Resonant and non-resonant driving of linearly-polarized excitons in Cd_3P_2 magic-size clusters

Domain: chemistry
Published: 2026-03-16
Editorial outcome: major_revision

Abstract
Coherent driving of optical transitions in matter using laser fields has been established as an important paradigm in quantum optics, quantum information processing and Floquet engineering. Recent studies extended coherent driving to solution-processed materials such as lead halide perovskites and colloidal semiconductor nanocrystals. Such systems, however, often feature broad and continuous absorption profiles that limit the studies mostly to below-bandgap driving, while a comprehensive understanding of coherent light-matter interaction requires both nonresonant and resonant driving. Here we utilize magic-size clusters of Cd3P2 in the limit of extreme quantum confinement to solve these issues. Their sharp, well-isolated and linearly-polarized band-edge exciton transition allows to coherently drive it at resonance, producing a Mollow-like lineshape using cross-polarized transient absorption measurements, which is also captured by our quantum simulation using the density matrix method. Analyzing the spectral responses under both nonresonant and resonant driving consistently yields a dipole moment exceeding 20 Debye.

Introduction
A long-lasting theme in the field of light-matter interaction is coherent driving of the matter states by laser fields. Such a topic has experienced different phases of research interest and motivation in the past decades, from early efforts of understanding the fundamental principles of light-matter interaction1,2 to recent interest in quantum information processing and quantum light sources3,4, and to the emerging field of Floquet engineering of nonequilibrium matter states5,6. The subject of research has also been greatly expanded in this journey. Early studies mostly focused on atoms and molecules in gas phase,7,8 which possess sharp optical transitions that are particularly suitable for optical driving and detection. The experiments and principles were then translated to studies of artificial systems with discrete transitions, such as epitaxially grown III-V group quantum dots (QDs; also called “artificial atoms”) and quantum wells (QWs) under cryogenic temperatures.9-12 More lately, monolayer or few-layer van der Waals materials have also been established as a versatile platform to explore various type of coherent driving phenomena, including but not limited to, valley-selective optical Stark effect (OSE),13,14 biexcitonic OSE,15,16 Autler-Towns splitting (ATS) under intraexciton driving,17 quantum interference,18 and Floquet engineering of band structures.19-22

A highly interdisciplinary field emerging in the past few years is coherent driving of solution-processed semiconductor materials, including lead halide perovskites and colloidal semiconductor nanocrystals.23-31 These materials allows for conducting coherent driving experiments in ambient conditions at room temperature, which is of direct relevance to applications such as optoelectronic devices and photochemical reactions.29 However, due to the broad and continuous optical transitions in these systems, these experiments were mostly restricted to observing OSE under non-resonant driving by negatively detuned laser pulses such as to avoid direct excitation of continuous transitions.23-27 A typical observation is that the lowest energy transition is transiently blue shifted by the driving pulse, whereas interesting physics beyond this phenomenon remains to be explored. Although colloidal QDs have relatively narrow and isolated transitions, their room temperature absorption profiles are often still dominated by strongly overlapped manifolds of transitions.24-26 In a recent study, colloidal nanoplatelets with three-level-like energy structure resembling that of QWs were used to demonstrate the ATS line shapes, which entails resonant/nonresonant driving of the “empty” intersubband transition using near-infrared pulses and probing of the interband absorption changes using visible pulses.28

Much richer physics will emerge if a two-level system can be driven both nonresonantly and resonantly. Beyond a laser-induced spectral shift, exotic phenomena such as Mollow triplet emission profile,32 quantum interference 33 and gain without inversion 34,35 can be observed. This has been nicely illustrated in a classical study on epitaxial QDs at 5 K;12 while strong driving of an exciton transition and probing of another coupled exciton transition resulted in expected ATS lineshapes, driving and probing the same exciton transition gave complex Mollow-like absorption profiles that include “negative” absorption region (i.e., optical gain). This optical gain originates from coherent exchange of energy in the pump and probe fields through the commonly coupled QD exciton transition, and it should be regarded as gain without inversion as there is no population inversion in this system.12 Inspired by these studies, a solution-processed material system that could be examined for comprehensive driving experiments at room temperature should satisfy the following criteria: (i) it should have narrow and well-isolated band-edge exciton transitions almost free from inhomogeneous broadening; (ii) it should have a large transition dipole moment that facilitates the realization of strong driving; and (iii) ideally the transition is highly polarized that enables rejection of unwanted signals (e.g., scattering) through manipulation of laser polarizations.

A potential candidate that satisfies the above requirements is the family of magic-size clusters (MSCs).36-43 Their atomically precise structures essentially eliminate inhomogeneous broadening encountered in typical colloidal nanocrystals,39,43,44 and their sizes (typically <2 nm) are situated in a stronger confinement regime than typical QDs.38,41,42 These features lead to molecular-like discrete electronic levels with sharp and well-isolated optical transitions at the ensemble level. Moreover, in this extreme confinement regime, even slight symmetric breaking will cause highly anisotropic, linearly polarized transitions, which has already been exploited to demonstrate aggregation-induced macroscopic chirality.45-47 Here we choose extremely-confined Cd3P2 MSCs, with strong optical transition in the visible region, to investigate their transient optical response under resonant and nonresonant driving conditions using femtosecond pump-probe transient absorption (TA) spectroscopy at room temperature. We observe not only the typical blue-shifted OSE lineshape under negatively detuned driving, but also peak splitting nature near time zero under resonant driving. A Mollow-like lineshape is extracted by a global fitting procedure, which is also successfully captured by our quantum simulation using the density matrix approach. Importantly, in the resonant case, the splitting of the Rabi sidebands scales linearly with the electric field strength, which is distinct from nonresonant OSE with energy shift scaling linearly with the light intensity. Resonant and nonresonant driving conditions consistently give a large transition dipole moment exceeding 20 Debye which underlies the observed strong optical responses.

Results
Principles and system design
Figure 1 summarizes the principles of coherent interaction between a two-level system (comprising the excited state |e⟩ and the ground state |g⟩ separated by ћωeg) and a driving light field with photon energy of ћωp. The coherent absorption and emission of photons yield a series of photon-dressed states (also called Floquet replica states), and those with strongest interaction with the equilibrium matter states are denoted as |e - ћωp⟩ and |g + ћωp⟩, as indicated in the figure. With a negatively detuned driving photon (i.e., Δ = ћωp - ћωeg <0), the photon-dressed states hybridize with the matter states to produce a blue shifted transition that carries the major oscillator strength of the original matter states (Figure 1a), thereby generating a derivative-like TA spectrum (Figure 1b). When ћωp becomes in resonance with ћωeg, the photon-dressed states are in degeneracy with the matter states, resulting in strongest hybridization. The results are four levels that are half-light and half-matter in nature (Figure 1c). The absorption profile should consist of three lines similar to the Mollow triplet in emission, with the strength of the center band equaling the sum of the two Rabi sidebands (Figure 1d). A hallmark feature of this resonant driving scheme is that the splitting of these two Rabi sidebands is proportional to the electric field strength,12 whereas for large-detuned nonresonant driving, the spectral shift is approximately proportional to the light intensity.23-25,27 Finally, when ћωp exceeds ћωeg, the light-matter interaction becomes similar to the negatively detuned driving condition, except that the transition that carries the major oscillator strength is now a redshifted one (Figure 1e). The TA spectrum is also derivative-like (Figure 1f). A realistic complexity associated with resonant and positively detuned driving is excitation of real populations that gives large ground-state bleach and excited-state absorption signals overwhelming the coherent driving signals. Due to this issue, such experiments have never been conducted for solution-processed semiconductor materials.

We note that the spectral lineshape illustrated for resonant driving in Figure 1d is a simplified one without considering the interference effect. In this scheme, the strong driving pulse couples to the transition and modifies the energy levels, whilst the weak probe pulse serves as a spectator to measure the modified transitions. In a complete picture, however, the electric field (ε) that interacts with the transition dipole (μ) is a sum of the driving and probe fields (ε = εd + εp).12 This is the fundamental reason that the pump and probe fields can coherently exchange energy through interacting with the common transition, resulting in complex interference spectral lineshapes.12

Cd3P2 MSCs were chosen for the present study as the small bandgap of bulk Cd3P2 (~0.55 eV) 48,49 allows the extremely-confined MSCs to still absorb in the visible spectral range that can be facilely accessed by pump-probe TA spectroscopy, whereas CdS and CdSe MSCs commonly studied in the synthesis literature typically absorb in the ultraviolet spectral range.38,39,44 Colloidal Cd3P2 MSCs were synthesized following the method reported in the literature,50 as detailed in the Methods section. Figure 2a presents the steady-state absorption spectrum of Cd3P2 MSCs dispersed in hexane, which is dominated by a sharp exciton peak at ~2.75 eV. This peak is almost completely isolated from higher-energy absorption bands, which is distinct from the spectra of typical colloidal QDs. The photoluminescence (PL) spectrum of Cd3P2 MSCs also shows a sharp peak centered at ~2.70 eV (Fig. 2b).

Previous transmission electron microscope (TEM) characterization of Cd3P2 MSCs gave a diameter <2 nm, which is complicated by the issue that electron beams can induce cluster aggregation.51 Even 2 nm is already 20-fold smaller than the Bohr exciton diameter of ~40 nm for bulk Cd3P2.49 Reliable determination of the exact structure and morphology of these Cd3P2 MSCs remains challenging. Here we carried out pair distribution function (PDF) analysis of Cd3P2 MSCs using synchrotron X-ray total scattering (see Methods). As shown in Supplementary Figure 1, the PDF in general resembles that of α-Cd37S20 MSCs reported in the literature,44 which suggests that the size and structure of these two types of MSCs are similar. The α-Cd37S20 MSCs were reported to take an anisotropic shape that leads to linearly polarized transition dipoles.52 Such a property enables the formation of chiral assemblies from these achiral MSCs.45 In principle, such a linearly polarized transition should also benefit our coherent driving experiments, which leads us to first investigate the polarization characteristics of our Cd3P2 MSCs.

Identification of linearly-polarized exciton transition
We first measured the PL anisotropy of Cd3P2 MSCs by rotating a polarizer placed in front of the detector. As shown in the inset of Figure 2b, when these MSCs are excited by linearly polarized light (at 2.88 eV), the PL intensity measured at 2.69 eV is strongly dependent on the polarization angle, consistent with a linear polarization of the band edge exciton. Considering that the emissive state is Stokes-shifted than the absorptive state, absorption anisotropy should be more relevant to our coherent driving experiments than emission anisotropy. However, unlike PL measurements in which we can select a subset of MSCs with their transition dipoles aligned along the electric field direction of the excitation light, direct measurements of absorption anisotropy would not work due to the random orientation of the MSCs in the ensemble. Instead, we can conduct absorption anisotropy measurements of Cd3P2 MSCs using polarization-controlled femtosecond pump-probe TA spectroscopy in which the linearly polarized pump beam can also select a subset of MSCs with their transition dipoles aligned along the electric field direction of the pump beam;53-55 see Methods for details. The linearly polarized laser pulses are labeled as V (vertical) and H (horizontal), and VV and VH stand for experiments with mutually parallel and perpendicular pump-probe polarizations, respectively. The pump photon energy is at 2.82 eV and the pump pulse energy is controlled low (~7 μJ cm-2). Figure 3a and 3b present the VV and VH TA spectra at indicated pump-probe delays from 1 ps to 1 ns. The spectra can be understood as a bleach of the lowest energy exciton peak superimposed on a broad absorption feature.56 The latter suggests that the biexciton states in these small-size MSCs are not bound states; otherwise, one would expect a red-shifted absorptive feature indicative of biexciton binding, as commonly observed on the TA spectra of colloidal QDs. 57 This allows us to indeed approximate these MSCs as a two-level system.

Importantly, although the spectral shapes of VV and VH spectra have no difference, the signal amplitude in the former configuration (Figure 3a) is about three times larger than that in the latter (Figure 3b). Figure 3c plots the bleach recovery dynamics averaged in the range of 2.73-2.79 eV (lower panel), from which the time-dependent anisotropy r(t) can be calculated (upper panel) according to: r(t) = (SVV(t)-SVH(t))/(SVV(t)+2SVH(t)), where SVV(t) and SVH(t) stand for VV and VH signal sizes, respectively. The obtained anisotropy is close to 0.4 at the early time, which indicates a near fully linearly polarized absorption dipole.58,59 Interestingly, r(t) shows only slight decay within 100 ps and then negligible decay in a few ns, which likely correspond to aggregation-free individual clusters and cluster aggregates, respectively. Using the anisotropy decay time of 100 ps, the individual cluster size is estimated to be ~1 nm; see Supplementary Note 1 for details.

Pump photon energy (ћωp) dependent TA anisotropy was further conducted. In this experiment, the pump photon energy was tuned from 2.8 eV to 4 eV, and the anisotropy of the TA signals near the band edge was calculated. The early time r(t) is plotted in Figure 3d along with the absorption spectrum of the MSCs. The anisotropic value drops substantially when the pump photon energy exceeds 3.0 eV, which is stabilized at ~0.2 for photon energy larger than 3.5 eV. There are two possible reasons accounting for the reduced TA anisotropy: (1) The higher energy transitions are linearly polarized but there is an angle of ~35° between their transition dipoles and the band-edge transition dipole; or (2) the higher energy transitions are not linearly polarized. Nevertheless, the anisotropic value close to 0.4 for pump photon energy below 3 eV establishes a highly linearly polarized band-edge exciton transition in these MSCs. By contrast, TA spectra of these Cd3P2 MSCs collected with co- and counter-circularly polarized pump-probe pulses have similar signal amplitudes (Supplementary Figure 2), indicating that the exciton transition is not circularly polarized.

In general, linearly polarized excitons in nanostructures can result from intrinsic electronic structure and/or extrinsic dielectric confinement effect.60 The latter typically requires highly anisotropic geometric shapes such as 1D nanorods61,62 or nanowires63 and 2D nanosheets64, which unlikely applies to our small clusters with only slight shape anisotropy. For the intrinsic electronic structure, short-range and long-range exchange interaction of the electron-hole pairs in QDs leads to a series of fine structure levels with well-defined quantum numbers and therefore well-defined linear or circular polarizations.65 In particular, the anisotropic long-range exchange interaction arising from morphologic and/or lattice asymmetry often creates linearly polarized excitons.66 However, the fine structure splitting in typical QDs ranges from μeV to meV,65 resulting in negligible polarization at room temperature due to thermal mixing of different states. In contrast, MSCs represent an extremely strong confinement region, which should give giant fine structure splitting. Qualitatively, one can understand the highly linearly-polarized transition dipole of the MSCs by treating them as small molecules. Most molecules have anisotropic molecular shape and their lowest energy transition is strongly linearly polarized, giving rise to transient absorption anisotropy also close to 0.4.52 This similarity between MSCs and molecules further corroborate the atomically precise structure of the MSCs, that is, the structural anisotropy is nearly identical for each MSC. As a comparison, we also conducted linearly polarized TA measurements for ~2.6 nm Cd3P2 QDs (Supplementary Figure 3), which gives a very small anisotropy value of ~0.05 (close to zero) standing in stark contrast to the Cd3P2 MSCs. Microscopically, each QD in the sample has a different size and morphology, and therefore each QD should also have a different degree of polarization for the band edge exciton transition, resulting in negligible anisotropy at the ensemble level.

Non-resonant and resonant coherent driving
We then conducted coherent driving experiments for Cd3P2 MSCs using two different photons energies, ћωp = 2.12 and 2.75 eV, corresponding to the negative detuning (Δ = -630 meV) and resonant (Δ ≈ 0) conditions, respectively. The spectra of these driving pulses are plotted in Figure 2a for comparison with the absorption spectrum of Cd3P2 MSCs. All the experiments were performed at room temperature and, unless otherwise specified, the VH pump-probe configuration was adopted in order to suppress the leakage of the pump pulse into the TA spectra, which is especially important for resonant driving.

2D pseudocolor TA spectra acquired with 2.12 and 2.75 eV driving are presented in Figure 4a and 4c, respectively, with the driving power density indicated in each panel. In these spectra, we focus on the data within ~200 fs around time zero, which are the spectral features arising from coherent driving. In Figure 4a, negatively detuned driving leads to a typical derivative-like (i.e., with negative and positive alternating amplitudes) TA signal due to normal OSE blueshift, and this signal fades away after the duration of the driving pulse. Importantly, for the resonant driving case in Figure 4c, we observe a narrowed negative feature sandwiched between two narrowed positive wings near time zero, which resembles the schematic spectrum presented in Figure 1d. This Mollow-splitting like spectrum implies we might have indeed achieved resonant coherent driving of the band-edge exciton in Cd3P2 MSCs. After the driving pulse duration, the spectrum evolves into a different one that is consistent with the real excitation spectra presented in Figure 3. This transformation could arise from dephasing of the coherent driving signals and/or direct real excitation of the band-edge exciton.29 Strictly speaking, real excitation also originates from coherent driving, but represents a case of very rapid dephasing (driving while dephasing).29 Considering that we are dealing with an ensemble sample at room temperature, distinct dephasing times should exist in the sample.

The kinetic trace of the bleach probed at 2.72 eV of Figure 4a is plotted in Figure 4b, and the response of this signal is comparable to the driving pulse duration (~170 fs). For resonant driving, the kinetic traces probed at 2.74 and 2.81 eV (i.e., the bleach and the blue-side positive wing, respectively, in Figure 4c) are shown in Figure 4d. The rapid decay of the signal within the pulse duration can be clearly observed, and the rest signals due to real population shows oscillations arising from electron–phonon coupling (see Supplementary Figure 4 for details).

In order to isolate the coherent driving signals from real excitation signals for resonant driving, we performed global fitting using a sequential kinetic model; see Supplementary Note 2. Details for the fits are provided in Figures S5-S7. In this global fitting, we can isolate three spectral species, a coherent one (Scoh.), S1 and S2 (Supplementary Figure 5). Scoh. is not symmetric due to the asymmetric lineshape of the band-edge exciton (Figure 2a), but its temporal profile is consistent with its coherent nature (Supplementary Figure 5). S1 and S2 can be assigned to multiparticle and single-exciton species, respectively, which are consistent with our previous study.56 The ~1.35 ps time for the decay of S1 and growth of S2 is the characteristic time for multiparticle Auger-like annihilation (Supplementary Figure 5).56 Note that the multiparticle species S1 actually contains contributions from both single-exciton and biexciton states. We cannot differentiate the spectra of clean single-exciton and biexciton species due to the complicated distribution of absorbed photons per MSC at time zero across the ensemble sample. In the case of excitation well above the band edge in QDs, this distribution is often assumed to be Poissonian, but here we are not in this excitation regime, and therefore the distribution remains unknown. In this case, our global fitting gives an “average” result that as time goes on the initial spectrum that contains both single-excitons and biexcitons (S1) gradually evolves into a spectrum that is dominated by single-excitons (S2). We have conducted the global fitting for several different driving intensities (Supplementary Figures 6 and 7), and obtained the spectra of S1 and S2 as shown in Supplementary Figure 8. The ratio between the spectral amplitudes of S1 and S2 indeed first increases with pump intensity and then gradually approaches saturation (Supplementary Figure 8), consistent with the increase of biexciton populations with increasing pump intensity.

Importantly, in order to substantiate the existence of Scoh. and its assignment, we have compared the results of global fitting with or without the inclusion of Scoh., whereas the rest components as well as their time evolution are kept identical. As shown in Figure 4e and 4f, the simulated spectrum without Scoh. shows a substantial discrepancy compared to the experimental spectrum, especially within ~200 fs around time zero. Additionally, the kinetic traces probed at 2.82 and 2.75 eV, for example, are compared in Figures S5e and S5f, from which the inclusion of Scoh. clearly reproduces the experimental traces much better than the one without Scoh.. This analysis further confirms the observation of a Mollow-splitting spectrum due to quantum coherent driving.

The driving power dependent coherent spectra isolated from direct observation and global fitting are plotted in Figure 5a and 5b negatively detuned and resonant driving, respectively. In both cases, the signal amplitudes increase with increasing driving powers. Importantly, these spectra can be well simulated by spectral shifting (for nonresonant driving) or splitting (for resonant driving) of the experimental band-edge exciton absorption according to the principles illustrated in Figure 1; see Supplementary Note 3. The simulated spectra are plotted in Figure 5c and 5d, with the shifting or splitting energy indicated in each panel. As noted above, the asymmetric spectra of resonant driving can be well reproduced by using the asymmetric absorption profile of Cd3P2 MSCs.

The driving power dependent coherent spectra can be used to quantify the spectral shifting or splitting, and therefore, the magnitude of transition dipole moment (μ) of Cd3P2 MSCs. For the case of negatively detuned driving, we adopt a well-established spectral-weight-transfer (SWT) method 13; see Supplementary Note 4. The obtained power-dependent SWT and OSE shift (δE) are both plotted in Figure 5e. The derived energy shift scales linearly with the driving power, which is in good agreement with eq 1:23

δE = sqrt(ℏ^2ΩR^2 + Δ^2) - Δ ≈ (ℏΩR)^2 / 2Δ = (με)^2 / 2Δ ∝ I (1),

where ℏΩR is the Rabi energy equaling με, ε is the amplitude of the electric field and I is the driving power. The value of μ quantified from Figure 5e using eq 1 is ~13.2 D; see Supplementary Note 5 for calculation details. Notably, however, this experiment is conducted with the VH pump-probe configuration, indicating that the calculated dipole moment is an underestimated one. For negatively detuned driving, it is also convenient to measure the OSE shift using the VV pump-probe configuration, and the results are provided in Supplementary Figure 9. Under an identical driving power, the signal amplitude ratio of VV and VH configurations is close to 3:1, which is consistent with our TA anisotropy measurements discussed above. Using the VV signals, we estimate μ to be ~22.8 D, satisfying the theoretical prediction of μVV^2 / μVH^2 = 3/1. This value (22.8 D) is comparable with those derived for much larger size colloidal QDs of various compositions (e.g., 24 D for 3.6 nm CdSe QDs;25 34 D for 5 nm CsPbI3 QDs 24). The extreme confinement in these Cd3P2 MSCs concentrates the oscillator strength to the band-edge exciton, which underlies the strong optical responses we observed in these coherent driving experiments.

As for the resonant driving condition, the splitting of the two Rabi sidebands with respect to the original absorption peak should satisfy the following equation: 12

ℏΩR = με ∝ sqrt(I) (2).

Experimentally, however, it is not straightforward to directly readout the Rabi splitting from the TA spectra, due to the spectral overlap between the two Rabi sidebands and the center band bleach. We therefore use the simulated TA spectra in Figure 5d to match the experimental spectra extracted from global fitting in Figure 5b. From the best match, we can estimate the splitting energy, which is found to scale linearly with the square root of the driving power intensity (i.e., the electric field strength) instead of the power itself (Figure 5f), consistent with eq 2. The calculated μVH is ~11.8 D. Considering that the calculation of the transition dipole moment relies critically on the pump intensity (I0) which in turn depends sensitively on beam-size and pulse-duration measurements, the consistency among the values of μ calculated from resonant and nonresonant driving conditions is remarkable. Moreover, although the signal amplitude at each probe energy increases with the driving power, the total integrated TA signal remains close to zero (Supplementary Figure 10). This behavior reflects the conservation of total oscillator strength, i.e., the original transition strength is redistributed to dressed states under strong optical driving, as schematically illustrated in Figure 1d. We have also conducted the positively detuned driving experiment in order to reproduce the schemes in Figure 1e and 1f, but the results are much more complex, as discussed in Supplementary Note 6 and Supplementary Figures 11 and 12.

Methods
Synthesis of Cd3P2 MSCs. The colloidal Cd3P2 MSCs were synthesized following the procedure reported by Yu et al.50 Briefly, 0.533 g (0.2 mmol) of Cadmium acetate dihydrate, 0.2 mmol oleic acid and 5 mL 1-octadecene were loaded in a 100-mL three-neck flask and then degassed at 100 °C for 2 hours. The phosphorus precursor prepared by mixing 0.05 mmol of Tris(trimethylsilyl)phosphine and 1 g of 1-octadecene, was quickly injected into the flask. The reaction was kept at 100 °C for 20 seconds under nitrogen flow, after which it was stopped by removing the heating mantle and was cooled down to room temperature. The obtained yellow solution was washed with excess ethanol to remove solvent, excessive ligands and unreacted precursors. The resulting precipitate was collected and dissolved in n-hexane or heptane for spectroscopic measurements.

Synthesis of Cd3P2 QDs. Cd3P2 QDs were synthesized according to the procedure developed by Peng et al.72 Briefly, a mixture of 77 mg of Cadmium Oxide, 0.3 mL of oleic acid and 9 mL of 1-octadecene was heated at 230 °C under nitrogen flow until it turned clear. 1 mL of 0.2 M Tris(trimethylsilyl)phosphine in 1-octadecene (with a small amount of hexane) was swiftly injected into the flask and the reaction was kept for 5 or 6 seconds at 230 °C. The crude solution was washed for several times by adding ethanol.

X-ray total scattering measurements. PDF data were collected at the PD beamline at the Australian Synchrotron using a Mythen II detector over a 2θ range of 120º. Samples were packed into 0.7 mm borosilicate capillaries and rotated during data collection. The wavelength, zero error and instrument contribution to the peak shape was refined to 0.59 Å using NIST LaB6 660b. Energy was calibrated with NIST LaB6 standard. We further convert the experimental integrated scattering data into the scattering function G(r) by using available software package PDFgetx3. The function [G(r)] of Cd3P2 MSCs is plotted against the interatomic distance, as demonstrated in Supplementary Figure 1. The coordination numbers of anions and cations in both cubic and tetragonal phases of Cd3P2 are inconsistent with those of the well-known InP or CdSe MSCs. The amount of calculation is beyond the scope of this study. Nevertheless, we find some similarities between PDFs of Cd3P2 MSCs and α-Cd37S20 MSCs.44 The major peak positions at 5-10 Å of Cd3P2 MSCs match reasonably well with those of α-Cd37S20 MSCs, and the strongest peak position is also close to that of α-Cd37S20 MSCs, indicating their similar overall sizes.

PL anisotropy measurements. Steady-state PL spectra are collected by a setup that consists of a monochromator (SpectraPro-2300i, Acton Research Co., USA) and a charge-coupled device camera (PIXIS 100, Princeton Instruments, USA), and the same microscope objective was used for excitation and PL collection. Polarization-dependent PL measurements were conducted by rotating a linear polarizer placed after the sample, while another linear polarizer for the excitation beam remained unchanged.

Femtosecond pump-probe spectroscopy. Femtosecond transient absorption measurements were conducted using a Pharos femtosecond laser system (Light Conversion; 1030 nm, 10 kHz repetition rate). The output laser was split in two beams with a beam splitter, one of which was directed to an optical parametric amplifier (Orpheus-HP; Light Conversion) to generate wavelength-tunable pump pulses, and the other being delayed by a motorized delay stage and then focused onto a BBO frequency-doubling crystal to generate 515 nm beam and then focused into a 2 mm thick sapphire window to generate a white light continuum (WLC) used for probe beam. The probe beam was focused with an Al parabolic reflector onto the sample. The probe beam was then collimated and focused into a fiber-coupled spectrometer with a line scan camera and detected at a frequency of 10 kHz. Circular and linear polarized pump and probe beams were obtained by separately inserting two sets of linear polarizers (Thorlabs) and quarter-wave or half-wave plates (Thorlabs) right before the sample. Samples were placed in a flow cell and were vigorously circulated during the measurements to avoid photodamaging, which was further confirmed by the same absorption spectra of samples before and after TA measurements. All measurements were carried out at room temperature. For above-bandgap and resonant cases, the optical density at excitonic absorption peak (2.75 eV) is around 0.31; and it was changed to 0.57 for below-bandgap case.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Yuan Liu et present an interesting study, where they observe the coherent interaction between an oscillating electric field (laser pulse) and extremely confined colloidal nanoclusters. This is done in three different energy regimes: below resonance, above resonance, and on resonance. As noted by the authors, the observation of a blue shift due to the optical Stark effect in the below resonance regime is not new (see for example, Unold et al., PRL, 2004) and the biggest novelty in the paper lies in the observation of a red shift in the above resonance case and a Mollow-like lineshape in the resonance case due to Autler-Townes Effect. The authors claim that this is possible due to the system they are investigating: magic-sized nanoclusters, in the extreme confinement regime, with almost no inhomogeneous broadening and linearly-polarized exciton transitioning. Overall, I believe this study is well-conducted and valuable for the community. Therefore, I believe it can be suitable for publication in Nature Communications. However, extracting the coherent dynamics in the resonant and above-resonance cases is tough because the coherent signal is convoluted within a strong population-induced signal. In my view, the authors need to undoubtedly show that the TA signal they are extracting (for the resonance and above resonance cases) comes from photon-dressed state effects, and not state-filling effects. In this regard, my concerns are the following: 1- The authors use a previously reported sequential kinetic model to fit their TA data (in time and wavelength) and separate coherent from population signals. This is essentially a form of 2D fitting with multiple parameters, which, as a result, extracts the dynamics of three (for the resonant case) and four (for the above resonant case) spectral species. However, from the data provided by the authors in Figure 5 and Figures S4 and S5, it is not clear to me if the coherent species (Scoh) is really needed to fit the data. Could the authors refit the data just with the population species and show that the coherent signal is needed to explain the observed signal? 2- In Figure S4, the authors show their fitting for two energies. Could the author provide the fittings and data for more energies? 2D plots comparing the fit and data could be useful to visualize how good the model fits the data, especially on how the spectrum shifts with time. 3- In Figures 5b and d the authors show the time evolution of different species. This is done in a plot of “Normalized Intensity” as a function of delay. What exactly is “normalized intensity”? Is it (eq S2) for a fixed lambda? Does it come from an integral of the population amplitude signal over lambda? Could the authors provide exactly how the normalized intensity is extracted from the fitting parameters described in Supplementary Note 2? 4- For the resonant case, apart from the coherent species, they extract two population signals, S1 and S2, which they attribute to “multiparticle” and “single-exciton” species, respectively. In Figure 5b, they show that at t=0, the single exciton population is 0, while there is already the formation of the multiparticle population. On their model, as the multiparticle population decays due to Auger annihilation, the single particle population is formed, following the rate equation S2. However, this situation is just true if the excitation power is so strong that all the excited particles have completely filled band edge states. If we are not on that limit, probabilistically speaking, some of the initial excited population should be formed by single excitons and some by multiple excitons. From the power dependence data shown in Figure S4, it is not clear that for lower powers, their fitting extracted an initial population formed by single excitons. Could the authors provide data on how the ratio between the single exciton and multi-exciton population varies with incident fluence in relation in early times? This would help to clarify how effective their methodology is in extracting the signals from different species. 5- Could the authors clarify the degeneracy of the band edge states of the studied material? This is also important to understand how the ratio between single and multi-exciton species should scale as a function of incident fluence. 6- I would expect the multi-excitons to be red-shifted in relation to single-exciton species, due to the different binding energies. Therefore, I would expect the plot in Figure 4c (and also 4b) to shift to the blue as the multi-exciton species convert to a single exciton. This also seems to be consistent with the fittings that the authors show in Figure 5a and c, where the multiparticle species seem to be redshifted in relation to the single exciton. However, what I see in Figures 4b and c is a red shift. Could the authors clarify that? 7- In Figures 5a and c, there is a positive feature for S1 and S2 around 2.85 eV, could the authors explain where this comes from? 8- In Figure S4 b, the authors show the difference between their fitting and the TA data. They claim that this difference comes from the signal of coherent phonons that are not contained in their model. However, this difference seems quite drastic to me, especially at t=0 (the most important part of the fitting to extract the coherent species). From the first oscillation to the second, the phonon signal decays from around 7 mOD to 0.5 mOD (~14 times), while it is almost constant from the second to the third and fourth oscillations. Interestingly, this is not the case for a different energy (2.74 eV) in Figure S4c. Could the authors explain these differences? I also have some minor suggestions: 9- On page 10, when discussing the signal from different species, the authors distinguish a “single-exciton” and a “mutiparticle” species. It is perfectly fine to understand what they meant by the context, but I suggest changing the multiparticle nomenclature to multi-exciton, because the term “particle” is sometimes also used in the literature to describe the nanoparticles. 10- Can the authors provide information on what optical density was used during the experiments? This information is relevant for reproducing these results. 11- In the title, the authors use the word “driving” to describe their work. However, in my opinion, the word driving does not differentiate between the generation of coherent and population processes. In my view, using this expression hides the novelty of the work. Using a nomenclature that shows that this work is observing coherent processes would make the title stronger.", "reviewer_2": "The manuscript discusses the coherent optical driving of solution-processed semiconductor materials, focusing on magic-size clusters (MSCs)—specifically Cd₃P₂ MSCs—as a promising platform for exploring light–matter interactions at room temperature. The study is motivated by a longstanding interest in coherent excitation of matter by laser fields, with evolving applications from fundamental physics to quantum information science and nonequilibrium materials engineering. As a conclusion I will recommend the manuscript to be accepted after revision. Here are my comments on the manuscript: 1. The energy levels in Figure 1a, 1b, and 1c are confusing. Please add breaks in hωeg or other modifications to make the scale bar more reasonable.The caption order is also confusing. 2. The steady absorption polarization property of the MSCs can be different from their emission polarization properties. It would be important to add such measurement results can provide useful information to the exciton behavior. Given knowledge of the one of absorption or emission, using fluorescence anisotropy is probably sufficient. More generally, if the perception of the origin of polarization is distortion of some type, what does this mean at a batch-to-batch or even ensemble level: symmetry is broken in the same way, stochastically, or something else? 3. Please provide more detailed explanation on why the 2.6nm QDs showed much smaller anisotropy compared to the MSCs. The quantum confinement is still significant at this size as it is much smaller than the Bohr radius of the material. 4. The authors claim that the Cd₃P₂ MSCs platform for exploring light–matter interactions at room temperature. On the other hand, the authors stated that the stability issue requires cryo-measurement. Please justify it. 5. As a general question: it is peculiar that the proposed experiments “work” here in terms of isolating a resonant stark effect response in situations with the coexistence of real carriers. The bandwidth of the samples is similar or worse to platelets which have been studied a lot in this respect, similar or worse than tmds; is the sample really special or is this more a function of looking for the right signals? 6. Derivative-type spectra are repeatedly described (as they should be) for the sub-gap case of OSE, but I don’t see the actual data: please show the derivative overlayed with the data. Related: the simulation spectra in Figure 6 have similar shape, but do not line up with the right energies as in the experimental data. Why? 7. the global analysis or singular value decomposition is a bit handwaving for the above gap case: the carrier cooling effect on the bandwidth is continuous with carrier temperature, not a different weighting of the spectrum. More generally, if multiparticle effects like Auger occur, this seems avoidable: lower powers should turn off these effects and simplify the model further. What is the average excitons/particle in the experiments? This should be clarified to show why MX is reasonable addition—and better, avoided at lower intensities.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "Liu et al. reported the resonant and non-resonant driving of excitons in nanoclusters. They attempt to demonstrated the system allows coherent driving of quantum states under various detuning conditions that may lead to blue-shift, redshift and even Mollow triplet lineshape. At first, this article looks promising to show the process, according to their abstract and introductory paragraph. However, I find the data hardly support their arguments and the quality of the work, including the analysis, hardly convince me this work can be accepted for publication in Nature Communications. The main issue with this article is that – the data is dominated by strong incoherent processes arising from exciton dephasing – phonon scatterings, exciton recombination, local inhomegeneity, auger effects. A coherent process, without the sophisticated treatment from Lindblad-formalism, in principle, can be observed directly from pump-probe spectrum with signal appears only when pump and probe pulses are overlapped in time-domain. Even at time very closed to the temporal overlap of their pump and probe profile, the incoherent processes can already be dominant. In the case when incoherent signal arising from real population transition is negligibly small, the effects will be promising. However, this is not the case in their system because the incoherent signal is simply too strong to distinguish any coherent response. The authors then perform spectral fitting process. This process itself is not robust and cannot provide any relevant coherent quantum optical process directly linked to the optical excitation. In-particular, when the pump is resonant or above the exciton transition, the signals are largely contaminated by various scattering processes, related to phonon scattering, linewidth broadening, and even resonant coupling to phonon Raman under SRS process in a pump-probe spectroscopy. All these processes can give rise to the lineshape observed in their spectrum, which cannot be disentangled from the pure electronic processes and they are not even considered properly in their analysis. For example, above resonance, you can have lattice heating by optical phonon which simply decays within femtosecond timescale and that gives you redshift. This is particularly prominent because you excite real population that drive electron-phonon coupling. The temporal resolution from the author setups is not sufficient to distinguish this process, not to say their spectra are already dominated by other incoherent processes. Another point that I feel this work is not relevant to coherent driving processes is that – when the spectra are dominated by the strong incoherent processes, why not perform a full quantum optics treatment that account all these processes together? This is a more robust method to help to visualize the impact – how dephasing diminishes the Rabi oscillation and how that leads to the expected spectrum? A comprehensive treatment accounting for the dominant processes is definitely necessary. This will then help to build the simple model back to Rabi oscillation. Without a proper treatment on these datasets, it is difficult to simply extract the coherent processes without considering any dephasing processes. The dipole moment extracted directly from your fitting is probably not quite relevant at the end. Another issue is that – the linewidth is extremely broad, comparable and even larger than the “Rabi frequency” the authors have claimed. In this case, you generally do not see clear Rabi splitting. The lineshape itself already imply that the incoherent process is too strong and coherent driving will require much stronger Rabi frequency. Within the pump-probe overlap, the incoherent dephasing processes already take place and their Fourier transform will then give you spectrum that is not relevant to the coherent phenomena, i.e., optical Stark effects. By over-interpreting the data within the coherent driving process, and by employing a spectral analysis process without comprehensive quantum optics treatment, it is impossible to understand how the data supports any coherent phenomena and further quantitative analysis based on the simple Rabi model becomes irrelevant. I suggest the authors to reinterpret their data again carefully, without over-claiming the phenomena that barely seen from their data."}, "author_response": "Response: We thank the reviewer for their summary of our work and the kind comments. Below we address their concerns point-by-point. 1. The authors use a previously reported sequential kinetic model to fit their TA data (in time and wavelength) and separate coherent from population signals. This is essentially a form of 2D fitting with multiple parameters, which, as a result, extracts the dynamics of three (for the resonant case) and four (for the above resonant case) spectral species. However, from the data provided by the authors in Figure 5 and Figures S4 and S5, it is not clear to me if the coherent species (Scoh) is really needed to fit the data. Could the authors refit the data just with the population species and show that the coherent signal is needed to explain the observed signal? Response: We are grateful to the reviewer for this valuable suggestion, as it further strengthens our conclusions. Below we compare the fitting results with or without the coherent species in the case of resonant driving (Figure R1). Figure R1. (a-c) 2D TA spectra of Cd P MSCs (a) measured with ћω of 2.75 eV (Δ = 0) and 3 2 p simulated spectra with (b) or without (c) the coherent species S . (d, e) Simulated transient coh species spectra (d) and their corresponding temporal dynamics (e) obtained from global fitting with S included. (f, g) Similar plots as d and e but without S included. (h, i) Comparison of coh coh TA dynamics at the indicated energy obtained from experiment (symbols) and spectral fits (solid and dashed lines for fitting with and without S , respectively). coh Figure R1 presents the comparison of experimental TA spectra and dynamics to fitting results with or without coherent species (S ). From the 2D spectra in panels a to c, coh the fitted spectrum without S shows a significant discrepancy compared to the coh experimental data, especially within ~200 fs around time zero. The fitting details are provided in panels d-g, from which we can see that the only difference between the two different fitting methods is the inclusion of S or not, whereas the rest coh components as well as their time evolution are identical. The specific kinetic traces probed at 2.82 and 2.75 eV, for example, are provided in panels h and i, from which the fitting with S clearly reproduces the experimental traces much better than the coh one without S , if we ignore the oscillations induced by coherent phonons. coh For the case of above-resonance driving, we have made the same comparison between fits with or without S (Figure R2). However, the difference between them is largely coh masked by the dominant contribution of the hot exciton spectral component (S₀) near time zero. Since this hot exciton species also has an induced absorption on the redside of the exciton bleach, it makes it difficult to unambiguously prove the need of including S in the fit. Although in term of the kinetics probed at the redside coh absorption feature (2.66 eV,) the fit with S appears to be slightly better than the one coh without S (panel 2i), the difference is too small for us to make an unambiguous coh conclusion. Figure R2. (a-c) 2D TA spectra of Cd P MSCs (a) measured with ћω of 3.05 eV (Δ = 300 meV) 3 2 p and simulated spectra with (b) or without (c) the coherent species S . (d, e) Simulated transient coh species spectra (d) and their corresponding temporal dynamics (e) obtained from global fitting with S included. (f, g) Similar plots as d and e but without S included. (h, i) Comparison of coh coh TA dynamics at the indicated energy obtained from experiment (symbols) and spectral fits (solid and dashed lines for fitting with and without S , respectively). coh We still strongly believe that the spectrum near time zero in the case of above-resonance driving results from the OSE, especially considering that transition dipole moment μ obtained from above-resonance driving is consistent with the values derived from below-resonance or resonant driving. However, considering that the global fitting cannot clearly differentiate the results with or without S , for the sake coh of scientific rigorousness, we have decided to remove the above-resonance driving results from the main body of this paper. In the revised manuscript, we focus on the results of below-resonance and resonant driving, which does not affect the novelty and significance of the work, as resonant and nonresonant cases are both still covered. The above-resonance driving results are now only briefly discussed in the SI, and we noted particularly that we have evidence for coherent driving but we cannot fully differentiate coherent driving or direct hot exciton excitation based on our current experiments. Revision: We have revised Figure 4 by including the comparison of global fitting spectra with or without the coherent species S . Additionally, we have moved the coh above-resonance driving results from the main text to the Supplementary Information (SI) and we have also added new Figures S5-S7 and S10 in the SI and made relevant discussions. Page 11 of the revised MS: “Importantly, in order to substantiate the existence of S coh and its assignment, we have compared the results of global fitting with or without the inclusion of S , whereas the rest components as well as their time evolution are kept coh identical. As shown in Figure 4e and 4f, the simulated spectrum without S shows a coh substantial discrepancy compared to the experimental spectrum, especially within ~200 fs around time zero. Additionally, the kinetic traces probed at 2.82 and 2.75 eV, for example, are compared in Figures S5e and S5f, from which the inclusion of S coh clearly reproduces the experimental traces much better than the one without S .” coh Page 13 of the revised MS: “We have also conducted the positively detuned driving experiment in order to reproduce the schemes in Figure 1e and 1f, but the results are much more complex, as discussed in Supplementary Note 6 and Figures S10 and S11.” Page 14 of the revised SI: “Supplementary Note 6. Positively detuned driving We also conducted the experiments under positively detuned driving with ћω = 3.05 eV p (Δ = 300 meV). The results are provided in Figures S10 and S11. Although ћω is p already chosen for the weakest absorption (Figure 2a), with the ratio between absorption at 3.05 and 2.75 eV being 1:38, the results are still complicated by hot exciton signals. Specifically, we have also compared the simulation results with or without a derivative-like S to the experimental result (Figure S10). However, the coh difference between them is largely masked by the dominant contribution of the hot exciton spectral component (S₀) near time zero. This species has an induced absorption on the redside of the exciton bleach, it makes it difficult to unambiguously prove the need of including S in the fit. Nevertheless, the transition dipole moment coh obtained from the analysis of the amplitude of S as a function of power intensity coh (μ ≈ 13.6 D; Figure S11) is in good agreement with those obtained from VH negatively-detuned and resonant drivings, suggesting that coherent driving has also likely been achieved for ћω = 3.05 eV.” p 2. In Figure S4, the authors show their fitting for two energies. Could the author provide the fittings and data for more energies? 2D plots comparing the fit and data could be useful to visualize how good the model fits the data, especially on how the spectrum shifts with time. Response: We thank the reviewer for this suggestion. We have now not only added the power-dependent fittings and data at two more probe energies (2.57 eV and 2.72 eV), but also provided 2D plots (for both 0.71 and 1.42 GW/cm2 driving intensity) to illustrate the quality of the fit. Revision: A new Figure S6 has been added to the SI. Figure S6. Fitting TA dynamics under resonant driving. The comparison of TA dynamics probed at (a) 2.57 eV and (b) 2.72 eV obtained from experiment (symbols) and global fitting (solid lines), with driving photon energy of 2.75 eV and various power intensities. 3. In Figures 5b and d the authors show the time evolution of different species. This is done in a plot of “Normalized Intensity” as a function of delay. What exactly is “normalized intensity”? Is it (eq S2) for a fixed lambda? Does it come from an integral of the population amplitude signal over lambda? Could the authors provide exactly how the normalized intensity is extracted from the fitting parameters described in Supplementary Note 2? Response: We apologize for the inaccurate use of the label “normalized intensity”, which may have caused confusion. It actually means the population (i.e., P(t) in eq i S2). In our fitting model, the initial total population at t = 0 is set to be 1, and it gradually decays and evolves into other populations. The signal amplitude for a given species S is expressed as the product of its fitted population dynamics (P(t) in eq S2) i i and its wavelength-dependent transient spectra (∆ε(λ) in eq S2). In this way, we can i reproduce the 2D TA spectra as function of both time and wavelength. Revision: The term \"Normalized Intensity\" in the relevant figure captions have been replaced with \"Relative Population\" and the term \"Pulse Intensity\" have also been introduced to explicitly emphasize the coherent part. 4. For the resonant case, apart from the coherent species, they extract two population signals, S1 and S2, which they attribute to “multiparticle” and “single-exciton” species, respectively. In Figure 5b, they show that at t=0, the single exciton population is 0, while there is already the formation of the multiparticle population. On their model, as the multiparticle population decays due to Auger annihilation, the single particle population is formed, following the rate equation S2. However, this situation is just true if the excitation power is so strong that all the excited particles have completely filled band edge states. If we are not on that limit, probabilistically speaking, some of the initial excited population should be formed by single excitons and some by multiple excitons. From the power dependence data shown in Figure S4, it is not clear that for lower powers, their fitting extracted an initial population formed by single excitons. Could the authors provide data on how the ratio between the single exciton and multi-exciton population varies with incident fluence in relation in early times? This would help to clarify how effective their methodology is in extracting the signals from different species. Response: We thank the reviewer for this suggestion. We would like to clarify that the term \"multi-particle\" here refers to a mixed population of single-exciton and biexcitons, consistent with our previous study (Sci. China Chem., 2023, 66, 3628-3635). Therefore, the spectrum of S actually contains contributions from both 1 single-exciton and biexciton species. The reason why we cannot differentiate the spectra of clean single-exciton and biexciton species is due to the complicated distribution of absorbed photons per MSC at time zero across the ensemble sample. In the case of excitation well above the band edge in QDs, this distribution is often assumed to be Poissonian. But here we are not in this excitation regime, and therefore the distribution remains unknown. In this case, our global fitting gives an “average” result that as time goes on the initial spectrum that contains both single-excitons and biexcitons (S ) 1 gradually evolves into a spectrum that is dominated by single-excitons (S ). 2 Nevertheless, following the reviewer’s suggestion, we have conducted global fits for several different driving intensities, and obtained the spectra of S and S as shown in 1 2 Figure R3. The ratio between the spectral amplitudes of S and S indeed first 1 2 increases with pump intensity and then gradually approaches saturation. This trend is consistent with the increase of biexciton populations with increasing pump intensity. However, due to the lack of the specific distribution function of absorbed photons per MSC, quantitative analysis is prohibited. Figure R3. (a) Global fitting derived spectra of S (solid lines) and S (dashed lines) at different 1 2 pump intensities in the case of resonant driving. (b) Ratio between the spectral amplitudes of S 1 and S (at ~2.75 eV) as a function of pump intensity. 2 Revision: Page 11 of the revised MS: “Note that the multiparticle species S actually contains 1 contributions from both single-exciton and biexciton states. We cannot differentiate the spectra of clean single-exciton and biexciton species due to the complicated distribution of absorbed photons per MSC at time zero across the ensemble sample. In the case of excitation well above the band edge in QDs, this distribution is often assumed to be Poissonian, but here we are not in this excitation regime, and therefore the distribution remains unknown. In this case, our global fitting gives an “average” result that as time goes on the initial spectrum that contains both single-excitons and biexcitons (S ) 1 gradually evolves into a spectrum that is dominated by single-excitons (S ). We have conducted the global fitting for several different driving intensities 2 (Figure S7), and obtained the spectra of S and S as shown in Figure S8. The ratio 1 2 between the spectral amplitudes of S and S indeed first increases with pump 1 2 intensity and then gradually approaches saturation (Figure S8), consistent with the increase of biexciton populations with increasing pump intensity.” 5. Could the authors clarify the degeneracy of the band edge states of the studied material? This is also important to understand how the ratio between single and multi-exciton species should scale as a function of incident fluence. Response: We thank the reviewer for this insightful comment. According to a previous transient absorption (TA) study on Cd P QDs (much weakly confined 3 2 compared to our current MSCs), the TA bleach is actually dominated by the contribution of band-edge electrons which have a two-fold spin degeneracy (J. Phys. Chem. A 2013, 117, 6362-6372). However, the electronic structure of the extremely-confined MSCs might differ from that of QDs. Indeed, as we pointed out in our manuscript, the TA spectra of MSCs can be understood as a bleach of the lowest energy exciton peak superimposed on a broad absorption feature, with the latter suggesting that the biexciton states in these small-size MSCs are not bound states. Otherwise, one would expect a red-shifted absorptive feature indicative of biexciton binding, which has been observed on the TA spectra of Cd P QDs (J. Phys. Chem. A 3 2 2013, 117, 6362-6372). 6. I would expect the multi-excitons to be red-shifted in relation to single-exciton species, due to the different binding energies. Therefore, I would expect the plot in Figure 4c (and also 4b) to shift to the blue as the multi-exciton species convert to a single exciton. This also seems to be consistent with the fittings that the authors show in Figure 5a and c, where the multiparticle species seem to be redshifted in relation to the single exciton. However, what I see in Figures 4b and c is a red shift. Could the authors clarify that? Response: We thank the reviewer for this valuable comment. However, it is important to note the spectral evolution observed on the 2D TA spectra within ~200 fs is induced by the interplay between the coherent species and the S species. Therefore, 1 the red or blue shift on this timescale is not related to conversion from multi-excitons to single-excitons; the latter occurs on a slower timescale of a few ps and the spectral change is actually quite small. Additionally, as we noted above, the TA spectrum of MSCs contains an exciton bleach feature superimposed on a broad absorption feature, with the latter suggesting that the biexciton states in these small-size MSCs are not bound states. So we should not expect typical spectral changes associated with conversion from multi-excitons to single-excitons observed in QDs to be present here. 7. In Figures 5a and c, there is a positive feature for S1 and S2 around 2.85 eV, could the authors explain where this comes from? Response: We thank the reviewer for this comment. In fact, S and S have positive 1 2 features not only on the blue side at ~2.85 eV but also on the red side at ~2.6 eV. As we emphasized above, this is actually an exciton bleach feature superimposed on a broad absorption feature suggestive of unbound biexciton states in these MSCs. Strong Coulombic scattering between excitons created in such a small volume generates such a broadened exciton-to-biexciton transition, which is distinct from the attractive biexciton interaction observed in typical QDs. 8. In Figure S4 b, the authors show the difference between their fitting and the TA data. They claim that this difference comes from the signal of coherent phonons that are not contained in their model. However, this difference seems quite drastic to me, especially at t = 0 (the most important part of the fitting to extract the coherent species). From the first oscillation to the second, the phonon signal decays from around 7 mOD to 0.5 mOD (~14 times), while it is almost constant from the second to the third and fourth oscillations. Interestingly, this is not the case for a different energy (2.74 eV) in Figure S4c. Could the authors explain these differences? Response: We thank the reviewer for this very insightful comment. The coherent phonon dynamics is not the focus of this work, and therefore we did not discuss it in detail. But we agree with the reviewer due to the presence of phonon dynamics near time zero which is the most important part of the fitting to extract the coherent species, we need to clarify the phonon dynamics further. First, the observation that the phonon signal decays from around 7 mOD to 0.5 mOD (~14 times), while it is almost constant from the second to the third and fourth oscillations is likely due to the presence of multiple phonon mode frequencies. Indeed, Figure R4 presents the fast Fourier transform (FFT) spectra of the dynamics in Figure S4b and S4c, which shows multiple phonon frequencies. Figure R4. FFT spectra of TA coherent phonon dynamics probed at (a) 2.82 eV and (b) 2.74 eV. Secondly, the phonon dynamics observed at 2.74 eV (the exciton bleach) is indeed different from that observed at 2.82 eV (the absorptive feature). This is likely due to different electron-phonon coupling mechanisms in the ground and excited states. The exciton bleach reflects information mainly on the ground state whereas the photoinduced absorption is mainly associated with the excited state, and these two can coupled to phonons in different ways. For example, in previous studies on gold nanoclusters (see Annu. Rev. Phys. Chem. 2021, 72, 121-142), different phonon dynamics have also been observed at different wavelengths (Figure R5, particularly panel e). Figure R5. Coherent phonon dynamics observed in Au nanoclusters. Redacted due to copyright reasons. Please refer to Figure 7 in Annu. Rev. Phys. Chem. 2021, 72, 121-142. Revision: Figure S4 in the SI has been revised. Figure S4. TA dynamics under resonant driving and coherent phonon dynamics. The comparison of power-dependent TA dynamics obtained from experiment (symbols) and global fitting (solid lines) at the indicated energies of 2.82 eV (a) and 2.74 eV (c). The pump photon energy is centered at 2.75 eV. The same lifetime parameters are maintained for all pump intensity. (b) The difference between experimental and global fitted dynamics in (a), which can be attributed to signal oscillations due to electron-phonon coupling. (d, e) Fast Fourier transform (FFT) spectra of TA coherent phonon dynamics probed at (a) 2.82 eV and (b) 2.74 eV. The exciton bleach at 2.74 eV reflects information mainly on the ground state whereas the photoinduced absorption at 2.82 eV is mainly associated with the excited state, and these two can coupled to phonons in different ways, giving rise to the different FFT spectra. I also have some minor suggestions: 9. On page 10, when discussing the signal from different species, the authors distinguish a “single-exciton” and a “mutiparticle” species. It is perfectly fine to understand what they meant by the context, but I suggest changing the multiparticle nomenclature to multi-exciton, because the term “particle” is sometimes also used in the literature to describe the nanoparticles. Response: We thank the reviewer for this suggestion. However, as we explained above, the word “multiparticle” in our study refers to a mixture of multi-excitons and single-exciton. We have now further explained this point in the main text to avoid confusions. Revision: Page 11 of the revised MS: “Note that the multiparticle species S actually contains 1 contributions from both single-exciton and biexciton states. We cannot differentiate the spectra of clean single-exciton and biexciton species due to the complicated distribution of absorbed photons per MSC at time zero across the ensemble sample. In the case of excitation well above the band edge in QDs, this distribution is often assumed to be Poissonian, but here we are not in this excitation regime, and therefore the distribution remains unknown. In this case, our global fitting gives an “average” result that as time goes on the initial spectrum that contains both single-excitons and biexcitons (S ) 1 gradually evolves into a spectrum that is dominated by single-excitons (S ). We have conducted the global fitting for several different driving intensities 2 (Figure S7), and obtained the spectra of S and S as shown in Figure S8. The ratio 1 2 between between the spectral amplitudes of S and S indeed first increases with pump 1 2 intensity and then gradually approaches saturation (Figure S8), consistent with the increase of biexciton populations with increasing pump intensity. ” 10. Can the authors provide information on what optical density was used during the experiments? This information is relevant for reproducing these results. Response: We thank the reviewer for this suggestion. For above-bandgap and resonant cases, the optical density at excitonic absorption peak (2.75 eV) is around 0.31; and it was changed to 0.57 for below-bandgap case. Revision: These details have now been added to the Methods. 11. In the title, the authors use the word “driving” to describe their work. However, in my opinion, the word driving does not differentiate between the generation of coherent and population processes. In my view, using this expression hides the novelty of the work. Using a nomenclature that shows that this work is observing coherent processes would make the title stronger. Response: We thank the reviewer very much for this kind suggestion. Ideally, we would also like to use coherent driving in the title. But as exactly pointed out by the reviewer, we have created both coherent species and real populations in our experiments. In order to be more conservative, we choose to simply use the term “driving” instead of “coherent driving”. Reviewer #2 (Remarks to the Author): The manuscript discusses the coherent optical driving of solution-processed semiconductor materials, focusing on magic-size clusters (MSCs)—specifically Cd₃P₂ MSCs—as a promising platform for exploring light–matter interactions at room temperature. The study is motivated by a longstanding interest in coherent excitation of matter by laser fields, with evolving applications from fundamental physics to quantum information science and nonequilibrium materials engineering. As a conclusion I will recommend the manuscript to be accepted after revision. Here are my comments on the manuscript: Response: We thank the reviewer very much for their kind comments on our work. Below we address their concerns point-by-point. 1. The energy levels in Figure 1a, 1b, and 1c are confusing. Please add breaks in hω eg or other modifications to make the scale bar more reasonable. The caption order is also confusing. Response: We thank the reviewer for pointing this out. We have revised Figure 1a–c to improve clarity. Revision: See the revised Figure 1. 2. The steady absorption polarization property of the MSCs can be different from their emission polarization properties. It would be important to add such measurement results can provide useful information to the exciton behavior. Given knowledge of the one of absorption or emission, using fluorescence anisotropy is probably sufficient. More generally, if the perception of the origin of polarization is distortion of some type, what does this mean at a batch-to-batch or even ensemble level: symmetry is broken in the same way, stochastically, or something else? Response: We thank the reviewer for this comment. We agree that absorption and emission polarization can in principle differ, which is the reason why we conducted both PL anisotropy measurements (Figure 2b) and transient absorption anisotropy measurements (Figure 3). In practice, however, due to the random orientation of molecular-like MSCs in isotropic solution—as opposed to deliberately self-assembled structures with preferential orientation, steady-state absorption polarization measurements would not work. This is fundamentally different from PL anisotropy measurements in which we use a linearly polarized laser to excite the sample, thereby preferentially exciting MSCs with dipole components along the laser electric field direction. Instead, we implemented polarization-controlled pump-probe TA spectroscopy to assess absorption anisotropy. This approach also involves selectively exciting MSCs using a linearly polarized pump beam, thereby photo-selecting those with dipole components aligned along the pump electric field. Regarding the origin of polarization, we envision that the symmetry breaking should be identical across the ensemble sample, which is the reason why we can obtain an anisotropy close to 0.4 (indicating fully linearly polarized transition dipole) in the case of resonant pumping (Figure 3c). Qualitatively, one can understand this behavior by treating the MSCs as small molecules. Most molecules have anisotropic molecular shape and their lowest energy transition is strongly linearly polarized, giving rise to transient absorption anisotropy also close to 0.4. This similarity between MSCs and molecules further corroborate the atomically precise structure of the MSCs. Revision: Page 6 of the revised MS: “However, unlike PL measurements in which we can select a subset of MSCs with their transition dipoles aligned along the electric field direction of the excitation light, direct measurements of absorption anisotropy would not work due to the random orientation of the MSCs in the ensemble. Instead, we can conduct absorption anisotropy measurements of Cd P MSCs using 3 2 polarization-controlled femtosecond pump-probe TA spectroscopy in which the linearly polarized pump beam can also select a subset of MSCs with their transition dipoles aligned along the electric field direction of the pump beam; see Methods for details.” Page 8-9 of the revised MS: “Qualitatively, one can understand the highly linearly-polarized transition dipole of the MSCs by treating them as small molecules. Most molecules have anisotropic molecular shape and their lowest energy transition is strongly linearly polarized, giving rise to transient absorption anisotropy also close to 0.4.52 This similarity between MSCs and molecules further corroborate the atomically precise structure of the MSCs, that is, the structural anisotropy is nearly identical for each MSC.” 3. Please provide more detailed explanation on why the 2.6 nm QDs showed much smaller anisotropy compared to the MSCs. The quantum confinement is still significant at this size as it is much smaller than the Bohr radius of the material. Response: We thank the reviewer for this comment, which is related to the comment above. Although the 2.6 nm QDs are also strongly confined, they are radically different from MSCs in that the QDs cannot be atomically precise. As a result, the absorption and PL peaks of the 2.6 nm QDs are much broader than those of MSCs. As each QD has different size and morphology in the ensemble, it should also have a different degree of polarization for the band edge exciton transition. Consequently, the TA anisotropy value is as small as 0.05 at the ensemble level. Revision: Page 9 of the revised MS: “Microscopically, each QD in the sample has a different size and morphology, and therefore each QD should also have a different degree of polarization for the band edge exciton transition, resulting in negligible anisotropy at the ensemble level.” 4. The authors claim that the Cd₃P₂ MSCs platform for exploring light–matter interactions at room temperature. On the other hand, the authors stated that the stability issue requires cryo-measurement. Please justify it. Response: We thank the reviewer for this comment. We would like to clarify that we propose cryo-measurement not to solve the stability issue of the MSCs but to explore more exotic quantum phenomena under strong driving (Rabi frequency >> linewidth). Besides the Rabi splitting observed here, quantum interference between the pump and probe beams and gain without inversion are expected under strong driving conditions. In order to attain the regime of Rabi frequency >> linewidth, we propose to conduct cryo-measurement to reduce the linewidth. 5. As a general question: it is peculiar that the proposed experiments “work” here in terms of isolating a resonant stark effect response in situations with the coexistence of real carriers. The bandwidth of the samples is similar or worse to platelets which have been studied a lot in this respect, similar or worse than tmds; is the sample really special or is this more a function of looking for the right signals? Response: We thank the reviewer for this insightful comment. Indeed, there are many other systems with narrow transition bandwidths, such as the nanoplatelets or TMDs mentioned by the reviewer. Our present system is not advantageous over these systems in terms of the bandwidth; rather, it is unique due to the linearly polarized exciton transition that is well isolated from other transitions. This linear transition allows us to conduct VH experiment to reject unwanted scattering signals under resonant driving. Although nanoplatelets also have linearly-polarized in-plane exciton transitions, there is a strong overlap between the heavy- and light-hole exciton transitions with the continuous band. Moreover, they have intersubband transitions that can also be driven, which strongly complicates the analysis of band-edge exciton Rabi splitting (see our previous work in Sci. Adv., 2024, 10, eabq2793). On the other hand, TMDs have circularly polarized exciton transitions. In this case, one might consider using σ+ pump and σ- probe to reject scattering signals, but it would introduce extra complications from the exciton-to-biexciton transition which is exactly of opposite circular polarization compared to the ground-to-exciton transition, as demonstrated in the case of monolayer MoS₂ (Nat. Phys., 2018, 14, 1092-1096). Overall, none of these previous systems can be approximated as a simple two-level system while allowing for the use of cross polarizations to obtain clean coherent driving signals. Revision: Page 14 of the revised MS: “Although several other systems such as colloidal nanoplatelets (NPLs) or monolayer transition metal dichalcogenides (TMDs) also exhibit narrow linewidths are room temperature, they each have their own complexity. NPLs have linearly-polarized in-plane exciton transitions, but there is a strong overlap between the heavy- and light-hole exciton transitions with the continuous band. Moreover, their intersubband transitions can also be driven,28 which strongly complicates the analysis of band-edge exciton Rabi splitting. For TMDs, the exciton-to-biexciton transition is exactly of opposite circular polarization compared to the ground-to-exciton transition,15 which prohibits the use of cross-polarized TA to conduct resonant driving of the ground-to-exciton transition.” 6. Derivative-type spectra are repeatedly described (as they should be) for the sub-gap case of OSE, but I don’t see the actual data: please show the derivative overlayed with the data. Related: the simulation spectra in Figure 6 have similar shape, but do not line up with the right energies as in the experimental data. Why? Response: We thank the reviewer for this comment. The term \"derivative-like TA spectrum\" used here does not literally refer to taking first or second derivatives of the absorption spectrum; rather, it emphasizes its positive and negative alternating features, which is derivative-like. Fundamentally, the OSE spectrum arises from a small spectral shift, which is qualitatively similar to taking a derivative of the steady-state absorption spectrum. We also thank the reviewer for noticing that the simulation spectra in Figure 6 do not fully line up with the experimental spectra in terms of transition energies. This is because the simulation spectra are obtained from shifting the peak position of the steady-state exciton absorption, whereas the TA spectra are acquired on the TA spectrometer. The spectral accuracy of these two systems are not perfectly calibrated, thereby resulting in small discrepancies in the absolute energy positions. Revision: Page 12 of the revised MS (Figure 5 caption): “The simulated spectra are slightly different from the experimental spectra in terms of transition energies, because the steady-state absorption and TA spectrometers are not perfectly calibrated in absolute wavelengths.” 7. The global analysis or singular value decomposition is a bit handwaving for the above gap case: the carrier cooling effect on the bandwidth is continuous with carrier temperature, not a different weighting of the spectrum. More generally, if multiparticle effects like Auger occur, this seems avoidable: lower powers should turn off these effects and simplify the model further. What is the average excitons/particle in the experiments? This should be clarified to show why MX is reasonable addition—and better, avoided at lower intensities. Response: We thank the reviewer for this comment. We acknowledge that the coherent signal for above-bandgap case is challenging to perfectly extract via global fitting due to its mixing with hot exciton cooling effects. This point has also been raised by Reviewer #1 in their comment 1, and we refer the reviewer to our response above. Due to this complexity, we therefore have decided to remove the above-resonance driving results from the main text. The difficulty of quantifying average exciton number has also been explained in our response to Reviewer #1’s comment 4. Revision: Page 13 of the revised MS: “We have also conducted the positively detuned driving experiment in order to reproduce the schemes in Figure 1e and 1f, but the results are much more complex, as discussed in Supplementary Note 6 and Figures S10 and S11.” Page 14 of the revised SI: “Supplementary Note 6. Positively detuned driving We also conducted the experiments under positively detuned driving with ћω = 3.05 eV p (Δ = 300 meV). The results are provided in Figures S10 and S11. Although ћω is p already chosen for the weakest absorption (Figure 2a), with the ratio between absorption at 3.05 and 2.75 eV being 1:38, the results are still complicated by hot exciton signals. Specifically, we have also compared the simulation results with or without a derivative-like S to the experimental result (Figure S10). However, the coh difference between them is largely masked by the dominant contribution of the hot exciton spectral component (S₀) near time zero. This species has an induced absorption on the redside of the exciton bleach, it makes it difficult to unambiguously prove the need of including S in the fit. Nevertheless, the transition dipole moment coh obtained from the analysis of the amplitude of S as a function of power intensity coh (μ ≈ 13.6 D; Figure S11) is in good agreement with those obtained from VH negatively-detuned and resonant drivings, suggesting that coherent driving has also likely been achieved for ћω = 3.05 eV.” p Reviewer #3 (Remarks to the Author): I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts. Response: Thank you very much for contributing to the review of our work. Reviewer #4 (Remarks to the Author): Liu et al. reported the resonant and non-resonant driving of excitons in nanoclusters. They attempt to demonstrated the system allows coherent driving of quantum states under various detuning conditions that may lead to blue-shift, redshift and even Mollow triplet lineshape. At first, this article looks promising to show the process, according to their abstract and introductory paragraph. However, I find the data hardly support their arguments and the quality of the work, including the analysis, hardly convince me this work can be accepted for publication in Nature Communications. Response: We sincerely thank the reviewer for the critical and constructive comments. While we acknowledge that our data are not perfect due to realistic experimental conditions, we believe the experimental results provide solid evidence of coherent driving of quantum states under various detuning conditions. As shown in Fig.4 in the main context, both time- and frequency-domain measurements reveal a typical derivative-like TA signal under non-resonant driving and the Mollow-like lineshape under resonant driving. The evaluations of nonresonant OSE shift and resonant Rabi splitting through our model analysis, as shown in Fig. 5e and 5f, scale linearly with the power intensity and electric field strength, respectively, confirming that these features originate from coherent light–matter coupling rather than artifacts. To further strengthen our observation, we have included more detailed fitting procedures and also a quantum simulation in the updated version, as detailed below in our response to the reviewer’s further comments. Also, while this reviewer has a strong concern for our conclusions, we would like to point out that the novelty and significance of this work have actually been well recognized by Reviewers #1 and #2; see above. We therefore sincerely hope that the reviewer can re-evaluate our work based on all reviewers’ comments and based on our detailed responses to all comments. The main issue with this article is that – the data is dominated by strong incoherent processes arising from exciton dephasing – phonon scatterings, exciton recombination, local inhomogeneity, auger effects. A coherent process, without the sophisticated treatment from Lindblad-formalism, in principle, can be observed directly from pump-probe spectrum with signal appears only when pump and probe pulses are overlapped in time-domain. Even at time very closed to the temporal overlap of their pump and probe profile, the incoherent processes can already be dominant. In the case when incoherent signal arising from real population transition is negligibly small, the effects will be promising. However, this is not the case in their system because the incoherent signal is simply too strong to distinguish any coherent response. Response: We thank the reviewer for this important point. We agree that the strong incoherent processes are indeed unavoidable and dominate the spectra for the resonant case. Nevertheless, our data clearly reveal characteristic signatures of coherent light-matter interaction during the pump–probe temporal overlap. We observed a Mollow-like lineshape at zero-time delay, which serves as a coherent fingerprint under resonant driving, and a derivative-like spectral profile under both above- and below-bandgap driving conditions, which is consistent with the typical OSE behavior. Furthermore, we have included a careful comparison of the global fitting results with and without including the coherent driving signals, as detailed in our response to Reviewer #1’s comment 1. The authors then perform spectral fitting process. This process itself is not robust and cannot provide any relevant coherent quantum optical process directly linked to the optical excitation. In-particular, when the pump is resonant or above the exciton transition, the signals are largely contaminated by various scattering processes, related to phonon scattering, linewidth broadening, and even resonant coupling to phonon Raman under SRS process in a pump-probe spectroscopy. All these processes can give rise to the lineshape observed in their spectrum, which cannot be disentangled from the pure electronic processes and they are not even considered properly in their analysis. For example, above resonance, you can have lattice heating by optical phonon which simply decays within femtosecond timescale and that gives you redshift. This is particularly prominent because you excite real population that drive electron-phonon coupling. The temporal resolution from the author setups is not sufficient to distinguish this process, not to say their spectra are already dominated by other incoherent processes. Response: We thank the reviewer for raising this critical point. We acknowledge that various mechanisms can possibly induced spectral profiles similar to the OSE or Rabi splitting features. However, as we responded above, the observation of a Mollow-splitting like TA spectrum of the Cd P MSCs is validated by comparing the 3 2 experimental TA spectra to the global fitting results with or without including the Mollow-splitting species (Figure R1), and by a linear scaling relationship between the splitting energy and the driving electric field strength instead of power intensity. These unique features allow us to conclude that while incoherent processes exist, the Mollow-splitting feature is not completely obscured. Nevertheless, we agree with the reviewer that the above resonance driving case could be particularly complex, as it introduces additional hot-electron effects. As we responded to reviewer #1’s comment 1, in this case we have also made the comparison between fits with or without coherent driving species (Figure R2). However, the difference between them is largely masked by the dominant contribution of the hot exciton spectral component near time zero. Since this hot exciton species also has an induced absorption on the redside of the exciton bleach, it makes it difficult to unambiguously prove the need of including the coherent species S in the coh fit. Although in term of the kinetics probed at the redside absorption feature (2.66 eV,) the fit with S appears to be slightly better than the one without S (Figure R2i), coh coh the difference is too small for us to make an unambiguous conclusion. In light of the issues above, we have decided to remove the above-resonance driving results from the main body of this paper for the sake of scientific rigorousness. In the revised manuscript, we focus on the results of below-resonance and resonant driving, which does not affect the novelty and significance of the work, as resonant and nonresonant cases are both still covered. Revision: Page 14 of the revised MS: “Nevertheless, a major advancement of the present study is the observation of a Mollow-splitting like TA spectrum of the Cd P MSCs under 3 2 resonant driving at room temperature, which is validated by comparing the experimental TA spectra to the global fitting results with or without including the Mollow-splitting species (Figure 4), and by a linear scaling relationship between the splitting energy and the driving electric field strength instead of power intensity (Figure 5f).” Another point that I feel this work is not relevant to coherent driving processes is that – when the spectra are dominated by the strong incoherent processes, why not perform a full quantum optics treatment that account all these processes together? This is a more robust method to help to visualize the impact – how dephasing diminishes the Rabi oscillation and how that leads to the expected spectrum? A comprehensive treatment accounting for the dominant processes is definitely necessary. This will then help to build the simple model back to Rabi oscillation. Without a proper treatment on these datasets, it is difficult to simply extract the coherent processes without considering any dephasing processes. The dipole moment extracted directly from your fitting is probably not quite relevant at the end. Another issue is that – the linewidth is extremely broad, comparable and even larger than the “Rabi frequency” the authors have claimed. In this case, you generally do not see clear Rabi splitting. The lineshape itself already imply that the incoherent process is too strong and coherent driving will require much stronger Rabi frequency. Within the pump-probe overlap, the incoherent dephasing processes already take place and their Fourier transform will then give you spectrum that is not relevant to the coherent phenomena, i.e., optical Stark effects. By over-interpreting the data within the coherent driving process, and by employing a spectral analysis process without comprehensive quantum optics treatment, it is impossible to understand how the data supports any coherent phenomena and further quantitative analysis based on the simple Rabi model becomes irrelevant. I suggest the authors to reinterpret their data again carefully, without over-claiming the phenomena that barely seen from their data. Response: We agree with the reviewer that the broad lineshape indeed represents a critical issue. However, a major advancement of the present study is the demonstration of coherent driving effects for a solution-grown sample at room temperature. Yes we need to fight with the linewidth issue, but it does not completely preclude the observation of coherent driving effects. As we pointed out in the discussion section, the large transition dipole and the intense femtosecond driving pulse has led to Rabi splitting reaching 20 meV, which is smaller than the room-temperature ensemble absorption linewidth of ~100 meV. Nevertheless, this still leads to measurable spectral modulations corresponding to the Mollow-like spectrum we extracted from the TA results. Ideally we would like to enter the regime in which Rabi splitting >> linewidth, but it is challenging at room temperature due to sample damage issue. In order to gain further physical insight into the coherent driving case, we also conducted a quantum simulation using the Lindblad master equation based a density matrix approach. While it is unrealistic to perform full quantum optical simulation for a system as complex as the Cd P MSCs here at room temperature, this Lindblad 3 2 equation simulation can capture the physical essence of a two-level system under laser driving and in the presence of dephasing, as established in our previous study (Nat. Photonics, 2024, 18, 1044-1051). For simplicity, we consider a coherently driven, resonant two-level system with an energy gap of 2.75 eV ( ℏ𝜔 ) to describe our MSC, the time-dependent evolution of 01 ⅆ𝜌(𝑡) −𝑖 which is governed by the Lindblad master equation: = [𝐻(𝑡),𝜌(𝑡)]+ ⅆ𝑡 ℏ 𝐷[𝜌(𝑡)]. Here, 𝐻(𝑡) is the time-dependent Hamiltonian, which is defined as: 0 𝜇 𝜀 01 𝑠𝑢𝑚 𝐻(𝑡) = ( ) 𝜇 𝜀 ℏ𝜔 10 𝑠𝑢𝑚 01 in which 𝜀 = 𝜀 𝑔 (𝑡)𝑐𝑜𝑠(𝜔 𝑡)+𝜀 𝑔 (𝑡)𝑐𝑜𝑠(𝜔 𝑡) is the total electric 𝑠𝑢𝑚 𝑝𝑢 𝑝𝑢 𝑝𝑢 𝑝𝑟 𝑝𝑟 𝑝𝑟 filed of the pump (𝜀 ) and probe (𝜀 ) pulses and 𝑔(𝑡) is the pulse envelop. For the 𝑝𝑢 𝑝𝑟 pump we took the same Gaussian pulse with FWHM around 100 fs. For the probe we took a broadband chirped pulse with linear group velocity delay, and the time zero of TA signals was later corrected according this group delay. The pump and probe pulse intensities are set to be 1 GW cm-2 and 10-3 GW cm-2, respectively. We set the transition dipoles 𝜇 = 𝜇 = 13 Debye. 10 01 𝐷[𝜌(𝑡)] accounts for depopulation and dephasing effects, which can be expressed as: 1−𝜌 (𝑡) −𝜌 (𝑡) 00 01 𝑇 1 𝑇 2 𝐷[𝜌(𝑡)] = −𝜌 (𝑡) 𝜌 (𝑡) 10 11 − 𝑇 𝑇 ( 2 1 ) T and T represent the depopulation and dephasing time constants, respectively. We 1 2 set T as infinitely long (ns time scale, much longer than the dephasing timescales 1 here), and T = 50 fs to account for the dephasing effects. The initial populations were 2 set to 𝜌 = 1 and 𝜌 = 0. the equation was then solved numerically. 00 11 The signal polarization induced by light filed can be obtained as: 𝑃 = 𝑡𝑟(𝜌(𝑡)𝜇) To calculate the TA signals, we need to evaluate three polarization components: P , P and P , which represent the polarization signals induced by the (pump+probe) (pump) (probe) combined pump-probe, pump-only, and probe-only laser fields, respectively. The TA signal can be calculated using: |𝐹𝑇[𝜀 (𝑡−𝜏)+𝜂𝑖𝑃(𝑝𝑟𝑜𝑏𝑒)|2 𝑝𝑟 ∆𝐴(𝜏) = 𝑙𝑜𝑔 |𝐹𝑇[𝜀 (𝑡−𝜏)+𝜂𝑖𝑃(𝑝𝑢𝑚𝑝+𝑝𝑟𝑜𝑏𝑒)−𝜂𝑖𝑃(𝑝𝑢𝑚𝑝)]|2 𝑝𝑟 Here 𝜏 is the time delay between the pump and probe pusles, FT represent Fourier transformation, and the factor 𝜂 is a scaling parameter that converts polarization to electric field. Figure R6. Quantum Simulation of resonant coherent driving. (a) Simulated 2D TA spectra of a single MSC. Details are provided in Supplementary Note 7. (b) Simulated TA spectra near time zero (averaged between -60 fs and 20 fs). (c) Simulated TA dynamics near the bleach centre (averaged between 2.748 eV and 2.752 eV). The final simulated results are presented in Figure R6. Note that the simulation is performed for one MSC instead of an ensemble sample. Consequently, while the laser pulse intensity used in simulation is comparable to experiments, the TA signal does not exactly match the experimental conditions. In the simulation, the phases of the pump and probe pulses are well-defined, enabling the observation of coherent phenomena beyond the Mollow triplet. These are manifested as distinct Rabi oscillations in both the frequency and time domains, as shown in Figure R6. However, these detailed features are typically obscured in experiments by the phase fluctuations of the pump and probe pulses. The overall Mollow-like TA spectral profile remains robust (Figure R6) and is successfully observed in experiments. Revision: Page 14 of the revised MS: “In order to gain further physical insight into the coherent driving case, we also conducted a quantum simulation using the Lindblad master equation based a density matrix approach. While it is unrealistic to perform full quantum optical simulation for a system as complex as the Cd P MSCs here at room 3 2 temperature, this Lindblad equation simulation can capture the physical essence of a two-level system under laser driving and in the presence of dephasing, as established in our previous study.29 Details of the simulation is provided in Supplementary Note 7. As shown in Figure S12, this simulation predicts fine structures exhibited as multiple fringes in both time and frequency domains, consistent with previous models.70 These features are interpreted as the interference between fields emitted at different times during the driving pulse, but they are typically obscured in experiments by the phase fluctuations of the pump and probe pulses and by many other competing processes.29,71 The overall Mollow-like TA spectral profile remains robust (Figure S12) and is successfully observed in experiments.” In addition, the simulation details presented above are provided as Supplementary Note 7 in the SI, and Figure R6 is included as Figure S12."}, {"round": 2, "reviewer_comments": {"reviewer_1": "Recommendation: Publish subjected to minor revisions The authors made a thorough response, addressing most of my previous comments. The main change in the manuscript was the removal of the “above resonance driving” from the main text. Even with this removal, I believe that this study is highly significant due to the observation of Autler-Townes (AT) splitting in these systems. I believe the work is suitable for publication in Nature Communications; however, I have three remaining points regarding the physical interpretation and data transparency that should be addressed prior to acceptance. Specific points: 1. As noted in my previous review, the primary significance of this work lies in the strong evidence for the Autler-Townes effect. In this regard, how the Scoh species scales with power is very important. The authors have shown in Figure 5 that the Rabi splitting scales linearly with the electric field (as expected). However, the behavior of the transition amplitudes remains unclear to me. In my view, while the splitting increases with pump power, the total oscillator strength (the integrated area of the split transitions) should be conserved (or remain conserved per available oscillator, accounting for real population created by the pump). It is currently unclear how the individual transition amplitudes (sketched as Gaussians in Figure 1d) evolve experimentally. Action: I recommend that the authors clarify in the main text how the sum of the individual amplitudes of the dressed states scales with power. A plot or discussion demonstrating that the data follow the expected trend would significantly strengthen the claim that this is indeed an AT signature and not another possible population effect. 2. Since at early times the population signal S1 is convoluted with Scoh, the correct attribution of S1 is important. One important feature of this study is that S1 and S2 have almost identical spectral signatures (Figure S5). This suggests that the net Coulombic interactions among carriers in the multiparticle states are very small. However, I find that this point is still underdeveloped in the manuscript. In their previous work (Sci. China Chem. 2023), the authors reported strong Coulombic scattering that suppresses biexciton coherence, resulting in a broad exciton-to-biexciton transition. While this observation is relevant, it does not fully explain why the multiparticle (S1) and single-exciton (S2) species display such similar spectral signatures in the present study. A physically plausible explanation is the extreme confinement regime of these clusters, in which the carrier wavefunctions are forced to overlap nearly identically. In this limit, the attractive electron–hole interaction and the repulsive electron–electron and hole–hole interactions may partially cancel, leading to minimal net Coulombic energy shifts. Action: I recommend that the authors clarify the physical origin of the similar spectral signatures of S1 and S2. In particular, citing previous studies on highly confined semiconductor nanostructures that demonstrate cancellation or suppression of multiparticle Coulombic interactions under strong confinement would strengthen this interpretation. 3. Given that the observation of the AT effect in this system relies on disentangling a complex, convoluted population signal at very early times (<200 fs), the reproducibility of the analysis is very important. While in my view the data supports the authors' conclusions, there is a possibility that other transient effects (currently unknown to me or the authors) could produce similar spectral signatures. Action: In line with the open science mandates of Nature Communications, I strongly recommend that the authors deposit the raw power-dependent Transient Absorption data (particularly for the resonant case) and the analysis scripts used for the global fitting/deconvolution in a public repository. In my view, allowing the community to independently access the data and analyses will greatly enhance the impact and longevity of these findings.", "reviewer_2": "I feel that the authors have given reasonable answers or taken reasonable action in revisions to be responsive to the reviews.", "reviewer_4": "The authors have replied to my concerns. The manuscript can be considered for publication in Nature Communications."}, "author_response": "Response: We sincerely thank the reviewer for positive and encouraging assessment of our work. We appreciate the reviewer’s thoughtful comments and constructive suggestions that have helped improve the manuscript. Below, we address the three remaining points concerning physical interpretation and data transparency in a point-by-point manner. 1. As noted in my previous review, the primary significance of this work lies in the strong evidence for the Autler-Townes effect. In this regard, how the Scoh species scales with power is very important. The authors have shown in Figure 5 that the Rabi splitting scales linearly with the electric field (as expected). However, the behavior of the transition amplitudes remains unclear to me. In my view, while the splitting increases with pump power, the total oscillator strength (the integrated area of the split transitions) should be conserved (or remain conserved per available oscillator, accounting for real population created by the pump). It is currently unclear how the individual transition amplitudes (sketched as Gaussians in Figure 1d) evolve experimentally. Action: I recommend that the authors clarify in the main text how the sum of the individual amplitudes of the dressed states scales with power. A plot or discussion demonstrating that the data follow the expected trend would significantly strengthen the claim that this is indeed an AT signature and not another possible population effect. Response: We thank the reviewer for this insightful comment. As schematically illustrated in Fig. 1d of the transient absorption measurement scheme (pump-on minus pump-off), the spectral splitting redistributes the oscillator strength of the original transition into two dressed states. Consequently, while the spectral amplitudes at specific probe energies—namely the two positive side peaks and the central negative bleach—grow with pump power, the total integrated spectral weight of the transient absorption band is expected to be conserved. In a transient absorption measurement, this implies that the net integrated signal remains approximately zero. To verify this behavior, we quantitatively analyzed the power dependence of the integrated transient absorption signal based on the data in Fig. 5b. The resulting integrated area, shown in Fig. R1 below, indeed remains close to zero over the entire range of pump powers. This behavior is verifies conservation of the total oscillator strength and supports our interpretation that the observed splitting originates from coherent driving. Figure R1. Power-dependent amplitudes of the coherent spectral features probed at three transition energies under resonant driving (hollow circles), together with the total integrated area (solid circles). Revision: In the revised manuscript, we add more discussion on page 13: “Moreover, although the signal amplitude at each probe energy increases with the driving power, the total integrated TA signal remains close to zero (Supplementary Figure 10). This behavior reflects the conservation of total oscillator strength, i.e., the original transition strength is redistributed to dressed states under strong optical driving, as schematically illustrated in Figure 1d.” Additionally, Figure R1 is added as the new Supplementary Figure 10 in the SI. 2. Since at early times the population signal S1 is convoluted with Scoh, the correct attribution of S1 is important. One important feature of this study is that S1 and S2 have almost identical spectral signatures (Figure S5). This suggests that the net Coulombic interactions among carriers in the multiparticle states are very small. However, I find that this point is still underdeveloped in the manuscript. In their previous work (Sci. China Chem. 2023), the authors reported strong Coulombic scattering that suppresses biexciton coherence, resulting in a broad exciton-to-biexciton transition. While this observation is relevant, it does not fully explain why the multiparticle (S1) and single-exciton (S2) species display such similar spectral signatures in the present study. A physically plausible explanation is the extreme confinement regime of these clusters, in which the carrier wavefunctions are forced to overlap nearly identically. In this limit, the attractive electron–hole interaction and the repulsive electron–electron and hole–hole interactions may partially cancel, leading to minimal net Coulombic energy shifts. Action: I recommend that the authors clarify the physical origin of the similar spectral signatures of S1 and S2. In particular, citing previous studies on highly confined semiconductor nanostructures that demonstrate cancellation or suppression of multiparticle Coulombic interactions under strong confinement would strengthen this interpretation. Response: We thank the reviewer for this comment. In strongly confined systems, the substantial overlap of exciton wavefunctions enhances both direct Coulomb and exchange interaction (Phys. Rev. B 87, 041304 (2013)). As shown previously (Sci. China Chem. 66, 3628–3635 (2023)), strong Coulomb scattering in ultrasmall volumes can broaden the exciton-to-biexciton transition to ~280 meV FWHM, much larger than typical exciton–exciton interaction energies (a few to tens of meV), effectively masking any spectral shifts. We are reluctant to claim that nearly identical spectral features of S and S imply 1 2 negligible Coulomb effects, which might be physically wrong. As emphasized in our previous round of responses. The spectrum of S actually contains contributions from 1 both single-exciton and biexciton species. Our global fitting gives an “average” result that as time goes on the initial spectrum that contains both single-excitons and biexcitons (S ) 1 gradually evolves into a spectrum that is dominated by single-excitons (S ). Similarity between S and S might simply suggest that the biexciton 2 1 2 contribution in S is quite small. 1 Revision: In the revised SI, we add more discussion on page 11-12: “Note that the spectrum of S actually contains contributions from both single-exciton 1 and biexciton species. Our global fitting gives an “average” result that as time goes on the initial spectrum that contains both single-excitons and biexcitons (S ) 1 gradually evolves into a spectrum that is dominated by single-excitons (S ). Spectral 2 similarity between S and S might simply suggest that the biexciton contribution in S 1 2 1 is very small. Quantitative understanding the spectra of S and S requires knowledge 1 2 on the complex Coulombic interactions in these ultrasmall clusters, which is beyond the scope of this work.” 3. Given that the observation of the AT effect in this system relies on disentangling a complex, convoluted population signal at very early times (<200 fs), the reproducibility of the analysis is very important. While in my view the data supports the authors' conclusions, there is a possibility that other transient effects (currently unknown to me or the authors) could produce similar spectral signatures. Action: In line with the open science mandates of Nature Communications, I strongly recommend that the authors deposit the raw power-dependent Transient Absorption data (particularly for the resonant case) and the analysis scripts used for the global fitting/deconvolution in a public repository. In my view, allowing the community to independently access the data and analyses will greatly enhance the impact and longevity of these findings. Response: We thank the reviewer for emphasizing the importance of reproducibility and data accessibility. In accordance with the open science policies of Nature Communications and in response to the reviewer’s suggestion, we will provide all the raw power-dependent transient absorption data in the final submission. Regarding the global fitting and deconvolution analysis, these were performed using a custom-built LabVIEW program that has been developed and refined by the co-author over several years. This software is not script-based, and translating the full functionality, workflow, and operational logic into standalone scripts would require a substantial effort and may not reliably reproduce the original fitting performance due to its highly integrated and hardware-dependent structure. To maintain transparency, however, the fitting software is available for verification and inspection upon request from the co-corresponding author J.Z. (jingyizh@dicp.ac.cn). Response: We sincerely thank the reviewer for positive and encouraging assessment of our work. We appreciate the reviewer’s thoughtful comments and constructive suggestions that have helped improve the manuscript. Response: We sincerely thank the reviewer for positive and encouraging assessment of our work. We appreciate the reviewer’s thoughtful comments and constructive suggestions that have helped improve the manuscript."}, {"round": 3, "reviewer_comments": {"reviewer_1": "I believe my main concerns were addressed. I am happy for this paper to be published as is and I would like to congratulate the authors for the nice results and to thank them for their comprehensive clarifications."}, "author_response": ""}]

Decision letter