Phonon-driven wavefunction localization enhances room-temperature single-photon purity in large hybrid lead halide perovskite quantum dots

Domain: neuroscience
Published: 2026-01-23
Editorial outcome: minor_revision

Abstract
In lead halide perovskites (APbX3), the effect of the A-site cation on optical and electronic properties has initially been thought to be marginal. Yet, evidence of beneficial effects on solar-cell performance and light emission is accumulating. Here, we report that the A-site cation in soft APbBr3 colloidal quantum dots (QDs) controls the phonon-induced localization of the exciton wavefunction. Insights from ab-initio molecular-dynamics simulations and single-particle fluorescence spectroscopy demonstrate that anharmonic crystal vibrations and the resulting disorder act as an additional confinement potential. Avoiding the trade-off between single-photon purity and optical stability faced by downsizing conventional QDs into the strong confinement regime, dynamical phonon-induced confinement in large organic-inorganic perovskite QDs enables bright (106 photons/s), stable (>1 h), and pure (>95%) single-photon emission tunable across a wide spectral range (495-745 nm). Strong electron-phonon interaction in soft perovskite QDs provides an unconventional route toward developing scalable room-temperature quantum-light sources.

Introduction
Reliable and scalable single-photon sources (SPSs) are essential for the broad adoption of quantum optical methods in computing1, communication2, and imaging3. So far, epitaxially-grown quantum dots (QDs) operated at liquid-helium temperatures excel with on-demand photon generation at exceptional single-photon purity, fluorescence rates, coherence times and indistinguishability4. Recently, coherent5, indistinguishable6 and fast single-photon emission7 was also demonstrated for colloidal cesium lead halide perovskite QDs at similar temperatures. Yet, capitalizing on the excellent performance of such sources is constrained by the need for cryogenic operation that limits their scalability and accessibility. Room-temperature SPSs alleviate the need for cryogenic cooling thus reducing significantly the system complexity. Amongst other room-temperature SPSs like doped organic crystals8 or defect-based emitters4, colloidal QDs exhibit scalable and low-cost syntheses based on simple solution chemistry, precise control over their optical properties, and ease of solution processing. Comprehensive structural and compositional engineering elevated the photoluminescence (PL) characteristics of colloidal QDs to near-unity quantum yields (QY), suppressed PL fluctuations, and narrow-band emission9–11. Already implemented in various classical optoelectronic technologies12–14, some of which are widely commercialized, colloidal QDs are becoming promising candidates for single-emitter applications15–17. However, colloidal QDs are yet to match the performance of cryogenically-cooled epitaxial QDs in terms of single-photon purity and PL stability over time.

For colloidal QDs, improvements of their single-photon purity can be obtained by reducing their physical size18–20. Such size confinement yields discretized energy levels, strong (multi-)exciton interaction and accelerated non-radiative Auger-Meitner21 recombination of multiexcitons22,23. The resulting single-photon purity (exceeding 90% in colloidal QDs)17,18,24 is inferred from the amplitude of the dip in g2(τ), the second-order photon-photon correlation function, at zero delay time (Fig. 1a). However, enhancing the single-photon purity via QD downsizing limits the spectral tunability and comes at a high cost: Firstly, vulnerability to surface defects and matrix effects that cause stochastic intensity fluctuations (blinking) and photobleaching is amplified in small QDs9,10,25. Secondly, increased coupling to surface vibrations broadens the emission spectrum of small QDs9,26. Finally, the volume scaling of the absorption cross sections translates into the need for high excitation densities, compromising the potential for on-chip optical pumping by LEDs and prompting uncorrelated background photons that contaminate single-photon emission. In conclusion, finding alternative ways toward high single-photon purities without the need for detrimental size confinement would improve the quality, stability, and versatility of colloidal QDs as room-temperature SPSs.

In search for alternatives to size confinement, we identified the coupling between electronic and vibrational degrees of freedom (vibronic coupling)27 as an underutilized route towards confining charge carriers in semiconductor sources. Vibronic coupling broadens the emission lines of semiconductors employed in display and lighting applications, limits the overall charge mobility in crystalline and amorphous materials used in photovoltaics, and causes the loss of exciton optical coherence, a key resource for photon-based quantum computing. Nevertheless, while generally considered a detrimental interaction, vibronic coupling can mediate fascinating phenomena, such as Bose-Einstein condensation of exciton-polaritons28, superconductivity29, and optical cooling30.

In crystalline materials at finite temperatures, crystal vibrations (phonons) dynamically displace atoms from their average lattice site (Fig. 1b), hereby inciting dynamic structural disorder that translates into electronic disorder through vibronic coupling31,32. Such dynamic disorder adds to any potentially present static disorder such as lattice site disorder in defected crystalline solids33 or structural disorder in amorphous solids34. The associated disordered deformation potential may serve as an effective confinement potential that causes contracted and localized electronic wavefunctions, in some aspects resembling Anderson localization32,35–38. We hypothesize that the combination of phonon-induced disorder and exciton-phonon coupling can act in addition to the quantum size effect, thereby enhancing quantum confinement of (multi-)excitons and consequently single-photon purity (Fig. 1c). This approach does not come with the risk of trap state formation which usually reduces PL quantum yield and the overall performance of single-photon sources, but requires the use of materials that combine high amplitude, anharmonic vibrations at room temperature with strong exciton-phonon coupling32.

A suitable platform to test our hypothesis is provided by lead halide perovskite (LHP) materials (APbX3; A=formamidinium, methylammonium, Cs; X=Cl, Br, I) that emerged as promising solar-cell materials in part due to their high defect tolerance and facile solution processing. Colloidal perovskite QDs share this defect-tolerance and therefore achieve near-unity PL QY and relatively weak blinking without the need for delicate core-shell engineering39,40. Perovskite QDs have demonstrated single-photon emission at room temperature41,42 as well as bright, coherent, and indistinguishable single-photon emission at cryogenic temperatures5,6. Initially, the A-site cation has largely been deemed a bystander in defining the photo-physics of LHPs due to its lack of direct electronic contribution to the band-edge states43. However, A-site cations can influence the structure of the PbX3- framework and associated crystal dynamics, which, via a strong coupling of the exciton to anharmonic vibrations, also impact the optoelectronic performance44,45. Consequently, the A-site cation’s importance for optical and electronic characteristics in LHPs is being recognized and exploited to improve, for example, LHP solar cell performance43,46–51.

In this work, we suggest that the large dynamic structural disorder in formamidinium lead bromide (FAPbBr3) QDs increases the effective quantum confinement and thereby renders even large QDs high-quality quantum emitters of single photons. Identifying disorder as an alternative route towards quantum confinement alleviates well-known size-dependent QD performance trade-offs. It extends the single-photon emission capability to QDs of sizes larger than the Bohr diameter, hereby straightforwardly improving both the source brightness and its stability. Ab-initio molecular-dynamics (AIMD) simulations pinpoint A-site-cation-controlled anharmonic vibrational modes as the origin of the structural disorder responsible for dynamic wavefunction localization and enhanced quantum confinement. The resulting enhanced Auger-Meitner recombination rate of multi-excitons even in large FAPbX3 QDs enables pure single-photon emission, which is also bright, stable, and finely spectrally tuneable across the visible spectrum.

Results
Influence of the A-site cation on structure and optical properties
We selected CsPbBr3 and FAPbBr3 QDs as candidates to study the effect of the A-site cation on structural disorder because their room-temperature crystal structures and dynamics are tuned by the choice of the A-site cation. The room-temperature crystal structure of CsPbBr3 is orthorhombic as determined by single-crystal X-ray diffraction (XRD; Fig. 2a). On the other hand, FAPbBr3 adopts an average cubic crystal structure (Fig. 2b). Figure 2c and d display magnified views of the Pb-Br-Pb bonding of the corner-shared PbBr6 octahedra. CsPbBr3 and FAPbBr3 differ in the Pb-Br-Pb angles as well as displacement ellipsoids that describe the static and dynamic disorder occurring at finite temperatures. In CsPbBr3, the average Pb-Br-Pb angle deviates from 180° as octahedral tilting compensates for the undersized Cs cation. Contrarily, a larger cation size and thus better size fitness of FA in the octahedral void stabilizes an average cubic structure, as evidenced by a Pb-Br-Pb angle of 180°. The shape of the displacement ellipsoids of Br reveal that in both systems octahedral tilting dominates disorder in the PbBr3- framework, albeit the cubic FAPbBr3 structure additionally exhibits larger displacement along the Pb-Br bond, 0.1910(16) Å compared to 0.157(3) Å in CsPbBr3.

Single-particle PL spectroscopy can optically probe the small changes in structure and dynamics of the PbBr3- framework that are introduced by the different A-site cations. Figure 2e displays representative PL spectra of CsPbBr3 QDs and FAPbBr3 QDs with edge lengths of 9.9(1.2) nm as determined by transmission electron microscopy (Supplementary Fig. 1). With edge lengths of 1.61(19) and 1.27(16) times their Bohr diameter (dB; 6.16 nm for CsPbBr3 and 7.76 nm for FAPbBr3 52), the samples should display similar size-induced quantum confinement. Samples were prepared following recently published room-temperature procedures (Supplementary Note 1)39. Compared to CsPbBr3, the PL peak center of a FAPbBr3 QD is redshifted by 72 meV and the PL peak width (full width at half maximum; FWHM) is increased by 13 meV. These spectral trends are consistent across a larger single-QD dataset (Supplementary Fig. 2) and with ensemble PL measurements (Supplementary Fig. 3). The red-shift is associated with the straightened and stretched Pb-Br-Pb bonding (Fig. 2a-d), and the increased peak width suggests stronger electron-phonon coupling in FAPbBr3 26,48.

Moreover, we observed an exponential low-energy (Urbach) tail in single-particle PL spectra, see Fig. 2f. Across a large size range, the tails are consistently steeper for CsPbBr3 than in comparably sized FAPbBr3 QDs (Fig. 2g). As elucidated in a prior study48, this suggests stronger exciton-phonon coupling in FAPbBr3 QDs, consistent with PL phonon replica at cryogenic temperatures 53,54 and optical-pump–electron-diffraction-probe measurements55. This trend is also confirmed by our accompanying density functional theory (Supplementary Fig. 4) and ensemble PL studies (Supplementary Fig. 3). Urbach tails are associated with exciton-phonon coupling through the formation of localized states that result from thermal disorder and form the low-energy tail32,34,36. They thus also serve as a first indication of the hypothesized wavefunction confinement which may be enhanced in pseudo-cubic FAPbBr3 QDs.

Phonon-driven wavefunction localization
An understanding of electron-phonon coupling can be gleaned from finite-temperature ab-initio molecular-dynamics (AIMD) simulations based on density functional theory (DFT). We performed simulations for 3.6 nm large ABr-terminated CsPbBr3 and FAPbBr3 QD models at the Perdew-Burke-Ernzerhof level of theory (see “Methods”)56. Previously, such a balance between computationally approaching experimental QD sizes yet ensuring a sufficiently high level of theory has successfully reproduced experimental results related to surface chemistry57 and exciton-phonon interaction7,26. To study wavefunction confinement, we computed the wavefunction of the highest occupied molecular orbital (HOMO) which represents the hole wavefunction. Figure 3a displays representative snapshots of the HOMO wavefunctions at 10 K and 300 K. At low temperatures, the wavefunctions fully delocalize across the entire QD volume for both CsPbBr3 and FAPbBr3 QDs. At 300 K, however, the wavefunction strongly contracts and localizes in the FAPbBr3 QD, while such a thermally activated wavefunction confinement is considerably weaker in the CsPbBr3 QD. Figure 3b shows the wavefunction sizes averaged across >120 snapshots (>12 ps; further details in Supplementary Note 7) in the CsPbBr3 and FAPbBr3 QDs along AIMD trajectories at 10, 100, and 300 K (further temperatures in Supplementary Fig. 5). In both materials, the wavefunction sizes remain between 1.6 and 1.8 nm at 10 and 100 K, limited by QD size. Above 100 K, the wavefunction size of the FAPbBr3 QD decreases to just 0.93 ± 0.17 nm, whilst the wavefunction size of the CsPbBr3 QD moderately decreases to 1.40 ± 0.05 nm. This observation is consistent across several QD diameters up to 5.4 nm (Supplementary Fig. 6-8). While the band gap in both systems is still influenced by the QD size (Supplementary Fig. 9), the wavefunction size for FAPbBr3 QDs at 300 K is largely limited by the disorder-induced wavefunction localization (Supplementary Fig. 6). In contrast, CsPbBr3 retains a QD-size dependent wavefunction extension. The electron wavefunction, represented by the lowest unoccupied molecular orbital (LUMO), experiences similar wavefunction localization as the HOMO (Supplementary Note 8, Supplementary Fig. 10, 11). Finally, temperature-induced localization even occurs in the respective bulk materials, specifically for FAPbBr3 (Supplementary Fig. 11, 12). Although bulk crystals will not experience Auger-Meitner relaxation due to vanishing inter-exciton coupling, such localization effects reduce radiative and non-radiative rates and limit carrier mobilities in lead halide perovskites45,58–60. Such a systematically observed wavefunction localization across model sizes ranging from 1.8 nm QDs to the bulk alongside the qualitative agreement in key experimental observations linked to exciton-phonon coupling (Supplementary Fig. 3, 4)7,26, affirm that such wavefunction localization also occurs in the perovskite QD sizes accessed experimentally (vide infra).

Importantly, the wavefunction localization is dynamic. Both the size and location inside the QD vary between snapshots along the MD trajectory conveying a transient localization behavior of the wavefunction (Supplementary Videos, Supplementary Fig. 13). Moreover, the time-averaged wavefunction, integrated across the entire AIMD trajectories (Supplementary Fig. 14), smoothly covers the entire volume which excludes static contributions to the localization. The highly dynamic wavefunction as well as the temperature-dependence suggest that anharmonic phonons are the primary cause for the transient structural disorder and the associated wavefunction localization. This mechanism is reinforced by temperature-dependent radial and angular distribution functions that support the existence of temperature-induced atomic displacement from their average position (Supplementary Figs. 15-17). Moreover, the structural disorder caused by these atomic displacements is discernible from temperature-dependent spatial correlation functions of the PbBr3- framework (Supplementary Fig. 18). Finally, the correlation of temperature-induced structural disorder and wavefunction localization with the population of phonon modes underpins allocating the disorder to phonons (Supplementary Fig. 19).

Further insight into the phonon-driven wavefunction localization is provided by the autocorrelation function of the HOMO wavefunction coefficients which describes the time evolution of the wavefunction (Supplementary Note 7, Supplementary Equation 9). Figure 3c depicts the normalized wavefunction autocorrelation functions at 300 K exhibiting decays within hundreds of femtoseconds that envelop oscillations with a period of roughly 200 fs. Notably, the loss of correlation is significantly faster for the FAPbBr3 QD than for the CsPbBr3 QD7. This observation resembles the bandgap-energy autocorrelations, which have previously been assessed to derive the phonon-driven electronic dephasing in these and other semiconductor materials26,61,62.

The power spectra of the wavefunction autocorrelation (Supplementary Note 7) in the inset in Fig. 3c reveal the dominant vibrational features driving the wavefunction localization in both systems. While the wavefunction-phonon coupling strength in CsPbBr3 essentially concentrates within a single peak at 17 meV, the coupling strength in FAPbBr3 QD additionally derives from a broad range of low-energy phonons. The population of these modes is confirmed by vibrational spectral density at these energies in our AIMD simulations as well as experimental Raman spectra with good agreement between the two methods (Fig. 3d). The broad signal in FAPbBr3, contrary to the narrow peak in CsPbBr3, attests a high degree of anharmonicity. Whilst the 17 meV peak is associated with a stretching mode along the Pb-Br bonds, the highly anharmonic tilting of PbBr6 octahedra contributes to the lower-energy modes53,55. Peculiar to average cubic perovskites like FAPbBr3, this anharmonic motion induces local and transient symmetry breaking, prompting their labelling as pseudo-cubic63–66. Because it also causes a significant bandgap renormalization63, local symmetry breaking is expected to affect the wavefunction localization of the excitons strongly. Consequently, tuning between the orthorhombic and pseudo-cubic structure through the A-site cation allows us to manipulate the disorder potential and the resulting wavefunction localization. The local symmetry breaking through octahedral tilting and the influence of the A-site cation on the structural disorder are apparent in X-ray diffuse scattering experiments of FAPbBr3 and CsPbBr3 single crystals (Supplementary Fig. 20)66–68. Based on structural similarities69, we expect other pseudo-cubic perovskites like methylammonium lead bromide (MAPbBr) or aziridinium lead bromide (AZPbBr3) to exhibit similarly pronounced wavefunction localization.

Purified single-photon emission
We now assess the single-photon purity of similarly sized (9.9(1.2) nm) CsPbBr3 QDs and FAPbBr3 QDs in single-particle PL measurements. In the CsPbBr3 QDs, g2(0) values are strongly size-dependent and increase drastically with increasing QD size18. Fig. 4a shows a representative second-order photon-photon correlation (g2(τ)) of a CsPbBr3 QD. The incomplete anti-bunching of g2(0)=0.29 translates into a single-photon purity of only 71%, with a high probability of biexciton emission. This observation is consistent with the behaviour of traditional semiconductors for which QD-size confinement is the major knob for tuning multiexciton quenching via Auger-Meitner recombination. When size confinement is progressively lost upon increasing the QD size, so is the Coulomb interaction mediating the Auger-Meitner recombination of multi-excitons. Consequently, for large QDs with a size exceeding the exciton Bohr diameter, the radiative decay of multi-excitons may increasingly become a competitive channel, hereby compromising the single-photon purity. Several reports have indeed shown that large CsPbBr3 QDs lose their characteristic anti-bunching due to multi-photon emission from biexcitons18,70,71.

In contrast, despite a similarly weak QD-size confinement, a representative FAPbBr3 QD achieves a drastically higher single-photon purity of 96% (g2(0)=0.04) (Fig. 4b). Figure 4c shows the size dependence of the single-photon purity including data from >150 single QDs as well as previously published results from our laboratory18. FAPbBr3 QDs exhibit systematically smaller g2(0) than comparably size-confined CsPbBr3 QDs supporting the postulated enhancement of quantum confinement in the cubic FAPbBr3 QDs. In fact, CsPbBr3 QDs with an edge length of 17.2 nm are only weakly anti-bunched (g2(0)=0.3-0.7), while the g2(0) from FAPbBr3 QDs of the same size remain well below 0.5 (Supplementary Fig. 21). Although the concept of Bohr diameters is not strictly valid at room temperature for materials exhibiting strong exciton-phonon coupling, we additionally corroborated the A-site-cation dependent single-photon purity for samples with identical sizes normalized by Bohr diameters (Supplementary Fig. 22).

In a low-excitation-density regime, g2(0) corresponds to the ratio of the biexciton QY to the exciton QY19. Given identical exciton QYs in CsPbBr3 and FAPbBr3 QDs (88% and 89%, respectively), our experiments point to quenched biexciton QYs in FAPbBr3 QDs either (i) due to a speed-up of non-radiative Auger-Meitner recombination of the biexciton and/or (ii) due to slower radiative decay of the biexciton. To disentangle the two contributions, we selectively probe the biexciton emission using heralded single-particle spectroscopy (Supplementary Note 5)72. Fig. 4d displays a representative biexciton PL decay trace of a CsPbBr3 QD with a biexciton lifetime of 0.8 ns. In a representative FAPbBr3 QD, the biexciton PL decays significantly faster with a lifetime of just 0.4 ns (Fig. 4e) despite a slower radiative decay of the exciton (Supplementary Fig. 23). Furthermore, the statistics from 77 QDs show that, while both mechanism (i) and (ii) contribute to quenched biexciton QYs, mechanism (i), i.e., an accelerated non-radiative recombination, is the main lever in quenching biexcitons and enhancing single-photon purity in FAPbBr3 QDs (Fig. 4f).

We introduced the phonon-induced wavefunction localization as a temperature-dependent effect in the previous section. Therefore, we assessed the temperature dependence of the photon anti-bunching (g2(0)) for FAPbBr3 and CsPbBr3 QDs of similar sizes (Supplementary Figs. 21, 24, 25). Second-order correlation functions are not anti-bunched at 4 K for single QDs of either material, attesting high biexciton PL QYs at cryogenic temperature. Matching the temperature dependence of the wavefunction localization (Fig. 3b), the single-photon purity increases with increasing temperature in both materials (Supplementary Figs. 21, 24). It is noteworthy that this temperature dependence could in parts be linked to single-photon superradiance, a process associated with a wavefunction delocalization that causes a rapid radiative decay and that is lost at higher temperatures due to thermal disorder7, or to Auger-Meitner recombination intrinsically requiring thermal activation73,74. Nevertheless, a stronger increase of single-photon purity observed in FAPbBr3 demonstrates that disorder-induced wavefunction localization plays a key role (Supplementary Figs. 21, 22, 24). These temperature and composition trends are also followed by the increasing PL peak width, an indicator for exciton-phonon coupling, and the increasing PL lifetime (Supplementary Fig. 3), which was recently linked with wavefunctions localized through vibrations58,59. All these effects reflect the disorder-induced wavefunction localization.

Our AIMD simulations uncovered that disorder-induced wavefunction localization is related to octahedral tilting causing local symmetry breaking in pseudo-cubic perovskites. Pseudo-cubic APbBr3 QDs can also be obtained for other organic A-site cations, such as methylammonium (MA) and aziridinium (AZ). Indeed, weakly size-confined MAPbBr3 QDs and AZPbBr3 QDs exhibit an average single-photon purity of 90(5)% and 89(3)%, comparable to FAPbBr3 QDs of similar size confinement (Supplementary Figs. 26, 27)39,69, attesting the universality of the proposed phonon-mediated mechanism.

FAPbX3 QDs as single-photon sources
Efficiently quenching multi-excitons via phonon-driven wavefunction confinement opens a new and straightforward avenue toward (i) stable, (ii) bright, and (iii) color-tunable single-photon sources - all enabled by the here-introduced route to circumvent the previous need for QD downsizing to achieve high single-photon purity.

Stability issues are predominantly associated with the QD surface, especially for smaller QDs with high surface-to-volume ratios, which exhibit notably poor photostability. Photodegradation has been attributed to surface-matrix reactions, eventually leading to a shrinkage of the QD core that manifests itself in a dynamical spectral blue-shift and a reduction in PL intensity75. Indeed, an individual QD from a large FAPbBr3 QD sample (10.1(9) nm) exhibits extreme photostability with spectrally stable emission during continuous irradiation for 75 minutes, see Fig. 5a. Figure 5b, c display the g2(τ) measured before and after irradiation of 1 hour, respectively. Strong anti-bunching with g2(0) of 7% and 5% conveys that single-photon purity does not deteriorate over the course of continuous irradiation for 1 hour. Moreover, the QD displays only very weak blinking, on par with sophisticated core-shell structures9,10 and remains largely unchanged. Before irradiation, we observe that the QD spends 98% in its bright (ON) state, while after 1 hour of continuous irradiation an ON fraction of 94% is retained (Supplementary Fig. 28).

Several quantum-engineering applications based on room-temperature single-photon emitters specifically demand bright emitters. For example, quantum key distribution generally requires emitters with high brightness at a single-photon purity >90%4. Fig. 5d displays the power-dependent PL intensity (top panel) and single-photon purity (bottom panel) of an individual large and weakly confined QD at a laser repetition rate of 10 MHz. We extract a saturation power density Psat of 70 W/cm2 and a maximum brightness of 1.1 Mcps suggesting that the radiative rate limits the brightness (details in Supporting Information). At a low excitation power (0.1 Psat), strong anti-bunching attests a single-photon purity of 95% (Fig. 5e). An increase of the excitation to 1.5 Psat only reduces the single-photon purity to 92% (Fig. 5f). In quantum emitters such as defects or molecules, single-photon purity in saturation can be deteriorated by strong background emission at high laser power. However, large perovskite QDs do not suffer from this limitation owing to their orders of magnitude larger absorption cross sections. This is illustrated by the agreement between the measured power dependence of the single-photon purity and the background-free prediction for a biexciton PL QY of 5% of the exciton PL QY (bottom panel; Fig. 5d).

Lastly, extending high single-photon purity from small to large FAPbX3 QDs also allows spectral fine-tuning of single-photon emission. Together with the already available compositional tuning through the halide identity, FAPbX3 QDs can now be precision-engineered to deliver high single-photon purity across the entire visible range, for example blue-emitting FAPb(Br/Cl)3 (Fig. 5g; 9.2(8) nm edge length), green-emitting FAPbBr3 (Fig. 5h, 10.2(1.2) nm edge length), and far-red emitting FAPbI3 (Fig. 5i, 11.3(1.3) nm edge length). All QDs display strong anti-bunching certifying single-photon purities >90%. Blue-emitting single-photon sources receive increasing interest for underwater communication76,77. Meanwhile, reaching the far-red and near-infrared spectrum with FAPbI3 QDs is particularly relevant due to reduced losses in fiber-optic communication2 and biological tissue78.

In summary, we discovered a strategy to enhance quantum confinement in colloidal QDs and improve their single-photon purity without the need for QD downsizing. Caused by anharmonic vibrations that couple to electronic degrees of freedom, dynamic disorder induces wavefunction confinement which increases with temperature and can be tuned through the control of crystal vibrations. In our implementation, transient wavefunction confinement is realized by deploying organic A-site cations in lead halide perovskite QDs yielding pseudo-cubic structures with a tendency towards dynamic local-symmetry breaking. Phonon-induced wavefunction localization presents a counterintuitive but beneficial effect of strong exciton-phonon coupling, contrasting the typically negative connotation of such coupling in optoelectronic materials. Owing to the resulting purified single-photon emission in large QDs, we can report competitive brightness, photostability and single-photon purity while retaining the broad spectral tunability that make colloidal QDs stand out amongst room-temperature single-photon sources. These characteristics are expected to further improve under more favourable conditions such as resonant excitation79,80, temporal and spectral filtering7,80,81, or integration into optical microcavities and nano antennae82,83. Taken together, large colloidal organic-inorganic perovskite QDs are a versatile and scalable platform for room-temperature single-photon sources. Considering the ubiquity of electron-phonon coupling, we are confident that actively tuning this interaction provides opportunities for functional material engineering beyond perovskite QDs.

Methods
Representation of errors
Throughout this work, standard deviations are reported in the text in rounded brackets, while 95% confidence intervals are reported after a ±-sign. Error bars in Figs. 3 and 4 represent 95% confidence intervals, as stated in the caption.

Single-crystal X-ray diffraction
Single-crystal growth was conducted using the inverse-temperature crystallization method for FAPbBr3 and CsPbBr3. Detailed descriptions are found Supplementary Note 3.
Single-crystal X-ray diffraction data for crystal-structure determination were collected on a Rigaku Synergy S diffractometer equipped with microfocus Mo K-alpha X-ray source, a Dectris Pilatus 300K hybrid pixel detector, and an Oxford Cryostream 800 for temperature control. Diffraction images were acquired every 0.2° of rotation, during continuous rotation of the crystal around the phi axis. The resulting data was processed with the software CrysAlis Pro (Rigaku Oxford Diffraction) for indexing, instrumental model refinement, and intensities integration. Descriptions of structure solutions and refinement are provided in the Supporting Information. Results are detailed in Tables S1-3, included as Supporting Data, and available in the CSD and ICSD databases under deposition numbers 2407691 (FAPbBr3) and 2407795 (CsPbBr3).
For X-ray total scattering, selected specimens were mounted on a MiTeGen kapton loop using inert NVH high viscosity oil. Diffraction data was collected by a Rigaku Synergy S diffractometer equipped with a Pilatus 300K hybrid pixel detector, Mo K-alpha microfocus source, and an Oxford Cryostream system set at a temperature of 300 K. Further details regarding data collection, processing and analysis are provided in Supplementary Note 3.

Colloidal quantum dot synthesis
FAPbBr3, CsPbBr3, MAPbBr3 and FAPb(Br/Cl)3 QDs were synthesized following the methods described in ref. 39. FAPbI3 QDs were prepared according to ref. 84. The synthesis of AZPbBr3 QDs followed ref. 69. Detailed descriptions of the syntheses are provided in Supplementary Note 1.

Transmission electron microscopy
Transmission electron microscopy (TEM) images were acquired either using a Hitachi HT7700 microscope operated at 100 kV or using a JEOL Model JEM-2200FS microscope operated at 200 kV. Scanning transmission electron microscopy (STEM) images were collected using a JEOL Model JEM-2200FS microscope operated at 200 kV. Particle sizes were determined using ImageJ.

Raman spectroscopy
Raman spectroscopy was performed using a confocal Raman microscope (Horiba LabRAM HR Evolution). FAPbBr3 and CsPbBr3 QD thin films were prepared by dropcasting respective QD colloids under ambient conditions onto a glass microscope slide, cleaned previously with ethanol. A 785 nm cw laser (Toptica XTRA II; about 30 mW) was focused onto the sample by an objective (Olympus MPlan N; NA=0.9, 100x magnification). The scattered light was collected by the same objective, Rayleigh-scattered light removed by a 785 nm long pass filter, and the remaining Raman-scattered light spectrally dispersed onto a CCD camera (Horiba Synapse) using a grating (300 g/mm, blazed at 600 nm). All Raman spectra are reported after wavenumber calibration (using a Si wafer) and correction for the spectral response of the detector. The samples were stable throughout the acquisition time of ~100 s and no sample damage was observed due to the Raman excitation.

Variable-temperature time-integrated and time-resolved PL measurements
CsPbBr3 and FAPbBr3 QD thin films were prepared via spin coating onto Si substrates with a 2 µm thick SiO2 surface oxide layer. The samples were mounted in a liquid-helium closed-cycle cold-finger cryostat (ARS; DE204AE; sample in vacuum) and probed by a PL spectrometer (Picoquant; FluoTime 300) using a pulsed 354.3 nm laser for excitation and a photomultiplier tube for detection. The laser power was adjusted to ensure low excitation densities, with less than one exciton per QD on average. Time-integrated PL spectra were acquired using a monochromator in the detection path, employing a bandwidth of 0.7 nm and 80 MHz repetition rate. Time-resolved PL decays were acquired at the PL maximum, with an emission bandwidth of 27 nm, to average out spectral dynamics, and employing a repetition rate of 2.5 MHz. To record temperature-dependent PL characteristics, PL spectra and decays were acquired at several temperatures between 17 K to 296 K. All PL spectra were corrected for dark counts and the spectrally varying detector sensitivity. All time-resolved PL decays were corrected for dark counts. Both PL spectra and time-resolved PL decays were normalized to the peak counts.

Single-particle optical spectroscopy
Samples for single-particle optical spectroscopy were prepared in nitrogen filled gloveboxes using anhydrous solvents. As-synthesized QD samples were diluted in multiple steps by three to five orders of magnitude in cyclohexane, toluene, or octane and spin-coated (100 µL, 150 revolutions per second, 60 s) onto clean cover glasses or, prior to spin-coating, diluted by additional one to two orders of magnitude in 3 mass% solutions of polymers (SEBS in cyclohexane or polystyrene in toluene) for additional protection from moisture and air. The resulting sparse QD films were placed in a nitrogen filled sample holder or measured under ambient conditions. Full details of the preparation of single-dot samples are provided in Supplementary Note 4.
Conventional single-particle spectroscopy was performed with a custom-built µPL setup. A 405 nm pulsed excitation laser (PicoQuant, 1–10 MHz repetition rate, <50 ps pulse width, <100 W/cm2) is focussed with an oil immersion objective (1/e2=1 µm, 1.3 NA) onto the sample that is mounted on XYZ translational stages (SmarAct, <1 nm resolution). Collected by the same objective, the light emitted from single QDs is passed through a dichroic mirror as well as a long-pass filter (both 450 nm cut-on wavelengths). The filtered light is either sent to a monochromator coupled to an EMCCD (Princeton Instruments, one frame per second) or to a Hanbury-Brown and Twiss setup consisting of a 50:50 beam splitter, two avalanche photodiodes (Excelitas, 250 ps time resolution), a time-correlated single-photon counting module (PicoQuant, HydraHarp) and a short-pass filter (cut-off wavelength 750 nm). PL spectra were recorded before and after measurements in the HBT setup to ensure that QDs did not blue shift under illumination. The raw data was processed in custom python codes or using SymPhoTime (PicoQuant). Complete details of the data analysis are provided in Supplementary Note 4.
Heralded single-particle spectroscopy utilized a SPAD array spectrometer previously described in refs. 72, 85 and built around a commercial inverted microscope (Eclipse Ti-U, Nikon). Excitation light from a pulsed laser source (470 nm, 5 MHz, LDH-P-C-470B, PicoQuant) is focused onto the sample with an oil immersion objective (×100, 1.3 NA, Nikon). The emitted light is collected by the same objective and passed through a dichroic mirror (FF484-FDi02-t3, Semrock) and a long-pass filter (BLP01-473R, Semrock). The magnified image plane (×150) serves as the input for a Czerny-Turner spectrometer that consists of a 4-f system (AC254-300-A-ML and AC254-100-A-ML, Thorlabs) with a blazed grating (53-*-201R, Richardson) at the Fourier plane. At the output image plane of the spectrometer, a 512-pixel on-chip linear SPAD array is placed. Photon pairs following the same excitation pulse are post-selected as heralded events and energy- as well as time-tagged to reconstruct spectra and PL decays of the first (biexciton) and second photon (exciton). Further details on the SPAD array spectrometer and data processing are found in Supplementary Note 5.
Single-particle spectroscopy at cryogenic temperatures was performed in a custom-built µPL setup equipped with an evacuated liquid-helium closed-loop cryostat (Montana Instruments). Single QDs were excited using a tuneable fs pulsed laser (set to 480 nm and 80 MHz, Toptica) that is fiber coupled and focused on the sample by a microscope objective (NA=0.8, 100x, dry). PL from single QDs was collected by the same objective, passed through a long-pass filter (cut-off 500 nm), and sent either into the monochromator coupled to back-illuminated CCD camera (Princeton Instruments) to record the spectra or towards the HBT setup with a 50:50 beam splitter and two SPDs (ca. 50 ps time resolution) to record the second-order intensity correlation. Signals from SPDs were fed into the TCSPC module (Picoharp, PicoQuant). PL spectra were recorded with a grating of 300 lines per mm, blaze at 500 nm, giving a spectral resolution of around 1 meV. Further experimental details and data processing are found in Supplementary Note 6.

AIMD simulations
Finite-size ab-initio molecular-dynamics (AIMD) simulations were performed using density functional theory (DFT). Previously published models26 of AX-terminated cuboidal CsPbBr3 and FAPbBr3 QDs with truncated corners and approximate edge lengths of 1.2, 2.4, 3.0, 3.6 and 5.4 nm were placed in vacuum cell (at least 1 nm on each cell). We used the Quickstep module in CP2K employing Gaussian and plane waves with a plane-wave cutoff of 280 Ry86, DZVP-MOLOPT basis sets87, Goedecker-Teter-Hutter pseudopotentials88 and Perdew-Burke-Ernzerhof exchange-correlation functionals56. Molecular dynamics (MD) simulations in the NVT ensemble employed time steps of 1 or 10 fs and a canonical sampling through velocity rescaling thermostat89 with a time constant of 15 or 250 fs. Trajectories with a duration of at least 12 ps were produced after equilibration at the respective temperature for 6 ps. To analyse the wavefunction confinement in real space, cube files for the highest occupied molecular orbital (HOMO) wavefunctions were extracted every 0.1 ps (>120 snapshots). The analyses of the trajectories yielding wavefunction sizes and descriptors of disorder are detailed in the Supporting Information. MD simulations in the NVE ensemble were performed after thermal equilibration of at least 6 ps in the NVT ensemble. Trajectories with time steps of 1 fs and a duration of at least 10 ps were prepared for to analyse nuclear vibrations and extract wavefunction dynamics through the molecular orbital coefficients, details of which are provided in the Supplementary Note 7.
Periodic-boundary AIMD simulations were performed using DFT. We used the Quickstep module in CP2K employing Gaussian and plane waves with a plane-wave cutoff of 280 Ry86, DZVP-MOLOPT basis sets87, Goedecker-Teter-Hutter pseudopotentials88 and Perdew-Burke-Ernzerhof exchange-correlation functionals56. As starting structures of MD simulations, 4x4x4 orthorhombic supercells of FAPbBr3 and CsPbBr3 were constructed. MD simulations in the NPT ensemble employed time steps of 1 or 10 fs, a canonical sampling through velocity rescaling thermostat89 with a time constant of 15 fs, and a barostat with a time constant of 1000 fs. Trajectories with a duration of at least 7 ps were produced after equilibration at the respective temperature for at least 3 ps. To analyse the wavefunction confinement, cube files for the HOMO and LUMO wavefunctions were extracted every 0.1 ps and their inverse participation ratio was calculated (details in Supplementary Note 7, 8).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this manuscript, the authors propose a novel strategy to enhance quantum confinement in colloidal quantum dots (QDs) and improve their single-photon purity, eliminating the need for QD downsizing. Based on combined theoretical and spectroscopic approaches, it is revealed that the A-site cation in soft APbBr3 colloidal QDs controls the phonon-induced localization of the exciton wavefunction. Detailed density functional theory (DFT) results demonstrate that the transient wavefunction confinement is realized via strong exciton-phonon coupling, contrasting the generally negative impact of such couplings in optoelectronic materials. In my opinion, this work presents a solid piece of work, which should be of interest to the community. It sheds new light on achieving bright, stable, and pure single-photon emission, while circumventing the optical stability issues associated with conventional QD downsizing. I would recommend its publication pending the authors' consideration of the following points:\n1.In the computational section, have the selected methods undergone benchmark work to verify their applicability to the target system? Additionally, how are the weak interactions accounted for?\n2.A key highlight of this work is the achievement of high-purity single-photon emission without the need for QD downsizing. What criteria determine the optimal QD size for this mechanism? Is there potential for this mechanism to be applicable to bulk materials as well?\n3.As clarified in this manuscript, two types of QDs exhibit distinct phonon-driven wavefunction localization. Figure S3 in the Supporting Information presents the PL spectra of these two QDs as a function of temperature. I recommend that the authors conduct a quantitative analysis of the relationship between wavefunction localization and PL energy. Given that the PL energy shifts are generally regarded as directly related to wavefunction confinement, such a correlation could further validate the novel mechanism.\n4.It is found that the loss of correlation is significantly faster for the FAPbBr QD than for the CsPbBr3 QD. Generally, the phonon-driven electronic dephasing will severely limit the prospects of QDs as single-photon sources. How should this issue be comprehensively addressed? While it does not undermine the novelty of the work.", "reviewer_2": "The manuscript “Phonon-driven wavefunction localization promotes room temperature, pure single-photon emission in large hybrid lead halide perovskite quantum dots” by Feld and coworkers presents experimental evidences of the positive role from the dynamic disorder associated with the A-cation thermal motion on the single photon emission properties of halide perovskite quantum dots (QD). Namely, measurements of the second-order autocorrelation function of the photon arrival times (g2(t)) at zero delay demonstrate single photon emission, at room temperature in CsPbBr3 and FAPbBr3 QDs, with single-photon purity of 71% and 96%, respectively. This is by itself an important results. The purity difference for the two compositions is then further rationalized using DFT-based Ab-Initio Molecular Dynamics (AIMD) simulations. The premise is that stronger confinement of the charge (as induced by reducing the QD size) improves single-photon purity (which I take as given, as I am not an expert myself in this specific field). From this premise, the authors provide convincing evidences from AIMD that the thermal motion in QD and related structural dynamic disorder localizes the electron/hole wavefunctions, ultimately acting as an additional confinement mechanism. Furthermore, they show that wavefunction localization is stronger in FAPbBr3, as due to its specific dynamics.\nThought I am not an expert in the field of single-photon emission, I enjoyed reading this manuscript, its shape well suiting the broad audience of the journal. The results are overall convincing and represent to my understanding a step forward in the current understanding of the problem. However, before recommending its publication on Nature Communication, I believe the authors should try to clarify an important point.\nAs stated above, I like the message of the positive role of dynamic disorder on the single particle wavefunction confinement. Actually a similar mechanism was proposed long time ago by Wang (Nano Lett. 15, 248) and De Angelis (PCCP, 17,9394) to explain the reduced electron/hole recombination in 3D bulk halide perovskites. This however does not reduce the originality of the present work, as the charge localization is discussed here in relation to single photon-emission. However, the authors report indicative measurements of the wavefunction spread in Figure 3a,b, which set around 1-2 nm. On the other hand, when introducing the idea of spatial confinement from the exciton extension itself, in contrast to the QD size, the authors mention radii, the authors mention Bohr radii for the exciton of 6.16 nm and 7.76 nm, for CsPbBr3 and FAPbBr3, respectively. Even more, the Bohr radius is smaller for CsPbBr3, while the results from AIMD point out stronger confinement from FAPbBr3. This discrepancy somehow cast some shadow the full argument of the charge localization. Actually, it gives the impression that one compares “apples” with “pears”. Could the authors comment a bit on this? Why this discrepancy?\nIn addition, I take the occasion to rise few additional remarks, which are absolutely secondary but may still add something to the text.\n1) I find the red signal in Figure 3 unexpectedly neat. For the sake of reference, the power spectrum of the band gap at 300 K of bulk halide perovskites was reported recently in Adv. Opt. Mater., 2301105, 2023, showing very messy signals in the frequency domain. Notice also that such clean signal somehow contrasts a lot with the importance of phonon anharmonicity that the authors bring up at page 7. Overall, what comes out is that there is this clean harmonic oscillation beating at 17 meV (in nice agreement with the electron-phonon coupling analysis of Giustino Nat. Comm. 7,11755), which actually points towards the lack of anharmonicities. Then, the power spectrum in Figure 3d, shows again broad features, pointing towards anharmonicities. So, is anharmonicity important or not? And why one gets cleaner signal from FAPbBr3, which is supposed to be more disordered?\nI personally suspect that there is just a mistake on the data treatment, as especially at 300 K one does not expect anything similar. The authors can compare with the power spectrum of the band gap, as a reference.\n2) I think the authors should reduce the emphasis on the general importance of the A-site and may rather insist on the fact that single-photon emission is found at room temperature. Dealing with the A-site, the 72 meV redshift in the PL of FAPbBr3 with respect to CsPbBr3 is indeed marginal, especially when compared with the 700 meV blueshift with respect of CsPbI3. However, I fully agree that this “marginal” effect has important implications in specific applications like single photon emission. In particular, an important outcome of the proposed localization mechanism is that it is effective at higher temperatures, where thermal motion is indeed more important. In this sense, the chemical message is partly “depleted” (what else can you use apart FA and MA?) but the mechanism itself of thermally induced charge localization may indeed guide the design of new materials, featuring this property.\n3) I have some pain from time to time to connect with the procedures and results in Supporting Information: these are very well detailed but I must jump from the text to the SI to better understand what the authors do. I recommend the authors to slightly adjust this. For instance:\n3.1) When mentioning the crystalline structure of materials (Figure 2a,b and Supporting Information), I recommend the authors to show the XRD patterns. I know the authors report a lot of data but sending the reader to the CCDC is very annoying.\n3.2) I am not sure the authors mention anything about the statistics of the analysed AIMD snapshots. How many snapshots did you study? This will give a clear impression about how robust the statistic is (I know it is robust, as the videos and all the SI information suggest few hundreds snapshots).\n3.3) When calling the results in Figure 3b and 3c, please add a reference to the employed procedure used to evaluate the size of the wavefunction and the autocorrelation function in the text (“see note #7” or “see Eq. SI8).\n3.4) How do you get the power spectra in the inset of Figure 3c? just Fourier transforming the time-domain signal or is there anything else? Especially if you do something more complex, please, add a short note, either in the text or in Supporting Information,", "reviewer_3": "The work by Feld et al. reports high-purity (g (2) (0) < 0.05), stable (> 1 hour), and bright (> 1Mcps) single-photon emission from individual colloidal FAPbX3 quantum dots with large spectral tunability. Through a combination of single-dot photoluminescence spectroscopy and ab initio molecular dynamics simulations, the authors attribute the suppressed biexciton emission to enhanced Auger recombination induced by charge-carrier localization. This study highlights the potential of organic–inorganic hybrid perovskite nanocrystals as high-performance quantum light sources operable under ambient conditions, and offers new fundamental insights into tailoring charge-carrier interactions in perovskite QDs beyond conventional size or shape control. I believe this work is of a quality suitable for publication in Nature Communications, pending clarification of several claims.\n(1) In the section “Influence of the A-site cation on structure and optical properties” (Page 5), the authors state that CsPbBr3 QDs and FAPbBr3 QDs of almost the same size exhibit similar quantum confinement, despite their different exciton Bohr diameters (6.16 nm vs. 7.76 nm). However, the larger exciton Bohr diameter in FAPbBr3 QDs would suggest stronger quantum confinement compared to CsPbBr3 QDs, which in turn could lead to stronger charge-carrier localization. I would like the authors to clarify whether this enhanced quantum confinement, in addition to that mediated by electron-phonon interactions, might also contribute to the more efficient biexciton Auger recombination and the higher single-photon purity observed here.\n(2) In the same section (Page 5), the authors attribute the localized states to thermal disorder, which is somewhat unclear. The larger FA cation could better stabilize the crosslinked Pb–Br octahedra in their cubic phase, leading to less lattice distortion. Thermal disorder would therefore be more prominent in CsPbBr3 QDs rather than in FAPbBr3 QDs. I wonder if the disorder and the flatter Urbach tail could originate from excited-state lattice deformation and strong electron–phonon interactions in FAPbBr3 QDs. Could the authors comment on this point?\n(3) In the section “Phonon-driven wavefunction localization” (Page 6), the authors use 3.6 nm, ABr-terminated CsPbBr3 and FAPbBr3 QD models, both of which fall within the strong confinement regime, to analyze wavefunction localization in these two types of QDs. It would be helpful to clarify whether such models can accurately describe wavefunction localization in the experimentally studied QDs, which are in the weak confinement regime. It is worth noting that the wavefunction of HOMO in CsPbBr3 extends notably to the QD surface even at 300 K (Fig. 3a and S8), indicative of strong quantum confinement. Clarification on how these models capture carrier localization under experimental conditions would strengthen the discussion.\n(4) In the section “Phonon-driven wavefunction localization” (Page 6), the authors mainly analyze the spatial distribution of the HOMO in their QD models. However, since the experimental photoluminescence spectroscopy reflects the combined behavior of electrons and holes, it would be more appropriate to consider both the HOMO and LUMO levels simultaneously. Could the authors provide such an analysis?\n(5) In the section “Purified single-photon emission” (Page 10), the authors attribute the quenched biexciton emission mainly to accelerated non-radiative (Auger) recombination in FAPbBr3 QDs. However, according to a recent study (Angew. Chem. Int. Ed., 2020, 59, 14292–14295) based on power-dependent transient absorption spectroscopy measurements, CsPbBr3 and FAPbBr3 QDs of similar sizes exhibit comparable biexciton Auger recombination rates at room temperature, which are shorter than biexciton lifetimes determined by heralded single-particle photoluminescence spectroscopy. Could the authors clarify whether this discrepancy might arise from differences in measurement techniques?\n(6) In the section “Purified single-photon emission” (Page 10–11), the authors attribute the reduced photon anti-bunching at lower temperatures to enhanced wavefunction delocalization, which is said to significantly increase the radiative recombination rate of exciton species. However, biexciton Auger recombination could also be suppressed at low temperatures due to its thermally activated nature, as this non-radiative recombination requires energy and momentum conservation and is likely to be phonon-assisted. Therefore, the increased biexciton emissivity observed at low temperature might also arise from this effect. I recommend the authors clarify this point.\n(7) In the section “FAPbX3 QDs as single-photon sources” (Page 13), the authors demonstrate the wide spectral tunability of single-photon emission from FAPbX3 QDs by controlling the halogen composition. However, phase segregation, as well as local environmental inhomogeneity even at the single-QD level, could introduce site-to-site variations in the emission properties of localized excitons, potentially leading to more significant spectral diffusion. Could the authors provide an outlook on possible technical approaches to suppress spectral diffusion in mixed-halide FAPbX3 QDs?"}, "author_response": "Response to the Referee Reports for Feld et al. (NCOMMS-25-40681-T)\nWe sincerely thank all the Reviewers for their detailed reading of the manuscript and their\nconstructive comments. We followed their suggestions in close detail to improve the\nmanuscript. Below, please find the revisions in response to the Reviewer's comments point-\nby-point, with the comments of the Reviewers in black, followed by our reply and action in\nblue.\nWe are convinced that our implemented changes have improved the manuscript and\naddressed the open points raised by the Reviewers.\nReviewer #1 (Remarks to the Author):\nIn this manuscript, the authors propose a novel strategy to enhance quantum confinement\nin colloidal quantum dots (QDs) and improve their single-photon purity, eliminating the need\nfor QD downsizing. Based on combined theoretical and spectroscopic approaches, it is\nrevealed that the A-site cation in soft APbBr3 colloidal QDs controls the phonon-induced\nlocalization of the exciton wavefunction. Detailed density functional theory (DFT) results\ndemonstrate that the transient wavefunction confinement is realized via strong exciton-\nphonon coupling, contrasting the generally negative impact of such couplings in\noptoelectronic materials. In my opinion, this work presents a solid piece of work, which\nshould be of interest to the community. It sheds new light on achieving bright, stable, and\npure single-photon emission, while circumventing the optical stability issues associated with\nconventional QD downsizing.\nOur response: We are grateful for the appreciation and the attentive reading of our work.\nI would recommend its publication pending the authors' consideration of the following\npoints:\n1.In the computational section, have the selected methods undergone benchmark work to\nverify their applicability to the target system? Additionally, how are the weak interactions\naccounted for?\nOur response: Our computational methodology is based on recent work on phonon-induced\ndephasing and emission broadening in perovskite QDs (Nat. Commun. 2022 ,13, 2587). We\nhave opted for this method as it provides a well-proven balance between a sufficiently high\nlevel of theory and a sufficiently large model size, to best reflect the experimentally\ninvestigated QDs. This balance has previously facilitated to, often quantitatively, reproduce\n1\nseveral experimental observations made for perovskite QDs, from the emission linewidth to\nthe carrier-cooling rates and the defect tolerance, see for example ACS Photonics 2020, 7, 5,\n1088–1095, Nat. Commun. 2022, 13, 2587, and ACS Energy Lett. 2019, 4, 1, 63–74. In Figure\nR1, we summarize recent studies that showcase the high compatibility of this computational\nmethodology and single-QD PL measurements. The agreement between experimental and\nsimulated phonon frequencies as well as QD-size-dependent electron–phonon coupling\ndemonstrates that this computational method is suitable for investigating electron–phonon\ncoupling in soft perovskite compounds (Adv. Optical Mater. 2024, 12, 2301534; Nat.\nCommun. 2022, 13, 2587). In the present work, we additionally benchmarked our\ncomputational approach by comparing bandgap fluctuations in MD simulations and PL\nbroadening in single-particle and ensemble PL studies (Supplementary Fig. 2-4). Also here,\nour computational models recover the experimentally observed temperature and A-site-\ncation dependence of phonon-induced energy fluctuations. Although still qualitative, given\nthe small QD sizes computationally accessible, these past and present works demonstrate\nthe suitability of the employed theoretical approach and its effectiveness in providing\nimportant insights into the photophysics governing exciton-phonon coupling.\nAs in the highlighted studies, we chose not to explicitly account for weak interactions in\nfavour of larger computational models and longer MD trajectories. This choice is\nunderpinned by reproducing various experimental trends at this level of theory and is\nadditionally justified by the limited role of such interactions at room temperature. While such\ninteractions affect geometry optimized (0 K) structures (ACS Omega 2020, 5, 40, 25723–\n25732) and are linked to the emergence of ordered nanodomains in otherwise disordered\nFAPbBr and MAPbBr crystals (Nat. Nanotechnol. 2025, 20, 755–763), such interactions\n3 3\ncannot impose order over extended spatial or temporal ranges. This also becomes apparent\nin our X-ray total scattering experiments (Supplementary Fig. 20). Even at cryogenic\ntemperatures, FAPbBr lacks long range order (J. Phys. Chem. Lett. 2023, 14, 5, 1288–1293),\n3\nsuggesting a relatively weak interaction between the A-cation and Pb-Br frame. Further\nevidence comes from nuclear magnetic resonance of structurally-related cubic lead iodide\nperovskites, where the FA cations were observed to rotate invariantly on the picosecond-\ntimescale (J. Am. Chem. Soc. 2023, 145, 2, 978–990).\nAIMD simulations. a, Representative single-particle PL spectrum from a sample of CsPbBr\nthe phonon replica with respect to the ZPL, extracted from spectra as shown in a. c, Intensity\nof the phonon replica relative to the intensity of the ZPL, representing the magnitude of the\nexciton-phonon coupling strength. d, Reorganization energy as a function of the phonon\nenergy obtained from AIMD simulations of a 1.8 nm CsPbBr QD. e, Size-dependence of the\n3\ncoupling strength of the three phonon modes and the extrapolated room-temperature PL\nbroadening. Note that the latter estimation neglects anharmonic effects that are relevant in\nthis system at room temperature. Panels a-c were reproduced from Adv. Optical Mater. 2024,\n12, 2301534; d,e were reproduced from Nat. Commun. 2022, 13, 2587.\n2.A key highlight of this work is the achievement of high-purity single-photon emission\nwithout the need for QD downsizing. What criteria determine the optimal QD size for this\nmechanism? Is there potential for this mechanism to be applicable to bulk materials as well?\nOur response: We appreciate the chance to address this point, because, there is indeed\nevidence that wavefunction localization exists in bulk and affects several photophysical\nproperties (J. Am. Chem. Soc. 2022, 144, 41, 19137–19149, Nano Lett. 2015, 15, 1, 248–253).\nemitters. While wavefunction localization relaxes momentum conservation rules, a first\nrequirement for Meitner-Auger recombination, excitons also need to be coupled to each\nother (Nano Lett. 2010, 10, 1, 313–317). A sizable interaction strength, or biexciton binding\nenergy, is required, which is not the case in bulk materials where excitons are independent.\nIn other words, excitons created in the bulk do not collide and annihilate each other\nefficiently. Hence, bulk materials do not act as single-photon emitters. The stochastic nature\nof the absorption process generates a random number of excitons, which subsequently\nradiate a random number of photons, thereby precluding the antibunching feature\ncharacteristic of single-photon emitters.\nOverall, we agree that there may be an optimal QD size for achieving stable and bright single-\nphoton emission. Increasing the QD size enhances the brightness (via enhanced light\nabsorption) and the photostability of the emitter. However, increasing the QD size eventually\nwill also compromise the single-photon purity, since bulk systems are not effective two-level\nsystems, as explained above. This suggests the existence of an optimal QD size for bright and\nstable QD-based single-photon emitters. While we refrain from providing here a precise\nnumber for the optimal QD size, our present work shows that good single-photon purity can,\nthanks to dynamic partial wavefunction localization, now be extended to QD sizes beyond 10\nnm, i.e., much larger than usual, yielding stable single-photon emission with a million counts\nper second (1 Mcps).\nOur action: We clarified this point in the main text:\n“Finally, temperature-induced localization even occurs in the respective bulk\nmaterials, specifically for FAPbBr (Supplementary Fig. 11, 12). Although bulk crystals\n3\nwill not experience Auger-Meitner relaxation due to vanishing inter-exciton coupling,\nsuch localization effects reduce radiative and non-radiative rates and limit carrier\nmobilities in lead halide perovskites.45,58-60”\n3.As clarified in this manuscript, two types of QDs exhibit distinct phonon-driven\nwavefunction localization. Figure S3 in the Supporting Information presents the PL spectra\nof these two QDs as a function of temperature. I recommend that the authors conduct a\nquantitative analysis of the relationship between wavefunction localization and PL energy.\nGiven that the PL energy shifts are generally regarded as directly related to wavefunction\nconfinement, such a correlation could further validate the novel mechanism.\nOur response: We appreciate this valuable suggestion. To follow it, we correlate the PL peak\nenergy, PL peak width, and PL lifetime from Supplementary Fig. 3 with the wavefunction sizes\nfrom Figure 3b and Supplementary Fig. 5. As apparent in Figure R2, all three experimentally\nobtained metrics correlate with the wavefunction sizes, suggesting that these effects are\n4\nlinked. PL peak width, PL lifetime and wavefunction localizations are directly linked to\nexciton-phonon coupling. We agree with your suggestion that the PL peak energy is sensitive\nto the wavefunction localization, and the analysis in Figure R2, indeed confirms our\nattribution that a stronger wavefunction localization is in place in FAPbBr QDs (larger PL\n3\nbroadening, longer lifetime, larger PL bandgap shift). Nevertheless, there are several other\ntemperature-sensitive influences that would need to be accounted for to fully describe PL\npeak energy temperature evolution. Those include optical and acoustic phonons (Nanoscale\n2020,12, 13113-13118, J. Phys. Chem. Lett. 2021, 12, 1, 569–575), structural distortions\n(Small Struct. 2024, 5, 2300264, J. Phys. Chem. Lett. 2020, 11, 7, 2490–2496, J. Phys. Chem.\nLett. 2021, 12, 1, 569–575), disorder (npj Comput. Mater. 2023, 9, 153), thermal expansion\ndue to anharmonicity (Phys. Chem. Chem. Phys. 2020,22, 26069-26087), as well as phases\ntransitions, for example occurring in FAPbBr between 100 and 250 K (Nano Lett. 2018, 18, 7,\n3\n4440–4446, Inorg. Chem. 2018, 57, 2, 695–701, J. Chem. Phys. 2023, 158, 201104). The\noptical bandgap is, unfortunately, a complicated probe for temperature-induced\nwavefunction localization. A quantitative analysis is hindered by multiple effects, some of\nwhich are intertwined, making the attribution of wavefunction localization through this\nmethodology possible only on a qualitative level.\nwavefunction size. a, PL peak energy extracted from ensemble PL measurements. Grey\narrows indicate phase transitions in FAPbBr QDs (Chem. Mater. 2025, 37, 9, 3443–3454). b,\nlifetime (1/e decay time) from ensemble time-resolved PL. d, Average HOMO wavefunction\nsize (given as FWHM) in MD simulations of 3.6 nm QD models.\n4.It is found that the loss of correlation is significantly faster for the FAPbBr QD than for the\nCsPbBr3 QD. Generally, the phonon-driven electronic dephasing will severely limit the\nprospects of QDs as single-photon sources. How should this issue be comprehensively\naddressed? While it does not undermine the novelty of the work.\nOur response: We absolutely agree that such a temperature- and phonon-mediated\nmechanism is incompatible with applications that require long coherence time and\n6\nindistinguishability, such as photonic quantum computing. Nevertheless, there exist several\nquantum-enhanced communication or imaging schemes based on pure single photon\nsources which do not require such a property, e.g. QKD using the BB84 protocol. Such\napplications are not affected by dephasing and can be targeted by room-temperature single-\nphoton sources (Opt. Express 2023, 31, 9437-9447, Nat. Photonics 2019, 116–122). If they\noffer sufficient photon count rates, single-photon purity and photostability for these\napplications, room-temperature single-photon sources may favour scalability and\nsustainability compared to cryogenically cooled single-photon sources. In this work, we\npropose and demonstrate a method that can enhance the single-photon purity in large,\nphotostable QDs. We believe that marks an important step for colloidal QDs as room-\ntemperature quantum light sources.\nReviewer #2 (Remarks to the Author):\nThe manuscript “Phonon-driven wavefunction localization promotes room temperature,\npure single-photon emission in large hybrid lead halide perovskite quantum dots” by Feld\nand coworkers presents experimental evidences of the positive role from the dynamic\ndisorder associated with the A-cation thermal motion on the single photon emission\nproperties of halide perovskite quantum dots (QD). Namely, measurements of the second-\norder autocorrelation function of the photon arrival times (g2(t)) at zero delay demonstrate\nsingle photon emission, at room temperature in CsPbBr3 and FAPbBr3 QDs, with single-\nphoton purity of 71% and 96%, respectively. This is by itself an important results. The purity\ndifference for the two compositions is then further rationalized using DFT-based Ab-Initio\nMolecular Dynamics (AIMD) simulations. The premise is that stronger confinement of the\ncharge (as induced by reducing the QD size) improves single-photon purity (which I take as\ngiven, as I am not an expert myself in this specific field). From this premise, the authors\nprovide convincing evidences from AIMD that the thermal motion in QD and related\nstructural dynamic disorder localizes the electron/hole wavefunctions, ultimately acting as\nan additional confinement mechanism. Furthermore, they show that wavefunction\nlocalization is stronger in FAPbBr3, as due to its specific dynamics.\nThought I am not an expert in the field of single-photon emission, I enjoyed reading this\nmanuscript, its shape well suiting the broad audience of the journal. The results are overall\nconvincing and represent to my understanding a step forward in the current understanding\nof the problem. However, before recommending its publication on Nature Communication, I\nbelieve the authors should try to clarify an important point.\nOur response: We really appreciate the positive assessment of our work.\nparticle wavefunction confinement. Actually a similar mechanism was proposed long time\nago by Wang (Nano Lett. 15, 248) and De Angelis (PCCP, 17,9394) to explain the reduced\nelectron/hole recombination in 3D bulk halide perovskites. This however does not reduce the\noriginality of the present work, as the charge localization is discussed here in relation to\nsingle photon-emission. However, the authors report indicative measurements of the\nwavefunction spread in Figure 3a,b, which set around 1-2 nm. On the other hand, when\nintroducing the idea of spatial confinement from the exciton extension itself, in contrast to\nthe QD size, the authors mention radii, the authors mention Bohr radii for the exciton of 6.16\nnm and 7.76 nm, for CsPbBr3 and FAPbBr3, respectively. Even more, the Bohr radius is\nsmaller for CsPbBr3, while the results from AIMD point out stronger confinement from\nFAPbBr3. This discrepancy somehow cast some shadow the full argument of the charge\nlocalization. Actually, it gives the impression that one compares “apples” with “pears”. Could\nthe authors comment a bit on this? Why this discrepancy?\nOur response: We appreciate the constructive feedback, and we acknowledge that\nintroducing the Bohr diameter could be misleading. We mentioned the Bohr diameters to\nindicate that both samples would be in the intermediate confinement regime if one excluded\nthe effect of disorder-induced wavefunction localization. Extracted from band structures,\nBohr diameters indicate the exciton wavefunction size at 0 K. In weakly electronically\ndisordered systems, the Bohr diameter can remain a good indicator of the exciton size or\nconfinement also at finite temperature. In the presence of pronounced disorder at higher\ntemperatures, however, this concept needs to be revised. The occupation of vibrational\nmodes and their coupling to electronic degrees of freedom generate a disordered\ndeformation potential that spatially and temporally fluctuates around a mean value (Entropy\n2024, 26, 7, 552, Phys. Rev. B 2022, 106, 054311). This deformation potential acts as an\nadditional effective confinement potential on the wavefunctions (Phys. Rev. Lett. 2016, 116,\n056602). Therefore, at elevated temperature, FAPbBr exhibits smaller wavefunctions than\n3\ninstantaneous value extracted from DFT calculations. Given the dynamic nature of this\nlocalization, integration over many time realizations will broaden the spatial extent. However,\nit is this instantaneous localization that is responsible for the enhanced Auger rate and,\nconsequently, the significantly improved single-photon emission.\nOur action: We have clarified this point in the main text:\n“Although the concept of Bohr diameters is not strictly valid at room temperature for\nmaterials exhibiting strong exciton-phonon coupling, we additionally corroborated the\nA-site-cation dependent single-photon purity for samples with identical sizes\nnormalized by Bohr diameters (Supplementary Fig. 22).”\nsecondary but may still add something to the text.\n1) I find the red signal in Figure 3 unexpectedly neat. For the sake of reference, the power\nspectrum of the band gap at 300 K of bulk halide perovskites was reported recently in Adv.\nOpt. Mater., 2301105, 2023, showing very messy signals in the frequency domain. Notice\nalso that such clean signal somehow contrasts a lot with the importance of phonon\nanharmonicity that the authors bring up at page 7. Overall, what comes out is that there is\nthis clean harmonic oscillation beating at 17 meV (in nice agreement with the electron-\nphonon coupling analysis of Giustino Nat. Comm. 7,11755), which actually points towards\nthe lack of anharmonicities. Then, the power spectrum in Figure 3d, shows again broad\nfeatures, pointing towards anharmonicities. So, is anharmonicity important or not? And why\none gets cleaner signal from FAPbBr3, which is supposed to be more disordered?\nI personally suspect that there is just a mistake on the data treatment, as especially at 300 K\none does not expect anything similar. The authors can compare with the power spectrum of\nthe band gap, as a reference.\nOur response: We appreciate this constructive comment on the data treatment and the\neffect of anharmonicity. We agree that the wavefunction autocorrelations and the associated\npower spectra are strikingly clean (they are obtained at 300 K) in comparison to bandgap\npower spectra in Adv. Opt. Mater. 2023, 2301105 or those reported by some of us (Nat.\nCommun. 2022, 13, 2587). By following the suggestion to construct such bandgap power\nspectra, we will show below that the displayed curves are not an artifact. However, we will\nfirst describe how the wavefunction autocorrelation and power spectra in Figure 3c were\nobtained. The absolute of the normalized molecular orbital coefficients 𝑐 (𝑡) of the HOMO\n𝑖\nwavefunction recorded in NVE simulations (300 K, >14’000 steps, 1 fs timesteps) were first\nshifted by their average and then autocorrelated, followed by summation over all\ncoefficients:\n∞\n𝐶 (𝜏) = ∫ 𝑑𝑡∑ (|𝑐 (𝑡)|−〈𝑐 〉 ) (|𝑐 (𝑡+𝜏)|−〈𝑐 〉 ) . (eq. S9)\nΦ −∞ 𝑖 𝑖 𝑖 𝑡 𝑖 𝑖 𝑡\nWe multiplied this autocorrelation function with a cos2-window function to minimize the\naccumulation of noise at long delay times: 𝐶′ (𝜏) = 𝐶 (𝜏)×cos2( 𝜋𝜏 ), where L is the length\nΦ Φ\n𝐿\nof the MD trajectory. Finally, we normalized the autocorrelation function as 𝐶 (𝜏) =\nΦ,norm\n𝐶′ (𝜏)/𝐶′ (0), see Figure 3c. The power spectrum (Figure 3c, inset) was obtained as the\nΦ Φ\nabsolute square of the Fourier transform of 𝐶′ (𝜏). In Figure R3a, we compare this analysis\nΦ\nwith and without applying a window function. This comparison convinced us that the window\nfunction does not cause the clean spectrum; it rather fulfills its purpose of enhancing the\nsignal-to-noise ratio without significantly broadening spectral features.\nand recorded the time-dependent bandgap (HOMO-LUMO gap). We recorded 2000 steps for\nboth systems, employing 1 fs timesteps for FAPbBr and 10 fs for CsPbBr . The corresponding\n3 3\nautocorrelations and power spectra of the bandgaps are plotted in Figure R3b. We observe a\nhigh degree of similarity with the complex spectra shown in Adv. Opt. Mater. 2023, 2301105.\nWe are therefore confident in the correctness of our procedure to obtain the power spectra\nof the bandgap and of the wavefunction coefficients. As for the difference between the power\nspectra of the bandgap and the wavefunction, we suspect that the multidimensionality of the\nwavefunction coefficients may enhance the signal-to-noise ratio.\nLead halide perovskites exhibit a high degree of anharmonicity, apparent for example in\ntemperature-dependent crystal structures (Chem. Mater. 2025, 37, 9, 3443–3454) or short\nphonon lifetimes (Proc. Natl. Acad. Sci. U.S.A. 2018, 115, 47, 11905-11910). Pseudo-cubic\nperovskites are considered the most anharmonic amongst lead halide perovskites due to a\nshallow potential energy surface allowing for spontaneous local symmetry breaking (npj\nComput. Mater. 2023,9, 153). The anharmonicity is also apparent by the broad low-frequency\nfeature in the wavefunction power spectrum for FAPbBr , in stark contrast to CsPbBr , in\n3 3\nwhich the less localized wavefunction is not influenced by an anharmonic mode, but by the\nPb-Br stretching. Overall, the much broader power spectrum of the wavefunction\nautocorrelation in FAPbBr , alongside the significantly stronger disorder-enhanced single-\n3\nphoton purity and thermally increased PL lifetime in this material, suggest that\nanharmonicity is indeed an important ingredient for the disorder-induced wavefunction\nlocalization.\nOur action: We have revised the description of the power spectrum analysis in the\nSupplementary Information (Supplementary Note 7.3), added Adv. Opt. Mater. 2023,\n2301105 to our reference list, and clarified our discussion of anharmonicity in the\ncomputational section of the main text:\n“The broad signal in FAPbBr , contrary to the narrow peak in CsPbBr , attests a high\n3 3\ndegree of anharmonicity.”\n3 3\nof the wavefunction coefficients and the effect of applying a cos2-window function, indicated\nby (w). b, Autocorrelation and power spectrum (inset) of the bandgap.\n2) I think the authors should reduce the emphasis on the general importance of the A-site\nand may rather insist on the fact that single-photon emission is found at room temperature.\nDealing with the A-site, the 72 meV redshift in the PL of FAPbBr3 with respect to CsPbBr3 is\nindeed marginal, especially when compared with the 700 meV blueshift with respect of\nCsPbI3. However, I fully agree that this “marginal” effect has important implications in\nspecific applications like single photon emission. In particular, an important outcome of the\nproposed localization mechanism is that it is effective at higher temperatures, where thermal\nmotion is indeed more important. In this sense, the chemical message is partly “depleted”\n(what else can you use apart FA and MA?) but the mechanism itself of thermally induced\ncharge localization may indeed guide the design of new materials, featuring this property.\nOur response: We appreciate this valuable comment, and we agree that the control offered\nby this mechanism is an important result of this work with implications beyond lead halide\nperovskites. We would like to stress, however, that the marginal difference between the\nelectronic structures of Cs- and FA-based QDs was indeed crucial for the present discovery\nand intended so. Although one may expect the two QD systems to exhibit similar electronic\nbehavior (with Cs and FA orbitals not even participating in the electronic structure), their\nsingle-photon emission properties are markedly different, which we traced back to their\ndiffering crystal softness and exciton–phonon coupling strength. Having had at hand two\nelectronically similar systems with markedly different structural dynamics and associated\neffects on the electronic wavefunction significantly strengthened our confidence in the main\nclaim of this manuscript, i.e. the attribution of improved single-photon purity to stronger\nwavefunction localization. We highlight that all known organic (AZ, MA, FA)-inorganic\n11\nperovskite QDs possess better single photon purity at room temperature compared to fully\ninorganic perovskite QDs. As nicely outlined by you, we indeed hope that the results\npresented here can serve as the basis for new design rules in emerging material systems,\nopening pathways toward highly efficient quantum light sources operating at room\ntemperature.\nOur action: We have included your suggestion in the conclusions section of the main text:\n“Considering the ubiquity of electron-phonon coupling, we are confident that actively\ntuning this interaction provides opportuniteis for functional material engineering\nbeyond perovskite QDs.”\n3) I have some pain from time to time to connect with the procedures and results in\nSupporting Information: these are very well detailed but I must jump from the text to the SI to\nbetter understand what the authors do. I recommend the authors to slightly adjust this. For\ninstance:\nOur response: We appreciate that these points are raised, and we have improved the\nreadability and accessibility of our manuscript and data.\n3.1) When mentioning the crystalline structure of materials (Figure 2a,b and Supporting\nInformation), I recommend the authors to show the XRD patterns. I know the authors report\na lot of data but sending the reader to the CCDC is very annoying.\nOur response: The crystal structures were obtained from single crystal X-ray scattering data,\nwhich we have reported as normalized total scattering in Supplementary Fig. 20. The\ncorresponding structure solutions are reported in Supplementary Table. 1-3. Such data is\nusually not represented as one-dimensional XRD patterns.\n3.2) I am not sure the authors mention anything about the statistics of the analysed AIMD\nsnapshots. How many snapshots did you study? This will give a clear impression about how\nrobust the statistic is (I know it is robust, as the videos and all the SI information suggest few\nhundreds snapshots).\nOur response: We appreciate raising this point. MD simulations in the NVT ensemble\nreported in Figure 3a,b employed timesteps of 1 fs. After an equilibration phase of 6 ps\nduration, we acquired trajectories for another 12-14 ps and printed cube files every 0.1 ps.\nThus, over 120 snapshots were averaged per datapoint in Figure 3b. Similar analyses reported\nin the Supplementary Information employed 10 fs timesteps for CsPbBr in some cases, e.g.,\n3\nbulk and 5.4 nm QDs. Simulations in the NVE ensemble reported in Figure 3c,d were\nperformed after equilibration in the NVT ensemble. We employed 1 fs timesteps and\nacquired trajectories for > 14 ps while printing the HOMO wavefunction coefficients in each\ntimesteps for FAPbBr and 10 fs timesteps for CsPbBr .\n3 3\nOur action: We followed the suggestion to indicate the statistics in the main text and in\nmethods section:\n“Figure 3b shows the wavefunction sizes averaged across >120 snapshots (>12 ps;\nfurther details in Supplementary Note 7) in the CsPbBr and FAPbBr QDs along AIMD\n3 3\ntrajectories at 10, 100, and 300 K (further temperatures in Supplementary Fig. 5).”\n“Trajectories with a duration of at least 12 ps were produced after equilibration at the\nrespective temperature for 6 ps. To analyse the wavefunction confinement in real\nspace, cube files for the highest occupied molecular orbital (HOMO) wavefunctions\nwere extracted every 0.1 ps (>120 snapshots).”\n3.3) When calling the results in Figure 3b and 3c, please add a reference to the employed\nprocedure used to evaluate the size of the wavefunction and the autocorrelation function in\nthe text (“see note #7” or “see Eq. SI8).\nOur response and action: We followed this suggestion and included such references in the\nmain text:\n“Further insight into the phonon-driven wavefunction localization is provided by the\nautocorrelation function of the HOMO wavefunction coefficients which describes the\ntime evolution of the wavefunction (Supplementary Note 7, Supplementary Equation\n9).”\n“The power spectra of the wavefunction autocorrelation (Supplementary Note 7) in\nthe inset in Figure 3c reveal the dominant vibrational features driving the wavefunction\nlocalization in both systems.”\n3.4) How do you get the power spectra in the inset of Figure 3c? just Fourier transforming the\ntime-domain signal or is there anything else? Especially if you do something more complex,\nplease, add a short note, either in the text or in Supporting Information,\nOur response and action: We have provided a detailed description of the procedure in\nresponse to point 1) above and revised the description of the power spectrum analysis in the\nSupplementary Information (Supplementary Note 7.3)\nReviewer #3 (Remarks to the Author):\nThe work by Feld et al. reports high-purity (g (2) (0) < 0.05), stable (> 1 hour), and bright (>\n1Mcps) single-photon emission from individual colloidal FAPbX3 quantum dots with large\n13\nspectral tunability. Through a combination of single-dot photoluminescence spectroscopy\nand ab initio molecular dynamics simulations, the authors attribute the suppressed\nbiexciton emission to enhanced Auger recombination induced by charge-carrier localization.\nThis study highlights the potential of organic–inorganic hybrid perovskite nanocrystals as\nhigh-performance quantum light sources operable under ambient conditions, and offers\nnew fundamental insights into tailoring charge-carrier interactions in perovskite QDs beyond\nconventional size or shape control. I believe this work is of a quality suitable for publication\nin Nature Communications, pending clarification of several claims.\nOur response: We are grateful for the positive evaluation of our work.\n(1) In the section “Influence of the A-site cation on structure and optical properties” (Page 5),\nthe authors state that CsPbBr3 QDs and FAPbBr3 QDs of almost the same size exhibit similar\nquantum confinement, despite their different exciton Bohr diameters (6.16 nm vs. 7.76 nm).\nHowever, the larger exciton Bohr diameter in FAPbBr3 QDs would suggest stronger quantum\nconfinement compared to CsPbBr3 QDs, which in turn could lead to stronger charge-carrier\nlocalization. I would like the authors to clarify whether this enhanced quantum confinement,\nin addition to that mediated by electron-phonon interactions, might also contribute to the\nmore efficient biexciton Auger recombination and the higher single-photon purity observed\nhere.\nOur response: We appreciate the opportunity to clarify this important point. In the absence\nof electron-phonon interaction, this small difference in size-confinement may lead to a small\ndifference in single-photon purity. However, such a size difference is insufficient to explain\nthe observed pronounced A-site-cation dependence of the single-photon purity: First, even\nat similar size-confinement d/d (dimensionless size; d: edge length, d : Bohr diameter),\nB B\nFAPbBr QDs exhibit a higher single-photon purity (Figure R4). Secondly, there exists a\n3\nstrikingly different size dependence of the single-photon purity in these two materials (Figure\n4c, Supplementary Fig. 21c). Both points demand an explanation that extends beyond the\nQD size, as we have provided with support of computational modelling, heralded single-\nparticle PL spectroscopy, temperature-dependent single-particle PL spectroscopy,\ntemperature-dependent time-resolved PL spectroscopy, transmission electron microscopy,\nand X-ray total scattering.\nOur action: We have clarified this point in the main text and added Figure R4 as\nSupplementary Fig. 22 to the Supplementary Information:\n“Although the concept of Bohr diameters is not strictly valid at room temperature for\nmaterials exhibiting strong exciton-phonon coupling, we additionally corroborated the\nA-site-cation dependent single-photon purity for samples with identical sizes\nnormalized by Bohr diameters (Supplementary Fig. 22).”\n3 3\nand 95% confidence intervals (error bars) at identical confinement.\n(2) In the same section (Page 5), the authors attribute the localized states to thermal disorder,\nwhich is somewhat unclear. The larger FA cation could better stabilize the crosslinked Pb–Br\noctahedra in their cubic phase, leading to less lattice distortion. Thermal disorder would\ntherefore be more prominent in CsPbBr3 QDs rather than in FAPbBr3 QDs. I wonder if the\ndisorder and the flatter Urbach tail could originate from excited-state lattice deformation and\nstrong electron–phonon interactions in FAPbBr3 QDs. Could the authors comment on this\npoint?\nOur response: We agree that H-bonding between the FA cation and the Pb-Br framework\nshould have a stabilizing effect. In fact, a recent report linked such interactions to the\nemergence of ordered nanodomains in FAPbBr and MAPbBr (Nat. Nanotechnol. 2025, 20,\n3 3\n755–763). Yet, such interactions could not impose long range order, which also becomes\napparent in our X-ray total scattering experiments (Supplementary Fig. 20). Even at cryogenic\ntemperatures, FAPbBr lacks long range order (J. Phys. Chem. Lett. 2023, 14, 5, 1288–1293),\n3\nwhich suggests a relatively weak interaction between the A-cation and Pb-Br frame. Further\nevidence comes from nuclear magnetic resonance of structurally-related cubic lead iodide\nperovskites, where the FA cations were observed to rotate invariantly on the picosecond-\ntimescale (J. Am. Chem. Soc. 2023, 145, 2, 978–990).\nSpontaneous, dynamic, and local symmetry breaking through octahedral tilting, which is\ncoupled with the A-site-cation reorientation in hybrid perovskites, appears as a global\nphenomenon in the high-temperature cubic phases of lead halide perovskites (Nat.\nNanotechnol. 2025, 20, 755–763, Joule 2023, 7, 1051-1066, ACS Energy Lett. 2016, 1, 4, 880–\n887, J. Phys. Chem. Lett. 2017, 8, 19, 4720–4726, J. Phys. Chem. C 2023, 127, 38, 19141–\n15\n19151). Moreover, thermal disorder prevails over static disorder in these systems at room\ntemperature (J. Phys. Chem. Lett. 2023, 14, 5, 1288–1293, J. Phys. Chem. C 2018, 122, 30,\n17473–17480). With this body of literature and supported by our own experiments and\nsimulations (Supplementary Fig. 15-18, 20), we are confident in our statement that FAPbBr\nUrbach tails in FAPbBr are also explained by a stronger exciton-phonon coupling. Electron-\n3\nphonon coupling translates the structural disorder into electronic disorder. Various reports\nsubstantiate that FAPbBr exhibits stronger electron-phonon coupling (Nat. Phys. 2024, 20,\n3\n47–53 , Nano Lett. 2022, 22, 18, 7674–7681, Adv. Optical Mater. 2024, 12, 2301534).\nConsequently, electronic disorder and the associated wavefunction localization in FAPbBr\nFinally, there is support for assigning the Urbach tails to phonon-induced wavefunction\nlocalization. Recent theoretical studies of disorder in structurally related MAPbI found that\n3\nthe deformations by phonon-induced disorder are of similar magnitude as those caused by\npoint defects (J. Am. Chem. Soc. 2022, 144, 41, 19137–19149). While polarons certainly play\na key role in the photophysics of lead halide perovskites, the influence of thermal disorder on\ncharge mobilities exceeds that of polarons (Phys. Rev. Lett. 2020, 124, 196601) to which\nreduced charge mobilities were often attributed (Sci. Adv. 2017, 3, e1701217, Nat. Commun.\n2016, 7, 12253).\nOur action: We clarified the discussion of the Urbach tail:\n“Moreover, we observed an exponential low-energy (Urbach) tail in single-particle PL\nspectra, see Figure 2f. Across a large size range, the tails are consistently steeper for\nCsPbBr than in comparably sized FAPbBr QDs (Figure 2g). As elucidated in a prior\n3 3\nstudy,48 this suggests stronger exciton-phonon coupling in FAPbBr QDs, consistent\n3\nwith PL phonon replica at cryogenic temperatures53,54 and optical-pump–electron-\ndiffraction-probe measurements.55 This trend is also confirmed by our accompanying\ndensity functional theory (Supplementary Fig. 4) and ensemble PL studies\n(Supplementary Fig. 3). Urbach tails are associated with exciton-phonon coupling\nthrough the formation of localized states that result from thermal disorder and form\nthe low-energy tail.32,34,36 They thus also serve as a first indication of the hypothesized\nwavefunction confinement which may be enhanced in pseudo-cubic FAPbBr QDs.”\n3\n(3) In the section “Phonon-driven wavefunction localization” (Page 6), the authors use 3.6\nnm, ABr-terminated CsPbBr3 and FAPbBr3 QD models, both of which fall within the strong\nconfinement regime, to analyze wavefunction localization in these two types of QDs. It would\nbe helpful to clarify whether such models can accurately describe wavefunction localization\n16\nin the experimentally studied QDs, which are in the weak confinement regime. It is worth\nnoting that the wavefunction of HOMO in CsPbBr3 extends notably to the QD surface even at\nmodels capture carrier localization under experimental conditions would strengthen the\ndiscussion.\nOur response: We appreciate your comment about the distinct size ranges covered by QD\nmodels and by experiments. While indeed unfortunate, such a remaining size disparity is the\nstate of the art in the field when comparing single-QD measurements and ab-initio methods.\nOn the one hand, experimentally measuring single QDs becomes increasingly more\nchallenging for decreasing QD size, especially for hybrid organic-inorganic variants, and is\ntypically only performed for perovskite QDs larger than about 4 nm. Moreover, ab-initio\nmethods, as required here to quantify wavefunction localization, intrinsically come with a\nvery high computational cost. For example, for each computational data point in Figure 3b,\nwe have committed nearly 106 CPU hours to obtain ~20 ps long trajectories of atomistic\nFAPbBr QD models in the canonical ensemble. The same effort was put towards the NVE\n3\nsimulations for Figure 3c. The high cost originates from the large system size (3652 atoms)\nand the small timesteps (1 fs) required to capture fast intramolecular vibrations of the\norganic A-cation. With currently available computational methods, there is no shortcut\ncircumventing this bottleneck, as our interest in the global wavefunctions of the QD\nprecludes cost-cutting methods such as linear-scaling DFT which imposes a fragmentation\nof the system (J. Phys. Chem. A 2023, 127, 3, 589–618).\nDespite the partially incomplete match between computationally and experimentally\nassessed QD size ranges, several recent studies successfully combined computational QD\nmodelling and single-QD PL spectroscopy, particularly to assess exciton-phonon coupling.\nFor instance, there is great agreement in electron-phonon coupling spectra between single-\nQD PL measurements at cryogenic temperatures and computational modelling, delivering\nmatching phonon mode frequencies and QD size dependencies (Figure R5; Adv. Optical\nMater. 2024, 12, 2301534; Nat. Commun. 2022, 13, 2587). In the present work, we further\nbenchmarked our computational approach by comparing bandgap fluctuations in MD\nsimulations and PL broadening in single-particle and ensemble PL studies (Figures S2, S3,\nand S4). We found that our computational models recover both the temperature and A-cation\ndependence of phonon-induced energy variations.\nTo account for the QD-size dependence within the given computational-cost constraints, our\noriginal submission quantified the extent and dynamics of the wavefunction for QD sizes up\nto 3.6 nm for FAPbBr and up to 5.4 nm for CsPbBr (Supplementary Fig. 6, 8). We found\n3 3\ntemperature-induced localization across all QD sizes. In previous studies (Nature 2024, 626,\n535–541, Nat. Commun. 2022, 13, 2587, Nano Lett. 2020, 20, 3, 1819–1829), such QD size\n17\ndependencies proved instrumental in establishing a qualitative agreement between AIMD\nsimulations and experimental observations. Here, we find that the wavefunction size in\nCsPbBr QDs depends on the QD size while that of FAPbBr QDs does not. This observation\n3 3\nsupports your suggestion of size confinement in CsPbBr , while it suggests a disorder-limited\n3\nwavefunction size in FAPbBr . It also aligns with the experimentally observed weakly size-\n3\ndependent single-photon purity in FAPbBr and a stronger size dependence in CsPbBr .\n3 3\nFinally, we have eliminated any potential overestimation of finite size effects by monitoring\nthe wavefunctions in simulations of bulk crystals, employing periodic boundary conditions.\nWe observe similar temperature and A-site-cation dependencies of the wavefunction\nlocalization (Supplementary Fig. 9). Such bulk simulations of wavefunction localization have\npreviously also explained long radiative and non-radiative charge carrier lifetimes (J. Am.\nChem. Soc. 2022, 144, 41, 19137–19149, Nano Lett. 2015, 15, 1, 248–253).\nTaken together, we have benchmarked our computational-experimental approach for the\nstudy of exciton-phonon coupling in perovskite QDs before its application to wavefunction\nlocalization. Additionally, there exist numerous studies that have successfully taken a similar\nroute to decode exciton-phonon coupling in general and wavefunction localization in\nparticular, demonstrating a solid link to experimental observations.\nOur action: We have now addressed this point in the main text of the revised manuscript:\n“Such a systematically observed wavefunction localization across model sizes\nranging from 1.8 nm QDs to the bulk alongside the qualitative agreement in key\nexperimental observations linked to exciton-phonon coupling (Supplementary Fig. 3,\n4),7,26 affirms that such wavefunction localization also occurs in the perovskite QD\nsizes accessed experimentally (vide infra).”\nAIMD simulations. a, Representative single-particle PL spectrum from a sample of CsPbBr\nthe phonon replica with respect to the ZPL, extracted from spectra as shown in a. c, Intensity\nof the phonon replica relative to the intensity of the ZPL, representing the magnitude of the\nexciton-phonon coupling strength. d, Reorganization energy as a function of the phonon\nenergy obtained from AIMD simulations of a 1.8 nm CsPbBr QD. e, Size-dependence of the\n3\ncoupling strength of the three phonon modes and the extrapolated room-temperature PL\nbroadening. Note that the latter estimation neglects anharmonic effects that are relevant in\nthis system at room temperature. Panels a-c were reproduced from Adv. Optical Mater. 2024,\n12, 2301534; d,e were reproduced from Nat. Commun. 2022, 13, 2587.\n(4) In the same section, the authors mainly analyze the spatial distribution of the HOMO in\ntheir QD models. However, since the experimental photoluminescence spectroscopy\nreflects the combined behavior of electrons and holes, it would be more appropriate to\nconsider both the HOMO and LUMO levels simultaneously. Could the authors provide such\nan analysis?\nOur response: We recognize that including the LUMO (electron wavefunction) in our analysis\nis crucial and we accept the challenge to do so. We identified two approaches to model the\n19\nnotoriously difficult LUMO wavefunction and demonstrate that the LUMO, similarly to the\nHOMO, indeed also experiences a thermally activated wavefunction localization, hereby\nsupporting the proposed link between thermal disorder and single-photon purity. We\nappreciate the constructive comment which strengthened the conclusions of our work by\nclarifying the role of the electron wavefunction.\nThe key to analyse the LUMO wavefunction lies in passivating the perovskite QD surface.\nPerovskite QD models without full surface passivation create surface-localized LUMOs due\nto undercoordinated bromine atoms, see Figure R6a as well as previous works on the topic,\ne.g. ACS Energy Lett. 2019, 4, 1, 63–74 and J. Am. Chem. Soc. 2022, 144, 25, 11059–11063.\nWhile the atomistic understanding of the surface termination has recently experienced help\nfrom nuclear magnetic resonance spectroscopy (J. Am. Chem. Soc. 2020, 142, 13, 6117–\n6127, Nature 2024, 626, 542–548), the exact ligand coverage and ligand arrangement\nremains elusive. This insufficient understanding of the precise surface chemistry hinders us\nfrom passivating the QD surface with organic surface-capping ligands in exact analogy to the\nexperiment. Currently, efforts are underway to introduce a chemically and physically\nsatisfying passivation in our models and experimental validation is ongoing.\nFor the time being, the electrostatic environment provided by surface ligands can be\nemulated by epitaxially shelling CsPbBr with a CsCaBr -shell, as shown in Figure R6b. This\n3 3\nwide-bandgap shell enables the delocalization of the LUMO wavefunction in the QD core, as\nwe could show in two recent studies (Nat. Commun. 2022, 13, 2587, Nature 2024, 626, 535–\n541). Such an approach also successfully reproduces the temperature-dependent\nwavefunction (de)localization observed by single-particle photoluminescence experiments\nin conjunction with k.p modelling (Extended Data Figure 4 and Supplementary Video 2 in\nNature 2024, 626, 535–541). Here, we employ this surface passivation for CsPbBr to provide\n3\na full analysis of the LUMO wavefunction size, summarized in Figure R6. As in the assessment\nof the HOMO wavefunction, we equilibrated the QD models at the target temperature for\n6 picoseconds and performed MD simulations in the canonical ensemble for another\n14 picoseconds. Figure R6c displays representative snapshots of the LUMO wavefunction at\nvideo of the LUMO density as Supplementary Video 8. It becomes clear that the LUMO\nwavefunction, like the HOMO, contracts as thermally induced dynamic disorder emerges.\nFurthermore, being in CsPbBr and not FAPbBr , the effect remains relatively small. The\n3 3\nrelative change in wavefunction size correlates well with that of the HOMO wavefunction in\na bare 2.4 nm CsPbBr QD (Supplementary Fig. 7) suggesting a roughly comparable vibronic\n3\ncoupling for electron and hole. While this data so far fully aligns with our theory, a\ncomparison between CsPbBr and FAPbBr is not possible. A shelling material that is\n3 3\ndielectric and lattice-matched, like CsCaBr for CsPbBr , is yet to be identified for FAPbBr .\n3 3 3\n3 3\nboundary conditions, mimicking the respective bulk materials. We reported in our original\nsubmission that the disorder-induced wavefunction localization also occurs under periodic\nboundary conditions. In line with QD models, the HOMO wavefunction size was temperature\nand A-cation dependent, experiencing stronger localization in FAPbBr than in CsPbBr\n3 3\n(Supplementary Fig. 12). Eliminating any surface effects, these bulk studies provide the\nopportunity to fully address your concern and assess the LUMO wavefunction localization in\nboth FAPbBr and CsPbBr . Our main results of the analysis are summarized in Figure R7,\n3 3\nwhile videos of the LUMO density are provided as Supplementary Videos 6 and 7.\nRepresentative snapshots of the LUMO wavefunctions in FAPbBr and CsPbBr at 100 K and\n3 3\nis stronger in FAPbBr than in CsPbBr (Figure R7a). This initial observation is confirmed by\n3 3\nquantifying the wavefunction localization as the inverse participation ratio (Figure R7b). We\ntherefore find that the temperature and the A-cation dependence of the disorder-induced\nwavefunction localization holds also for the LUMO wavefunction.\nOur action: We have included Figure R6 and R7 as Supplementary Fig. 10 and 11 in the\nupdated Supplementary Information alongside Supplementary Note 8. Videos of LUMO\nwavefunctions are provided as Supplementary Videos 6-8. The LUMO wavefunction\nlocalization is cited in the main text:\n“The electron wavefunction, represented by the lowest unoccupied molecular orbital\n(LUMO), experiences similar wavefunction localization as the HOMO (Supplementary\nNote 8, Supplementary Fig. 10, 11). Finally, temperature-induced localization even\noccurs in the respective bulk materials, specifically for FAPbBr (Supplementary Fig.\n3\n11, 12).”\nof a 3.6 nm CsPbBr /CsCaBr core/shell QD. a, Representative LUMO wavefunction in a\n3 3\n3.6 nm CsPbBr QD at 100 K. b, Cross-section of a 3.6 nm CsPbBr /CsCaBr core/shell QD\n3 3 3\nwith a 2.4 nm CsPbBr core. c, Representative LUMO wavefunctions of a 3.6 nm\n3 3\nwidth at half maximum, markers) and 95% confidence intervals (error bars) at 100 K and 300\nK.\nof bulk FAPbBr and CsPbBr . a, Representative LUMO wavefunctions at 100 and 300 K. b,\n3 3\nMean inverse participation ratio (IPR) as a function of temperature. Error bars indicate 95%\nconfidence intervals.\n(5) In the section “Purified single-photon emission” (Page 10), the authors attribute the\nquenched biexciton emission mainly to accelerated non-radiative (Auger) recombination in\nFAPbBr3 QDs. However, according to a recent study (Angew. Chem. Int. Ed., 2020, 59, 14292–\n14295) based on power-dependent transient absorption spectroscopy measurements,\nCsPbBr3 and FAPbBr3 QDs of similar sizes exhibit comparable biexciton Auger\nrecombination rates at room temperature, which are shorter than biexciton lifetimes\ndetermined by heralded single-particle photoluminescence spectroscopy. Could the\nauthors clarify whether this discrepancy might arise from differences in measurement\ntechniques?\nOur response: We thank you for this important point. While it is difficult to comment on other\nworks, we do share your opinion that such apparent discrepancy could arise from the\ndifferent technique or averaging effect at the ensemble level.\n3\nbiexciton lifetime in both systems via heralded single-QD PL spectroscopy. Heralded\nspectroscopy, by the fact that it post-selects for emitted pairs, naturally excludes\ncontributions from dim emitting states and from charged states which may affect ensemble\ntransient absorption measurements and would appear as shorter Auger decay times.\n(6) In the section “Purified single-photon emission” (Page 10–11), the authors attribute the\nreduced photon anti-bunching at lower temperatures to enhanced wavefunction delocalization,\nwhich is said to significantly increase the radiative recombination rate of exciton species.\nHowever, biexciton Auger recombination could also be suppressed at low temperatures due to\nits thermally activated nature, as this non-radiative recombination requires energy and\nmomentum conservation and is likely to be phonon-assisted. Therefore, the increased\nbiexciton emissivity observed at low temperature might also arise from this effect. I recommend\nthe authors clarify this point.\nOur response: We appreciate this comment. We agree that the temperature dependence of\nthe biexciton quantum yield could be affected also by a thermally activated nature of Auger-\nMeitner recombination (J. Phys. Chem. C 2010, 114, 41, 17550–17556, J. Phys. Chem.\nLett. 2014, 5, 1, 99–105), although we want to point out that a consensus on the universal\npresence of such effects has not been reached, see e.g. J. Phys. Chem. Lett. 2020, 11, 16,\n6513–6518. Disentangling the relative contributions to the temperature-dependent single-\nphoton purity is challenging, as these effects exhibit a similar temperature dependence.\nHowever, acquired at the same temperature, the A-site-cation dependent experiments\nsubstantiate our conclusions.\nOur action: We have refined the discussion of the temperature dependence to account for\nthe combined effects of thermally activated Auger recombination and disorder-induced\nwavefunction localization:\n“Matching the temperature dependence of the wavefunction localization (Figure 3b),\nthe single-photon purity increases with increasing temperature in both materials\n(Supplementary Fig. 21, 24). It is noteworthy that this temperature dependence could\nin parts be linked to single-photon superradiance, a process associated with a\nwavefunction delocalization that causes a rapid radiative decay and that is lost at\nhigher temperatures due to thermal disorder,7 or to Auger-Meitner recombination\nintrinsically requiring thermal activation.73,74 Nevertheless, a stronger increase of\nsingle-photon purity observed in FAPbBr demonstrates that disorder-induced\n3\nwavefunction localization plays a key role (Supplementary Fig. 21, 22, 24). These\ntemperature and composition trends are also followed by the increasing PL peak\nwidth, an indicator for exciton-phonon coupling, and the increasing PL lifetime\n24\n(Supplementary Fig. 3), which was recently linked with wavefunctions localized\nthrough vibrations.58,59 All these effects reflect the disorder-induced wavefunction\nlocalization.”\n(7) In the section “FAPbX3 QDs as single-photon sources” (Page 13), the authors\ndemonstrate the wide spectral tunability of single-photon emission from FAPbX3 QDs by\ncontrolling the halogen composition. However, phase segregation, as well as local\nenvironmental inhomogeneity even at the single-QD level, could introduce site-to-site\nvariations in the emission properties of localized excitons, potentially leading to more\nsignificant spectral diffusion. Could the authors provide an outlook on possible technical\napproaches to suppress spectral diffusion in mixed-halide FAPbX3 QDs?\nOur response: We acknowledge that the wavefunction localization, subject of this work, in\nliaison with other forms of heterogeneities can give rise to altered optical characteristics. The\nmentioned heterogeneity in mixed halide perovskites could lead to particularly complex and\nfascinating features. To the best of our knowledge, there currently exists no consensus\nregarding the halide distribution in perovskite QDs in equilibrium or under irradiation.\nMoreover, we could not identify conclusive proof of phase segregation as observed in the\nbulk. We would thus need to speculate on the interplay of phonon-induced wavefunction\nlocalization and heterogeneity in halide distribution. While a deeper analysis would be\nrequired to provide a definitive conclusion, we could so far not spot significantly enhanced\nspectral diffusion in our data. It is noteworthy that spectral diffusion is largely masked by the\nhomogeneous linewidth, which dominates PL peak broadening of perovskite QDs in single-\nparticle PL measurements at room-temperature (Nat. Commun. 2022, 13, 2587).\n25"}, {"round": 2, "reviewer_comments": {"reviewer_1": "Thank you for submitting the revised version of your manuscript. I appreciate the authors’ careful and thoughtful responses to my previous comments. The revisions have improved the clarity and overall quality of the work.\nAlthough a few of the issues I previously pointed out could not be fully resolved, the authors have provided reasonable explanations showing that these points fall outside the scope of the current study. I agree that addressing them in depth would exceed the intended focus of the manuscript and therefore does not affect its suitability for publication.\nOverall, I am satisfied with the revisions and have no further concerns.", "reviewer_2": "The authors nicely addressed all my comments.\nThe manuscript, which was already excellent before the revision round where I was involved, further improved and reached a quality and clarity level worth of Nature Communication.\nI congratulate with the author for the very good job", "reviewer_3": "I thank the authors for their clear and detailed responses to my comments. The revisions are thorough and significantly strengthen the manuscript. I believe the authors have addressed all concerns within the current technical limitations, and I recommend acceptance of the manuscript in its present form."}, "author_response": ""}]

Decision letter