 512 nm to single exciton-to-ground state transitions. The red-shift would reflect the biexciton binding energy, of 23 meV in this case. The red shift observed for other superlattices ranges between 10 and 29 meV (Supplementary Fig. 23), which is well within the range of binding energies reported for the CsPbBr₃ QD materials[66-77]. On the other hand, we acknowledge that the contribution of the low-energy shoulder is relatively high, and that other explanations of the PL lineshape might be possible, such as exciton-phonon coupling specific to the superlattices that is not observed for the constituting QDs.

Methods
Synthesis of CsPbBr₃ quantum dots

CsPbBr₃ quantum dots (QDs) are synthesized using a modified room-temperature synthesis procedure based on a previously reported method[40]. In a typical synthesis, 0.08 mmol CsBr and 0.08 mmol PbBr₂ were mixed with 2 mL of N,N-dimethylformamide (DMF). After stirring for 45 minutes, 200 µL of oleic acid (OAC) and 100 µL of oleylamine (OAm) were added sequentially to stabilize the precursor solution. A total of 650 µL of the precursor solution was then added to 10 mL of toluene and stirred for 3 minutes. To purify the sample and minimize the size distribution of the QDs, the solution was centrifuged at 12,000 rpm for 10 minutes. After centrifugation, the supernatant was discarded, and the precipitate was re-dissolved in toluene for a second centrifugation at 12,000 rpm for 10 minutes. Finally, the precipitate was discarded, and the supernatant was collected for the fabrication of superlattices.

Fabrication of the superlattices

CsPbBr₃ quantum dot superlattices were prepared on 2.4 cm × 2.4 cm glass substrates by a self-assembly method. The glass substrates were first cleaned through a series of ultrasonic treatments: 5 minutes in acetone, followed by 5 minutes in ethanol, and finally 5 minutes in deionized water. After cleaning, a glass Petri dish was filled with toluene, and a glass stage approximately 3 cm × 3 cm × 2 cm in size was placed inside. The cleaned glass substrate was positioned on stage and 150 µL of the quantum dot dispersion in toluene at the concentration of 2.29 x 10-7 mol/L was spread onto the glass substrate. The Petri dish was then covered with a glass lid to allow the toluene to evaporate slowly. During the evaporation of toluene, the quantum dots gradually self-assembled into superlattices on the glass substrate.

Experimental setup

The imaging, PL spectra and intensity fluctuation measurements were carried out using a wide-field inverted fluorescence microscope (IX 71, Olympus) with dry objective lens (UMPlanFl 100 ×, N.A. 0.95, Olympus). The samples were excited by a continuous-wave 360 nm laser (UV-FN-360-100 mW-5%-LED, CNI Laser) and the signal was detected with an electron-multiplying (EM) charge-coupled device (CCD) camera (iXon, Andor Technology). A 373 nm dichroic mirror and a 380 nm long pass filter were used to reject the excitation light. The excitation intensity was kept at 1 W/cm2 for all experiments, except for the excitation intensity dependence measurements. For the spectral measurements, the PL signal was further dispersed by using an imaging spectrograph (CLP-50LD, Bunkou Keiki) placed before the EM-CCD camera.

The blinking trajectories were obtained with integration times of 30 ms. The trajectories were constructed by identifying a superlattice emission spot in the image, cropping a small area around the spot and integrating the PL intensity over the area. Area of the same size was then cropped and integrated near the superlattice spot where there was no emission visible, and this area was used as a background that was subtracted from the integrated superlattice signal. This process was repeated for all images in the series and the background-subtracted intensity was plotted as the blinking trajectory.

The time resolved PL measurements were carried out using the same microscope in a confocal mode, the Hanbury-Brown Twiss detection scheme and a time-correlated single photon counting (TCSPC) system (Pico Quant HydraHarp 400). The samples were excited with a picosecond 375 nm laser (LDH-D-C375, Pico Quant) focused to approximately 500 nm spot with the excitation intensity of 1 W/cm2 and repetition rate within the range of 1 x 106 ~ 1 x 107 Hz. The PL signal from the samples was sent through a non-polarizing 50/50 beam splitter to two avalanche photodiodes (APD, Excelitas SPCM-AQRH-14). A 435 nm dichroic mirror and a 400 nm long pass filter were used to reject the excitation light. The arrival times of the photons relative to the laboratory time and the laser pulse time were recorded, and the PL time trajectory, the PL decay curve and the second-order correlation g(2)(t) function were constructed. The resolution limiting factor in this setup is the APD jitter (uncertainty in the photon-to-electron conversion). Such timing resolution for the SPCM-AQRH-14 is product-specified as 225 ps. However, the real resolution is longer, on the sub-ns order. The degree of bunching was calculated based on the ratio of peak intensities after background subtraction. Representative data were also analyzed by integrating the peak areas, and presented in the Supplementary Fig. 8.

The blinking trajectories were acquired using the TTTR acquisition mode, and the HydraHarp software used for the analysis enables integration times (binning) as short as 0.2 ms. To determine the appropriate binning time, we analyzed the blinking traces using a series of binning times between 0.2 ms and 10 ms. Two such examples are shown in the Supplementary Fig. 24 and Supplementary Fig. 25. With the shorter times (< 2 ms), the individual intensity levels in the histogram get smeared out due to the increased noise. At the same time, shortening of the binning time does reveal any additional features such as unresolved OFF or grey states. We therefore used the 10 ms bin as the optimum compromise between the S/N ratio and the resolution. To acquire the background, identical measurements were carried out from nearby locations without detectable emission. All blinking trajectories presented are background-subtracted in this way.

Super-resolution localization analysis

Super-resolution localization methods typically rely on the stochastic switching of individual emitters and determining their centroid positions in the photoluminescence (PL) image. A diffraction-limited PL spot, whose size is similar to or slightly larger than the point spread function (PSF) of the microscope, may contain multiple nanoscale emitters. To determine the shape and central location of the emission profile, a rotational 2D Gaussian function is commonly used for fitting:

𝐺(𝑥, 𝑦, 𝜃) = 𝐴 𝑒𝑥𝑝 (− (𝑎(𝜇𝑥 − 𝑥)2 + 2𝑏(𝜇𝑥 − 𝑥)(𝜇𝑦 − 𝑦) + 𝑐(𝜇𝑦 − 𝑦)2))

where

𝑎 = (𝑐𝑜𝑠2𝜃) / (2𝜎𝑥2) + (𝑠𝑖𝑛2𝜃) / (2𝜎𝑦2), 𝑏 = (𝑠𝑖𝑛2𝜃) / (4𝜎𝑥2) + (𝑠𝑖𝑛2𝜃) / (4𝜎𝑦2),

𝑐 = (𝑠𝑖𝑛2𝜃) / (2𝜎𝑥2) + (𝑐𝑜𝑠2𝜃) / (2𝜎𝑦2).

Here, A represents the amplitude, (µx, µy) denote the mean position in the x-y plane, (σx, σy) correspond to the widths along the major and minor axes, and  indicates the angle of rotation relative to a reference point in the x-y plane. If multiple emitting objects exist within the diffraction-limited region and operate independently, the centroid position fluctuates over time depending on the PL-active and inactive states of the objects.

In the CsPbBr system studied here, individual superlattice, aggregates or isolated nanocrystals may emit separately, especially when charge or energy transfer among them is minimal. By tracking the Gaussian centroid position across consecutive frames of a recorded sequence, one can determine the spatial emission coordinates, referred to as the localization position. The spatial distribution of these positions, along with intensity correlations, enables the identification of the number of emitting objects and their relative alignment. If a single localization position is detected within a spot, it suggests that only one emitter is present, within which excitons diffuse freely. Conversely, the presence of multiple localization positions implies multiple emission centers likely originating from more than one superlattices or aggregates to the overall PL signal.

Power spectral density estimation

The power spectral density (PSD) of blinking traces is estimated by applying the Fourier transform to the intensity autocorrelation function, following the Weiner-Khinchin theorem. Here, PSD models intensity fluctuations involving a two-level system within the framework of the multiple recombination center model[86]. Details of the method can be found elsewhere[63].

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Tan et al. report on the optical properties of self-assembled perovskite QD superlattices at room temperature and observe collective blinking and bunched photon statistics, which are the key results of the study. The origin of the collective behavior is attributed to exciton migration followed by emission through the exciton-biexciton cascade. Considering the potential utility of this material system for a variety of applications, this study presents phenomena and proposed mechanisms that will be important for future work in this material system. Overall, the manuscript is well written, clear, and cites relevant literature. However, in its present form, I don’t recommend publication in Nature Communications. In particular, the following points need to be addressed:\n1. The power-dependence of the biexciton emission should increase quadratically at low power (see ref. [61] Nano Lett. 16, 2349-2362 2016), but the data shown in Fig. 4(c) shows a linear increase. This apparent discrepancy needs to be addressed.\n2. The authors also use the saturation behavior in Fig. 4(c) as evidence of biexciton emission. However, both exciton and biexciton emission exhibit saturation behavior in ref. [61]. Can the authors provide more detail on why they expect the emission from biexcitons to saturate, but not excitons?\n3. Similarly, biexciton emission should result in a faster component to the PL emission lifetime (see ref. [61] and many other studies across different material systems), but the authors report an extended lifetime that they attribute (as far as I can tell) to the exciton migration to an emitting site. It seems that this would result in delayed emission followed by fast emission rather than what is observed. A quantitative model (e.g., Monte-Carlo with rate- equation modeling) of the proposed emission process is important to confirm the proposed emission mechanisms.\n4. It would be useful, particularly for the readers of Nature Communications, to connect with a broader community by referencing other material systems (e.g., epitaxial quantum dots, molecules, defects in materials, colloidal nanocrystals). Some of these materials are more developed as quantum light sources (including demonstrations of collective effects) and have also had significant research devoted to exciton/biexciton dynamics.\n5. The supplement has 11 figures, some without any description (aside from the brief description in the main text). More detail describing the importance of the data shown is required.\nMinor comments:\nLine 31 (and at various places in the text): It would be better to use more precise language (i.e., bunching with a g^((2) ) (0) up to 2.75).\nLine 88: Awkward phrasing (“Such and other requirements…”). Alternative: “These results…”. Also, references to “other results” should be included, or the phrase shouldn’t be used.", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "In this manuscript, Tan et al. demonstrate the fabrication of CsPbBr3 superlattices and their unique optical properties with potential applications in quantum light generation for quantum photonic technologies. First, the authors present the synthesis of CsPbBr3 QDs and their self-assembly into CsPbBr3 superlattices. The ~10 nm cubic CsPbBr3 QDs self-assemble in a toluene rich environment to make 100-500 nm-sized superlattices on a glass substrate. These CsPbBr3 superlattices exhibit emergent optical properties compared to a random ensemble of CsPbBr3 QDs, including narrower emission linewidths, longer emission lifetimes, multi-level blinking behavior, and photon bunching. The degree of photon bunching and emission lifetime modification varies superlattice-to-superlattice, as well as with the superlattice blinking intensity. Additionally, a fluorescence lifetime intensity distribution (FLID) analysis was employed. The authors conclude the FLID map suggests blinking results from the charging of the superlattices. Super-resolution imaging was used to show that the emission is spatially confined to regions tens of nanometers in size, suggesting excitation migration to localized energy traps. The authors argue that the increased exciton density at these traps leads to higher biexciton formation rates. The combined mechanism of exciton migration and biexciton formation results in the observed bunching behavior and longer lifetimes in the CsPbBr3 superlattices.\nWhile the observation of coherent superlattice blinking along with photon bunching is indeed an unexpected and impactful result, there are discrepancies within the model proposed by the author which are not well addressed within the manuscript. Furthermore, the experimental techniques used to identify photon bunching appear to be flawed. These concerns are detailed in the list below. Finally, a key property of quantum light generation is photon indistinguishability which is related to the exciton coherence time and exciton lifetime. To achieve high photon indistinguishability, the coherence time and the lifetime must be on similar timescales. The emission linewidth (inversely related to the coherence time) of these CsPbBr3 superlattices suggests that the coherence time is many orders of magnitude faster than the lifetime, meaning indistinguishability is extremely low. Therefore, the motivation provided for why the authors observations are impactful is lacking. For these reasons, I believe that publication in any form is premature at this time.\nSpecific Comments:\n1) The experimental conditions of the cross-correlation (g2(t)) measurements appear to be inconsistent with the assumptions and criteria presented when the method was initially derived in three fundamental ways (DOI: 10.1021/nl104054t). The excitation densities used to measure the biexciton-exciton correlations appear to be too high. To ensure second-order correlations within the same excitation pulse are a result of a biexciton-exciton cascade process, the excitation density should be low enough such that negligible multiexcitons that are higher-order than the biexciton are generated. Even at an excitation density N = 0.75, the number of triexcitons generated is 25% that of biexciton generation which will lead to misinterpretations of the biexciton behavior in the QD superlattice system. Relatedly, the authors do not make it clear if they measure the CsPbBr3 superlattices at similar powers as single CsPbBr3 QDs, assuming similar excitation densities. The absorption cross-section of the superlattices should scale with the number of QDs within the lattice. If the superlattices are excited at similar powers as the single QD, the excitation density of the superlattice system may be significantly underestimated in this study leading to many higher-order multiexcitons being generated in this system, greatly complicating the picture of the excited system. The authors should measure the absorption cross-section of the CsPbBr3 superlattices and measure the correlations at low excitation densities such that minimal multiexcitons higher than the biexciton are generated within the entire superlattice.\nThe repetition rate of the laser should be slow enough such that the emissive system is guaranteed to be in the ground state before the next excitation pulse. Otherwise, multiple excitons/multiexcitons may be generated at a faster rate than Poissonian statistics predict, breaking down the above-mentioned assumption of the biexciton-exciton correlation measurement. With lifetimes as long as 69 ns, I would expect repetition rates of 1 MHz or lower would be necessary to ensure the excited state fully decays within the excitation window. The authors utilized repetition rates of 2.5 MHz and faster. The authors should either justify the validity of their chosen repetition rates or repeat the measurements to conform with the assumptions of the methodology.\n2) The resolution of the correlation figures throughout the paper are low. Each peak within the g2(t) appears to have ~10-15 data points which is not sufficient to identify relevant features within the correlations, such as bunching. Typically, correlations are binned with a similar resolution as a lifetime (1 ns). The authors should work up the correlation data with a finer time resolution to better resolve the bunching effects.\n3) The authors do not report the concentration of the QD toluene solutions that are used to perform the self-assembly. It is further unclear how, when they measure the emission of single CsPbBr3 superlattices, they ensure that the emission is from only one superlattice. The authors should elucidate how they identify isolated superlattices in their spectroscopic methods. Additionally, if possible, the authors should present raster images of the emission from the CsPbBr3 films to show that the superlattices are sufficiently isolated from one another.\n4) Some superlattices have irregular shapes as shown in Fig. S2. The authors should report what percentage of the superlattices have this irregular shape. The authors should present widefield TEM images of the films CsPbBr3 superlattices to confirm these distributions. The authors should further explain how they confirm whether they are looking at an irregular shaped superlattice vs. a cubic superlattice when measuring the emission from the superlattices.\n5) In Fig. 2, the authors filter the emission by intensity and categorize the high and low intensity sections as the ON-state and grey-state emission, respectively. I would argue that the ON-state is broader than is suggested by the filtering cut-off around 0.2 kc/ms, and is partially captured within the grey-state. Because of this it appears possible that the grey-state is actually an OFF-state. The authors should utilize a range of cutoffs to clearly demonstrate the presence of a grey state rather than an artifact due to thresholding.\n6) On pg. 8, the authors suggest that the double Gaussian fit to the CsPbBr3 superlattice emission spectrum (Fig. 4a) with an energy splitting of 23 meV is evidence of biexciton-exciton cascade emission. It is difficult to believe in this evidence, because given the low excitation densities in these measurements, biexciton emission should contribute less than 5% of the total emission, before accounting for reduced emission efficiencies of multiexcitons due to Auger recombination. The integrated intensities of the two Gaussians in the emission spectrum in Fig. 4a appear to have approximately equal areas which would be unreasonable if the lower energy Gaussian was biexciton emission. To the best of our knowledge, the prevailing wisdom within the community ascribes the lower energy shoulder of the room temperature ensemble emission spectrum of CsPbBr3 to the inelastic phonon scattering of optical phonons. The authors should provide more evidence that the lower energy shoulder results from biexciton emission via power dependent emission spectra.\n7) As the authors show in Fig. S8, the strong antibunching in the g2(t) of a single CsPbBr3 indicates the biexciton quantum yield of single CsPbBr3 QDs is very low. However, the center peak in the CsPbBr3 superlattice correlation is strongly bunched. These results appear entirely contradictory. Since the biexciton quantum yield is low for single CsPbBr3 QDs, the authors need to justify why formation of assemblies drastically changes the single QD multiexciton properties.\n8) The authors show in Fig. 4c that the degree of bunching increases with decreasing PL intensity. It is unclear why bunching would increase with lower PL intensities, and the authors do not provide further explanation. At lower PL intensities, it is possible for background counts and/or laser scatter to introduce artifacts in the correlations, artificially increasing or decreasing the correlation contributions to the center peak. Given the concern of potential artifacts, the authors should justify the anti-correlation of the bunching and PL intensity as it is unintuitive why this behavior occurs.", "reviewer_4": "Response to Reviewers\nFirst of all, we sincerely thank the editor and reviewers for their time to read the manuscript and\nfor their effort to provide valuable and constructive feedback. All the comments and suggestions\nare to the point and have helped us significantly improve the manuscript. Below, we provide a\npoint-by-point response to the comments raised. The original reviewers’ comments are in italics,\nour replies in normal fonts and the changes to the manuscript in blue. Changes in the marked\nversion of the submitted revised manuscript are also highlighted in blue.\nIn particular, we carried out additional experiments on the photon correlation at different laser\nrepetition rates, and on the time evolution of single superlattice PL spectra. Further, we carried out\nanalysis of the photon bunching dynamics which provided us with the biexciton – exciton lifetime.\nBased on this we reorganized the discussion part to strengthen the biexciton model proposed to\nexplain the observed phenomena.\nWe believe that the revised version sufficiently addresses all reviewers’ concerns, and we hope\nthat it now meets the standards for publication in Nature Communications. Detailed answers to the\nindividual comments are provided below.\nReviewer #1\nTan et al. report on the optical properties of self-assembled perovskite QD superlattices at room\ntemperature and observe collective blinking and bunched photon statistics, which are the key\nresults of the study. The origin of the collective behavior is attributed to exciton migration followed\nby emission through the exciton-biexciton cascade. Considering the potential utility of this\nmaterial system for a variety of applications, this study presents phenomena and proposed\nmechanisms that will be important for future work in this material system. Overall, the manuscript\nis well written, clear, and cites relevant literature. However, in its present form, I don’t recommend\npublication in Nature Communications.\nWe thank the reviewer for taking interest in our work.\nIn particular, the following points need to be addressed:\n1. The power-dependence of the biexciton emission should increase quadratically at low power\n(see ref. [61] Nano Lett. 16, 2349-2362 2016), but the data shown in Fig. 4(c) shows a linear\nincrease. This apparent discrepancy needs to be addressed.\nThe quadratic increase of biexciton emission in ref. [67,74,75] is caused by direct formation of\nbiexcitons by two-photon absorption. In contrast, here we put forward a model in which one-\nphoton absorption creates single excitons which then migrate to an emission center where the\nincreased exciton density leads to the formation of biexcitons via attractive interaction. As such,\nthe power dependence reflects the one-photon absorption process and increases linearly.\nWe rearranged the description of the model on page 11 to emphasize this point.\n2. The authors also use the saturation behavior in Fig. 4(c) as evidence of biexciton emission.\nHowever, both exciton and biexciton emission exhibit saturation behavior in ref. [67]. Can the\nauthors provide more detail on why they expect the emission from biexcitons to saturate, but not\nexcitons?\nThank you for this comment which lead us to reconsider the interpretation of the power\ndependences. It is not the emission from biexcitons only but the total emission that saturates. In\nref. [67], the onset of saturation of the emission from single excitons is close to N = 1. In our\nexperiment on single QDs (Fig. 4b) the onset of PL saturation shows around the 100 W/cm2\nintensities, corresponding to N = 0. 75. However, for a typical superlattice containing 1000\nindividual QDs, the number N under 1 W/cm2 excitation is 7.5. Most of these excitons are funneled\nto and accumulate in the emission center, where the resulting N is higher than the onset of\nsaturation. On the other hand, such funneling is not present in the random QD ensemble where\nindividual QDs emit independently and the linearity is maintained.\nThe discussion on page 11 has been modified to address this point.\n3. Similarly, biexciton emission should result in a faster component to the PL emission lifetime\n(see ref. [67] and many other studies across different material systems), but the authors report an\nextended lifetime that they attribute (as far as I can tell) to the exciton migration to an emitting\nsite. It seems that this would result in delayed emission followed by fast emission rather than what\nis observed. A quantitative model (e.g., Monte-Carlo with rate- equation modeling) of the proposed\nemission process is important to confirm the proposed emission mechanisms.\nPL emission lifetime of biexcitons mostly reflects the Auger recombination lifetime, and lifetime\nvalues ranging from tens to hundreds of ps have been reported (in ref. [67, 70, 77] and elsewhere).\nWe have not resolved such short components in our experiments. However, in our photon\ncorrelation experiments we are concerned with the mechanism of cascade-like biexciton-exciton\nemission. Analysis of the photon bunching dynamics can in principle provide information on the\ncascade-like process [84]. Specifically, fitting of the central 𝑔(2)(𝜏) peak and comparison with the\nfitting of the side peaks enables extraction of a fast component which corresponds to the time delay\nbetween the emission of the first (biexciton) and second (exciton) photons in the biexciton-exciton\ncascade, and can be interpreted as the biexciton-exciton emission lifetime. We carried out such\nanalysis on the 𝑔(2)(𝜏) of several superlattices, and present an example in the Fig. 4d. The central\npeak is well fit with a two-exponential function with lifetimes of 1.3 ns and 27 ns, respectively.\nThe side peaks show single exponential decay with a lifetime of 27 ns. We can therefore attribute\nthe 1.3 ns component to the biexciton-exciton delay (or biexciton lifetime). We note that the 1.3\nns value is very likely resolution-limited, and that the actual delay is shorter.\nThe fitting has been added as Fig. 4d and the above discussion including ref [84] added on page\n12.\n4. It would be useful, particularly for the readers of Nature Communications, to connect with a\nbroader community by referencing other material systems (e.g., epitaxial quantum dots, molecules,\ndefects in materials, colloidal nanocrystals). Some of these materials are more developed as\nquantum light sources (including demonstrations of collective effects) and have also had\nsignificant research devoted to exciton/biexciton dynamics.\nWe appreciate this suggestion and added a note in the introduction, including 3 representative\nreferences.\nA note on page 2 ad references [1-3] have been added.\n5. The supplement has 11 figures, some without any description (aside from the brief description\nin the main text). More detail describing the importance of the data shown is required.\nWe agree, the description of the figures has been expanded.\nMinor comments:\nLine 31 (and at various places in the text): It would be better to use more precise language (i.e.,\nbunching with a g^((2) ) (0) up to 2.75).\nThank you, we corrected the expressions.\nLine 88: Awkward phrasing (“Such and other requirements…”). Alternative: “These results…”.\nAlso, references to “other results” should be included, or the phrase shouldn’t be used.\nCorrected, thank you.\nReviewer #2\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is\npart of the Nature Communications initiative to facilitate training in peer review and to provide\nappropriate recognition for Early Career Researchers who co-review manuscripts.\nWe thank the reviewer for using this occasion for rigorous peer-review training of young\nresearchers which is very important for the future of science.\nReviewer #3\nIn this manuscript, Tan et al. demonstrate the fabrication of CsPbBr3 superlattices and their\nunique optical properties with potential applications in quantum light generation for quantum\nphotonic technologies. First, the authors present the synthesis of CsPbBr3 QDs and their self-\nassembly into CsPbBr3 superlattices. The ~10 nm cubic CsPbBr3 QDs self-assemble in a toluene\nrich environment to make 100-500 nm-sized superlattices on a glass substrate. These CsPbBr3\nsuperlattices exhibit emergent optical properties compared to a random ensemble of CsPbBr3\nQDs, including narrower emission linewidths, longer emission lifetimes, multi-level blinking\nbehavior, and photon bunching. The degree of photon bunching and emission lifetime modification\nvaries superlattice-to-superlattice, as well as with the superlattice blinking intensity. Additionally,\na fluorescence lifetime intensity distribution (FLID) analysis was employed. The authors conclude\nthe FLID map suggests blinking results from the charging of the superlattices. Super-resolution\nimaging was used to show that the emission is spatially confined to regions tens of nanometers in\nsize, suggesting excitation migration to localized energy traps. The authors argue that the\nincreased exciton density at these traps leads to higher biexciton formation rates. The combined\nmechanism of exciton migration and biexciton formation results in the observed bunching\nbehavior and longer lifetimes in the CsPbBr3 superlattices.\nWhile the observation of coherent superlattice blinking along with photon bunching is indeed an\nunexpected and impactful result, there are discrepancies within the model proposed by the author\nwhich are not well addressed within the manuscript. Furthermore, the experimental techniques\nused to identify photon bunching appear to be flawed. These concerns are detailed in the list below.\nFinally, a key property of quantum light generation is photon indistinguishability which is related\nto the exciton coherence time and exciton lifetime. To achieve high photon indistinguishability, the\ncoherence time and the lifetime must be on similar timescales. The emission linewidth (inversely\nrelated to the coherence time) of these CsPbBr3 superlattices suggests that the coherence time is\nmany orders of magnitude faster than the lifetime, meaning indistinguishability is extremely low.\nTherefore, the motivation provided for why the authors observations are impactful is lacking. For\nthese reasons, I believe that publication in any form is premature at this time.\nWe thank the reviewer for the overall evaluation and understanding of the manuscript. Regarding\nthe lifetime, the values presented in the manuscript (tens of ns) reflect mainly the long-distance\nmigration of single excitons. In the revised version we added analysis of the photon bunching\ndynamics which can, in principle, provide information on the cascade-like biexciton-exciton\nemission process [84]. Fitting of the central 𝑔(2)(𝜏) peak and comparison with the fitting of the\nside peaks enables extraction of a fast component which can be interpreted as the biexciton-exciton\nemission lifetime. Such biexciton lifetime has sub-ns values, and is thus closer to the coherence\ntime mentioned by the reviewer.\nHowever, we would like to stress that our primary motivation is not to demonstrate a light source\nof highly indistinguishable photons, but rather to report on the discovery of the room temperature\ncollective phenomena and offer explanation of these novel observations that may have\nimplications in future quantum light generation.\nThe fitting has been added as Fig. 4c and the above discussion added on page 12 including ref [84].\nSpecific Comments:\n1) The experimental conditions of the cross-correlation (g2(t)) measurements appear to be\ninconsistent with the assumptions and criteria presented when the method was initially derived in\nthree fundamental ways (DOI: 10.1021/nl104054t).\nThe excitation densities used to measure the biexciton-exciton correlations appear to be too high.\nTo ensure second-order correlations within the same excitation pulse are a result of a biexciton-\nexciton cascade process, the excitation density should be low enough such that negligible\nmultiexcitons that are higher-order than the biexciton are generated. Even at an excitation density\nN = 0.75, the number of triexcitons generated is 25% that of biexciton generation which will lead\nto misinterpretations of the biexciton behavior in the QD superlattice system.\nThe value of N = 0.75 exciton/QD refers to the extreme power of 100 W/cm2 in the power\ndependence in Fig. 4b. For the photon correlation experiments we typically use the power of 1\nW/cm2 which results in N = 0.0075 exciton/QD. In addition, as mentioned in the reply to the\nquestion 1 of Reviewer #1, we assume one-photon absorption creating single excitons (which then\nmigrate to an emission center and form biexcitons), leading to the linear power dependence in Fig.\n4b. We do not see any deviation from the linearity which would indicate direct formation of\nbiexcitons or higher order multiexcitons.\nThe discussion on page 11 has been modified to address this point.\nRelatedly, the authors do not make it clear if they measure the CsPbBr3 superlattices at similar\npowers as single CsPbBr3 QDs, assuming similar excitation densities. The absorption cross-\nsection of the superlattices should scale with the number of QDs within the lattice. If the\nsuperlattices are excited at similar powers as the single QD, the excitation density of the\nsuperlattice system may be significantly underestimated in this study leading to many higher-order\nmultiexcitons being generated in this system, greatly complicating the picture of the excited system.\nThe authors should measure the absorption cross-section of the CsPbBr3 superlattices and\nmeasure the correlations at low excitation densities such that minimal multiexcitons higher than\nthe biexciton are generated within the entire superlattice.\nThank you for pointing out this important aspect. Unfortunately, we are unable to directly measure\nthe absorption cross-section of individual superlattices because we do not know the size of a\nparticular superlattice emitting under the microscope. The size information has been obtained\nindependently using SEM, and correlating the two techniques is beyond our experimental\ncapabilities. Again, the superlattice is measured with the power of 1 W/cm2 corresponding to N =\n0.0075 exciton/QD. This would result in N = 7.5 excitons per superlattice. However, we assume\nhere that the absorption proceeds spatially randomly within the superlattice and occurs\nindependently on individual QDs within the structure. Therefore, the predominant process is linear\nabsorption creating a randomly distributed ensemble of single exciton which then accumulate in\nthe emission center.\nThe discussion on page 11 has been modified to address this point.\nThe repetition rate of the laser should be slow enough such that the emissive system is guaranteed\nto be in the ground state before the next excitation pulse. Otherwise, multiple\nexcitons/multiexcitons may be generated at a faster rate than Poissonian statistics predict,\nbreaking down the above-mentioned assumption of the biexciton-exciton correlation measurement.\nWith lifetimes as long as 69 ns, I would expect repetition rates of 1 MHz or lower would be\nnecessary to ensure the excited state fully decays within the excitation window. The authors utilized\nrepetition rates of 2.5 MHz and faster. The authors should either justify the validity of their chosen\nrepetition rates or repeat the measurements to conform with the assumptions of the methodology.\nYes, this is another very important point. We carried out additional experiments using the\nrepetition rates of 1 MHz and 5 MHz, and the results show that the effect of photon bunching is\nclearly observed at these two frequencies. The results are shown in Supplementary Fig. 6.\nThe bunching experiment with 1 MHz rate is has been mentioned on page 5 and Supplementary\nFig. 6 has been added.\n2) The resolution of the correlation figures throughout the paper are low. Each peak within the\ng2(t) appears to have ~10-15 data points which is not sufficient to identify relevant features within\nthe correlations, such as bunching. Typically, correlations are binned with a similar resolution as\na lifetime (1 ns). The authors should work up the correlation data with a finer time resolution to\nbetter resolve the bunching effects.\nAgain, thank you for pointing this out. We redraw all the data in the figures with the 1 ns resolution.\nThe Fig. 2, and Supplementary Figs. 5, 7, 8, 9 have been modified.\n3) The authors do not report the concentration of the QD toluene solutions that are used to perform\nthe self-assembly. It is further unclear how, when they measure the emission of single CsPbBr3\nsuperlattices, they ensure that the emission is from only one superlattice. The authors should\nelucidate how they identify isolated superlattices in their spectroscopic methods. Additionally, if\npossible, the authors should present raster images of the emission from the CsPbBr3 films to show\nthat the superlattices are sufficiently isolated from one another.\nThe information on the concentration (2.29 x 10-7 mol/L ) has been added in the Methods section.\nThe individual superlattices are identified in wide-field imaging mode in the CCD fluorescence\nimages as isolated bright spots. An examples presented in Supplementary Fig. 3 shows several\nbright spots that are well separated, ensuring sufficient isolation of individual particles.\nThe Methods section has been modified. The Supplementary Fig. 3 has been added and described\non page 3.\n4) Some superlattices have irregular shapes as shown in Fig. S2. The authors should report what\npercentage of the superlattices have this irregular shape. The authors should present widefield\nTEM images of the films CsPbBr3 superlattices to confirm these distributions. The authors should\nfurther explain how they confirm whether they are looking at an irregular shaped superlattice vs.\na cubic superlattice when measuring the emission from the superlattices.\nWe used SEM to characterize the superlattice shape as it proved difficult to grow larger numbers\nof well separated superlattices directly on the TEM grids. We measured the shape in total of 20\nparticles, 9 of which (approximately 50 %) showed the regular cubic shapes. We added more data\nin the SI in Supplementary Fig. 2. As mentioned above, the SEM and fluorescence experiments\nare carried out independently and correlating the two techniques is beyond our experimental\ncapabilities. We therefore cannot directly relate the shape and the PL properties on individual\nparticles, and we base our results on the assumption that the (mostly) two-level blinking originates\nfrom the cubic-shaped superlattices. Also, the irregular-shaped superlattices are significantly\nlarger that the regular ones, and as such emit with higher intensities and small (if any) degree of\nbunching. We do not include such superlattices in the statistical analysis presented in the paper.\nThe Supplementary Fig. 2 has been replaced.\n5) In Fig. 2, the authors filter the emission by intensity and categorize the high and low intensity\nsections as the ON-state and grey-state emission, respectively. I would argue that the ON-state is\nbroader than is suggested by the filtering cut-off around 0.2 kc/ms, and is partially captured within\nthe grey-state. Because of this it appears possible that the grey-state is actually an OFF-state. The\nauthors should utilize a range of cutoffs to clearly demonstrate the presence of a grey state rather\nthan an artifact due to thresholding.\nWe agree that assigning the grey state based on the intensity level only can be ambiguous. Here,\nwe also confirmed that the grey state has PL lifetime characteristic of perovskite QDs, and based\non the suggestion of the reviewer #4 we also verified that the PL spectrum of the grey state\ncorresponds to the perovskite spectrum. None of these would be expected for completely OFF\nstate where the signal would be originating from background scattering or impurity fluorescence.\nThe classification of grey state has been added on page 4, 6 and 12.\n6) On pg. 8, the authors suggest that the double Gaussian fit to the CsPbBr3 superlattice emission\nspectrum (Fig. 4a) with an energy splitting of 23 meV is evidence of biexciton-exciton cascade\nemission. It is difficult to believe in this evidence, because given the low excitation densities in\nthese measurements, biexciton emission should contribute less than 5% of the total emission,\nbefore accounting for reduced emission efficiencies of multiexcitons due to Auger recombination.\nThe integrated intensities of the two Gaussians in the emission spectrum in Fig. 4a appear to have\napproximately equal areas which would be unreasonable if the lower energy Gaussian was\nbiexciton emission. To the best of our knowledge, the prevailing wisdom within the community\nascribes the lower energy shoulder of the room temperature ensemble emission spectrum of\nCsPbBr3 to the inelastic phonon scattering of optical phonons. The authors should provide more\nevidence that the lower energy shoulder results from biexciton emission via power dependent\nemission spectra.\nWe based our explanation of the PL spectral shape of the superlattices on the fact that neither the\nrandom ensemble nor single QDs show the low-energy shoulder. We added examples of single\nQD spectra in the SI (Supplementary Fig. 17), and these are characterized by mostly Lorentzian\nlineshapes. Also, as we do not directly excite the biexcitons, the biexciton contribution can be\nlarger than the 5% the reviewer mentions, and would depend on the probability of the biexciton\nformation in the emitting site. This interpretation is also not contradicted by the comparison of PL\nspectra between the ON and grey states (measured upon the suggestion of the reviewer #4) which\nshows that there is no difference between the two because the GREY state in many cases does\nshow bunching similar to the ON state. This is a consequence of the fact that (as mentioned in the\nmain text) the photon bunching and collective blinking are separate independently occurring\nphenomena.\nGiven the above, we believe that the exciton binding energy is one of the plausible explanations\nof the spectral shape. However, we admit that the contribution of the low-energy shoulder is\nrelatively high, and that other explanations might be possible, such as the exciton-phonon coupling\nproposed by the reviewer. However, such phenomenon would have to be specific to the\nsuperlattices as it is not observed for the constituting QDs.\nThe above discussion and Supplementary Fig. 17 have been added in the rearranged description\nof the model on page 13.\n7) As the authors show in Fig. S8, the strong antibunching in the g2(t) of a single CsPbBr3\nindicates the biexciton quantum yield of single CsPbBr3 QDs is very low. However, the center\npeak in the CsPbBr3 superlattice correlation is strongly bunched. These results appear entirely\ncontradictory. Since the biexciton quantum yield is low for single CsPbBr3 QDs, the authors need\nto justify why formation of assemblies drastically changes the single QD multiexciton properties.\nWe agree that these results may appear initially counterintuitive. Our model assumes that even\nthough direct biexciton formation in individual QDs is very low (also because of the low excitation\nintensity), the individual QDs in the superlatice work as an antenna and funnel individual excitons\ninto the emission center where the increased exciton density and attractive interaction lead to the\nbiexciton formation. We also note that in literature there has been nice demonstration of the\ntransition from antibunching to bunching when the number of coupled QDs increased from 1 to 3\n[49]. Also, very recently both single-exciton anti-bunching and multiexciton bunching have been\ndemonstrated in single large perovskite nanocrystals [50].\nThe above point has been mentioned on page 6 and references [49,50] were added.\n8) The authors show in Fig. 4c that the degree of bunching increases with decreasing PL intensity.\nIt is unclear why bunching would increase with lower PL intensities, and the authors do not\nprovide further explanation. At lower PL intensities, it is possible for background counts and/or\nlaser scatter to introduce artifacts in the correlations, artificially increasing or decreasing the\ncorrelation contributions to the center peak. Given the concern of potential artifacts, the authors\nshould justify the anti-correlation of the bunching and PL intensity as it is unintuitive why this\nbehavior occurs.\nThis phenomenon of increasing degree of bunching with decreasing excitation power (Fig. 4a) has\nbeen first observed on GaAs quantum dots [78], and later confirmed on InGaAs QDs [81], as well\nas perovskites [10], and recently [50]. It is caused by the fact that at high excitation intensities\n(high number of biexcitons) the probability of individual photon emission (from biexciton to\nexciton) surpasses the correlated photon pair emission from the biexciton-exciton cascade. The\nprocess has been theoretically described in the ref. [78] and we present the explanation in the\nSupplementary information.\nThis discussion has been added on page 9.\nReviewer #4\nRoom temperature collective blinking and photon bunching from CsPbBr3 quantum dot\nsuperlattice, Tan et al. This manuscript reports on observed collective blinking and photon\nbunching from individual perovskite quantum dot superlattices. The Reviewer finds the topic and\nacquired data interesting. However, the Reviewer feels that the study could use more work to form\na more impactful study. In its current state, the study lacks depth and does not appear fully formed.\nPurported novelty comes from both collective blinking and photon bunching. However, by the\nauthors’ own admission, blinking and bunching are not linked. Hence, we have two separate\nphenomena to explain. Neither is done to the sufficient depth required (from the Reviewer\nstandpoint) for publication in the Journal.\nMinor comments\n• Repeated issue: \"Occurence\" (misspelled) instead of “Occurrence”\nThank you, corrected\n• Line 161: Replace “the most of them” with a specific number or percentage to provide a\nclearer and more quantitative description.\nThe number is 193. The text has been corrected.\n• Figure 2a: The Reviewer recommends that the authors label all states explicitly in the trajectory\nto make more apparent the states being invoked. Add the labels “ON” and “GREY”. The same\ncomment applies to all the trajectories provided in the SI and especially the trajectory showing\nmultiple “ON” states. There it is even more important to label the states “ON 1”, “ON 2”, “ON\n3” and “GREY”.\nThank you for the suggestion.\nLabels were added in all relevant figures.\n• Figure 2a: the “GREY” state signal appears to be centered around 0. The Reviewer’s\nunderstanding is that “GREY” does not equal “off”. Perhaps a zoom in to show this would be\nuseful. Or if this is a mistake with the y-axis labels that this be corrected. Along the same lines, the\nReviewer notes that the green line drawn through the “grey” state signal does not align with the\npeak in the occurrence histogram. Please address this.\nYes, this is a mistake in the y-axis of the blinking trace (the histogram is correct). We thank the\nreviewer for pointing this out and apologize for the oversight. The Figure has been corrected.\nThe Fig. 2a was replaced with corrected version.\n•Figure 2c: yaxis label has a typo. It should be bunching, not “buncing.”\nDone, thank you.\n• Figure 4a: The fit line overlaps the spectrum, making it difficult to distinguish the two.\nConsider using a different style, such as a dashed line for the fit and markers for the spectrum.\nWe changed the figure as suggested.\n• Figure 5a: There is a typo there. “Cite” should be site.\nCorrected, thank you.\nMajor comments\n• The Reviewer would like the authors to provide more details of their experiment. For example,\nwhat was the integration time of their EMCCD when acquiring the trajectories being shown? What\nis the integration time in Figure 2a? If the integration time can be lowered, then it might be\ninteresting to observe what happens to the appearance of the trajectory and whether the bimodal\n“ON” and “GREY” designations still hold. This comment also applies to the cases where there\nare multiple “ON” states. It also applies to the non-blinking superlattice in Figure S7. Presumably\nlowering the integration time in Figure S7 would show that there are actually “GREY” periods.\nThe integration times using the EMCCD camera were 30 ms, the times using the APD were 10 ms.\nThe blinking trace in Fig. 2a was obtained using the APD with the 10 ms integration time. For\nboth detectors these are practically the limiting values, making it difficult to lower the integration\ntime.\nThe details were added in the Methods section.\n• Related to this, the Reviewer requests that the authors explicitly state the excitation intensities\nused to acquire all blinking trajectories. Additionally, (1) the authors should state how\nsuperlattices are excited. Are they being illuminated as a whole (illumination covers the entire\nsuperlattice) or does the laser focus to the center of the superlattice. In short, report the spot size\nof the laser. (2) What do trajectories look like at lower excitation intensities? Do they show say\nmore “GREY” behavior? Or do they look self-similar at all excitation intensities.\nThe excitation intensity was kept at 1 W/cm2 for all data throughout the manuscript except the\nexcitation intensity dependence. The PL lifetime and photon correlation experiments were carried\nout in confocal mode with the laser diameter of approximately 500 nm. For imaging and\nspectroscopy a wide-field fluorescence mode was used.\nWe have not specifically looked into the intensity dependence of the blinking traces. We do have\nsome data for 0.7 W/cm2 excitation which look similar but it is difficult to make any conclusions\ndue to the lack of statistics.\nThe details were added in the Methods section.\n• Continuing the thought, the Reviewer would like to know how long trajectories are acquired for.\nFigures 2a, S5, S6, and S7 are for 180 seconds. Figures S3 and S4 are for 60 seconds. Is 180\nseconds the longest one collects data for? Why? Do the particles, whether in superlattice form or\nsingle particle form, undergo photobleaching? If so, provide a timescale for photodegradation for\nsingle particles, uncorrelated ensembles, and superlattices under near UV illumination.\nRegarding near UV illumination, the Reviewer wonders whether the observed blinking and\npossible photodegradation is influenced by the excitation color. Why wouldn’t one excite the\nsuperlattices and QDs closer to their band edges? This would avoid any possible UV-induced\nphotochemistry.\nThe large majority of the superlattices measured and presented in the manuscript did not show\nphotobleaching during the measurement time of approx. 10 minutes. We limited many of the\nblinking traces to 180 s because we wanted to measure on the same superlattice the lifetimes and\nthe spectra as well. But we do present blinking traces lasting 10 min., e.g., in the Fig. 3a and\nSupplementary Figs. 13 and 14, where most of the superlattices do not show degradation over this\nperiod of time. As for the Fig. S3 and S4 (newly Supplementary Figs. 4 and 5), we replaced the\n60s traces with 180 s ones to unify the data presentation. Overall, the photodegradation is a\ncomplex issue. Apart from the stable ones presented in the manuscript, we occasionally observed\nthat depending on the preparation conditions, the stability of superlattices can be limited and some\nof them do not last enough to complete the experiments. A systematic study of the correlation\nbetween the preparation and stability will be needed in the future. As for the choice of the\nexcitation source, we used the one that was available for us in the setup.\nA note on photobleaching has been added on page 3. The Fig. S3 and S4 have been replaced as\nSupplementary Figs. 4 and 5.\n• Continuing the thought more. The authors should state more clearly how their blinking\ntrajectories are acquired. It would seem that raw (total) intensities are used. The samples are\nexcited with the near UV laser. A long pass filter is then used to block the laser and all emission\n(irrespective of color) is collected by the EMCCD camera. Is this correct? If so, please state this\nmore explicitly in the main text methods section. Alternatively, provide a detailed SI section on\nhow measurements were conducted.\nYes, the reviewer is correct. We apologize for not providing sufficient details of the experimental\nsetup and conditions to enable other reproduce the results. Detailed description is now provided in\nthe Methods part.\nThe details were added in the Methods section.\n• The reason the question arises is because the Reviewer is curious about the color of the photons\nduring “ON” versus “GREY” periods. Because of issues associated with reabsorption-induced\nredshifting and intra superlattice energy transfer, it is possible that the color of the photons during\n“ON” and “GREY” periods differ (or if the authors care to invoke the biexciton argument then\nthis is recast as biexciton versus non-biexciton emission). The Reviewer would therefore prefer if\nthe authors use their spectrometer and EMCCD camera to collect spectral trajectories and from\nthere generate intensity trajectories. This is the spectral analog of using the picoquant photon\ncounting electronics to generate the “ON” versus “GREY” TCSPC decays in Figure 2d.\nThank you for this suggestion. We followed it and measured the spectra during the blinking ON\nand GREY periods. The result presented in Supplementary Fig. 15 shows no significant difference\nbetween the ON and GREY state spectra. On one hand, this result confirms the assignment of the\nlow-intensity level to the GREY state (as opposed to an OFF state). At the same time, the result\ndoes not contradict the interpretation of the low-energy spectral shoulder as originating due to the\nbiexciton coupling, because the GREY state in many cases does show bunching similar to the ON\nstate. This is a consequence of the fact that (as mentioned in the main text) the photon bunching\nand collective blinking are separate independently occurring phenomena.\nThe Supplementary Fig. 15 has been added and mentioned on page 12.\n• Related to the above point is whether any evidence of spectral diffusion is observed during\nblinking trajectories. Are there any spectral dynamics one should be aware of? Consider the\nfollowing limiting case scenario. The authors acquire trajectories using a bandpass filter (recall\nthat no details were given). The superlattice spectrum is diffusing. Sometimes the emission is\ncentered in the middle of the bandpass. Other times it switches to a different mean energy and\nmoves away from the bandpass center. The intensity recorded by the EMCCD drops. This could\ngive rise to spurious “ON” states and/or “GREY” levels. Presumably this is not happening but\nthe authors should (a) check whether the emitting state remains the same during trajectories and\n(b) report more details about the experiment so that these sorts of questions don’t arise.\nThank you for this important suggestion. We followed this and monitored the PL spectra of several\nsuperlattices for 180 seconds (comparable in length to the blinking traces). We found no evidence\nof spectral diffusion or spectral jumps within these intervals (as shown in Supplementary Fig. 16),\nallowing us to exclude spectral changes as the origin of apparent GREY states.\nSupplementary Fig. 16 has been added and mentioned on page 12.\n• The Reviewer would like the authors to estimate the apparent quantum yield (QY) of the\nsuperlattice during “ON” and “GREY” periods. The authors know the size of the superlattice.\nThey know the cross section of a single particle. They can therefore estimate the number of dots\ncontributing to the signal. Knowing the collection efficiency of their microscope, they can then\nestimate the QY. The authors show a single QD trajectory in Figure S3a. Surprisingly, the single\nQD does not appear to exhibit any “OFF” periods. Is this typical? If so, one might naively assume\nthat single QDs also possess “GREY” states. What is the nature of the perovskite “GREY” state?\nWe agree that it would be helpful for the interpretation to know the PLQY of individual\nsuperlattices, However, as mentioned in replies to other reviewers, the shape and size of the\nsuperlattices were measured using SEM. The SEM and fluorescence experiments are carried out\nindependently and correlating the two techniques is beyond our experimental capabilities. We\ntherefore cannot directly relate the shape/size and the PL properties on individual particles. Since\nthe volume of individual superlattices can easily differ by a factor of 10 or more, we have no\nreliable way to determine the PLQY of a concrete superlattice observed in the fluorescence\nmicroscope.\nIn the revised version of the SI we replaced the 60 s single QD blinking trace in Fig. S3 with a\nlonger one (180 s) in Supplementary Fig. 4. Here, a clear OFF state can be identified around 130\ns. For clarity, we labeled the GREY and OFF state directly in the figure. As for the nature of the\ngrey states in perovskite QDs, the prevailing consensus is that these are charged excitons (positive\nor negative trions) [65].\nThe Fig. S3 has been replaced with Supplementary Fig. 4, and text on page 8 modified, including\nthe new ref. [65].\n• The authors show Figure S4a as an example of the emission trajectories exhibited by random\nQD ensembles. The Reviewer notices some periodicity in the data. The Reviewer would like the\nauthors to take a Fourier transformation of the trajectory to see whether there is a characteristic\nfrequency. It would look to be something like a 0.5 Hz signal. If yes, what is the origin of this\nperiodicity?\nThank you for pointing out this peculiar feature. We reproduced the experiment carefully and\nfound that the periodicity disappeared. The supplementary figure was replaced accordingly.\nThe Fig. 4 has been replaced with Supplementary Fig. 5.\n• In Figure 2b, the Reviewer notes that prior studies showing g2 of superlattices have sub-ps\ntemporal resolution. The importance of this is that with such temporal resolution, one observed\nsingle photon events. In the current study, the temporal resolution is ~6 order of magnitude coarser.\nHence, the g2 shown is likely a classical intensity-intensity correlation rather than a photon arrival\ntime correlation.\nThe photon correlation experiments were carried out using the standard Hanbury Brown – Twiss\nconfiguration with a picosecond pulsed laser (PicoQuant LDH-D-C375), two avalanche\nphotodiodes (Excelitas SPCM-AQRH-14) and a time-correlated single photon counting module\n(PicoQuant HydraHarp 400). The resolution limiting factor in this setup is the APD jitter\n(uncertainty in the photon-to-electron conversion). This timing resolution for the SPCM-AQRH-\n14 is product-specified as 225 ps (even though the real resolution might be longer). Further, we\nmake sure that the experimental conditions are adjusted so that we are in the regime of single\nphoton counting. We note that setups of comparable parameters have been used regularly for\nphoton correlation measurements in many other works reported in literature.\nThe details were added in the Methods section.\n• Regarding the intensity anticorrelation in Figure 2c, the Reviewer notes that the authors have\nstudied 46 individual superlattices. Among them are sub-ensembles of blinking and nonblinking\nsuperlattices. This begs the question as to whether there is a statistical difference between\nbunching contrast between blinking and non-blinking superlattices.\nThe non-blinking superlattices are rare appearances, constituting less than 5% of all measured ones,\nand this does not make it possible to meaningfully evaluate a statistical difference between the two\ntypes.\nThe fraction was added on page 4.\nNext, the authors suggest (late in the paper) that the anticorrelation is due to superlattice size.\nLarger QD superlattices absorb more light (the authors should see an earlier comment about how\nexactly they are exciting their superlattices. This would suggest that the entire superlattice is being\nexcited). They therefore create more excitons. When all are funneled to the same active site, a\nlarger (local) exciton density results, which suppresses bunching. If true, the Reviewer suggests\nthat the authors should lower their laser excitation intensity to see what the corresponding change\nin bunching contrast is. Presumably it increases if the hypothesis is true.\nHere, the reviewer’s logical deduction is exactly what we were trying to convey in the manuscript\n(but apparently did not succeed to explain sufficiently). Actually, the effect of lowering the\nexcitation intensity on the bunching contrast was the subject of the excitation power dependence\nof the degree of bunching, presented in Fig. 4a. Here, as the reviewer predicts, we do see the\nincrease of the bunching contrast.\nThe explanation is on page 9.\n• Regarding the lifetimes in Figure 2d, it is unclear what the short and long components are\nattributed to. The same question applies to the data in Figure S10. From reading later in the text,\nthe authors seem to suggest that the long component is a result of interparticle energy migration,\ni.e., an exciton hopping timescale.\nYes, the reviewer is correct, we assign the longer component to the long-distance exciton migration\nacross the superlattice.\nThe description is a part of the rearranged explanation of the model on page 12.\nBut if so, the Reviewer asks why would one prefer to undergo hopping if through space energy\ntransfer could be done? This would be a very fast process as no barriers are involved. The authors\nare arguing that hopping is preferred over Forster. Why?\nWe apologize for the confusing terminology. By hopping mechanism we did mean dipole-dipole\nForster energy transfer. The term hopping is often used to describe incoherent Forster transfer in\norganic aggregates or light-harvesting complexes to distinguish it from coherent energy\ndelocalization in Frenkel excitons. In the revised version we corrected the explanation to avoid\nmisunderstanding.\nThe text on page 12 has been corrected.\nNext, what is the short component of the decay? Presumably the authors will suggest that this is\nthe exciton recombination lifetime for the most part. But in a superlattice where one sees\n“collective blinking” why is that (short component, ca 17 ns) lifetime longer that that seen in a\nrandom ensemble of single QD (short component, ca 2.4 ns)? The authors should explain this as\none might naively think that a collective state would exhibit a shorter, not longer, lifetime than the\nuncorrelated case. One idea that come to mind is that the authors argue that the localized state\nhas a larger QY than other places in the lattice. Hence, during an “ON” state, the lifetime appears\nlonger due to suppressed non-radiative recombination. Conversely, during the time that the\nunspecified quencher is active, the active site QY drops below that of the random ensemble and\ngives rise to a “GREY” state. This again highlights the substance of the earlier QY question.\nThe long lifetime component observed in the superlattices which is assigned to the long-distance\nexciton migration is a new component not present in the random ensemble. We therefore assume\nthat the shorter superlattice component of 17 ns corresponds to the longer component of the\nensemble (12 ns), and the shorter component of the ensemble (2.4 ns) which is caused by non-\nradiative recombination process in the ensemble is negligibly small (disappears) in the superlattice.\nIn other words, formation of the superlattice and protection of the individual QD surfaces\nsuppresses the non-radiative recombination.\nA note has been added on page 11-12.\n• On a related note, indicate to readers that the instrument response function (IRF) has been\ndeconvoluted from Figure 2d data. Alternatively, add the IRF curve.\nThe IRF has been added in Fig. 2d.\nThe Fig. 2 has been modified.\n• Regarding the super localization study in Figure 3, why is the analysis conducted on this sort of\ntrajectory. Why not on something more representative like the trajectory in Figure 2a. Explain to\nreaders in the main text. At the very least, show other localization maps to assuage the Reviewer\nand reader that this data is representative.\nThe reviewer is absolutely correct, we should have provided more examples of localization in\nrepresentative trajectories from the beginning. We added 2 more examples in the SI\n(Supplementary Fig. 13), which show emission from localized nm scale area of a representative\nblinking trace (Supplementary Fig. 13a), and of a blinking trace that underwent partial\nphotobleaching (Supplementary Fig. 13b).\nThe Supplementary Fig. 13 has been added.\n• Similarly, show other power spectral densities in the SI to show that the conclusions reached are\nreproducible and representative. As far as the Reviewer sees, there is only one PSD shown.\nYes, again, we should have elaborated on this before. We added more blinking and PSD data in\nthe Supplementary Information (Supplementary Fig. 14). The frequency f dependent PSDs are fit\nusing a stretched Lorentzian function\n𝐴\nPSD(𝑓) = \n1+(𝑓/𝑓 )𝛽\n0\nwhich consists of a constant at the low frequency range (PSD = A, f ≪ f\n0\n) and a power law at high\nfrequencies (PSD ≈ f −β, f ≫ f ). Here,  is the characteristic time scale of PL fluctuations and is\n0\nrelated to the characteristic frequency f of the cross-over from saturation to power law as\n0\n2𝜋𝑓 = 1/𝜏\n0\nThe majority of the collective blinking traces examined show saturation at the lower frequencies\nand the value of  above 1.3. This stretched Lorentzian behavior suggests that a well-defined type\nof phototactivable quencher controls the collective blinking behavior. For  above 1.5 a single\ntwo-level photoactivated quencher with a specific switching rate is responsible for the blinking.\nFurther deviation of beta towards 1 suggests participation of additional fast two-level quenching\nprocesses. About 20% of the blinking traces showed power-law type behavior without the low-\nfrequency saturation, indicating that multiple two-level photoactivated quenchers with a\ndistribution of switching rates contribute to the collective blinking.\nOther examples of blinking traces and PSD were added as Supplementary Fig. 14. The above\nexplanation was added in the text on page 8.\n• Personally, the Reviewer would have preferred to have seen a low temperature measurement\nwhere linewidths are narrower. This would also (likely) make clearer the argued biexciton picture\nwhere the authors have estimated a biexciton binding energy of about 23 meV.\nWe would be also much interested in low-temperature measurements, but currently do not have\nthe means to do so. On the other hand, it is encouraging to be able to observe a rich variety of\nintriguing physical phenomena even at room temperatures.\n• Regarding Figure 4c and Equation 1, the Reviewer had the following comments:\n(1) The Equation would appear to predict negative values of the ratio for nominal values of N. For\nexample, if N = 0.75, the entire ratio will be negative. Given that the x axis of the plot is\nproportional to N, the reviewer asks that the authors provide corresponding N values to justify\nwhy the Equation is valid and simultaneously why the ratio never goes negative.\nWe sincerely thank the reviewer for noticing this mistake. The correct form of the eq. (1) is\n𝑔(2)(0) = 1\n1−𝑒−𝑁\nWe apologize for this oversight.\nThe equation has been corrected in the text.\n(2) For readers, define Px and PN^0 since they are never introduced.\nThe definition has been added in the Supplementary Information.\n(3) The derivation of Equation 1 in the SI assumes that the presence of at least 2 excitons in the\nsystem at t=0 will result in the simultaneous emission of 2 photons. However, correctly deriving\ng2(0) requires the calculation of the probability of simultaneous emission. The correct formula for\ng2(0) is\nwhere 𝑎† and 𝑎 are the photon creation and annihilation operators (refer to a Quantum Optics\ntextbook e.g., M. Fox, Quantum Optics: An Introduction or Mandel and Wolf, Optical Coherence\nand Quantum Optics.) If the formula used in the paper and derived in the SI is equivalent to the\ntextbook definition of 𝑔2(𝜏), then this equivalence should be clarified for the readers.\nIf we model the biexciton as a two-photon state, the state can be approximated as\n𝜌 = 𝑝 |0⟩⟨0|+𝑝 |2⟩⟨2|.\n0 2\nSince the denominator and nominator of 𝑔(2)(0) = 〈𝑎†𝑎†𝑎𝑎〉 = 〈𝑛(𝑛−1)〉 can be expressed using\n〈𝑎†𝑎〉2 〈𝑛〉2\n〈𝑛〉 = ∑𝑛𝑝 = 2𝑝 and 〈𝑛(𝑛−1)〉 = ∑𝑛(𝑛−1)𝑝 = 2𝑝 ,\nn 2 n 2\nwe obtain 𝑔(2)(0) = 1 .\n2𝑝2\nFurther, if we assume a Poissonian distribution of excitons, and the fact that all the excitons form\nbiexciton by reducing the number of excitons by a factor of 2,\n𝑝 = 1 ∑∞ 𝑃𝑛 = 1 (1−𝑒−𝑁)\n2 𝑛=1 𝑁\n2 2\nand thus\n𝑔(2)(0) = 1\n1−𝑒−𝑁\nis obtained, which qualitatively explains the phenomenon in the same way as the model of eq. 1.\nThis quantum optical derivation has been added in the SI.\nResponse to Reviewers\nWe sincerely thank the reviewers for their additional time and work spent on our\nmanuscript. We appreciate the reviewers’ honest effort to help improve the work. Below,\nwe provide a point-by-point response to the comments raised. We include only the new\ncomments that require further response. The original reviewers’ comments are in italics\ngrey, our previous replies in italics blue, the new comments on the revision in italics black,\nour new responses in normal fonts and the changes to the manuscript in blue. Changes in\nthe marked version of the submitted revised manuscript are also highlighted in blue.\nAs part of the second revision we carried out further experiments to fully address the\nreviewers’ comments, and added 7 new Supplementary figures. We believe that in the two\nrounds of extensive revisions, both of which included substantial new experimental\nevidence wherever required by the reviewers, we succeeded in addressing all the\nreviewers’ criticism. Also, we provided further convincing arguments, based on\nexperimental facts, theoretical model, published literature and widely accepted consensus\nin the field, to support our mechanism and interpretation of the phenomena, and to\nalleviate the Reviewers’ concerns. We hope that the manuscript now meets the standards\nfor publication in Nature Communications. Detailed answers to the individual comments\nare provided below.\nReviewer #1\nI am satisfied with the responses to my comments, as well as the changes made to the\nmain text and supplement. I have also read through the other referee reports and find that\nthe authors have adequately addressed the concerns in their responses and through\ncorresponding changes to the manuscript.\nReviewer #3\nSpecific Comments:\n1) The experimental conditions of the cross-correlation (g2(t)) measurements appear to\nbe inconsistent with the assumptions and criteria presented when the method was initially\nderived in three fundamental ways (DOI: 10.1021/nl104054t). The excitation densities\nused to measure the biexciton-exciton correlations appear to be too high. To ensure\nsecond-order correlations within the same excitation pulse are a result of a biexciton-\nexciton cascade process, the excitation density should be low enough such that negligible\nmultiexcitons that are higher-order than the biexciton are generated. Even at an\nexcitation density N = 0.75, the number of triexcitons generated is 25% that of biexciton\ngeneration which will lead to misinterpretations of the biexciton behavior in the QD\nsuperlattice system.\nThe value of N = 0.75 exciton/QD refers to the extreme power of 100 W/cm2 in the power\ndependence in Fig. 4b. For the photon correlation experiments we typically use the power\nof 1 W/cm2 which results in N = 0.0075 exciton/QD. In addition, as mentioned in the\nreply to the question 1 of Reviewer #1, we assume one-photon absorption creating single\nexcitons (which then migrate to an emission center and form biexcitons), leading to the\nlinear power dependence in Fig. 4b. We do not see any deviation from the linearity which\nwould indicate direct formation of biexcitons or higher order multiexcitons.\nThank you for clarifying the fluences used in the excitation of single CsPbBr3\nsuperlattices. Although the excitation density N = 0.0075/QD (assuming the absorption\ncross-section scales linearly with the number of NCs in the superlattice) should primarily\nlead to single exciton generation, the total excitation density N = 7.5 of single CsPbBr3\nsuperlattices remains a serious concern. If the exciton funneling mechanism proposed by\nthe authors is true, then it is likely that there are many higher-order multiexcitons that\nare formed as excitons funnel to emission centers. As the number of higher-order\nmultiexcitons present in the system increases, it becomes difficult to interpret the second\norder cross-correlation measurement 𝑔(2)(𝜏). There is no way to distinguish a biexciton-\nexciton correlation from a triexciton-biexciton correlation from a triexciton-exciton\ncorrelation (and so on). We do not believe it is reasonable to conclude that bunching is a\nresult of the biexciton-exciton cascade process while there are likely many higher-order\nmultiexcitons present in the system. Further studies should be performed at much lower\nexcitation densities (N < 0.1; N < 0.0001/QD) to ensure that there is a negligible\nlikelihood of three or more excitons present in the CsPbBr3 superlattice at any given time.\nWe understand the reviewer’s concern. The total exciton density of N = 7.5 per single\nsuperlattice at 1 W/cm2 is an ideal estimate. Assuming a realistic value of PLQY of 80%,\nthis number would drop to N = 6, as 20% of the excitons would decay non-radiatively\nbefore reaching the center. Further, on the funneling pathways from different parts of the\nsuperlattice the actual density would be much smaller, meaning that higher-order\nmultiexcitons would unlikely form on the way to the emission center. To obtain more\nexperimental evidence, we measured the bunching characteristics with 0.5 W/cm2, which\ncorresponds to N ~ 3 excitons (or 0.003 exciton/QD), as proposed by the reviewer. The\npower of 0.5 W/cm2 represents the lowest possible limit of the excitation intensity to\nobtain reliable 𝑔(2)(𝜏) data with reasonable S/N ratio. The result for the photon bunching\nobtained on a superlattice at 0.5 W/cm2 and 1.0 W/cm2 excitation intensities are shown\nas an example in the Supplementary Fig. 18. The superlattice shows significantly higher\ndegree of bunching for the lower intensity (2.07 vs 1.42). This result indicates that\nincreasing the intensity does not create more multiexcitons that would contribute to the\nincrease in bunching (such multiexciton-induced increase in bunching with excitation\nintensity was reported, e.g., in [PRL 94, 087403 (2005)]), and that the trend is in very\ngood agreement with the Fig. 4a and Eq. (3), i.e., with the excitation power dependence\nexpected for biexciton cascade emission. The result thus fully confirms our proposed\nmodel.\nDescription of the bunching at low excitation power and the above reference (as [84])\nwere added on page 9, and the Supplementary Fig. 18 was added in the SI.\n2) The resolution of the correlation figures throughout the paper are low. Each peak\nwithin the g2(t) appears to have ~10-15 data points which is not sufficient to identify\nrelevant features within the correlations, such as bunching. Typically, correlations are\nbinned with a similar resolution as a lifetime (1 ns). The authors should work up the\ncorrelation data with a finer time resolution to better resolve the bunching effects.\nAgain, thank you for pointing this out. We redraw all the data in the figures with the 1 ns\nresolution.\nWe appreciate the effort made to change the resolution of the correlation figures. The\nadded resolution enables further insight and analysis of the correlation dynamics. It is\ninteresting to see that the bunching effects are primarily within a small number of data\npoints in the center peak. Regarding Supplementary Fig. 6, is the degree of bunching of\n1.3 calculated based on the ratio of the peak intensities, or the integrated areas of the\ncenter and side peak?\nThe degree of bunching is based on the ratio of peak intensities.\nWe added this information in the Methods section.\n4) Some superlattices have irregular shapes as shown in Fig. S2. The authors should\nreport what percentage of the superlattices have this irregular shape. The authors should\npresent widefield TEM images of the films CsPbBr3 superlattices to confirm these\ndistributions. The authors should further explain how they confirm whether they are\nlooking at an irregular shaped superlattice vs. a cubic superlattice when measuring the\nemission from the superlattices.\nWe used SEM to characterize the superlattice shape as it proved difficult to grow larger\nnumbers of well separated superlattices directly on the TEM grids. We measured the\nshape in total of 20 particles, 9 of which (approximately 50 %) showed the regular cubic\nshapes. We added more data in the SI in Supplementary Fig. 2. As mentioned above, the\nSEM and fluorescence experiments are carried out independently and correlating the two\ntechniques is beyond our experimental capabilities. We therefore cannot directly relate\nthe shape and the PL properties on individual particles, and we base our results on the\nassumption that the (mostly) two-level blinking originates from the cubic-shaped\nsuperlattices. Also, the irregular-shaped superlattices are significantly larger that the\nregular ones, and as such emit with higher intensities and small (if any) degree of\nbunching. We do not include such superlattices in the statistical analysis presented in the\npaper.\nThank you to the authors for providing additional information on the cubic vs. irregularly\nshaped CsPbBr3 superlattice structures. The authors make a reasonable justification to\ndetermine whether they are measuring the emission coming from a cubic vs. irregularly\nshaped superlattice, however it would be helpful to the readers to understand roughly\nhow much brighter the irregularly shaped superlattices are compared to the cubic shaped\nsuperlattices.\nThe maximum emission intensity of the multi-level blinking particles is about twice or\nmore than that of the two-level blinking particles.\nThis information was added in the description of Supplementary Fig. 2.\n5) In Fig. 2, the authors filter the emission by intensity and categorize the high and low\nintensity sections as the ON-state and grey-state emission, respectively. I would argue\nthat the ON-state is broader than is suggested by the filtering cut-off around 0.2 kc/ms,\nand is partially captured within the grey-state. Because of this it appears possible that\nthe grey-state is actually an OFF state. The authors should utilize a range of cutoffs to\nclearly demonstrate the presence of a grey state rather than an artifact due to\nthresholding.\nWe agree that assigning the grey state based on the intensity level only can be ambiguous.\nHere, we also confirmed that the grey state has PL lifetime characteristic of perovskite\nQDs, and based on the suggestion of the reviewer #4 we also verified that the PL spectrum\nof the grey state corresponds to the perovskite spectrum. None of these would be expected\nfor completely OFF state where the signal would be originating from background\nscattering or impurity fluorescence.\nWe want to thank the authors for their response. However, we don’t believe our initial\ncomment was addressed. We apologize if our initial comment was not clear so we will\nattempt to restate it below. We were curious to know what the grey-state lifetime would\nlook like if the threshold was set at an even lower value such as 0.05 kc/ms (marked by\nthe black cutoff line in the below figure). We were initially concerned that some of the\nON-state was captured within the grey-state due to the arbitrarily created thresholding\nvalue around 0.22 kc/ms. We have marked this value on Fig. 2a for clarity.\nThank you for suggesting this important confirmation. We re-analyzed the trace for ‘pure’\ngrey state (below 0.05 kc/ms) and for ‘pure’ ON-state (for 0.375 - 0.45 kc/ms). The data\nare presented in the Supplementary Fig. 14, and show that there is no difference in the\nlifetimes from the ON-states. The grey state does show slight shortening of the long\ncomponent from 22 ns to 18 ns. This means that there might be some ON-states admixed\nwith the grey state at the original 0.22 kc/ms threshold (as the reviewer proposed), but it\nalso shows that the intensity values below 0.05 kc/ms correspond to a true grey state and\nnot and OFF-state.\nA note was added on page 6 and the Supplementary Fig. 14 was added in the SI.\nThe integrated intensities of the two Gaussians in the emission spectrum in Fig. 4a appear\nto have approximately equal areas which would be unreasonable if the lower energy\nGaussian was biexciton emission. To the best of our knowledge, the prevailing wisdom\nwithin the community ascribes the lower energy shoulder of the room temperature\nensemble emission spectrum of CsPbBr3 to the inelastic phonon scattering of optical\nphonons. The authors should provide more evidence that the lower energy shoulder\nresults from biexciton emission via power dependent emission spectra.\nWe based our explanation of the PL spectral shape of the superlattices on the fact that\nneither the random ensemble nor single QDs show the low-energy shoulder. We added\nexamples of single QD spectra in the SI (Supplementary Fig. 17), and these are\ncharacterized by mostly Lorentzian lineshapes. Also, as we do not directly excite the\nbiexcitons, the biexciton contribution can be larger than the 5% the reviewer mentions,\nand would depend on the probability of the biexciton formation in the emitting site. This\ninterpretation is also not contradicted by the comparison of PL spectra between the ON\nand grey states (measured upon the suggestion of the reviewer #4) which shows that there\nis no difference between the two because the GREY state in many cases does show\nbunching similar to the ON state. This is a consequence of the fact that (as mentioned in\nthe main text) the photon bunching and collective blinking are separate independently\noccurring phenomena. Given the above, we believe that the exciton binding energy is one\nof the plausible explanations of the spectral shape. However, we admit that the\ncontribution of the low-energy shoulder is relatively high, and that other explanations\nmight be possible, such as the exciton-phonon coupling proposed by the reviewer.\nHowever, such phenomenon would have to be specific to the superlattices as it is not\nobserved for the constituting QDs.\nThank you to the authors for their insightful response. The language added to the\nmanuscript acknowledges the various plausible explanations. In this context, it seems\nreasonable to propose the biexciton binding energy as one of the possibilities. We will\nnote however, if we assume that biexciton formation is much higher than what I was\nproposing in our initial comment (5% of total emission), then we have a similar concern\nas stated in 1) regarding the exciton/multiexciton distribution in the single CsPbBr3\nsuperlattices that have implications on the accurate interpretations of the 𝑔(2)(𝜏)\nbunching experiments.\nThe multiexciton question was addressed in response to Comment 1).\n7) As the authors show in Fig. S8, the strong antibunching in the g2(t) of a single CsPbBr3\nindicates the biexciton quantum yield of single CsPbBr3 QDs is very low. However, the\ncenter peak in the CsPbBr3 superlattice correlation is strongly bunched. These results\nappear entirely contradictory. Since the biexciton quantum yield is low for single\nCsPbBr3 QDs, the authors need to justify why formation of assemblies drastically\nchanges the single QD multiexciton properties.\nWe agree that these results may appear initially counterintuitive. Our model assumes that\neven though direct biexciton formation in individual QDs is very low (also because of the\nlow excitation intensity), the individual QDs in the superlattice work as an antenna and\nfunnel individual excitons into the emission center where the increased exciton density\nand attractive interaction lead to the biexciton formation. We also note that in literature\nthere has been nice demonstration of the transition from antibunching to bunching when\nthe number of coupled QDs increased from 1 to 3 [49]. Also, very recently both single-\nexciton anti-bunching and multiexciton bunching have been demonstrated in single large\nperovskite nanocrystals [50].\nThank you to the authors for their response. We maintain concern about the mechanistic\nconclusions of the bunching phenomenon observed in these CsPbBr3 superlattices for a\nfew reasons. First, references [10,49,50,78,81] are on coherent systems at cryogenic\ntemperatures (4 Kelvin) where indistinguishability of photon emission can be reasonably\nachieved. Indistinguishability is attributed as a key property in order to achieve the\nladder-like emission that results in bunching. Reference [49] shows the difference in\nbunching as 2 QDs are tuned in and out of indistinguishability. To our knowledge, photon\nindistinguishability has never been demonstrated in any QD system at room temperature,\nsuggesting it is unlikely the CsPbBr3 superlattices undergo the same process to produce\nbunched 𝑔(2)(𝜏) measurements. Second, the QD systems in references [49,50,78]\nrequired specific engineering solutions to achieve indistinguishability, and therefore\nbunching. For ref [49], the emission from the 3 coupled QDs were not intrinsically\nindistinguishable. The emission energy was tuned via strain and electric field effects to\nplace the 3 QD emission energies in resonance with one another. For ref [50], the single\nlarge perovskite nanocrystal size was tuned so that the biexciton-to-exciton transition\nwould be in resonance with the exciton-to-optical-phonon transition. Additionally, the\nexciton emission and trion emission were filtered in order to observe bunching. For ref\n[78], the Hanbury Brown Twiss 𝑔(2)(𝜏) spectroscopic setup was spectrally filtered such\nthat one detector only accepted biexciton emission and the other detector only accepted\nexciton emission, leading to the bunching effect. If the spectroscopic setup was not\nspectrally filtered, bunching would not be expected. The observed bunching on the\nCsPbBr3 superlattices is extremely interesting, however I do not believe there is sufficient\nevidence at this time to propose a precise mechanism of bunching. As shown in the\nreferences cited by the authors, coherent ladder-like biexciton-exciton cascade requires\nstringent design in other single quantum dot or small number of coupled quantum dot\nsystems at cryogenic temperatures. Therefore, it seems highly unlikely that the room\ntemperature CsPbBr3 superlattice (comprised of ~1000 coupled NCs) is undergoing a\ncoherent ladder-like emissive process that could result in the proposed bunching process.\nWe believe further investigation into the mechanism of bunching is required before\nrecommending publication.\nThe reviewer’s original question was that ‘… strong antibunching in the g2(t) of a single\nCsPbBr indicates the biexciton quantum yield of single CsPbBr3 QDs is very low.\n3\nHowever, the center peak in the CsPbBr3 superlattice correlation is strongly bunched.\n3\nThese results appear entirely contradictory.’ We replied to explain the contradiction and\nreiterated that in our model direct biexciton formation in individual QDs maybe indeed\nlow but the superlattice works as an antenna and funnels the excitons into the emission\ncenter where the increased exciton density and attractive interaction lead to the\nbiexciton formation.\nFor illustration, we cited a few examples of literature that did show transition from\nantibunching to bunching with increasing size, but by no means we meant to imply that\nour phenomena are equivalent to those observed in the references. As the reviewer notes,\nthe low-temperature experiments required filtering or energy tuning because of the clear\nseparation of the exc"}, "author_response": "Response to Reviewers\nFirst of all, we sincerely thank the editor and reviewers for their time to read the manuscript and\nfor their effort to provide valuable and constructive feedback. All the comments and suggestions\nare to the point and have helped us significantly improve the manuscript. Below, we provide a\npoint-by-point response to the comments raised. The original reviewers’ comments are in italics,\nour replies in normal fonts and the changes to the manuscript in blue. Changes in the marked\nversion of the submitted revised manuscript are also highlighted in blue.\nIn particular, we carried out additional experiments on the photon correlation at different laser\nrepetition rates, and on the time evolution of single superlattice PL spectra. Further, we carried out\nanalysis of the photon bunching dynamics which provided us with the biexciton – exciton lifetime.\nBased on this we reorganized the discussion part to strengthen the biexciton model proposed to\nexplain the observed phenomena.\nWe believe that the revised version sufficiently addresses all reviewers’ concerns, and we hope\nthat it now meets the standards for publication in Nature Communications. Detailed answers to the\nindividual comments are provided below.\nReviewer #1\nTan et al. report on the optical properties of self-assembled perovskite QD superlattices at room\ntemperature and observe collective blinking and bunched photon statistics, which are the key\nresults of the study. The origin of the collective behavior is attributed to exciton migration followed\nby emission through the exciton-biexciton cascade. Considering the potential utility of this\nmaterial system for a variety of applications, this study presents phenomena and proposed\nmechanisms that will be important for future work in this material system. Overall, the manuscript\nis well written, clear, and cites relevant literature. However, in its present form, I don’t recommend\npublication in Nature Communications.\nWe thank the reviewer for taking interest in our work.\nIn particular, the following points need to be addressed:\n1. The power-dependence of the biexciton emission should increase quadratically at low power\n(see ref. [61] Nano Lett. 16, 2349-2362 2016), but the data shown in Fig. 4(c) shows a linear\nincrease. This apparent discrepancy needs to be addressed.\nThe quadratic increase of biexciton emission in ref. [67,74,75] is caused by direct formation of\nbiexcitons by two-photon absorption. In contrast, here we put forward a model in which one-\nphoton absorption creates single excitons which then migrate to an emission center where the\nincreased exciton density leads to the formation of biexcitons via attractive interaction. As such,\nthe power dependence reflects the one-photon absorption process and increases linearly.\nWe rearranged the description of the model on page 11 to emphasize this point.\n2. The authors also use the saturation behavior in Fig. 4(c) as evidence of biexciton emission.\nHowever, both exciton and biexciton emission exhibit saturation behavior in ref. [67]. Can the\nauthors provide more detail on why they expect the emission from biexcitons to saturate, but not\nexcitons?\nThank you for this comment which lead us to reconsider the interpretation of the power\ndependences. It is not the emission from biexcitons only but the total emission that saturates.\nIn\nref. [67], the onset of saturation of the emission from single excitons is close to N = 1. In our\nexperiment on single QDs (Fig. 4b) the onset of PL saturation shows around the 100 W/cm2\nintensities, corresponding to N = 0. 75. However, for a typical superlattice containing 1000\nindividual QDs, the number N under 1 W/cm2 excitation is 7.5. Most of these excitons are funneled\nto and accumulate in the emission center, where the resulting N is higher than the onset of\nsaturation. On the other hand, such funneling is not present in the random QD ensemble where\nindividual QDs emit independently and the linearity is maintained.\nThe discussion on page 11 has been modified to address this point.\n3. Similarly, biexciton emission should result in a faster component to the PL emission lifetime\n(see ref. [67] and many other studies across different material systems), but the authors report an\nextended lifetime that they attribute (as far as I can tell) to the exciton migration to an emitting\nsite. It seems that this would result in delayed emission followed by fast emission rather than what\nis observed. A quantitative model (e.g., Monte-Carlo with rate- equation modeling) of the proposed\nemission process is important to confirm the proposed emission mechanisms.\nPL emission lifetime of biexcitons mostly reflects the Auger recombination lifetime, and lifetime\nvalues ranging from tens to hundreds of ps have been reported (in ref. [67, 70, 77] and elsewhere).\nWe have not resolved such short components in our experiments. However, in our photon\ncorrelation experiments we are concerned with the mechanism of cascade-like biexciton-exciton\nemission. Analysis of the photon bunching dynamics can in principle provide information on the\ncascade-like process [84]. Specifically, fitting of the central 𝑔(2)(𝜏) peak and comparison with the\nfitting of the side peaks enables extraction of a fast component which corresponds to the time delay\nbetween the emission of the first (biexciton) and second (exciton) photons in the biexciton-exciton\ncascade, and can be interpreted as the biexciton-exciton emission lifetime. We carried out such\nanalysis on the 𝑔(2)(𝜏) of several superlattices, and present an example in the Fig. 4d. The central\npeak is well fit with a two-exponential function with lifetimes of 1.3 ns and 27 ns, respectively.\nThe side peaks show single exponential decay with a lifetime of 27 ns. We can therefore attribute\nthe 1.3 ns component to the biexciton-exciton delay (or biexciton lifetime). We note that the 1.3\nns value is very likely resolution-limited, and that the actual delay is shorter.\nThe fitting has been added as Fig. 4d and the above discussion including ref [84] added on page\n12.\n4. It would be useful, particularly for the readers of Nature Communications, to connect with a\nbroader community by referencing other material systems (e.g., epitaxial quantum dots, molecules,\ndefects in materials, colloidal nanocrystals). Some of these materials are more developed as\nquantum light sources (including demonstrations of collective effects) and have also had\nsignificant research devoted to exciton/biexciton dynamics.\nWe appreciate this suggestion and added a note in the introduction, including 3 representative\nreferences.\nA note on page 2 ad references [1-3] have been added.\n5. The supplement has 11 figures, some without any description (aside from the brief description\nin the main text). More detail describing the importance of the data shown is required.\nWe agree, the description of the figures has been expanded.\nMinor comments:\nLine 31 (and at various places in the text): It would be better to use more precise language (i.e.,\nbunching with a g^((2) ) (0) up to 2.75).\nThank you, we corrected the expressions.\nLine 88: Awkward phrasing (“Such and other requirements…”). Alternative: “These results…”.\nAlso, references to “other results” should be included, or the phrase shouldn’t be used.\nCorrected, thank you.\nReviewer #2\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is\npart of the Nature Communications initiative to facilitate training in peer review and to provide\nappropriate recognition for Early Career Researchers who co-review manuscripts.\nWe thank the reviewer for using this occasion for rigorous peer-review training of young\nresearchers which is very important for the future of science.\nReviewer #3\nIn this manuscript, Tan et al. demonstrate the fabrication of CsPbBr3 superlattices and their\nunique optical properties with potential applications in quantum light generation for quantum\nphotonic technologies. First, the authors present the synthesis of CsPbBr3 QDs and their self-\nassembly into CsPbBr3 superlattices. The ~10 nm cubic CsPbBr3 QDs self-assemble in a toluene\nrich environment to make 100-500 nm-sized superlattices on a glass substrate. These CsPbBr3\nsuperlattices exhibit emergent optical properties compared to a random ensemble of CsPbBr3\nQDs, including narrower emission linewidths, longer emission lifetimes, multi-level blinking\nbehavior, and photon bunching. The degree of photon bunching and emission lifetime modification\nvaries superlattice-to-superlattice, as well as with the superlattice blinking intensity. Additionally,\na fluorescence lifetime intensity distribution (FLID) analysis was employed. The authors conclude\nthe FLID map suggests blinking results from the charging of the superlattices. Super-resolution\nimaging was used to show that the emission is spatially confined to regions tens of nanometers in\nsize, suggesting excitation migration to localized energy traps. The authors argue that the\nincreased exciton density at these traps leads to higher biexciton formation rates. The combined\nmechanism of exciton migration and biexciton formation results in the observed bunching\nbehavior and longer lifetimes in the CsPbBr3 superlattices.\nWhile the observation of coherent superlattice blinking along with photon bunching is indeed an\nunexpected and impactful result, there are discrepancies within the model proposed by the author\nwhich are not well addressed within the manuscript. Furthermore, the experimental techniques\nused to identify photon bunching appear to be flawed. These concerns are detailed in the list below.\nFinally, a key property of quantum light generation is photon indistinguishability which is related\nto the exciton coherence time and exciton lifetime. To achieve high photon indistinguishability, the\ncoherence time and the lifetime must be on similar timescales. The emission linewidth (inversely\nrelated to the coherence time) of these CsPbBr3 superlattices suggests that the coherence time is\nmany orders of magnitude faster than the lifetime, meaning indistinguishability is extremely low.\nTherefore, the motivation provided for why the authors observations are impactful is lacking. For\nthese reasons, I believe that publication in any form is premature at this time.\nWe thank the reviewer for the overall evaluation and understanding of the manuscript. Regarding\nthe lifetime, the values presented in the manuscript (tens of ns) reflect mainly the long-distance\nmigration of single excitons. In the revised version we added analysis of the photon bunching\ndynamics which can, in principle, provide information on the cascade-like biexciton-exciton\nemission process [84]. Fitting of the central 𝑔(2)(𝜏) peak and comparison with the fitting of the\nside peaks enables extraction of a fast component which can be interpreted as the biexciton-exciton\nemission lifetime. Such biexciton lifetime has sub-ns values, and is thus closer to the coherence\ntime mentioned by the reviewer.\nHowever, we would like to stress that our primary motivation is not to demonstrate a light source\nof highly indistinguishable photons, but rather to report on the discovery of the room temperature\ncollective phenomena and offer explanation of these novel observations that may have\nimplications in future quantum light generation.\nThe fitting has been added as Fig. 4c and the above discussion added on page 12 including ref [84].\nSpecific Comments:\n1) The experimental conditions of the cross-correlation (g2(t)) measurements appear to be\ninconsistent with the assumptions and criteria presented when the method was initially derived in\nthree fundamental ways (DOI: 10.1021/nl104054t).\nThe excitation densities used to measure the biexciton-exciton correlations appear to be too high.\nTo ensure second-order correlations within the same excitation pulse are a result of a biexciton-\nexciton cascade process, the excitation density should be low enough such that negligible\nmultiexcitons that are higher-order than the biexciton are generated. Even at an excitation density\nN = 0.75, the number of triexcitons generated is 25% that of biexciton generation which will lead\nto misinterpretations of the biexciton behavior in the QD superlattice system.\nThe value of N = 0.75 exciton/QD refers to the extreme power of 100 W/cm2 in the power\ndependence in Fig. 4b. For the photon correlation experiments we typically use the power of 1\nW/cm2 which results in N = 0.0075 exciton/QD. In addition, as mentioned in the reply to the\nquestion 1 of Reviewer #1, we assume one-photon absorption creating single excitons (which then\nmigrate to an emission center and form biexcitons), leading to the linear power dependence in Fig.\n4b. We do not see any deviation from the linearity which would indicate direct formation of\nbiexcitons or higher order multiexcitons.\nThe discussion on page 11 has been modified to address this point.\nRelatedly, the authors do not make it clear if they measure the CsPbBr3 superlattices at similar\npowers as single CsPbBr3 QDs, assuming similar excitation densities. The absorption cross-\nsection of the superlattices should scale with the number of QDs within the lattice. If the\nsuperlattices are excited at similar powers as the single QD, the excitation density of the\nsuperlattice system may be significantly underestimated in this study leading to many higher-order\nmultiexcitons being generated in this system, greatly complicating the picture of the excited system.\nThe authors should measure the absorption cross-section of the CsPbBr3 superlattices and\nmeasure the correlations at low excitation densities such that minimal multiexcitons higher than\nthe biexciton are generated within the entire superlattice.\nThank you for pointing out this important aspect. Unfortunately, we are unable to directly measure\nthe absorption cross-section of individual superlattices because we do not know the size of a\nparticular superlattice emitting under the microscope. The size information has been obtained\nindependently using SEM, and correlating the two techniques is beyond our experimental\ncapabilities. Again, the superlattice is measured with the power of 1 W/cm2 corresponding to N =\n0.0075 exciton/QD. This would result in N = 7.5 excitons per superlattice. However, we assume\nhere that the absorption proceeds spatially randomly within the superlattice and occurs\nindependently on individual QDs within the structure. Therefore, the predominant process is linear\nabsorption creating a randomly distributed ensemble of single exciton which then accumulate in\nthe emission center.\nThe discussion on page 11 has been modified to address this point.\nThe repetition rate of the laser should be slow enough such that the emissive system is guaranteed\nto be in the ground state before the next excitation pulse. Otherwise, multiple\nexcitons/multiexcitons may be generated at a faster rate than Poissonian statistics predict,\nbreaking down the above-mentioned assumption of the biexciton-exciton correlation\nmeasurement. With lifetimes as long as 69 ns, I would expect repetition rates of 1 MHz or lower\nwould be necessary to ensure the excited state fully decays within the excitation window. The\nauthors utilized repetition rates of 2.5 MHz and faster. The authors should either justify the validity\nof their chosen repetition rates or repeat the measurements to conform with the assumptions of\nthe methodology.\nYes, this is another very important point. We carried out additional experiments using the\nrepetition rates of 1 MHz and 5 MHz, and the results show that the effect of photon bunching is\nclearly observed at these two frequencies. The results are shown in Supplementary Fig. 6.\nThe bunching experiment with 1 MHz rate is has been mentioned on page 5 and Supplementary\nFig. 6 has been added.\n2) The resolution of the correlation figures throughout the paper are low. Each peak within the\ng2(t) appears to have ~10-15 data points which is not sufficient to identify relevant features within\nthe correlations, such as bunching. Typically, correlations are binned with a similar resolution as\na lifetime (1 ns). The authors should work up the correlation data with a finer time resolution to\nbetter resolve the bunching effects.\nAgain, thank you for pointing this out. We redraw all the data in the figures with the 1 ns resolution.\nThe Fig. 2, and Supplementary Figs. 5, 7, 8, 9 have been modified.\n3) The authors do not report the concentration of the QD toluene solutions that are used to perform\nthe self-assembly. It is further unclear how, when they measure the emission of single CsPbBr3\nsuperlattices, they ensure that the emission is from only one superlattice. The authors should\nelucidate how they identify isolated superlattices in their spectroscopic methods. Additionally, if\npossible, the authors should present raster images of the emission from the CsPbBr3 films to show\nthat the superlattices are sufficiently isolated from one another.\nThe information on the concentration (2.29 x 10-7 mol/L ) has been added in the Methods section.\nThe individual superlattices are identified in wide-field imaging mode in the CCD fluorescence\nimages as isolated bright spots. An examples presented in Supplementary Fig. 3 shows several\nbright spots that are well separated, ensuring sufficient isolation of individual particles.\nThe Methods section has been modified. The Supplementary Fig. 3 has been added and described\non page 3.\n4) Some superlattices have irregular shapes as shown in Fig. S2. The authors should report what\npercentage of the superlattices have this irregular shape. The authors should present widefield\nTEM images of the films CsPbBr3 superlattices to confirm these distributions. The authors should\nfurther explain how they confirm whether they are looking at an irregular shaped superlattice vs.\na cubic superlattice when measuring the emission from the superlattices.\nWe used SEM to characterize the superlattice shape as it proved difficult to grow larger numbers\nof well separated superlattices directly on the TEM grids. We measured the shape in total of 20\nparticles, 9 of which (approximately 50 %) showed the regular cubic shapes. We added more data\nin the SI in Supplementary Fig. 2. As mentioned above, the SEM and fluorescence experiments\nare carried out independently and correlating the two techniques is beyond our experimental\ncapabilities. We therefore cannot directly relate the shape and the PL properties on individual\nparticles, and we base our results on the assumption that the (mostly) two-level blinking originates\nfrom the cubic-shaped superlattices. Also, the irregular-shaped superlattices are significantly\nlarger that the regular ones, and as such emit with higher intensities and small (if any) degree of\nbunching. We do not include such superlattices in the statistical analysis presented in the paper.\nThe Supplementary Fig. 2 has been replaced.\n5) In Fig. 2, the authors filter the emission by intensity and categorize the high and low intensity\nsections as the ON-state and grey-state emission, respectively. I would argue that the ON-state is\nbroader than is suggested by the filtering cut-off around 0.2 kc/ms, and is partially captured within\nthe grey-state. Because of this it appears possible that the grey-state is actually an OFF-state.\nThe authors should utilize a range of cutoffs to clearly demonstrate the presence of a grey state rather\nthan an artifact due to thresholding.\nWe agree that assigning the grey state based on the intensity level only can be ambiguous. Here,\nwe also confirmed that the grey state has PL lifetime characteristic of perovskite QDs, and based\non the suggestion of the reviewer #4 we also verified that the PL spectrum of the grey state\ncorresponds to the perovskite spectrum. None of these would be expected for completely OFF\nstate where the signal would be originating from background scattering or impurity fluorescence.\nThe classification of grey state has been added on page 4, 6 and 12.\n6) On pg. 8, the authors suggest that the double Gaussian fit to the CsPbBr3 superlattice emission\nspectrum (Fig. 4a) with an energy splitting of 23 meV is evidence of biexciton-exciton cascade\nemission. It is difficult to believe in this evidence, because given the low excitation densities in\nthese measurements, biexciton emission should contribute less than 5% of the total emission,\nbefore accounting for reduced emission efficiencies of multiexcitons due to Auger recombination.\nThe integrated intensities of the two Gaussians in the emission spectrum in Fig. 4a appear to have\napproximately equal areas which would be unreasonable if the lower energy Gaussian was\nbiexciton emission. To the best of our knowledge, the prevailing wisdom within the community\nascribes the lower energy shoulder of the room temperature ensemble emission spectrum of\nCsPbBr3 to the inelastic phonon scattering of optical phonons. The authors should provide more\nevidence that the lower energy shoulder results from biexciton emission via power dependent\nemission spectra.\nWe based our explanation of the PL spectral shape of the superlattices on the fact that neither the\nrandom ensemble nor single QDs show the low-energy shoulder. We added examples of single\nQD spectra in the SI (Supplementary Fig. 17), and these are characterized by mostly Lorentzian\nlineshapes. Also, as we do not directly excite the biexcitons, the biexciton contribution can be\nlarger than the 5% the reviewer mentions, and would depend on the probability of the biexciton\nformation in the emitting site. This interpretation is also not contradicted by the comparison of PL\nspectra between the ON and grey states (measured upon the suggestion of the reviewer #4) which\nshows that there is no difference between the two because the GREY state in many cases does\nshow bunching similar to the ON state. This is a consequence of the fact that (as mentioned in\nthe main text) the photon bunching and collective blinking are separate independently occurring\nphenomena.\nGiven the above, we believe that the exciton binding energy is one of the plausible explanations\nof the spectral shape. However, we admit that the contribution of the low-energy shoulder is\nrelatively high, and that other explanations might be possible, such as the exciton-phonon\ncoupling proposed by the reviewer. However, such phenomenon would have to be specific to the\nsuperlattices as it is not observed for the constituting QDs.\nThe above discussion and Supplementary Fig. 17 have been added in the rearranged description\nof the model on page 13.\n7) As the authors show in Fig. S8, the strong antibunching in the g2(t) of a single CsPbBr3\nindicates the biexciton quantum yield of single CsPbBr3 QDs is very low. However, the center\npeak in the CsPbBr3 superlattice correlation is strongly bunched. These results appear entirely\ncontradictory. Since the biexciton quantum yield is low for single CsPbBr3 QDs, the authors need\nto justify why formation of assemblies drastically changes the single QD multiexciton properties.\nWe agree that these results may appear initially counterintuitive. Our model assumes that even\nthough direct biexciton formation in individual QDs is very low (also because of the low excitation\nintensity), the individual QDs in the superlatice work as an antenna and funnel individual excitons\ninto the emission center where the increased exciton density and attractive interaction lead to the\nbiexciton formation. We also note that in literature there has been nice demonstration of the\ntransition from antibunching to bunching when the number of coupled QDs increased from 1 to 3\n[49]. Also, very recently both single-exciton anti-bunching and multiexciton bunching have been\ndemonstrated in single large perovskite nanocrystals [50].\nThe above point has been mentioned on page 6 and references [49,50] were added.\n8) The authors show in Fig. 4c that the degree of bunching increases with decreasing PL intensity.\nIt is unclear why bunching would increase with lower PL intensities, and the authors do not\nprovide further explanation. At lower PL intensities, it is possible for background counts and/or\nlaser scatter to introduce artifacts in the correlations, artificially increasing or decreasing the\ncorrelation contributions to the center peak. Given the concern of potential artifacts, the authors\nshould justify the anti-correlation of the bunching and PL intensity as it is unintuitive why this\nbehavior occurs.\nThis phenomenon of increasing degree of bunching with decreasing excitation power (Fig. 4a) has\nbeen first observed on GaAs quantum dots [78], and later confirmed on InGaAs QDs [81], as\nwell as perovskites [10], and recently [50]. It is caused by the fact that at high excitation intensities\n(high number of biexcitons) the probability of individual photon emission (from biexciton to\nexciton) surpasses the correlated photon pair emission from the biexciton-exciton cascade. The\nprocess has been theoretically described in the ref. [78] and we present the explanation in the\nSupplementary information.\nThis discussion has been added on page 9.\nReviewer #4\n(Remarks to the Author)\nSee attached pdf file."}, {"round": 2, "reviewer_comments": {"reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "See attached document", "reviewer_4": "The authors have addressed the reviewer's concerns.", "reviewer_5": "My comments to the editor and author are attached as a PDF."}, "author_response": "Response to Reviewers\nWe sincerely thank the reviewers for their additional time and work spent on our\nmanuscript. We appreciate the reviewers’ honest effort to help improve the work. Below,\nwe provide a point-by-point response to the comments raised. We include only the new\ncomments that require further response. The original reviewers’ comments are in italics\ngrey, our previous replies in italics blue, the new comments on the revision in italics black,\nour new responses in normal fonts and the changes to the manuscript in blue. Changes in the marked\nversion of the submitted revised manuscript are also highlighted in blue.\nAs part of the second revision we carried out further experiments to fully address the\nreviewers’ comments, and added 7 new Supplementary figures. We believe that in the two\nrounds of extensive revisions, both of which included substantial new experimental\nevidence wherever required by the reviewers, we succeeded in addressing all the\nreviewers’ criticism. Also, we provided further convincing arguments, based on\nexperimental facts, theoretical model, published literature and widely accepted consensus\nin the field, to support our mechanism and interpretation of the phenomena, and to\nalleviate the Reviewers’ concerns. We hope that the manuscript now meets the standards\nfor publication in Nature Communications. Detailed answers to the individual comments\nare provided below.\nReviewer #1\nI am satisfied with the responses to my comments, as well as the changes made to the\nmain text and supplement. I have also read through the other referee reports and find that\nthe authors have adequately addressed the concerns in their responses and through\ncorresponding changes to the manuscript.\nReviewer #3\nSpecific Comments:\n1) The experimental conditions of the cross-correlation (g2(t)) measurements appear to\nbe inconsistent with the assumptions and criteria presented when the method was initially\nderived in three fundamental ways (DOI: 10.1021/nl104054t). The excitation densities\nused to measure the biexciton-exciton correlations appear to be too high. To ensure\nsecond-order correlations within the same excitation pulse are a result of a biexciton-\nexciton cascade process, the excitation density should be low enough such that negligible\nmultiexcitons that are higher-order than the biexciton are generated. Even at an\nexcitation density N = 0.75, the number of triexcitons generated is 25% that of biexciton\ngeneration which will lead to misinterpretations of the biexciton behavior in the QD\nsuperlattice system.\nThe value of N = 0.75 exciton/QD refers to the extreme power of 100 W/cm2 in the power\ndependence in Fig. 4b. For the photon correlation experiments we typically use the power\nof 1 W/cm2 which results in N = 0.0075 exciton/QD. In addition, as mentioned in the\nreply to the question 1 of Reviewer #1, we assume one-photon absorption creating single\nexcitons (which then migrate to an emission center and form biexcitons), leading to the\nlinear power dependence in Fig. 4b. We do not see any deviation from the linearity which\nwould indicate direct formation of biexcitons or higher order multiexcitons.\nThank you for clarifying the fluences used in the excitation of single CsPbBr3\nsuperlattices. Although the excitation density N = 0.0075/QD (assuming the absorption\ncross-section scales linearly with the number of NCs in the superlattice) should primarily\nlead to single exciton generation, the total excitation density N = 7.5 of single CsPbBr3\nsuperlattices remains a serious concern. If the exciton funneling mechanism proposed by\nthe authors is true, then it is likely that there are many higher-order multiexcitons that\nare formed as excitons funnel to emission centers. As the number of higher-order\nmultiexcitons present in the system increases, it becomes difficult to interpret the second\norder cross-correlation measurement 𝑔(2)(𝜏). There is no way to distinguish a biexciton-exciton correlation from a triexciton-biexciton correlation from a triexciton-exciton\ncorrelation (and so on). We do not believe it is reasonable to conclude that bunching is a\nresult of the biexciton-exciton cascade process while there are likely many higher-order\nmultiexcitons present in the system. Further studies should be performed at much lower\nexcitation densities (N < 0.1; N < 0.0001/QD) to ensure that there is a negligible\nlikelihood of three or more excitons present in the CsPbBr3 superlattice at any given time.\nWe understand the reviewer’s concern. The total exciton density of N = 7.5 per single\nsuperlattice at 1 W/cm2 is an ideal estimate. Assuming a realistic value of PLQY of 80%,\nthis number would drop to N = 6, as 20% of the excitons would decay non-radiatively\nbefore reaching the center. Further, on the funneling pathways from different parts of the\nsuperlattice the actual density would be much smaller, meaning that higher-order\nmultiexcitons would unlikely form on the way to the emission center. To obtain more\nexperimental evidence, we measured the bunching characteristics with 0.5 W/cm2, which\ncorresponds to N ~ 3 excitons (or 0.003 exciton/QD), as proposed by the reviewer. The\npower of 0.5 W/cm2 represents the lowest possible limit of the excitation intensity to\nobtain reliable 𝑔(2)(𝜏) data with reasonable S/N ratio. The result for the photon bunching\nobtained on a superlattice at 0.5 W/cm2 and 1.0 W/cm2 excitation intensities are shown\nas an example in the Supplementary Fig. 18. The superlattice shows significantly higher\ndegree of bunching for the lower intensity (2.07 vs 1.42). This result indicates that\nincreasing the intensity does not create more multiexcitons that would contribute to the\nincrease in bunching (such multiexciton-induced increase in bunching with excitation\nintensity was reported, e.g., in [PRL 94, 087403 (2005)]), and that the trend is in very\ngood agreement with the Fig. 4a and Eq. (3), i.e., with the excitation power dependence\nexpected for biexciton cascade emission. The result thus fully confirms our proposed\nmodel.\nDescription of the bunching at low excitation power and the above reference (as [84])\nwere added on page 9, and the Supplementary Fig. 18 was added in the SI.\n2) The resolution of the correlation figures throughout the paper are low. Each peak\nwithin the g2(t) appears to have ~10-15 data points which is not sufficient to identify\nrelevant features within the correlations, such as bunching. Typically, correlations are\nbinned with a similar resolution as a lifetime (1 ns). The authors should work up the\ncorrelation data with a finer time resolution to better resolve the bunching effects.\nAgain, thank you for pointing this out. We redraw all the data in the figures with the 1 ns\nresolution.\nWe appreciate the effort made to change the resolution of the correlation figures. The\nadded resolution enables further insight and analysis of the correlation dynamics. It is\ninteresting to see that the bunching effects are primarily within a small number of data\npoints in the center peak. Regarding Supplementary Fig. 6, is the degree of bunching of\n1.3 calculated based on the ratio of the peak intensities, or the integrated areas of the\ncenter and side peak?\nThe degree of bunching is based on the ratio of peak intensities.\nWe added this information in the Methods section.\n4) Some superlattices have irregular shapes as shown in Fig. S2. The authors should\nreport what percentage of the superlattices have this irregular shape. The authors should\npresent widefield TEM images of the films CsPbBr3 superlattices to confirm these\ndistributions. The authors should further explain how they confirm whether they are\nlooking at an irregular shaped superlattice vs. a cubic superlattice when measuring the\nemission from the superlattices.\nWe used SEM to characterize the superlattice shape as it proved difficult to grow larger\nnumbers of well separated superlattices directly on the TEM grids. We measured the\nshape in total of 20 particles, 9 of which (approximately 50 %) showed the regular cubic\nshapes. We added more data in the SI in Supplementary Fig. 2. As mentioned above, the\nSEM and fluorescence experiments are carried out independently and correlating the two\ntechniques is beyond our experimental capabilities. We therefore cannot directly relate\nthe shape and the PL properties on individual particles, and we base our results on the\nassumption that the (mostly) two-level blinking originates from the cubic-shaped superlattices.\nAlso, the irregular-shaped superlattices are significantly larger that the regular ones, and\nas such emit with higher intensities and small (if any) degree of bunching. We do not\ninclude such superlattices in the statistical analysis presented in the paper.\nThank you to the authors for providing additional information on the cubic vs. irregularly\nshaped CsPbBr3 superlattice structures. The authors make a reasonable justification to\ndetermine whether they are measuring the emission coming from a cubic vs. irregularly\nshaped superlattice, however it would be helpful to the readers to understand roughly\nhow much brighter the irregularly shaped superlattices are compared to the cubic shaped\nsuperlattices.\nThe maximum emission intensity of the multi-level blinking particles is about twice or\nmore than that of the two-level blinking particles.\nThis information was added in the description of Supplementary Fig. 2.\n5) In Fig. 2, the authors filter the emission by intensity and categorize the high and low\nintensity sections as the ON-state and grey-state emission, respectively. I would argue\nthat the ON-state is broader than is suggested by the filtering cut-off around 0.2 kc/ms,\nand is partially captured within the grey-state. Because of this it appears possible that\nthe grey-state is actually an OFF state. The authors should utilize a range of cutoffs to\nclearly demonstrate the presence of a grey state rather than an artifact due to thresholding.\nWe agree that assigning the grey state based on the intensity level only can be ambiguous.\nHere, we also confirmed that the grey state has PL lifetime characteristic of perovskite\nQDs, and based on the suggestion of the reviewer #4 we also verified that the PL spectrum\nof the grey state corresponds to the perovskite spectrum. None of these would be expected\nfor completely OFF state where the signal would be originating from background\nscattering or impurity fluorescence.\nWe want to thank the authors for their response. However, we don’t believe our initial\ncomment was addressed. We apologize if our initial comment was not clear so we will\nattempt to restate it below. We were curious to know what the grey-state lifetime would\nlook like if the threshold was set at an even lower value such as 0.05 kc/ms (marked by\nthe black cutoff line in the below figure). We were initially concerned that some of the\nON-state was captured within the grey-state due to the arbitrarily created thresholding\nvalue around 0.22 kc/ms. We have marked this value on Fig. 2a for clarity.\nThank you for suggesting this important confirmation. We re-analyzed the trace for ‘pure’\ngrey state (below 0.05 kc/ms) and for ‘pure’ ON-state (for 0.375 - 0.45 kc/ms). The data\nare presented in the Supplementary Fig. 14, and show that there is no difference in the\nlifetimes from the ON-states. The grey state does show slight shortening of the long\ncomponent from 22 ns to 18 ns. This means that there might be some ON-states admixed\nwith the grey state at the original 0.22 kc/ms threshold (as the reviewer proposed), but it\nalso shows that the intensity values below 0.05 kc/ms correspond to a true grey state and\nnot and OFF-state.\nA note was added on page 6 and the Supplementary Fig. 14 was added in the SI.\nThe integrated intensities of the two Gaussians in the emission spectrum in Fig. 4a appear\nto have approximately equal areas which would be unreasonable if the lower energy\nGaussian was biexciton emission. To the best of our knowledge, the prevailing wisdom\nwithin the community ascribes the lower energy shoulder of the room temperature\nensemble emission spectrum of CsPbBr3 to the inelastic phonon scattering of optical\nphonons. The authors should provide more evidence that the lower energy shoulder\nresults from biexciton emission via power dependent emission spectra.\nWe based our explanation of the PL spectral shape of the superlattices on the fact that\nneither the random ensemble nor single QDs show the low-energy shoulder. We added\nexamples of single QD spectra in the SI (Supplementary Fig. 17), and these are\ncharacterized by mostly Lorentzian lineshapes. Also, as we do not directly excite the\nbiexcitons, the biexciton contribution can be larger than the 5% the reviewer mentions,\nand would depend on the probability of the biexciton formation in the emitting site. This\ninterpretation is also not contradicted by the comparison of PL spectra between the ON\nand grey states (measured upon the suggestion of the reviewer #4) which shows that there\nis no difference between the two because the GREY state in many cases does show\nbunching similar to the ON state. This is a consequence of the fact that (as mentioned in\nthe main text) the photon bunching and collective blinking are separate independently\noccurring phenomena. Given the above, we believe that the exciton binding energy is one\nof the plausible explanations of the spectral shape. However, we admit that the\ncontribution of the low-energy shoulder is relatively high, and that other explanations\nmight be possible, such as the exciton-phonon coupling proposed by the reviewer.\nHowever, such phenomenon would have to be specific to the superlattices as it is not\nobserved for the constituting QDs.\nThe above discussion and Supplementary Fig. 17 have been added in the rearranged description\nof the model on page 13.\nThank you to the authors for their insightful response. The language added to the\nmanuscript acknowledges the various plausible explanations. In this context, it seems\nreasonable to propose the biexciton binding energy as one of the possibilities. We will\nnote however, if we assume that biexciton formation is much higher than what I was\nproposing in our initial comment (5% of total emission), then we have a similar concern\nas stated in 1) regarding the exciton/multiexciton distribution in the single CsPbBr3\nsuperlattices that have implications on the accurate interpretations of the 𝑔(2)(𝜏)\nbunching experiments.\nThe multiexciton question was addressed in response to Comment 1).\n7) As the authors show in Fig. S8, the strong antibunching in the g2(t) of a single CsPbBr3\nindicates the biexciton quantum yield of single CsPbBr3 QDs is very low. However, the center\npeak in the CsPbBr3 superlattice correlation is strongly bunched. These results appear\nentirely contradictory. Since the biexciton quantum yield is low for single CsPbBr3 QDs,\nthe authors need to justify why formation of assemblies drastically changes the single QD\nmultiexciton properties.\nWe agree that these results may appear initially counterintuitive. Our model assumes that\neven though direct biexciton formation in individual QDs is very low (also because of the\nlow excitation intensity), the individual QDs in the superlattice work as an antenna and\nfunnel individual excitons into the emission center where the increased exciton density\nand attractive interaction lead to the biexciton formation. We also note that in literature\nthere has been nice demonstration of the transition from antibunching to bunching when\nthe number of coupled QDs increased from 1 to 3 [49]. Also, very recently both single-exciton anti-bunching and multiexciton bunching have been demonstrated in single large\nperovskite nanocrystals [50].\nThank you to the authors for their response. We maintain concern about the mechanistic\nconclusions of the bunching phenomenon observed in these CsPbBr3 superlattices for a\nfew reasons. First, references [10,49,50,78,81] are on coherent systems at cryogenic\ntemperatures (4 Kelvin) where indistinguishability of photon emission can be reasonably\nachieved. Indistinguishability is attributed as a key property in order to achieve the\nladder-like emission that results in bunching. Reference [49] shows the difference in\nbunching as 2 QDs are tuned in and out of indistinguishability. To our knowledge, photon\nindistinguishability has never been demonstrated in any QD system at room temperature,\nsuggesting it is unlikely the CsPbBr3 superlattices undergo the same process to produce\nbunched 𝑔(2)(𝜏) measurements. Second, the QD systems in references [49,50,78]\nrequired specific engineering solutions to achieve indistinguishability, and therefore\nbunching. For ref [49], the emission from the 3 coupled QDs were not intrinsically\nindistinguishable. The emission energy was tuned via strain and electric field effects to\nplace the 3 QD emission energies in resonance with one another. For ref [50], the\nsingle large perovskite nanocrystal size was tuned so that the biexciton-to-exciton transition\nwould be in resonance with the exciton-to-optical-phonon transition. Additionally, the\nexciton emission and trion emission were filtered in order to observe bunching. For ref\n[78], the Hanbury Brown Twiss 𝑔(2)(𝜏) spectroscopic setup was spectrally filtered such\nthat one detector only accepted biexciton emission and the other detector only accepted\nexciton emission, leading to the bunching effect. If the spectroscopic setup was not\nspectrally filtered, bunching would not be expected. The observed bunching on the\nCsPbBr3 superlattices is extremely interesting, however I do not believe there is sufficient\nevidence at this time to propose a precise mechanism of bunching. As shown in the\nreferences cited by the authors, coherent ladder-like biexciton-exciton cascade requires\nstringent design in other single quantum dot or small number of coupled quantum dot\nsystems at cryogenic temperatures. Therefore, it seems highly unlikely that the room\ntemperature CsPbBr3 superlattice (comprised of ~1000 coupled NCs) is undergoing a\ncoherent ladder-like emissive process that could result in the proposed bunching process.\nWe believe further investigation into the mechanism of bunching is required before\nrecommending publication.\nThe reviewer’s original question was that ‘… strong antibunching in the g2(t) of a single\nCsPbBr3 indicates the biexciton quantum yield of single CsPbBr3 QDs is very low.\nHowever, the center peak in the CsPbBr3 superlattice correlation is strongly bunched.\nThese results appear entirely contradictory.’ We replied to explain the contradiction and\nreiterated that in our model direct biexciton formation in individual QDs maybe indeed\nlow but the superlattice works as an antenna and funnels the excitons into the emission\ncenter where the increased exciton density and attractive interaction lead to the\nbiexciton formation. For illustration, we cited a few examples of literature that did show\ntransition from antibunching to bunching with increasing size, but by no means we meant\nto imply that our phenomena are equivalent to those observed in the references. As the\nreviewer notes, the low-temperature experiments required filtering or energy tuning because\nof the clear separation of the exciton-biexciton spectra. At room temperature the two spectra\nlargely overlap, and filtering is not required. It might be that ‘photon indistinguishability has\nnever been demonstrated in any QD system at room temperature’, but ordered (ladder-\nlike) photon emission from multiexciton states of a QD at room-temperature measured\nby photon correlation techniques has been reported more than 20 years ago and many\ntimes since then. [PRL 94, 087403 (2005)]. This fact well supports our interpretation of\ncascade-like biexciton emission.\nWe agree with the reviewer that it is unlikely that the whole CsPbBr3 superlattice\ncomprised of ~1000 coupled NCs would undergo coherent ladder-like emission at room\ntemperature, but that is not what we are claiming. Again, the whole superlattice functions\nas an antenna and the emission originates from very small region in the superlattice. This\nregion may contain ~ 10 or less NCs, and the cascade-like emission is confined to this\nwell-ordered part of the superlattice. The strongest evidence for the cascade-like emission\nis the excitation intensity dependence in Fig. 4a (and indirectly also the PL intensity\ndependence in Fig. 2c). This dependence follows a theoretical model (derived by two\nindependent ways in the SI) that explains the trend based on the biexciton cascade-like\nemission. To the best of our knowledge there is no other model that would explain\ndecrease of photon bunching with increasing excitation intensity. We would be of course\nopen to other interpretations of the photon bunching phenomenon, but so far found none\nthat would explain all the experimental observations; we also believe that our model of\nthe bunching mechanism is not contradicting any existing knowledge in the field.\nReviewer #4\nIn this study, Tan et al. report on observed collective blinking and photon bunching from\nindividual perovskite quantum dot superlattices. The Reviewer has previously suggested\nthat the paper has an issue in that two phenomena are observed: collective blinking and\nphoton bunching. Were both phenomena to be connected, this would be rather interesting.\nHowever, the authors do not find a connection between them. Instead, separate studies\nare done to try and explain observed blinking and bunching. The problem with this is that\nthere is a certain depth that would ideally be met so that both separate sub studies would\nbe deep and truly impactful. Unfortunately, the Reviewer is still of the opinion that this\nbar has not been cleared.\nThank you for bringing up this aspect of the work that we have not sufficiently stressed\nin the current version of the manuscript. The impression might be that the blinking and\nphoton bunching are two separate phenomena happening by coincidence on the same\nsuperlattice, but they are in fact both deeply related via the presence of the confined\nemission center. Both are manifestations and consequences of the antenna effect of the\nsuperlattice and of funneling of the excitons to this center. The high density of the excitons\nin the small confined area (comprising ~ 10 or less QDs) is then responsible for both the\ncollective blinking and the photon bunching.\nWe have not delved into the mechanism of blinking of CsPbBr as this has been\ncovered extensively in literature, and instead concentrated on describing the novel aspect\nof the collective blinking. But we followed the reviewer’s recommendation and\nelaborated more on the mechanism by examining the blinking at lower excitation\nintensities. We believe that with respect to the photon bunching we carried out extensive\ncharacterization of this novel observation on a perovskite superlattice, and put forward a\nmechanism (including theoretical model) that succeeds in explaining all the experimental\ndata and is not contradicting any existing knowledge in the field.\nWe have rearranged and expanded the conclusion part as a Discussion section on\npages 13-14, and summarized our explanation of the phenomena, including the\nrelationship between the collective blinking and photon bunching.\nComment 1. The Reviewer would like the authors to provide more details of their\nexperiment. For example, what was the integration time of their EMCCD when acquiring\nthe trajectories being shown? What is the integration time in Figure 2a? If the integration\ntime can be lowered, then it might be interesting to observe what happens to the\nappearance of the trajectory and whether the bimodal “ON” and “GREY” designations\nstill hold. This comment also applies to the cases where there are multiple “ON” states.\nIt also applies to the non-blinking superlattice in Figure S7. Presumably lowering the\nintegration time in Figure S7 would show that there are actually “GREY” periods.\nAuthor Response: The integration times using the EMCCD camera were 30 ms, the times\nusing the APD were 10 ms. The blinking trace in Fig. 2a was obtained using the APD\nwith the 10 ms integration time. For both detectors these are practically the limiting\nvalues, making it difficult to lower the integration time.\nReviewer new comment: The Reviewer understands that the authors are using a Picoharp\ninstrument. In this case, it is possible to run the experiment using time tagged time\nresolved mode (TTTR) where each received photon is time stamped. Consequently, photon\ntrajectories can be constructed with arbitrary bin widths. For example, one can bin the\nreceived photons with a 10 ms overall integration or a 30 ms integration etc… The\nReviewer therefore questions the authors assertion that 10 ms is a lower limit when using\ntheir stated photon counting system.\nThe reviewer is absolutely correct and we apologize for the somewhat misleading\nresponse. In the confocal mode we did use the TTTR acquisition mode, and the\nHydraHarp software used for the analysis enables integration times as short as 0.2 ms. In\nthe Supplementary Figs. 24 and 25 we re-analyzed two of the original blinking traces\nusing several integration times. It is clear that with the shorter times (< 2 ms), the\nindividual intensity levels in the histogram get smeared out due to the increased noise. At\nthe same time, shortening of the bin time did not reveal any additional features such as\nunresolved OFF or grey states. We therefore used the 10 ms bin as the optimum\ncompromise between the S/N ratio and the resolution.\nThis explanation was added in the Methods section on page 15, and the Supplementary\nFigs. 24 and 25 were added in the SI.\nComment 2. Related to this, the Reviewer requests that the authors explicitly state the\nexcitation intensities used to acquire all blinking trajectories. Additionally, (1) the\nauthors should state how superlattices are excited. Are they being illuminated as a whole\n(illumination covers the entire superlattice) or does the laser focus to the center of the\nsuperlattice. In short, report the spot size of the laser. (2) What do trajectories look like\nat lower excitation intensities? Do they show say more “GREY” behavior? Or do they\nlook self-similar at all excitation intensities.\nAuthor response: The excitation intensity was kept at 1 W/cm2 for all data throughout the\nmanuscript except the excitation intensity dependence. The PL lifetime and photon\ncorrelation experiments were carried out in confocal mode with the laser diameter of\napproximately 500 nm. For imaging and spectroscopy a wide-field fluorescence mode\nwas used. We have not specifically looked into the intensity dependence of the blinking\ntraces. We do have some data for 0.7 W/cm2 excitation which look similar but it is difficult\nto make any conclusions due to the lack of statistics.\nReviewer new comment: The Reviewer would still prefer that the authors analyze their\nblinking behavior at different excitation intensities, especially lower intensities. This is\nbecause GREY states are suggested to be due to positively or negatively charged trions.\nIt would therefore seem that one way to indirectly test for this is to establish that the\nprevalence of such GREY states decreases with decreasing photogenerated carrier\ndensity/excitation intensity.\nWe followed the Reviewer’s suggestion and analyzed blinking data for a series of\nexcitation powers from 0.05 W/cm2 to 5.0 W/cm2. The data are presented in the\nSupplementary Fig. 17, and do show a trend of increase of grey states fraction with\nincreasing power. However, we present this as an illustration, as more systematic study\nwould be required to make a conclusive statement, and this would be outside the scope\nof the current manuscript.\nThe Supplementary Fig. 17 and its description were added on page 8.\nComment 4. Continuing the thought more. The authors should state more clearly how\ntheir blinking trajectories are acquired. It would seem that raw (total) intensities are used.\nThe samples are excited with the near UV laser. A long pass filter is then used to block\nthe laser and all emission (irrespective of color) is collected by the EMCCD camera. Is\nthis correct? If so, please state this more explicitly in the main text methods section.\nAlternatively, provide a detailed SI section on how measurements were conducted.\nAuthor response: Yes, the reviewer is correct. We apologize for not providing sufficient\ndetails of the experimental setup and conditions to enable other reproduce the results.\nReviewer new comment: The Reviewer still requests that the authors provide more\ninformation on how EMCCD-based trajectories were constructed. This information is not\nprovided in the revised methods section. Presumably, images are acquired with 30 ms\nintegration times. Then something is done to the image to construct the trajectory. This\ncould involve summing all the pixels associated with the superlattice. It could also be that\nonly one pixel is used. The reader or Reviewer does not know. Such information should\nbe provided for complete transparency. The same comments apply to how data for Figure\nS15 was acquired.\nThe blinking traces were measured either using EM-CCD camera with acquisition time\nof 30 ms (as stated in the Methods section) or using the APD with acquisition time of 10\nms (also in the Methods section). The EM-CCD based trajectories were constructed by\nidentifying a superlattice emission spot in the image, cropping a small area around the\nspot and integrating the intensity over the area. Area of the same size was then cropped\nand integrated near the superlattice spot where there was no emission visible, and this\narea was used as a background that was subtracted from the integrated superlattice signal.\nThis process was repeated for all images in the series and the background-subtracted\nintensity was plotted as the blinking trajectory.\nThe APD based trajectories are measured using the TTTR mode and constructed using\nthe HydraHarp software. To acquire the background, identical measurements are carried\nout from nearby locations without detectable emission. All blinking trajectories presented\nare background-subtracted in this way.\nThe measurement method used was added to the caption of each blinking figure. Detailed\ndescription of the construction of the blinking trajectories was added in the Methods\nsection on page 15.\nComment 5. The reason the question arises is because the Reviewer is curious about the\ncolor of the photons during “ON” versus “GREY” periods. Because of issues associated\nwith reabsorption-induced redshifting and intra superlattice energy transfer, it is possible\nthat the color of the photons during “ON” and “GREY” periods differ (or if the authors\ncare to invoke the biexciton argument then this is recast as biexciton versus non-biexciton\nemission). The Reviewer would therefore prefer if the authors use their spectrometer and\nEMCCD camera to collect spectral trajectories and from there generate intensity\ntrajectories. This is the spectral analog of using the picoquant photon counting\nelectronics to generate the “ON” versus “GREY” TCSPC decays in Figure 2d.\nAuthor response: Thank you for this suggestion. We followed it and measured the spectra\nduring the blinking ON and GREY periods. The result presented in Supplementary Fig.\n15 shows no significant difference between the ON and GREY state spectra. On one hand,\nthis result confirms the assignment of the low-intensity level to the GREY state (as\nopposed to an OFF state). At the same time, the result does not contradict the\ninterpretation of the low-energy spectral shoulder as originating due to the biexciton\ncoupling, because the GREY state in many cases does show bunching similar to the ON state.\nThis is a consequence of the fact that (as mentioned in the main text) the photon bunching\nand collective blinking are separate independently occurring phenomena.\nReviewer new comment: The Reviewer would prefer to see that the data in Figure S15 is\nrepresentative. As far as the Reviewer understands only one such trajectory exists. Please\nprovide additional trajectories to establish that GREY is indeed spectrally the same as\nON (at least at room temperature).\nWe are not sure what lead the Reviewer to understand that we measured only one\ntrajectory. We have many such data and added more in the revised SI.\nThe re-numbered Supplementary Fig. 20 was revised with two more examples.\nComment 6. Related to the above point is whether any evidence of spectral diffusion is\nobserved during blinking trajectories. Are there any spectral dynamics one should be\naware of? Consider the following limiting case scenario. The authors acquire trajectories\nusing a bandpass filter (recall that no details were given). The superlattice spectrum is\ndiffusing. Sometimes the emission is centered in the middle of the bandpass. Other times\nit switches to a different mean energy and moves away from the bandpass center. The\nintensity recorded by the EMCCD drops. This could give rise to spurious “ON” states\nand/or “GREY” levels. Presumably this is not happening but the authors should (a) check\nwhether the emitting state remains the same during trajectories and (b) report more\ndetails about the experiment so that these sorts of questions don’t arise.\nAuthor response: Thank you for this important suggestion. We followed this and\nmonitored the PL spectra of several superlattices for 180 seconds (comparable in length\nto the blinking traces). We found no evidence of spectral diffusion or spectral jumps within\nthese intervals (as shown in Supplementary Fig. 16), allowing us to exclude spectral\nchanges as the origin of apparent GREY states.\nReviewer new comment: The Reviewer had several minor comments here. (1) It would be\nbetter to normalize the individual spectra so that such heat maps exclude effects due to\nbrighter portions of the trajectory. (2) As with the previous comment about Figure S15,\nit would seem that only one such heat map exists. The Reviewer would be more confident of\nthe authors’ observations if there were more such heat maps to ensure reproducibility.\nWe truly appreciate the Reviewer’s rigorous approach to the peer-review process, as there\nare so many papers in literature recently that report irreproducible results. We would like\nto assure the Reviewer that both the Leuven and Tokyo groups have more than 20 years\nof experience in single-molecule and nanoscale spectroscopy research, and we are well\naware of the necessity of statistical confirmation of any phenomena before making a\nconcluding statement. We are prepared to provide more data upon request within the Data\navailability policy.\nThe re-numbered Supplementary Fig. 21 was revised with three more examples, and with\nnormalized spectra added.\nComment 8. The authors show Figure S4a as an example of the emission trajectories\nexhibited by random QD ensembles. The Reviewer notices some periodicity in the data.\nThe Reviewer would like the authors to take a Fourier transformation of the trajectory to\nsee whether there is a characteristic frequency. It would look to be something like a 0.5\nHz signal. If yes, what is the origin of this periodicity?\nAuthor response: Thank you for pointing out this peculiar feature. We reproduced the\nexperiment carefully and found that the periodicity disappeared. The supplementary\nfigure was replaced accordingly.\nReviewer new comment: The Reviewer is satisfied with this response. However, the\nReviewer suggests that spurious signals usually mean that there is something not under\ncontrol in the measurements. This looks poorly on the experimentalist. The origin of\nspurious signals should always be tracked down and eliminated to ensure that there is no\ncorruption of acquired data.\nWe fully agree with the reviewer. We tracked this artefact to a temporal instability of the\nvibration isolator on which the microscope is placed. We have since then taken\nprecautions to rectify this problem, and the oscillations no longer appear in the intensity\ntraces.\nReviewer #3\nReviewer Comment #3: Thank you for measuring the photon bunching at lower fluxes, however\nwe still suggest that the excitation density remain too high to conclude bunching is result of the\nbiexciton-exciton cascade process. If the excitation density for the whole superlattice is N ~ 3, the\npopulation of higher-order multiexcitons is not negligible. Based on this, using the g(2) to measure\nbiexciton properties or infer information about the biexciton properties specifically is not valid.\nWe reiterate here that if the population of higher-order multiexcitons was not negligible, we would\nbe observing increase of the bunching degree with increasing power. We observe exactly the\nopposite, which leads us to conclude that though we cannot exclude presence of multiexcitons,\ntheir probability is very low and their effect in the observed phenomena is negligible. This has\nbeen stated already, on the request from the reviewer, in the second revision of the manuscript on\npage 8. Also, as stressed in the manuscript, even in the emission center where most of the excitons\naccumulate, the exciton density is on the order of 0.6 per QD.\nReviewer Comment #3: For accurate quantification of bunching, the integrated area of each peak\nshould be used (https://doi.org/10.1021/nl104054t). We therefore conclude that the biexciton properties described in\nthe manuscript in its current form are inaccurate and publication in any form at this time is premature.\nWe suggest the authors return to the data and examine the background subtracted (see comment below)\nand pulse integrated properties before resubmission.\nWe believe that using the peak intensities in the analysis is sufficient to qualitatively characterize\nthe bunching, and this way of analysis is being widely used in literature (e.g., in ref. [10], [75] in\nthe ms). All analyses are done after background subtraction. But we also analyzed the\nrepresentative data by integrating the peak areas, and present that in the Supplementary\nInformation in the revised Fig. 8. While the contrast is inevitably lower because of the shorter\nlifetime of the XX exciton (as mentioned below), the figure shows unambiguously the presence of\nphoton bunching from the individual superlattices.\nThe Supplementary Fig. 8 has been modified, and a note on the analysis by integrating the peak\nareas was added in the Methods section on page 12.\nReviewer Comment #3: In our previous comment, we point out that the cited examples of literature\nthat exhibit bunching either because of coherent ladder-like emission and/or spectroscopic design.\nFirst, it does not seem reasonable to suggest that there is coherent emission in room temperature\nNCs. It is true that ladder-like photon emission from multiexciton states has been demonstrated\nfor room temperature QDs, however it is not the case that coherent ladder-like emission has been\ndemonstrated, which is a requirement for bunching. In many of the cited examples, the\nspectroscopic design was such that it would promote bunching in the correlation measurement. In\nthe absence of specific filtering, it is hard to believe that the room temperature NC superlattices\nare exhibiting a similar phenomenon. In the paper that the author’s cited [PRL 94, 087403 (2005)],\nthe center peak to side peak ratio increases as the excitation density is increased. However, at all\nexcitation intensities the center to side peak ratio is less than 1 and all emission is antibunched\ndue to Auger recombination. The authors claim of bunched emission would be a major discovery,\nhowever they have yet to provide plausible context or evidence that they are observing true\ncoherent emission, or an explanation for why bunching is a reasonable expectation in the absence\nof coherent emission.\nIt is not really clear to us what the reviewer means here. The center peak increase from single QD\nin [PRL 94, 087403 (2005)] is a consequence of emission of time-correlated photons from\nmultiexciton states, measured by the Hanbury Brown Twiss setup. In our case the same emission\nof correlated photons originating from biexcitons leads to the same increase of the center peak.\nThe difference is that in single QDs in the absence of the multiexciton emission the g(2)(0) = 0\n(complete antibunching), while in systems with multiple emitters such as the superlattice the\nabsence of multiexciton emission results in g(2)(0) = 1. We note that other works report true photon\nbunching (g(2)(0) > 1) in CsPbBr due to ladder-type biexciton emission without invoking the need\n3\nof phase coherence of the photons (e.g., ref. [50]).\nReviewer Comment #3: We thank the authors for their additional efforts. Based on the author’s\ncomments related to how they analyzed their g(2) data (max intensity vs integrated area), and the\nfact most if not all of the ‘bunching’ they observe is in the fastest time bin, and is specifically\nenhanced in the grey/off state, suggests the source of this signal is from a background component\nand not nanocrystal emission. Data of the type the authors show can easily be obtained from\ncoverslip emission. Laser scatter, or other related phenomena. The structure of this ‘bunched’\nsignal appears to stay constant with blinking state, and it is the nanocrystal emission which is\nmodulated in intensity. That is to say when the nanocrystal emits, it overwhelms the signal the\nauthors are ascribing to bunching. Without extensive further analysis, background measurements,\nand proper correlation calculations, the claim to have bunched emission appears to be erroneous.\nMost of the bunching is in the fastest bin because, as explained when analyzing of the photon\nbunching dynamics, the XX to X transition is very fast (resolution-limited in our case). Even when\nusing the fastest time bin that the software supports (1ns) the bunching is within one or a few\npoints. This is consistent with the observation in ref. [10] on CsPbBr QD that the radiative rate of\n3\nthe XX exciton is more than two orders faster than that of the X exciton.\nIt is not true that bunching is specifically enhanced in the grey/off state; we never mention that in\nthe manuscript and it is not clear to us where the reviewer obtained this information. The only\nindication might be the Supplementary Fig. 9 where the weakest ON state shows bunching, unlike\nthe stronger ON states. But none of them are grey states, and this behavior is an exception meant\nto show the complex dynamics of a presumably structurally diverse superlattice.\nWe took great care throughout the work to verify that the phenomena we are observing are not\nartefacts originating from laser scattering or substrate emission. We have sufficient data of\ncomparable PL intensity from superlattices and QD aggregates that show bunching for the\nsuperlattice and not for the aggregates. To further strengthen our point, in the revised version we\nadded a few new datapoints in Fig. 2c that show even stronger bunching up to 3.9. The\ncorresponding g(2)(t) was added in the Supplementary Fig. 8b.\nA note on the verification of the artefacts was added on page 5. The Fig. 2c was revised with three\nnew datapoints, and the Supplementary Fig. 8b was replaced with g(2)(t) showing the bunching\nvalue of 3.9.\nReviewer #5\n(Remarks to the Author)\nReviewer report for NCOMMS-25-40932B\nDear Editor,\nAccording to your request, and after reading two rounds of exchanges between the authors\nand 4 other reviewers, my report primarily focuses on the photon bunching claim.\n1. Iappreciatetheauthor’shonestresponsestoallreviewercommentsandthesubstantial\nadditions and changes they made to the manuscript. The experimental data presented\ninthemanuscriptandSIisclearlyextensiveanddetailed. Mycommentsbelowaddress\nthe interpretation of g(2) data in terms of the exciton-biexciton model.\n2. Derivation of the relationship between exciton number and g(2)(τ) makes too many\nunstated and unrealistic assumptions and therefore lacks theoretical accuracy. Specif-\nically, the presence of two excitons in the system at t = 0 does not imply that both\nexcitons will emit simultaneously. Generally, a Poisson distribution of excitons in the\nsample is not equivalent to a photon field density operator ρˆ = (cid:80) Pn |n⟩⟨n| as the\nN\nauthors suggest. A Poisson-distributed photon field corresponds to a uniform electric\nfield, which is theoretically only produced by an ideal laser. The model presented by\nthe authors ignores the fact the emission occurs over a period of time characterized by\nthe spontaneous emission lifetime.\n3. To solve the above problem correctly one needs to [see quantum optics texts such as M.\nO. Scully and M. Zubairy, Quantum Optics, (Cambridge University Press, 2012) or L.\nMandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University\nPress, 1995).]:\n• Start with the superlattice (exciton) density operator ρˆ , which is in a Poisson-like\ns\nˆ\nstate and a exciton Hamiltonian H (usually modeled as simple 2-level systems).\ns\n• Thephotonfielddensityoperatorρˆ maybeassumedtobeinathermalorvacuum\nb\nˆ\nstate for spontaneous emission, and the boson Hamiltonian H .\nb\n• The interaction Hamiltonian is a dipole-field interaction H = eA·µˆ, where µˆ is\nint\nthe exciton transition dipole moment operator and A is the vector potential.\n• Then,onetime-evolvesthephotondensityoperatorusingtime-evolutionequation,\ndρˆ = −i[H ˆ ,ρˆ], where ρˆ is the total (system+photon) density operator and H ˆ is\ndt ¯h\nthe total Hamiltonian.\n• Solving the above equation analytically or numerically, one obtains the time-\ndependent photon field density operator ρˆ (t), from where one can obtain g(2)(t)\nb\nfor any t.\n4. Due to the above considerations, the author’s assumption that a Poisson distribution\nof excitons simply leads to a Poisson distribution of photons is incorrect, and so is the\nresulting expression [Main text, equation (3), line number 380]\ng(2)(0) 1\n= . \ng(2)(τ) 1−e−N\nIf you pause to think about what this equation says, you will notice that\ng(2)(0)\nis\ng(2)(τ)\n(Remarks to the Author)\nτ-independent. This is in clear contradiction to all of the g(2)(t) plots presented in\nthe manuscript and supplementary information. As a result, a decreasing g(2) with in-\ncreasing N (i.e., increasing excitation intensity) is not evidence of the exciton-biexciton\nmodel proposed by the authors.\n5. I now address the fact that the authors point to references 78 and 82 as the source of\nequation (3) in the main text [line 378] (Ref. 82 is an arXiv preprint by the many of the\nsame authors as Ref. 78 and presents the same equation). Ref. 78 clearly states that\nthe equation is meant for a qualitative understanding of when bunching peaks may be\nvisible over the background (i.e., over accidental background photon coincidences). In\nother words, the equation only describes how the g(2) S/N ratio gets weaker as there\nare increasing background coincidences at high intensities. It is not a verification of\nthe biexciton-exciton cascade model.\n6. The experimental data showing blinking and bunching is important, and if the other\nreviewers agree, may on its own be suitable for publication. A detailed theoretical\nexplanation of every observation is not a prerequisite for publication in Nature Com-\nmunications, so my criticism of the theory should not be interpreted as advice against\npublication.\nFebruary 20, 2026\n2"}]

Decision letter