Long Exciton Dephasing Time and Coherent Phonon Coupling in CsPbBr$_{2}$Cl Perovskite Nanocrystals

Fully-inorganic cesium lead halide perovskite nanocrystals (NCs) have shown to exhibit outstanding optical properties such as wide spectral tunability, high quantum yield, high oscillator strength as well as blinking-free single photon emission and low spectral diffusion. Here, we report measurements of the coherent and incoherent exciton dynamics on the 100 fs to 10 ns timescale, determining dephasing and density decay rates in these NCs. The experiments are performed on CsPbBr$_{2}$Cl NCs using transient resonant three-pulse four-wave mixing (FWM) in heterodyne detection at temperatures ranging from 5 K to 50 K. We found a low-temperature exciton dephasing time of 24.5$\pm$1.0 ps, inferred from the decay of the photon-echo amplitude at 5 K, corresponding to a homogeneous linewidth (FWHM) of 54$\pm$5 {\mu}eV. Furthermore, oscillations in the photon-echo signal on a picosecond timescale are observed and attributed to coherent coupling of the exciton to a quantized phonon mode with 3.45 meV energy.

composition of the NCs. This results from a "giant oscillator strength" in the intermediate confinement regime with an exciton Bohr radius of 5 -7 nm for cesium lead bromide-chloride (CsPbBr2Cl) NCs and a bright lowest triplet state manifold 5 . Furthermore, almost blinking-free single photon emission and marginal spectral diffusion have been reported for CsPbX3 quantum dots at low temperature. 3,6 These remarkable features make perovskite-type lead halide NCs a prime candidate for the observation of strong light-matter interaction, e.g., showing coherent cooperative emission 7 or creating exciton-polaritons by embedding them in high-finesse optical cavities, as shown for CVD-grown fully-inorganic lead halide perovskite nanowires 8,9 and nanoplatelets. 10 However, the timescale of such coherent coupling is limited by the exciton dephasing, which is still unknown for this material.
Transient four-wave-mixing (TFWM) spectroscopy is a powerful method allowing to directly measure the loss of quantum coherence characterized by a dephasing time T2. It has been applied on various materials such as ruby crystals 11 , atoms 12 , molecules 13,14 and semiconductor nanostructures. [15][16][17] In general, the dephasing of excitons in semiconductor materials is caused by elastic and inelastic scattering processes with phonons and charge carriers, and by radiative population decay. 18 We studied the exciton dephasing and population dynamics using three-pulse degenerate TFWM spectroscopy on an ensemble of CsPbBr2Cl NCs. Previous measurements on colloidal CdSe-based NCs 19,20 and nanoplatelets 21 revealed a strong dependence of the 2 time on the material and its shape and size. The investigated cubic CsPbBr2Cl NCs with edge lengths of 10 ± 1 nm were synthesized as discussed in the Supplementary Information (SI), and possess the 3D-perovskite orthorhombic crystal structure (Pnma space group), shown in Figure 1a. Single quantum dot (QD) spectroscopy at cryogenic temperatures revealed that the emission of individual CsPbBr2Cl NCs exhibits a PL full-width at half-maximum (FWHM) below 1 meV, and a fine structure with an average energy splitting around 1 meV. 3,22 The exciton decay, measured at 5 K using non-resonant excitation, is mostly radiative with decay times of 180 -250 ps. This is 1000 times faster than in CdSe/ZnS QDs 23 at cryogenic temperatures, and attributed to high oscillator strength due to larger exciton coherence volume, and the absence of a low-energy dark state. 5 We performed TFWM experiments on films prepared by drop-casting a solution of NCs and polystyrene in toluene on c-cut quartz substrates (see SI). At room-temperature, the PL emission (see Figure 1c) is centered at a photon energy of 2.63 eV, and exhibits a Stokes shift of about 70 meV with respect to the ground-state exciton absorption resonance. At 5 K, the PL emission redshifts to 2.54 eV, which is a known feature of lead-based semiconductor NCs like PbS and PbSe 24 , and the PL FWHM decreases from 85 meV to 20 meV. The TFWM experiments have been performed by resonant excitation of the NCs at 2.6 eV with femtosecond pulses (120 fs intensity FWHM) from the second harmonic of a Ti:Sapphire oscillator with 76 MHz repetition rate (for details of the experimental setup see Naeem et al.,ref. 21). The first excitation pulse (P1) with wavevector 1 induces a coherent polarization of the emitters in the inhomogeneous sample, which is then subject to dephasing. After a time delay 12 , a second pulse (P2) converts the polarization into a population density grating. The third pulse (P3), that arrives on the sample after a time delay 23 , is diffracted by the density grating, creating a FWM signal with a wavevector of = 3 + 2 − 1 (refs. 20,21). To investigate the exciton population dynamics, we set the time delay between the first and the second pulse to zero ( 12 = 0 ps), and measure the FWM signal as a function of the time delay 23 . We use a spatial selection geometry to suppress the transmitted excitation pulses, and then further discriminate the FWM signal from the exciting pulses using a heterodyne technique, in which the pulse train Pi is radio-frequency shifted by i (i=1, 2, 3), resulting in a frequency-shifted FWM field which is detected by its interference with a reference pulse. 21 In Figure 2a, the measured FWM field amplitude (black) and phase (blue) at 5 K with their respective fits are shown. We fit amplitude and phase with a bi-exponential response function to quantify the population dynamics with decay rates Γ 1 > Γ 2 , as explained in the SI. Superimposed onto the population decay, we observe in the initial dynamics damped oscillations with a period of about 1.2 ± 0.1 ps, which we interpret as coherent phonon interactions, as we will discuss in more detail further below.
At 5 K, the FWM field amplitude decays with two distinct time constants. The fast decay time 1 = 28.2 ± 0.8 ps with a relative amplitude of 1 1 + 2 = 0.89 ± 0.07 (see Figure S2 in the SI) corresponds to a decay rate Γ 1 = 1 1 = 35.5 ± 1.0 ns -1 , i.e. a linewidth of ℏΓ 1 = 23.4±2 µeV ( Figure 2b). This rate is independent of excitation power (see Figure S3 in the SI), and we note that the decay rate is higher compared to non-resonantly excited PL. 5  The second decay component is four orders of magnitude slower, Γ 2 = 0.037 ± 0.009 ns -1 at 5 K, which is below the repetition rate in the experiment, and has a low weight of about 10% of the first one. In contrast to the first decay component, Γ 2 is increasing with temperature, as shown in Figure 2c. We assign this component to trap or defect states present in a small fraction of the NCs, increasing their decay rate with temperature by thermal activation. The relative amplitude of the components is temperature independent within error (see SI). The highest excitation density is estimated to excite up to 0.08 excitons per excitation pulse per NC, ruling out significant multiexciton effects.
The dephasing time can be extracted from the decay of the photon echo, which we measured using three-pulse FWM spectroscopy in a heterodyne detection scheme (see SI). We scan the time delay 12 between the first and the second pulse, while choosing a positive time delay 23 = 1 ps to avoid instantaneous non-resonant non-linearities. 21 The photon echo is then emitted at time 12 after the third pulse P 3 , as depicted in the inset of Figure 3a. The time-integrated FWM field amplitude as a function of delay time 12 is shown in Figure 3a for various temperatures. The FWM amplitude shows a bi-exponential behavior up to 41 K. For higher temperatures, it can be described with a fast mono-exponential decay. The initial fast decay of the amplitude proportional to exp (−2 1 12 ) in the time domain corresponds to a linewidth 2ℏ 1 = 4.37 ± 0.16 meV at 5 K.
It is attributed to phonon-assisted transitions and a quantum beat of the fine-structure split states, which show a distribution of splittings in the meV energy range as they vary from NC to NC. Since the exciton-exciton interaction is significant between any of the states, the density gratings of the states are adding up, constructively interfering at 12 = 0, and the beat with a wide distribution of frequencies results in a decay over the timescale of the inverse splitting. Assuming we excite the three bright states uniformly, a decay of the signal by a factor of three over the timescale of about a picosecond, given by the inverse energy splitting, is expected. Additionally, phonon-assisted transitions will contribute, as observed in other 3D confined systems. 19,20,25 However, the large extension of the excitons inside the individual NCs, which is the origin of the giant oscillator strength, is reducing their weight in the signal. This is confirmed in low-temperature single NC PL spectra, which do not show significant phonon-assisted emission, with an estimated zero phonon line (ZPL) weight of 0.93 (see SI). At 5 K, we find a linewidth of 2ℏ 1 = 4.37 ± 0.16 meV, which increases slowly as a function of temperature ( Figure 3b). In general, the homogeneous linewidth of each fine-structure transition in the spectral domain is composed of a broad acoustic phonon band that corresponds to the fast initial dephasing, which is superimposed on a sharp Lorentzianshaped ZPL, corresponding to the long exponential dephasing in the time domain. 25 From the second decay component of the photon echo, we can therefore deduce the ZPL width 2ℏ 2 = 54 ± 5 μeV, corresponding to a dephasing time 2 = 24.5 ± 1.0 ps at 5 K. In PL measurements of single NCs at 5 K, linewidths of typically a few hundred μeV are found for CsPbBr2Cl NCs 3 (see SI). The value obtained by FWM is consistent with this, considering that spectral diffusion is typically affecting single NC PL. 26 The temperature dependence of the homogeneous linewidth is plotted in Figure 3c. The solid red line is a temperature-activated fit of the ZPL width using constant up to 10 K and then starts to decrease with increasing temperature. As discussed above we attribute the initial decay of the photon echo mostly to an overdamped beat between the finestructure states, so that the reported nominal ZPL weight is lower than the real ZPL weight of the individual bright transitions. For equal weights of three transitions in the fine-structure beat, the amplitude will decay by a factor of three. Taking into account this decay results in a ZPL weight of 0.63 at 5 K. If we furthermore assume that the upper two fine-structure states are dephasing fast due to phonon-assisted transitions to the lowest state, there is an additional decay by a factor of three. Taking all these corrections into account results in a ZPL weight of 0.91 at 5 K, as is shown in Figure 3d on the right y-axis. This weight is consistent with the ZPL weight 0.93 extracted from single NC PL (see SI). The overall temperature dependence of the corrected ZPL weight is similar to the one observed for epitaxial InGaAs QDs. 27 In CdSe colloidal quantum dots 19 Figure 4b as a function of time delay 12 , exhibiting an oscillating behavior. The red curve displays a fit with a damped squared-sine function. From this, an oscillation period of 1.22 ± 0.02 ps is obtained, which concurs with the period of the initial oscillations in the density decay. We therefore attribute these oscillations to coherent exciton-phonon coupling, resulting from the modulated polarization as a function of the harmonic nuclei displacement, as previously reported for CdSe 28,29 and PbS 30 NCs. The resulting vibrational energy of 3.45 ± 0.14 meV is in good agreement with the measured phonon energies of phonon-assisted transitions in single NC PL measurements (see SI), which have been attributed to TO-phonon modes in bulk CsPbCl3 31 and for individual CsPbBr3 NCs. 22 The oscillation damping could be due to decay of the phonon mode decay into acoustic phonons 29 and inhomogeneous broadening of the phonon mode in the QD ensemble, as observed in FAPbBr3 NCs. 32 The three-pulse photon-echo signal ( 12 , 13 ) in the direction = − 1 + 2 + 3 of an inhomogeneously broadened two-level system coupled to a single harmonic mode of angular frequency can be calculated as: 29,33 ( 12 , 13 ) ∼ exp[−4Δ 2 ( ( ) + 1) ⋅ (1 − cos ( 12 ))(1 − cos ( 13 ) For better comparison with experiment, the calculated photon-echo signals for the experimental time delays are plotted in Figure 4d and Figure S5 in the SI, from which good agreement with measurements can be inferred.
In conclusion, we have investigated the coherence and density dynamics in fully inorganic CsPbBr2Cl NCs at cryogenic temperatures. Using three-beam FWM, we obtain a dephasing time and a density decay time of several tens of picoseconds at 5 K. Furthermore, we find excitation of

Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Notes
The authors declare no competing financial interest.   1 ) was swiftly injected. The reaction was stopped after 10 seconds by immersing the flask into a water-bath. The solution was centrifuged (4 min, 13,750 g) and the supernatant discarded. Hexane (0.3 ml) was added to the precipitate to disperse the NCs, and the mixture was then centrifuged again. The supernatant was collected separately, and 0.9 mL toluene was added. The NCs were precipitated by adding 0.24 mL acetonitrile and centrifuged (3 min, 6,740 g). The obtained precipitate was re-dispersed in 2 mL toluene and filtrated. For the sample preparation, we added 5 m% polystyrene in toluene in a 1:2 ratio to the solution of nanocrystals and immediately drop-casted on c-cut quartz substrates. The film had a thickness of 17 μm, and showed a certain degree of agglomeration, as can be seen in Figure S1. However, scattering was not an issue in the alignment of the box geometry in threepulse FWM (see SI: Transient Four-Wave Mixing and Heterodyne Detection). Figure S1. Fluorescence (left, excitation 410 -440nm) and transmission (right, condenser with numerical aperture NA=0.8) microscope image using a 100× NA=1.3 oil immersion objective.

Transient Four-Wave Mixing and Heterodyne Detection
The homogeneous linewidth of the spectral absorption is inversely proportional to the dephasing time 2 . In an ensemble measurement, the absorption linewidth is inhomogeneously broadened, thus making it impossible to deduce the microscopic 2 time from the absorption lineshape. Here, we make use of the third-order non-linearity of a material and perform four-wave mixing in the transient coherent domain after pulsed excitation to measure the 2 time in presence of inhomogeneous broadening. For the heterodyne detection scheme, the pulse train of the excitation laser is divided into the excitation pulses and a reference pulse. In the transient degenerate threebeam FWM configuration, three excitation pulses resonant to the absorption of CsPbBr2Cl ( = 2.6 eV) with a repetition rate Ω 2 = 76.11 MHz are used. To distinguish between the different non-linear orders, a so-called box geometry of the three excitation beams is used. According to the phase-matching conditions (i.e., momentum conservation), the third-order nonlinear signal is emitted in the direction = 3 + 2 − 1 , and is spatially selected by means of an iris. The signal is detected in a balanced heterodyne detection scheme, where the signal interferes with a reference beam at a beam-splitter, and the intensities of the two resulting beams are measured with two photodiodes. The photo-current is proportional to the square of the interfering incoming complex fields of the reference and signal pulse train. Furthermore, a frequency selection scheme allows to discriminate the order of the non-linear polarization. Hereby, the optical frequencies are slightly shifted by radio-frequency amounts Ω using acousto-optic modulators. Using pulse trains that exhibit controlled phase variations given by − Ω , the FWM signal can be detected at the frequency Ω = Ω 3 + Ω 2 − Ω 1 − Ω using lock-in amplifiers. This frequency selection scheme together with the interferometric detection of the FWM field amplitude constitutes the heterodyne detection scheme. For further information about the heterodyne detection scheme and the optical setup, we refer to refs. 21,35 .
Single CsPbBr2Cl NC spectroscopy: Fine-structure splitting and phonon replica.
Knowing the emission properties of single NCs helps to interpret the results of the FWM experiments on ensembles of CsPbBr2Cl NCs. Single NC spectroscopy of lead halide perovskite nanocrystals reveals a bright triplet emission with three orthogonal fine-structure states. On average, the fine-structure splitting for three emission peaks is Δ = 1.15 ± 0.26 meV with a large distribution ranging from several hundred eV up to meV. 5 In Figure S1, a spectrum of a single CsPbBr2Cl NC at 5 K is shown (see ref. 5 for details on the experiment) that exhibits two emission peaks with a fine-structure splitting of 2 meV. Additionally, we observe two phonon replica, red-shifted by 3.1 and 6.6 meV with Huang-Rhys factors of 0.038 and 0.033, respectively. In literature, these phonon replica were attributed to TO-phonon replica. 22 Figure S2. Single CsPbBr2Cl NC PL measurement. The nanocrystal exhibits a fine structure with two emission peaks split by Δ = 2.0 meV. Furthermore, the phonon replica at Δ 1 = 3.1 meV and Δ 2 = 6.6 meV are observed with Huang-Rhys factors of 1 = 0.038 and 2 = 0.033, respectively, resulting in a ZPL weight of 0.93.

Heterodyne-Detected Four-Wave Mixing: Complex Fit to the Exciton Population Dynamics
For a quantitative analysis, the population decay data is fitted with the complex multi-exponential response function, as explained in detail in ref. 35 : Here, , and are amplitude, phase and decay time of the n-th decay process. is a nonresonant instantaneous component to account for effects like two-photon absorption and Kerr effect. The above equation describes the FWM as a coherent superposition of exponential decay components with their respective phases, given by their relative effect on absorption and refractive index. Slow drifts of the setup, for example due to room temperature changes, can affect the relative phase of the reference and probe pulses over a timescale of minutes. The fitting procedure accounts for this phase drift with a prefactor − ( 0 + 0 ′ ) , corresponding to a linear timedependence of the phase, where is the real time during the measurement. The excitation pulse is taken into account by convoluting the response with a periodic Gaussian of full-width halfmaximum 2√ln(2) 0 in amplitude, given by the laser pulse auto-correlation. Furthermore, we include a pile-up of the signal ∼ ( − 1) −1 due to the finite repetition rate of the excitation pulses of 13 ns, relevant for decay components with lifetimes similar or longer than . Note that for lifetimes similar or longer than the modulation period of 1  1 − 2 = 160 ns, the pilep-up would be saturated; a regime which is not considered explicitly in the fit formula as it is not relevant to the presented fits. In the measurements, initial oscillations in the density decay can be seen. We model this by multiplying the decay components, excluding pile-up, by (1 + − ⋅ cos( )) with amplitude , damping rate and angular frequency of the oscillations . The fit function of the density decay uses two exponential decay processes, yielding ] .
Furthermore, we measure the excitation-power dependent density grating decay at 5 K, shown in Figure S3. Again, we fit the data with the above described complex fit. The resulting decay times are independent of the excitation density within error. From the complex fits, we obtain an average initial decay time 1 = 35.6 ± 1.5 ps. Figure S3. Excitation-power dependent FWM Field amplitude.
In Figure 2 in the main text, the temperature dependence of the first and the second decay component is shown. In Figure S4, we additionally plot the relative amplitude of the first decay component as a function of temperature, which stays almost constant within the temperature range measured. Figure S4. Relative amplitude of the first decay component of the bi-exponential complex fit as a function of temperature.
One can also use three exponential decay processes, resulting in a better fit at intermediate delay times around 100 ps. However, this leads to two time-constants in the 11 ps and 62 ps range, and is evidence for an inhomogeneous distribution of decay constants in the sample. We have therefore opted to use only two time-constants to extract the average decay, consistent with the analysis of the exciton dephasing.  For a better comparison, we plot the data of Figure 4a and d again in semi-log scale next to each other in Figure S5. As discussed in the main text, we use the model proposed by Mittelmann and Schoenlein et al. 28,29 to calculate the electric fields ( 12 , 23 )~√ ( 12 , 23 ) of the three-pulse photon-echo measurement in the presence of a single phonon mode. We added a damping term exp (− 12 ) exp (− 13 ) to the oscillations, and a rise-term