Fig. 1: Photon echoes in CsPbI3 nanocrystals.![\"Fig.](\"https:\/\/www.newsbeep.com\/ca\/wp-content\/uploads\/2026\/05\/41467_2026_73506_Fig1_HTML.png\")
The alternative text for this image may have been generated using AI.<\/p>\n
a Typical photoluminescence (PL) spectrum for excitation with photon energy of 2.33 eV. Vertical dashed line indicates the highest photon energy used in the four-wave mixing FWM experiment. The scale on top corresponds to the nanocrystal (NC) diameter. b Schematic representation of the experimental geometry, where the laser pulses hit the sample with wave vectors k1,\u00a0\u2009k2. The photon echo is detected along the direction 2k2\u00a0\u2212\u00a0k1 using a heterodyning technique by overlapping it with a reference pulse. Inset shows schematically a CsPbI3 perovskite nanocrystal. c Two-dimensional plot of the FWM electric field amplitude \\({{\\mathcal{E}}}_{{\\rm{FWM}}}^{*}\\) as function of delay between the two pulses \u03c412 and the reference time \u03c4ref. The photon echo (PE) signal forms at the time \u03c4ref\u00a0=\u00a02\u03c412. The amplitude of the PE shows oscillations during the initial evolution when \u03c412 is scanned. The signal is recorded in linearly co-polarized configuration. Photon energy h\u03bd\u00a0=\u00a01.736eV. d Decay of two-pulse photon echo amplitudes for excitation with different photon energies. The data are measured with ps excitation pulses, which gives better spectral resolution (see also Supplementary Note 3 for more details). Black dashed lines are fits with exponential functions, from which the labeld exciton coherence times T2 are extracted. Temperature T\u00a0=\u00a02 K.<\/p>\n
We perform transient four-wave mixing (FWM) experiments in transmission geometry as shown schematically in Fig.\u00a01<\/a>b. The NCs are resonantly excited with a sequence of two 120 fs laser-pulses with photon energies h\u03bd\u00a0<\u00a01.76 eV in the low energy flank of the PL band. The sample temperature is kept at T\u00a0=\u00a02 K. The electric field amplitude of the FWM signal \\({{\\mathcal{E}}}_{{\\rm{FWM}}}(t)\\) is resolved in time using heterodyne detection where the signal field is temporally overlapped with a strong reference pulse with amplitude \\({{\\mathcal{E}}}_{{\\rm{ref}}}(t-{\\tau }_{{\\rm{ref}}})\\) (see \u201cMethods\u201d and refs. 56<\/a>,57<\/a>,58<\/a>,59<\/a> for details). Here, time t\u00a0=\u00a00 corresponds to excitation with the first pulse, while \u03c4ref is the delay of the reference pulse with respect to the first pulse in the excitation sequence. The resulting FWM signal is shown in Fig.\u00a01<\/a>c, which gives a two-dimensional plot of \\({{\\mathcal{E}}}_{{\\rm{FWM}}}\\) as function of the delay time between the first and second pulses \u03c412 (vertical axis) and the reference delay time \u03c4ref (horizontal axis). The FWM signal demonstrates the expected peak centered at time \u03c4ref\u00a0=\u00a02\u03c412, corresponding to emission of a photon echo (PE) from the NC ensemble60<\/a>. This behavior arises from the inhomogeneous broadening of the optical transitions resulting from fluctuations of the NC size as demonstrated for similar halide perovskite NCs35<\/a>,46<\/a>. Interestingly, during the first tens of ps the PE amplitude shows pronounced high frequency oscillations with increasing \u03c412 which we will discuss below.<\/p>\n On a longer time scale, the amplitude of the photon echo decays exponentially with increasing delay time \u03c412 as shown in Fig.\u00a01<\/a>d. Using exponential decay fits, shown with the dashed lines, we obtain T2\u00a0=\u00a0150\u00a0\u2212\u00a0330\u2009ps for photon energies in the range 1.724\u00a0\u2212\u00a01.765 eV, showing a smooth decrease with increasing h\u03bd (for details on the spectral dependence of T2 see Suplementary Section 3). The population relaxation time T1, measured in three-pulse experiments, is found to be T1\u00a0=\u00a0600\u00a0\u2212\u00a0800\u2009ps, and is comparable to the exciton lifetime measured by the time-resolved PL on a similar sample61<\/a>. Therefore, we conclude that the signal at \u03c4ref\u00a0>\u00a050\u2009ps is due to the zero-phonon transition with long-lived optical coherence time T2 and exciton recombination lifetime T1. If the coherent dynamics were governed only by the population decay we would expect the relation T2\u00a0=\u00a02T1 to hold, which is the case for excitons in self-assembled InGaAs quantum dots at T\u00a0=\u00a02 K62<\/a>. Here, we observe a different situation, where elastic scattering processes (pure dephasing) with a decay constant of \\({T}_{{\\rm{p}}}={[1\/{T}_{2}-1\/(2{T}_{1})]}^{-1}=330\\) ps mainly govern the exciton coherence. Nevertheless, to the best of our knowledge, the homogeneous linewidth of the zero-phonon exciton of \u03932\u00a0=\u00a02\u210f\/T2\u00a0=\u00a04.8\u03bceV demonstrated here has a record low value for perovskite nanocrystals (see Supplementary Table\u00a01<\/a> in Supplementary Note\u00a03<\/a>).<\/p>\n Coherent exciton-phonon dynamics<\/p>\n As follows from Fig.\u00a01<\/a>c for short delay times \u03c412 \u2272 10 ps, oscillations of the PE signal with multiple frequencies are observed. For deeper insight into the origin of the oscillations, we analyze polarization-resolved PE signals. These results are summarized in Fig. 2<\/a>a\u2013c. We employ two configurations, where the first and second pulses are linearly co- or cross- polarized, while the detection polarization in all cases coincides with that of the first pulse (see \u201cMethods\u201d). The PE amplitudes A\u2225 and A\u00d7 as functions of \u03c412 in co-(\u2225) and cross-( \u00d7 ) polarized configurations are shown in Fig. 2<\/a>a. The oscillations can be grouped into two categories: slow oscillations with a period of 1\u201310\u2009ps and fast oscillations with a period of less than 1\u2009ps.<\/p>\n Fig. 2: Quantum beats of exciton-polarons. a Initial range of the PE dynamics in the co-linear (\u2225) and cross-linear ( \u00d7 )polarization configurations shown with blue and red lines, respectively. Photon energy h\u03bd\u00a0=\u00a01.746 eV. The amplitude of the \u00a0\u00d7\u00a0signal is multiplied by two for clarity. b Dynamics of the polarizaton-dependent combination \u03c1 (black) and polarization-independent one \u03a3\u00a0(yellow) as defined by Eqs. (2<\/a>) and (3<\/a>), respectively. The dashed blue curve is a fit using the exciton-polaron model with Eq. (5<\/a>) with the following parameters for two phonon modes: \u210f\u03a91\u00a0=\u00a03.2\u2009meV, SHR1\u00a0=\u00a00.097,\u00a0\u2009\u03c4ph1\u00a0=\u00a05.1\u2009ps; \u210f\u03a92\u00a0=\u00a05.1\u2009meV, SHR2\u00a0=\u00a00.032,\u00a0\u2009\u03c4ph2\u00a0=\u00a010\u2009ps. c Fast Fourier transform (FFT) amplitude spectra of \u03c1 (black), \u03a3 (yellow), and the fit curves in panel b. Vertical dashed lines mark the peaks corresponding to optical phonons. d Raman spectrum measured at photon energy h\u03bd\u00a0=\u00a01.734 eV. Vertical dashed lines indicate the positions of peaks corresponding to the optically active optical phonon modes that couple most strongly to the exciton.<\/p>\n It is evident that the fast oscillations are in phase and nearly identical for both polarization configurations. We will show below that these oscillations correspond to quantum beats between exciton-polaron states. By contrast, the slow oscillations are out of phase. The co-polarized A\u2225-signal starts at its maximum value, while the A\u00d7-signal starts from zero. Subsequently, the minimum of the slow oscillations in \u2225-configuration around \u03c412\u00a0=\u00a05\u2009ps corresponds to a maximum in the \u00a0\u00d7-configuration. This out-of-phase behavior is related to quantum beats between the orthogonally polarized bright exciton states46<\/a>. In perovskite nanocrystals, the bright exciton fine structure comprises three linearly polarized states17<\/a>,63<\/a>,64<\/a>, split by the energies \u03b41 and \u03b42 as shown in Fig.\u00a03<\/a>a. This energy level scheme is similar to the fine structure splitting in self-assembled quantum dots65<\/a>,66<\/a> and in II-VI colloidal NCs67<\/a>,68<\/a>,69<\/a>. We do not consider dark states with total angular momentum zero, as they cannot be optically addressed and, owing to the slow energy relaxation at low temperatures, do not contribute to the coherent dynamics. In order to isolate the oscillations due to the fine structure splitting in the data, we model the expected polarization dependence of the PE signal, accounting also for the random orientation of the nanocrystals (for details see Supplementary Note\u00a04<\/a>). We obtain the following equations for the amplitude of the phonon echo signal in two polarization configurations: <\/p>\n $$\\begin{array}{rcl}{A}_{\\parallel } &=& \\left[\\frac{18}{15}+\\mathop{\\sum }\\limits_{i}\\frac{4}{15}\\cos \\left({\\delta }_{i}{\\tau }_{12}\\right){e}^{-\\frac{2{\\tau }_{12}}{{t}_{i}^{*}}}\\right]{\\Psi }_{0}({\\tau }_{12}),\\\\ {A}_{\\times } &=& \\left[\\frac{6}{15}-\\mathop{\\sum }\\limits_{i}\\frac{2}{15}\\cos \\left({\\delta }_{i}{\\tau }_{12}\\right){e}^{-\\frac{2{\\tau }_{12}}{{t}_{i}^{*}}}\\right]{\\Psi }_{0}({\\tau }_{12}),\\end{array}$$<\/p>\n \n (1)\n <\/p>\n where the index i\u00a0=\u00a01,\u00a02,\u00a03 corresponds to the beats between the three optical transitions (\u03b43\u00a0=\u00a0\u03b41\u00a0+\u00a0\u03b42), and \\({t}_{i}^{*}\\) is the dephasing time of the beats caused by the dispersion of the splitting energies \u210f\u03b4i in the ensemble. The common factor \u03a80(\u03c412) is the same for both polarization configurations and will be discussed below, see also Supplementary Note\u00a04<\/a>. Following Eqs. (1<\/a>) it is possible to distinguish between quantum beats related to the exciton fine structure and other polarization insensitive contributions to the PE signal by introducing the polarization contrast <\/p>\n $$\\rho=\\frac{{A}_{\\parallel }-3{A}_{\\times }}{{A}_{\\parallel }+2{A}_{\\times }},$$<\/p>\n \n (2)\n <\/p>\n and the polarization sum <\/p>\n $$\\Sigma=\\frac{{A}_{\\parallel }+2{A}_{\\times }}{2}={\\Psi }_{0}({\\tau }_{12})\\,.$$<\/p>\n \n (3)\n <\/p>\n It follows that the temporal evolution of \u03c1 exhibits oscillations at frequencies corresponding to the energy splittings \u03b4i of the exciton fine structure and is independent of \u03a80(\u03c412). In contrast, the polarization sum \u03a3 is given by the intrinsic coherent dynamics \u03a80(\u03c412) only.<\/p>\n Figure\u00a02<\/a>b shows the dynamics of these quantities, calculated from the data in Fig.\u00a02<\/a>a. In full accord with our expectations we obtain that the high-frequency oscillations are absent in the dynamics of \u03c1, while the low-frequency oscillations are still present. On the other hand, the low-frequency oscillations vanish in the \u03a3 transient in contrast to the high-frequency components. This becomes even more clear from the fast Fourier transform (FFT) spectra of \u03c1 and \u03a3 shown in Fig.\u00a02<\/a>c. From the fit of the \u03c1 transient we evaluate \u03b41\u00a0=\u00a00.25\u2009meV and \u03b42\u00a0=\u00a00.55\u2009meV, which are in agreement with the exciton fine structure splitting evaluated by two-dimensional Fourier spectroscopy on CsPbI3 NCs of similar size: cube-shaped with side length of about 9 nm46<\/a>. A detailed discussion of the fine exciton structure combining pump-probe and PE studies will be published elsewhere. In what follows, we focus on the dynamics of \u03a80(\u03c412), which is independent of the spin level structure of the exciton.<\/p>\n Fig. 3: Energy level schemes. a Energy levels of the bright exciton fine structure. b Energy level structure of the excitons interacting with optical phonons. We assume this level structure for each phonon mode, independent of the exciton polarization. The probability of the diagonal transitions is proportional to the Huang-Rhys factor SHR. \u03b3ph is the phonon decay rate and \u03b30 is the exciton decay rate, dominated by the radiative decay. c Configurational coordinate diagram for a given phonon mode with adiabatic potentials in the ground nanocrystal (NC) state and a NC state with an exciton shown as solid parabolas. The dashed parabola indicates the adiabatic potential for a NC state with an exciton in the case when the exciton-phonon coupling is neglected. The solid parabolas are separated by the energy difference EX corresponding to the energy of zero-phonon optical transition. \u210f\u03a9 denotes the energy of the optical phonon.<\/p>\n The FFT spectrum of \u03a3, shown by the yellow line in Fig.\u00a02<\/a>c, exhibits several spectrally narrow features. These features are consistent with the peaks in the Raman spectrum in Fig.\u00a02<\/a>d. The latter originates from light scattering on optically active phonons. The features at energies below 0.5\u2009meV are attributed to confined acoustic phonons70<\/a>. Below we focus on the two most prominent features marked by the vertical dashed lines with energies of \u210f\u03a91\u00a0=\u00a03.2\u2009meV and \u210f\u03a92\u00a0=\u00a05.1\u2009meV, corresponding to the energies of optically active optical phonons in the vicinity of the \u0393 point. Indeed optical phonon modes with energies close to 3 and 5 meV in CsPbI3 were observed in ref. 23<\/a> and calculated in ref. 8<\/a>. Optical phonons with similar energies were also detected by Raman spectroscopy in other lead halide perovskites such as FAPbI371<\/a>,72<\/a>, CsPbBr321<\/a>, and CsPbCl319<\/a>,73<\/a>. Thus, we conclude that the high frequency THz oscillations in the two-pulse coherent optical response are due to the coherent evolution of the coupled, hybridized exciton-phonon system.<\/p>\n Exciton-polaron quantum beats<\/p>\n In order to describe the quantum dynamics of the coupled exciton-phonon system we consider the structure of the optically excited energy levels in a single NC, which requires the involvement of exciton-polaron states. To that end, we introduce the four-level scheme shown in Fig.\u00a03<\/a>b which comprises the state with zero excitons and zero phonons in the NC \\(\\left|0\\right\\rangle\\), the state with zero excitons and one optical phonon \\(\\left|{0}^{\\prime} \\right\\rangle\\), the ground state exciton-polaron (i.e., the state with one exciton and no phonons, but with account of lattice relaxation) \\(\\left|X\\right\\rangle\\), and the state with one exciton and one phonon \\(\\left|X{\\prime} \\right\\rangle\\) (excited exciton-polaron state, see details in the Supplementary Note\u00a05<\/a>). Due to the relatively small Huang-Rhys factor one may neglect the modes with many phonons. We neglect the exciton fine structure splitting to concentrate on the derivation of \u03a80(\u03c412) in Eq. (1<\/a>). Initially, we consider the exciton coupling to a single phonon mode in a NC. At the next stage, to compare with the experimental data, we perform summation over different phonon modes \u03a9i where we neglect the interaction between them <\/p>\n $${\\Psi }_{0}({\\tau }_{12})=\\mathop{\\sum }\\limits_{i}{\\Psi }_{0}({\\tau }_{12};{\\Omega }_{i})\\,.$$<\/p>\n \n (4)\n <\/p>\n The optical transitions between all energy levels are allowed using the same polarization. In the exciton-polaron model74<\/a>,75<\/a>, the transition probability amplitude is proportional to the product of the dipole matrix element and the overlap of the wavefunctions associated with vibronic modes shifted due to polaron formation (see Fig.\u00a03<\/a>c and Supplementary Note\u00a05<\/a>), which depends on the strength of the exciton-phonon interaction given by the Huang-Rhys factor SHR (Condon principle). The pulse duration is assumed to be short as compared with all other characteristic times in the system, i.e., we can assume the \u03b4-pulse limit. This is justified because the laser pulse duration is \u03c4d\u00a0\u2248\u00a0120 fs (see \u201cMethods\u201d) and the fit to experimental data gives relaxation times in the order of 10\u2009ps. The spectral width of the laser pulse \u00a0\u2248\u00a00.44h\/\u03c4d is about 15\u2009meV, which is larger than the phonon energies \u210f\u03a9i. Therefore, we assume that all four transitions between states \\(\\left|0\\right\\rangle,\\,\\left|{0}^{\\prime} \\right\\rangle\\) and \\(\\left|X\\right\\rangle,\\,\\left|X^{\\prime} \\right\\rangle\\) are covered by the spectral width of the laser. Although multiphonon states (e.g., 2\u210f\u03a91) lie within the laser spectral bandwidth, they can be neglected owing to the relatively small Huang-Rhys factor. We stress that the observation of pronounced quantum beats and the applicability of the model strongly relies on the long exciton coherence times T2, well resolved phonon frequencies, and intermediate Huang-Rhys factors. The choice of material is particularly important because in bromides, the optical phonon modes are relatively broad22<\/a>,76<\/a>, which may lead to rapid dephasing of excited exciton-polaron states.<\/p>\n The anharmonicity of optical phonon modes, critically important at room temperature30<\/a>,52<\/a>,77<\/a>, is known to have a minor effect at cryogenic temperatures72<\/a>. Even though the signal contains peaks at double phonon frequencies, this is an internal feature of nonlinear FWM, being present in the optical response of the sample due to interference between the \\({0}^{\\prime} \\to X\\) and \\(0\\to X^{\\prime}\\) transitions (see Supplementary Note\u00a05<\/a>). Also, we neglect the possible difference of phonon energy in the crystal ground and optically excited states as we do not find corresponding frequency changes in our experimental data and in previous reports in other perovskites71<\/a>,72<\/a>. Similar assumptions were made in the model developed for describing 2D spectroscopy in ref. 78<\/a>.<\/p>\n In the exciton-polaron model, we trace the evolution of the density matrix components which contribute to the PE. The result arises from the coherent off-diagonal terms and all levels contribute to the result in first order in the Huang-Rhys factor SHR, which quantifies the exciton-phonon interaction. Solution of the Lindblad equation in Supplementary Note\u00a05<\/a> gives the following expression for the amplitude of the phonon echo signal from a NC with single phonon mode \u03a9i<\/p>\n $${\\Psi }_{0}({\\tau }_{12};{\\Omega }_{i})=\t {{\\rm{e}}}^{-(1+{S}_{{\\rm{HR}}}){\\gamma }_{0}{\\tau }_{12}}\\left[1+{S}_{{\\rm{HR}}}{{\\rm{e}}}^{-({\\gamma }_{{\\rm{ph}}}-{S}_{{\\rm{HR}}}{\\gamma }_{0}){\\tau }_{12}}\\right.\\\\ \t+{S}_{{\\rm{HR}}}\\cos ({\\Omega }_{i}{\\tau }_{12}){{\\rm{e}}}^{-\\frac{{\\gamma }_{{\\rm{ph}}}-2{S}_{{\\rm{HR}}}{\\gamma }_{0}}{2}{\\tau }_{12}}\\left[2+{{\\rm{e}}}^{-{S}_{{\\rm{HR}}}{\\gamma }_{0}{\\tau }_{12}}+{{\\rm{e}}}^{-{\\gamma }_{{\\rm{ph}}}{\\tau }_{12}}\\right]\\\\ \t+\\left.{S}_{{\\rm{HR}}}\\cos (2{\\Omega }_{i}{\\tau }_{12}){{\\rm{e}}}^{-({\\gamma }_{{\\rm{ph}}}-{S}_{{\\rm{HR}}}{\\gamma }_{0}){\\tau }_{12}}\\right].$$<\/p>\n \n (5)\n <\/p>\n Here T2,0\u00a0=\u00a01\/\u03b30 is the coherence time associated with the zero-phonon optical transition, and \u03c4ph\u00a0=\u00a01\/\u03b3ph is the lifetime of the optical phonon. In the simplified model we do not include other sources of decoherence. The analytical expression fully supports the experimental observations for \u03a3\u00a0=\u00a0\u03a80(\u03c412) shown in Fig.\u00a02<\/a>b. The first term in Eq. (5<\/a>) corresponds to the long-lived zero-phonon coherence which decays exponentially with the time T2\u00a0=\u00a0T2,0\/(1\u00a0+\u00a0SHR) \u223c300\u2009ps, as evaluated above. The last two terms correspond to high frequency oscillations at the single and double frequency of the phonon mode \u03a9i, respectively. They appear due to quantum interference of excitations of different exciton-polaron states, i.e., due to quantum beats of exciton-polarons. These oscillations are superimposed on the long-lived signal and decay with a shorter time \u03c4ph. The relative amplitude of the oscillatory signal and the long-lived plateau allows one to measure the Huang-Rhys factor SHR.<\/p>\n Using Eqs. (4<\/a>) and (5<\/a>) for two independent phonon modes with \u210f\u03a91\u00a0=\u00a03.2\u2009meV and \u210f\u03a92\u00a0=\u00a05.1\u2009meV we obtain excellent agreement with the experimental data. The phonon frequencies are taken from the peak positions in the Fourier spectrum of Fig.\u00a02<\/a>c. The Huang-Rhys factors and phonon lifetimes are evaluated from the best fit to the experimentally measured transients, see Fig.\u00a02<\/a>b. We emphasize that in contrast to previous reports39<\/a>,40<\/a>,41<\/a>,42<\/a>,43<\/a>, our system demonstrates long-lived coherent dynamics, which is attributed to an exceptionally long zero-phonon exciton coherence (\u223c300\u2009ps) and a relatively long phonon lifetime (\u223c10\u2009ps). Here, we stress that the low-temperature regime with T\u00a0=\u00a02 K is essential, as the coherence of the zero-phonon excitons rapidly vanishes with increasing temperature. Furthermore, the optical phonons each are represented by well-defined frequencies due to their flat dispersion around the \u0393-point, as confirmed by the Raman spectrum in Fig.\u00a02<\/a>d, making CsPbI3 NCs embedded in a glass matrix particularly interesting for investigating coupled exciton-phonon dynamics. The peak widths in the Raman spectrum can be recalculated into phonon lifetimes using the following relation \u03c4ph\u00a0=\u00a02\u210f\/\u0394ER79<\/a>, where \u0394ER is the full width at half maximum in energy units. This yields phonon lifetimes of \u223c7\u2009ps and 11\u2009ps for the 3.2 and 5.1\u2009meV modes, respectively. These values are in excellent agreement with the decay times of the PE oscillatory signals. It should be noted that the Raman peaks in Fig. 2<\/a>d appear narrower than those in the FFT spectra in Fig.\u00a02<\/a>c, which originates from the difference between the power spectral and the amplitude spectral representations, respectively.<\/p>\n The proposed exciton-polaron model (i) provides an intuitive picture of the microscopic processes involved, (ii) has a simple analytical solution, and (iii) accounts explicitly for the coherence decay through simple equations. Equation (5<\/a>) gives a result that closely matches that in refs. 39<\/a>,80<\/a> and is in effect equivalent within the accuracy limits of the models. Note that a direct comparison between the results obtained by the two approaches is not straightforward: Eq. (5<\/a>) is obtained from the exact solution of the Lindblad equations for a four-level system, while ref. 80<\/a> considers the quasi-classical evolution of a complex system averaged over the phonon subsystem. For a detailed discussion and comparison of the two approaches see e.g., ref. 81<\/a>.<\/p>\n NC size dependence<\/p>\n Above, we discussed the PE dependence measured for the fixed laser pulse photon energy of h\u03bd\u00a0=\u00a01.746 eV and explained how the phonon parameters can be evaluated from the PE signal. However, the sample under study contains NCs with different sizes which can be selectively excited by tuning the laser photon energy. We measured PE transients at photon energies in the range of 1.72\u00a0\u2212\u00a01.76 eV. The photon energy can be recalculated into a NC diameter using the empirical fit \\(D=\\sqrt{16.93\/({E}_{{\\rm{X}}}-1.652)-4.31}\\) (where EX is the position of exciton peak in eV) to the results from ref. 51<\/a>, where the same sample was studied (sample #3, see also Supplementary Note\u00a02<\/a>). Figures\u00a04<\/a>a, b show the dependence of the evaluated Huang-Rhys factors SHR and phonon lifetimes \u03c4ph as function of the NC diameter, for phonon modes with energies \u210f\u03a91\u00a0=\u00a03.2\u2009meV and \u210f\u03a92\u00a0=\u00a05.1\u2009meV. We note that the frequencies of these modes do not depend on NC size within the accuracy of the experiment (see Supplementary Note\u00a06<\/a>). The Huang-Rhys factors for both phonon modes exhibit a clear size dependence, increasing with decreasing NC size. Correspondingly, the phonon lifetimes decrease with decreasing a. An increase in the electron-phonon coupling strength with decreasing NC size was previously reported for CdSe nanocrystals82<\/a>, PbS quantum dots83<\/a>, and perovskite nanocrystals, both for optical and acoustic phonons18<\/a>,21<\/a>. However, to the best of our knowledge, direct measurements of the associated phonon relaxation times have not been reported. It is also important to note that, in contrast to previous studies, our PE experiments selectively probe the coherent dynamics of each individual phonon mode.<\/p>\n Fig. 4: Nanocrystal (NC) size dependence. Dependence of a the Huang-Rhys factor, SHR, and b the phonon lifetime, \u03c4ph, for the two phonon modes with energies \u210f\u03a91\u00a0=\u00a03.2\u2009meV (blue dots) and \u210f\u03a92\u00a0=\u00a05.1\u2009meV (red dots) on NC diameter. Dashed lines in a are fits proportional to a\u22123. Dashed lines in b are guides to the eye. The horizontal error bar represents the uncertainty associated with the spectral width of the PE signal, corresponding to \u00a0\u00b1\u00a05\u2009meV. Vertical bars represent standard deviation.<\/p>\n In ref. 21<\/a>, the dominant mechanism of electron-phonon coupling is assigned to the optical deformation potential84<\/a>. In this case, the size dependence of the interaction can be estimated from the phonon normalization condition7<\/a> which leads to SHR \u223c a\u22123, where a is the NC radius. For completeness, let us discuss the size dependence of the interaction between charge carriers and optical phonons for the Fr\u00f6hlich mechanism following Takagahara85<\/a>. In the original work, the interaction of charge carriers with phonons is rewritten as potential for an electron in the field induced by the phonon mode polarization P(r). This potential is found from the Poisson equation \u22072\u03c6(r)\u00a0=\u00a04\u2009\u03c0\u2207 \u22c5 P(r) which leads to <\/p>\n $$\\varphi ({{\\bf{r}}}_{e})=\\int \\,d{\\bf{r}}\\,\\frac{{\\boldsymbol{\\nabla }}\\cdot {\\bf{P}}({\\bf{r}})}{| {\\bf{r}}-{{\\bf{r}}}_{e}| }=-\\int \\,d{\\bf{r}}\\,{\\bf{P}}({\\bf{r}})\\cdot {\\boldsymbol{\\nabla }}\\frac{1}{| {\\bf{r}}-{{\\bf{r}}}_{e}| }.$$<\/p>\n \n (6)\n <\/p>\n From Eq. (6<\/a>) the strength of electron-phonon interaction \u0394e \u223c a\u22121 which leads to \\({S}_{{\\rm{HR}}}^{e}={\\Delta }_{e}^{2}\/2 \\sim {a}^{-2}\\). This estimate is valid also for excitons in the strong confinement regime.<\/p>\n For weakly confined excitons, the change of exciton energy \u0394 is proportional to the difference of the electrostatic potential for electron and hole \u03c6(re)\u00a0\u2212\u00a0\u03c6(rh). It is found to be the sum of Eq. (6<\/a>) for electron and hole: <\/p>\n $$\\Delta \\sim \\varphi ({{\\bf{r}}}_{e})-\\varphi ({{\\bf{r}}}_{h})=\\int \\,d{\\bf{r}}\\,{\\bf{P}}({\\bf{r}})\\cdot {\\boldsymbol{\\nabla }}\\left(\\frac{1}{| {\\bf{r}}-{{\\bf{r}}}_{h}| }-\\frac{1}{| {\\bf{r}}-{{\\bf{r}}}_{e}| }\\right).$$<\/p>\n \n (7)\n <\/p>\n In the weak confinement regime when the exciton Bohr radius aB is small compared with the NC radius aB \u226a a, <\/p>\n $$\\frac{1}{| {\\bf{r}}-{{\\bf{r}}}_{h}| }\\approx \\frac{1}{| {\\bf{r}}-{{\\bf{r}}}_{e}| }+({{\\bf{r}}}_{e}-{{\\bf{r}}}_{h})\\cdot {\\boldsymbol{\\nabla }}\\frac{1}{| {\\bf{r}}-{{\\bf{r}}}_{e}| }.$$<\/p>\n \n (8)\n <\/p>\n Substituting Eq. (8<\/a>) into Eq. (7<\/a>) we obtain <\/p>\n $$\\varphi ({{\\bf{r}}}_{e})-\\varphi ({{\\bf{r}}}_{h})\\approx \\int \\,d{\\bf{r}}\\,{\\bf{P}}({\\bf{r}})\\cdot {\\boldsymbol{\\nabla }}\\left[({{\\bf{r}}}_{e}-{{\\bf{r}}}_{h})\\cdot {\\boldsymbol{\\nabla }}\\frac{1}{| {\\bf{r}}-{{\\bf{r}}}_{e}| }\\right].$$<\/p>\n \n (9)\n <\/p>\n As a result, in the weak confinement regime, the size dependence of this expression is estimated as <\/p>\n $$\\Delta \\sim \\varphi ({{\\bf{r}}}_{e})-\\varphi ({{\\bf{r}}}_{h})\\propto {a}^{3}\\cdot {a}^{-3\/2}\\cdot \\frac{{a}_{{\\rm{B}}}}{{a}^{3}}\\propto {a}^{-3\/2}.$$<\/p>\n \n (10)\n <\/p>\n Here, the first size factor comes from the integral, the second one from the normalization of the phonon mode, and the last one from the gradient of the expression in the square brackets. As a result the Huang-Rhys factor scales as SHR\u00a0=\u00a0\u03942\/2 \u223c a\u22123.<\/p>\n Thus, in the weak confinement regime both mechanisms, the deformation potential and the Fr\u00f6hlich interaction, lead to similar dependences of the Huang-Rhys factor on the NC size SHR \u223c a\u22123. In the strong confinement regime, the Fr\u00f6hlich interaction is expected to have a weaker size dependence, while the deformation potential mechanism should show the same scaling. The dashed lines in Fig.\u00a04<\/a> represent fits of the experimental data by the empirical relation SHR \u221d a\u22123. The experimental data in Fig.\u00a04<\/a>a even shows a steeper dependence of the Huang-Rhys factor on NC size, with pronounced deviation from the a\u22123 dependence, particularly for small NCs. This observation suggests that further analysis is required for smaller NCs to identify the mechanisms of electron-phonon interaction in perovskite NCs.<\/p>\n","protected":false},"excerpt":{"rendered":"Photon echo and long coherence of zero-phonon exciton The ensemble of CsPbI3 NCs is synthesized in a fluorophosphate…\n","protected":false},"author":2,"featured_media":700045,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[24],"tags":[49,48,1099,1100,37424,314,35708,66,34458],"class_list":["post-700044","post","type-post","status-publish","format-standard","has-post-thumbnail","category-physics","tag-ca","tag-canada","tag-humanities-and-social-sciences","tag-multidisciplinary","tag-nanoparticles","tag-physics","tag-quantum-dots","tag-science","tag-ultrafast-photonics"],"_links":{"self":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts\/700044","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/comments?post=700044"}],"version-history":[{"count":0,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts\/700044\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/media\/700045"}],"wp:attachment":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/media?parent=700044"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/categories?post=700044"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/tags?post=700044"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"Fig.](\"https:\/\/www.newsbeep.com\/ca\/wp-content\/uploads\/2026\/05\/41467_2026_73506_Fig2_HTML.png\")
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