Ultrafast infrared nano-imaging of far-from-equilibrium carrier and vibrational dynamics

Ultrafast infrared nano-imaging has demonstrated access to ultrafast carrier dynamics on the nanoscale in semiconductor, correlated-electron, or polaritonic materials. However, mostly limited to short-lived transient states, the contrast obtained has remained insufficient to probe important long-lived excitations, which arise from many-body interactions induced by strong perturbation among carriers, lattice phonons, or molecular vibrations. Here, we demonstrate ultrafast infrared nano-imaging based on excitation modulation and sideband detection to characterize electron and vibration dynamics with nano- to micro-second lifetimes. As an exemplary application to quantum materials, in phase-resolved ultrafast nano-imaging of the photoinduced insulator-to-metal transition in vanadium dioxide, a distinct transient nano-domain behavior is quantified. In another application to lead halide perovskites, transient vibrational nano-FTIR spatially resolves the excited-state polaron-cation coupling underlying the photovoltaic response. These examples show how heterodyne pump-probe nano-spectroscopy with low-repetition excitation extends ultrafast infrared nano-imaging to probe elementary processes in quantum and molecular materials in space and time. Ultrafast infrared nano-imaging has enabled the study of nanoscale dynamics, but has been limited to probing short-lived carrier lifetimes. Here, the authors present pump-probe nano-spectroscopy with enhanced sensitivity to image both carrier and vibrational dynamics associated with long-lived excitations.

F unctional materials offer intriguing applications based on their unique optical and electronic properties, such as quantum phase transition based Mott transistors 1 , polaronic carrier transport in lead halide perovskite photovoltaics 2 , coherent phonons and vibrations driving singlet fission 3 , or electronic energy transfer in light-harvesting complexes 4 . These properties emerge from the interplay of the elementary electronic, vibrational, and phononic quantum states. By exciting one of these degrees of freedom out of equilibrium and probing their dynamic response, ultrafast spectroscopy disentangles mode-coupling and competing interactions that are otherwise convoluted in static spectroscopy 5 . In addition, when the system is perturbed farfrom-equilibrium, it can be driven into new photoinduced quantum states, enabling ultrafast optical control of metallic, superconductive, and polaronic functionalities 6 . Further, strong and ultra-short laser fields lead to extreme nonlinear optical phenomena with applications from high harmonic generation 7,8 to light-field petahertz electronics 9 .
However, these ultrafast processes and material functions often exhibit spatial heterogeneities associated with, e.g., lattice defects, strain, grain boundaries, and nonuniform doping from atomic to device scales [10][11][12] . To address the associated spatiotemporal dynamics, a variety of ultrafast nanoimaging techniques have been developed, including ultrafast transmission electron microscopy (TEM) 13 , photoemission electron microscopy (PEEM) 14 , or X-ray microscopy 15 . Yet, these techniques often cannot readily resolve the low-energy electronic and lattice coupling and dynamics that critically control the material properties.
Important long-lived transients in materials arise from manybody interactions induced by a strong perturbation, such as photoinduced phases in correlated electron systems [30][31][32][33] or polaron dynamics in organic-inorganic hybrid photovoltaics 34,35 . In these systems, cooperative dynamics among the many degrees of freedom 36,37 result in the formation of nonequilibrium states with nano-to microsecond lifetimes. However, high-repetition excitation precludes quantitative probing of the ultrafast dynamics of those states. Full spatiotemporal-spectral resolution with lowrepetition excitation has only been achieved recently 21,23 , while the reduced duty cycle limits signal intensity and contrast. This calls for a generalized approach with enhanced excited-state contrast at low-repetition-rate excitation.
Here, we demonstrate non-degenerate heterodyne pump-probe infrared scattering scanning near-field optical microscopy (HPP IR s-SNOM) with low-repetition-rate modulated excitation, which provides simultaneous space, time, frequency, and phase resolutions with high sensitivity. A modulated optical pump excites the system into an excited state, followed by infrared heterodyne probing of the transient low-energy electronic and vibrational response. The induced third-order nano-localized polarization is isolated by sideband lock-in detection and directly detected interferometrically in the time domain, enabling ultrafast nanoimaging with high contrast even with the low-repetition excitation rate of 1 MHz. The excitation-modulated HPP IR s-SNOM thus provides the analog of ground-state nano-FTIR spectroscopy, resolving the transient and nano-localized excited-state response.
The spectrally resolved near-field pump-probe signal is modeled to quantitatively extract the spatiotemporal evolution of the transient dielectric function of the material based on a combination of finite dipole and four-layer reflection model.
As a representative application to quantum materials, we perform nanoimaging of the electron dynamics associated with the ultrafast photoinduced insulator-to-metal transition (IMT) in vanadium dioxide (VO 2 ), resolving transient domain dynamics that is distinct from the established strain-induced heterogeneity in the thermally induced transition 10,16,17,38 . In another application to soft molecular materials, based on transient nanospectroscopy with the spectral resolution, we directly resolve heterogeneity in polaron-cation coupling that controls the photovoltaic response by probing the vibrational dynamics in a triple cation perovskite 34,35 .
The low-repetition-rate excitation with highly sensitive detection leads to the study of photoinduced phase transitions as well as soft photovoltaic materials with long-lived carrier and vibrational responses. The approach thus holds promise to establish the missing links between the elementary processes at the nanoscale and the associated macroscopic optical, photophysical, catalytic, or electronic properties of a wide range of functional materials.
Results Figure 1a-c shows the schematics of HPP IR s-SNOM with femtosecond pump (Yb:KGW amplified laser with~185 fs fullwidth-at-half-maximum (FWHM) pulse duration, 1030 nm center wavelength,~1 MHz repetition rate, Pharos, Light Conversion), and broadband infrared nano-probe spectroscopy (tunable at 5-10 μm,~170 fs FWHM pulse duration, Orpheus OPA/DFG, Light Conversion), with overall time resolution of~200 fs (see Supplementary Fig. S1), and~40 nm spatial resolution as given by the apex radius of the metallic scanning probe tip ( Supplementary  Fig. S4). IR s-SNOM is implemented with an asymmetric Michelson interferometer, consisting of sample and reference arms as established 39,40 . An atomic force microscope (Innova AFM, Bruker) in the sample arm is operated in tapping mode with a tip-tapping frequency ω t (for details, see Supplementary Note 1). Pump and probe pulses, separated by time delay T, are collinearly directed onto the AFM tip (ARROW-NCPt, NanoAndMore USA) with an off-axis parabolic mirror (NA = 0.45). The fundamental or frequency-doubled pump source is modulated at frequency Ω M by an acousto-optic modulator (AOM), or a mechanical chopper. The pump-induced excitedstate population and corresponding ground-state bleach (Fig. 1b) are detected by pump-probe s-SNOM spectroscopy. Here, the tipscattered near-field probe signal E NF is then optically heterodyned with the local oscillator field E LO from the reference arm, with variable delay t, and detected by a HgCdTe (MCT) detector. As in conventional IR s-SNOM, lock-in demodulation (HF2LI, Zurich Instruments) at nω t (n = 1, 2, 3. . . ), combined with the interferometric heterodyne detection, isolates I NF;Het $ E NF E Ã LO that provides background-free nano-localized imaging contrast 39,40 .
In contrast to previous pump-probe IR s-SNOM implementations with un-modulated pump excitation 19,24 , with excitation modulation at Ω M we perform sideband lock-in demodulation at nω t ± Ω M to directly detect the photoinduced change in the near-field signal ΔI NF (T), similar to a recent implementation in ultrafast THz-EOS s-SNOM 29 . This detection scheme enhances the signal-to-noise ratio by a factor of >4 in comparison to the conventional scheme, corresponding to a reduction in the data acquisition time by more than one order of magnitude ( Supplementary Fig. S9). By scanning E LO in time t, we then acquire the heterodyne pump-probe interferogram ΔI HPP (t, T) in the time domain. ΔI HPP (t, T) is then Fourier transformed with respect to t and deconvolved with E LO ðνÞ to yield the real and imaginary parts of ΔE NF ðν; TÞ (Fig. 1c), and with much higher signal-to-noise ratio compared to the pump un-modulated case (see Supplementary  Fig. S9 for examples). ΔE NF ðν; TÞ quantitatively relates to the transient photoinduced change in the complex dielectric function, which can be retrieved through the model fitting. HPP IR s-SNOM thus provides the full 4D characterization of the transient material response with spatial (X, Y), temporal (T), and spectral (ν) resolutions, with high excited-state contrast enabled by the selective detection of the pump-induced near-field response.
As an example, Fig. 1d shows a representative pump-probe interferogram ΔI HPP (t, T) of a germanium reference sample, acquired by continuously scanning the E LO delay t at selected pump-probe time delays T. The t-dependent profiles of ΔI HPP (t, T) is determined and limited by the spectral profile of the infraredprobe pulse due to the nearly instantaneous and spectrally broad transient free carrier Drude response of germanium 41 . To determine the spectrally averaged amplitude relaxation R HPP (T), ΔI NF is recorded as T is scanned for sets of two distinct E LO phase values, ϕ = 0 and π (see Fig. 1d) for constructive and destructive interference near zero-path difference (ZPD), to yield the amplitude decay of the HPP signal as R HPP (T) = ΔI NF (ϕ = 0, T) − ΔI NF (ϕ = π, T) (Fig. 1e). This approach of extracting the spectrally averaged amplitude based on the two-phase measurement is applicable due to the narrower laser bandwidth of~100 cm −1 FWHM compared to the much broader Drude response in germanium (see Supplementary Note 1).
The HPP signal relaxation R HPP (T) (red) probing the recombination of photoinduced carriers is notably distinct from the self-homodyne pump-probe (SHPP) decay ΔI SHPP (T) (blue), recorded without interference with E LO . ΔI SHPP (T) shows a small initial rise followed by a slow decay 41 , while R HPP (T) only shows a decay yet faster than ΔI SHPP (T). We attribute this difference to the convolution of the time-dependent far-field background in ΔI SHPP (T) 22 . The tip-enhanced pump field results in higher local carrier density compared to the far-field pumped background (Fig. 1e, inset). With its pure local probe character, R HPP (T) thus exhibits a faster relaxation, reflecting the enhanced recombination and scattering induced by the higher excited carrier density. This observation critically highlights the necessity of interferometric heterodyne detection to quantify the nano-localized pump-probe dynamics, particularly in the high fluence regime where the observed dynamics are highly sensitive to the local pump intensity.
Theory of HPP IR s-SNOM. To evaluate excited-state absorption resonances at the nanoscale 18,21,23,24 , which contain critical information associated with many-body interactions, we extend the dipole model in combination with multilayer reflection, established for both ground-state 42-44 and ultrafast s-SNOM 18,24 , to quantitatively relate the experimentally observed sidebanddemodulated pump-probe interferogram ΔI HPP (t) to the underlying transient dielectric function Δε NF ðνÞ. Importantly, we find that ΔE NF ðνÞ as the direct observable in transient vibrational nano-spectroscopy is generally the convolution of the excitedstate absorption and Fano-type interference. This underscores the importance of the theoretical framework to retrieve Δε NF ðνÞ that purely encodes the excited-state absorption to distinguish these two contributions.
We illustrate the application for the case of a transient molecular vibrational response coupled to photoinduced carriers 35,45 , with a 600-nm thick sample film coated on a substrate (Fig. 2a). We assume ground-state and excited-state dielectric functions,ε ð0Þ ðνÞ and Δε NF ðνÞ, with vibrational resonances centered at ν gs and ν ex , respectively (Fig. 2b). The pump excitation modifies the dielectric function of the sample film toε ðeÞ ¼ε ð0Þ þ Δε NF , down to a depth d 1 = 100 nm determined by the absorption coefficient. The remaining depth of d 2 = 500 nm is left unperturbed atε ð0Þ . probe, and interferometric heterodyne near-field detection. AOM acousto-optic modulator, DFG difference-frequency generation, OAP off-axis parabolic mirror, OPA optical parametric amplifier, Osc .+RA oscillator and regenerative amplifier laser system. b Far-fromequilibrium excitation followed by mid-infrared probe of the transient low-energy vibrational and electronic response coupled to the excited state. c The tip-localized time domain signal of ΔE NF ðνÞ, from which the pump-induced change in the nano-localized complex dielectric function Δε NF ðνÞ is retrieved with spatiotemporal-spectral resolution. d Sideband-demodulated pump-probe interferogram ΔI NF (t, T) for a Ge reference sample. e T-dependent pumpprobe transients from two different E LO phase values ΔI NF (ϕ = 0, T) and ΔI NF (ϕ = π, T), the derived frequency-averaged heterodyne pump-probe amplitude relaxation R HPP (T), and the self-homodyne pump-probe signal relaxation ΔI SHPP (T). Inset: tip-enhanced pump excitation in nano-localized probe volume, leading to faster relaxation from higher excited carrier density.
The near-field scatter with and without pump excitation is calculated based on a combination of finite dipole 42 and fourlayer models for a sinusoidally modulated tip-sample distance and is demodulated with the second harmonic tip-tapping frequency. While the finite dipole model is known to quantitatively retrieve vibrational resonances 42 , the four-layer reflection model 46 accounts for partial excitation of a material layer. For details, see Supplementary Note 2. Figure 2c shows the calculated ground-state s-SNOM interferogram I NF (t) (black). The pump-induced response Δε NF ðνÞ, which consists of a broadband carrier response and a vibrational excited-state absorption resonance, then gives rise to the pumpprobe interferogram ΔI HPP (t) (red). As expected, the nonresonant term inε ð0Þ ðνÞ leads to a nearly instantaneous center burst of I NF (t) limited by the duration of the probe pulse followed by a vibrational free-induction decay (FID) 40 . ΔI HPP (t) is phaseshifted due to the complex broadband response of photoinduced carriers, with the FID from the frequency-shifted photoinduced vibrational resonance.
The Fourier transforms of I NF (t) and ΔI HPP (t), after deconvolution with E LO ðνÞ, yield the spectral profiles of the ground-state E NF ðνÞ (black) and its photoinduced change ΔE NF ðνÞ (red) as shown in Fig. 2d. As can be seen, E NF ðνÞ approximates the ground-state complex dielectric functionε ð0Þ ðνÞ as established 40,42 . In contrast, the spectral profile of ΔE NF ðνÞ is a complex convolution of ground-and excited-state dielectric responses. While ΔE NF ðνÞ exhibits a resonance corresponding to the excitedstate absorption at ν ex , it is also compounded by Fano-type interference of the vibrational response with the pump-induced broadband carrier response. In the analysis of the experimental data presented below, with the application of the finite dipole model 42  We note that Fano-type interference is generally expected to give rise to a nontrivial lineshape in ΔE NF whenever spectrally distinct and narrow resonances interfere with pump-induced broadband response, as is generally the case for visible-to-UV pump and infrared-to-THz probe nano-spectroscopy of a wide range of molecular and soft materials. The model is generalizable to other types of resonant excitation and is universally applicable to retrieve the underlying ground-and excited-state dielectric functions in HPP s-SNOM, enabling the quantitative separation of excited-state absorption and Fano-type interference.
Phase-controlled imaging of quantum IMT in vanadium dioxide nanobeam. We first apply HPP IR s-SNOM to nanoimaging heterogeneity in the photoinduced quantum phase transition and associated electron dynamics in a nanobeam of the correlated electron material vanadium dioxide (VO 2 ). VO 2 exhibits a thermal insulator-to-metal transition (IMT) at~340 K, which involves a bandgap collapse with a change from an insulating monoclinic to a metallic rutile phase 10,38,47,48 . The IMT can also be induced optically 10,16,17,30,32,33 with promising applications for, e.g., ultrafast photoswitches 48 .
The phase transition is believed to be caused by a complex interplay between electron-electron correlation and electron-phonon coupling, with details regarding the exact mechanism still remaining elusive 49,50 . As a manifestation of its intricacy, both the thermally induced and photoinduced transition of VO 2 exhibit spatial heterogeneity and are susceptible to local strain and chemical heterogeneity 10,16,17,38,47 . Recent optical pump/infraredprobe s-SNOM of the photoinduced IMT of VO 2 revealed a spatial profile distinct from that of thermal strain-induced heterogeneity and was attributed to possible stoichiometric zoning 16 . However,  that measurement was performed in SHPP, and the influence of the far-field background on the measured dynamics could not be ruled out 22 .
Here, we apply HPP IR s-SNOM to VO 2 nanobeams 51 to quantify the heterogeneous electron dynamics associated with the IMT in a background-free and phase-resolved manner (Fig. 3a). The micro-Raman spectrum of the nanobeams on a silicon substrate shows the VO 2 initially in the M2 phase at room temperature 52 (see Supplementary Fig. S4). Following the NIR pump excitation above the bandgap at 1.2 eV (1030 nm,~2 mJ/ cm 2 fluence), we measure the HPP interferogram ΔI HPP (t) with a 0.2 eV (6 μm, 1670 cm −1 ,~50 μJ/cm 2 fluence) mid-infrared probe at T = 0 ps (Fig. 3b). We adjust the phase of E LO relative to that of the photoinduced response ΔI HPP (t) as indicated by ϕ = 0 and π in the inset. Figure 3c (top) then shows the time-resolved HPP amplitude relaxation R HPP (T), acquired at three selected nanobeam locations based on this two-phase measurement.
While at position 2 and 3 the photoinduced infrared response fully relaxes within~2 ps and signifies excited carrier relaxation, at position 1 the pump-induced signal plateaus beyond 10 ps, characteristic for the formation of the metastable metallic state 10,16,30,33 . By adjusting the pump fluence, we can locally switch the behavior between ps-transient carrier relaxation in the photo-doped insulating state and the photoinduced IMT with its ultrafast nucleation of the metallic state followed by a slower transient domain growth (Supplementary Fig. S4). Figure 3d then shows the corresponding ultrafast HPP nanoimaging, derived from imaging the sideband-demodulated intensity ΔI NF for the two E LO phase values ϕ = 0 and π (example shown in Fig. 3e for T ps) as R HPP ({X, Y}) = ΔI NF (ϕ = 0, {X, Y}) − ΔI NF (ϕ = π, {X, Y}). In addition to the nonuniformity of the decay at individual representative locations as shown in Fig. 3c (bottom), the line profiles across and along the c R axis of the VO 2 crystal both demonstrate dynamically evolving spatial disorder in the HPP signal amplitude (Fig. 3f).
While the heterogeneity along the c R axis is particularly pronounced in this example, in other cases a heterogeneity perpendicular to the c R axis stands out (see Supplementary Fig.  S5). By measuring multiple nanobeams with different orientations, we verify that the nonuniform spatial profile of the pump beam does not account for the observed transient heterogeneity, being due to intrinsic heterogeneities in each nanobeam ( Supplementary Fig. S6).
As is apparent from Fig. 3b (inset) comparing I NF (t) and ΔI HPP (t) in time, the transient near-field signal ΔE NF is phaseshifted from the ground-state response E NF , due to the different dielectric response between the insulating and metallic phases as is known for the thermally induced IMT of a VO 2 film (see Fig. 3a, right) 53 . By fitting both the amplitude (|ΔE NF |/|E NF |~0.2) and the optical phase shift (Δϕ = ϕ(ΔE NF ) − ϕ(E NF )~57 ∘ ), we retrieve the actual photoinduced change in the dielectric constant of Δε NF $ 0:25 þ 1:0i at the probe energy of ν probe ¼ 1670 cm −1 . In comparison withε therm: metal % 3 þ 84i for the thermally induced metallic phase in an extended film sputtered on a silicon substrate 53 , our finding of jΔε NF j << jε metal j implies that the photoinduced excited state is only partially metallic at the pump fluence of~2 mJ/cm 2 . Im½Δε NF > Re½Δε NF is in agreement with the case of a thermally induced metallic phase, with the extracted nano-localized transient dielectric phase argðΔε NF Þ ¼ 76 ± 2 slightly smaller than that of a thermally induced metallic phasẽ ϵ therm: metal ¼ 95 ± 16 , which is derived from ellipsometry literature values for VO 2 films. With the samples prepared under different conditions in the literature 47,[53][54][55][56] , the deviation might arise from different morphology, strain, or doping, but also a possibly distinct quantum nature of the photoinduced phase in comparison to a thermally induced metallic phase.
A pronounced heterogeneity along the c R axis (Fig. 3d, f) has been established for thermally induced metallic VO 2 nanobeams and is attributed to nonuniform local strain 10  predominantly perpendicular with respect to the c R axis and is consistent with previous work that implied a distinct origin such as intrinsic stoichiometric zoning 16 . The co-existence of the two types of the dynamic heterogeneities both along and across the c R axis highlights the complexity of the photoinduced IMT, the exact mechanism of which remains unsolved to date. HPP IR s-SNOM, with its space and time resolutions, is thus applicable to address the intracrystalline heterogeneity of a photoinduced IMT in VO 2 to guide the development of device applications of VO 2 nanostructures with ultrafast control. With quantitative phase resolution, it also lays the groundwork to address the unsolved question regarding the distinct nature of quantum states underlying the photoinduced vs. thermally induced metallic phases 30,32,49 .
Transient vibrational nano-spectroscopy of the polaron-cation coupling in lead halide perovskites. In the extension of HPP IR s-SNOM to soft and molecular materials, we demonstrate ultrafast vibrational nano-spectroscopy of lead halide perovskites (Fig. 4a) to resolve electron-vibration coupling and its spatial heterogeneity. Lead halide perovskites exhibit an extraordinary optoelectronic response, characterized by the spontaneous formation of long-lived free carriers and long diffusion lengths 57 . Their unusual photovoltaic performance is believed to arise from polaron formation 58 , where the charge-phonon coupling extends across multiple unit cells and results in high defect tolerance and coherent carrier transport 2 .
As another unique aspect, lead halide perovskites exhibit heterogeneity over multiple length scales in their optoelectronic responses in, e.g., photoluminescence intensity, carrier lifetime, or open circuit voltage 11 . While several works have addressed the nanoscale heterogeneity in the lattice strain and elasticity in the electronic ground-state 59,60 , the direct relationship between the nonuniform optoelectronic response and the underlying polaronic heterogeneity has not yet been established. Ultrafast infrared vibrational spectroscopy has previously elucidated the coupling between a molecular cation and a photoinduced polaron in perovskites 35,61 (Fig. 4a), yet was unable to address the underlying spatial heterogeneity due to the diffractionlimited resolution.
Using HPP IR s-SNOM, we aim to resolve the polaron-cation coupling on the nanoscale. To establish the ground and excited-state vibrational responses and their expected coupling to the polaron, we first perform conventional far-field visible (2.4 eV)-pump/infrared (0.2 eV)-probe transmission spectroscopy as shown in Fig. 4b for a thin film of the triple cation perovskite FAMACs, with chemical composition ½ðFA 0:83 MA 0:17 Þ 0:95 Cs 0:05 PbðI 0:83 Br 0:17 Þ 3 . We estimate an injected carrier density of~10 19 cm −3 (for experimental details, see Supplementary Note 4). The pump-induced change in transmission (−ΔT/T) at the pump-probe delay of T = 0.5 ps exhibits the transient vibrational signature arising from the CN anti-symmetric stretch mode of the formamidinium (FA) cation 61 in addition to the spectrally broad background from polaron absorption 34 . The vibrational excited-state absorption (ΔA vib ) is compared to the ground-state absorption (A vib ) in Fig. 4b (bottom), exhibiting a blue-shift as well as an absorptive lineshape, suggesting an enhancement in the transition dipole moment. These two observations are consistent with previous observations on similar perovskites 35,61 . As established 35 , these two features signify the coupling of the molecular vibration to polaron absorption with a large transition dipole moment and lower resonance energy. This gives rise to the blue-shift and enhanced transition dipole moment of the hybridized polaroncoupled vibration (Fig. 4a). Based on the reported resonance frequency of polaron absorption at 1100-1200 cm −134,35 , the observed vibrational blue-shift of~5 cm −1 corresponds to a spatially averaged polaron-cation coupling strength of 50 cm −1 (see Supplementary Fig. S8 for details).
In ultrafast HPP IR s-SNOM, we then explore the associated nanoscale heterogeneity in polaron-cation coupling (Fig. 5a). Figure 5b shows ΔI HPP (t) for a pump-probe delay of T = 2 ps with the center burst arising from the instantaneous polaron absorption 34 and the long-lived coherence associated with the transient vibrational response. Figure 5c then shows the Fourier transformed spectral profile of the pump-induced ΔE NF ðνÞ and ground-state E NF ðνÞ. The transient vibrational resonant peak at ν ex in Im½ΔE NF ðνÞ is blue-shifted from the ground-state peak position ν gs in Im½E NF ðνÞ by~5 cm −1 , in agreement with the polaron-cation coupling observed in the far-field measurement.
We then apply and fit the data to the model described above (Fig. 2) to retrieve the transient complex dielectric function Δε NF ðνÞ. The resonant spectral profile in the retrieved Δε NF ðνÞ (Supplementary Fig. S7) is essentially identical to ΔE NF ðνÞ, with the background carrier response (Δε NF $ À0:1 þ 0:3i) qualitatively consistent with polaron absorption. The contribution from Fano-type interference to an apparent shift is negligible in this case compared to the transient vibrational response, due to the relatively small carrier background (Fig. 5d). We thus subtract the carrier background from Im½ΔE NF ðνÞ to extract the nano-localized excited-state absorption Im½ΔE NF;vib ðνÞ (see Supplementary Note 4).   (Fig. 5e, inset), obscured in the spatially averaged far-field spectroscopy above. Such disordered polaron-cation coupling is likely associated with a non-uniformity in the dynamic lattice elasticity. This interpretation is supported by other recent experimental and theoretical investigations 59,60,62,63 , which have identified a heterogeneity in chemical composition and resulting local lattice disorder and strain in perovskite films. With HPP IR s-SNOM probing the excited-state vibrational absorption, we resolve spatial heterogeneities in polaron-cation coupling arising from lattice disorder, which directly impacts polaron formation, lifetime, transport and, as such, photovoltaic device performance.

Discussion
Recent ultrafast infrared nanoimaging demonstrated the nanoscale probing of a range of low-energy phenomena in semiconductor, 2D, and other quantum materials [16][17][18][19][20]22,24,26 , yet mostly with highrepetition excitation that provides sufficient contrast against the simultaneously detected unpumped ground-state response. This conventional approach has therefore been limited primarily to the detection of short-lived nonequilibrium states. This has hampered the application of the technique to access long-lived transient states that often arise from cooperative dynamics associated with manybody interactions 36,37 represented by, e.g., photoinduced phase transitions in correlated electron materials or polaron formation in molecular materials.
In the adaptation of modulated excitation with sideband detection for HPP IR s-SNOM, we facilitate the isolation of the excited-state response from the unperturbed ground-state response, establishing nano-FTIR spectroscopy of the purely transient and nano-localized response with low-repetition excitation. HPP IR s-SNOM thus universally endows ultrafast infrared nanoimaging with the ability to quantitatively resolve ultrafast dynamics associated with long-lived perturbations. We note that HPP s-SNOM, which is compatible with the full range of probe frequencies from visible to far-infrared together with its relatively facile implementation, is complementary to ultrafast EOS s-SNOM probing nanoscale dynamics in the THz regime 29 .
As demonstrated in VO 2 , HPP IR s-SNOM isolates the nearfield pump-probe from the ground-state response to achieve ultrafast nanoimaging of the IMT dynamics with high contrast (Fig. 3d). Eliminating the convoluted far-field background contribution by heterodyning ΔE NF with a phase-controlled E LO 22 , HPP IR s-SNOM also accurately determines the timescale of the purely nano-localized carrier dynamics (Fig. 1e). Further, by simultaneous and phase-locked recording of I NF and ΔI NF interferograms, HPP IR s-SNOM quantifies the transient complex dielectric response on the nanoscale (Fig. 3a, b), thus essentially performing ultrafast nano-ellipsometry based on interferometric heterodyne detection. The observed transient spatial heterogeneity suggests intricate co-existence of competing mechanisms such as nonuniform strain and stoichiometric zoning.
In extensions to transient vibrational nano-spectroscopy, we resolve the generally weak excited-state vibrational coherence of a lead halide perovskite in the time domain (Fig. 5b) with sensitive and selective detection of ΔI NF (t) enabled by pump modulation. The resulting phase and spectrally resolved excited-state vibrational response quantifies the spatially varying polaron-cation coupling (Fig. 5e), which is central to the photovoltaic response of perovskites. Thus the combination of pump modulation and heterodyne detection in HPP IR s-SNOM provides pure transient and nano-localized contrast, extending the applicability of ultrafast infrared nanoimaging from the weakly perturbed to the far-from-equilibrium regime.  c Phase-and frequency-resolved nano-localized pump-probe ΔE NF ðνÞ and ground-state response E NF ðνÞ. d Decomposition of Im½ΔE NF ðνÞ into the transient vibrational (Im½ΔE NF;vib ðνÞ) and carrier (Im½ΔE NF;car ðνÞ) contributions with a minor feature from Fano-type interference. e The nano-localized transient vibrational signal Im½ΔE NF;vib ðνÞ at different sample locations shows nanoscale spatial heterogeneity in the polaron-cation coupling.
More generally, ultrafast nanoimaging based on an electronic excitation and a low-energy probe has a unique advantage in studying the spatial heterogeneity of electron-phonon coupling, a central topic in, e.g., two-dimensional materials 64 , hybrid photovoltaics 2 , or nanoscale thermal transport 65 . Ultrafast heterodyne infrared nanoimaging, with its direct access to vibational, phononic, and polaronic modes, can thus play an important role not only in mapping the inherent disorder in electron-phonon coupling 11 , but also the optical control of such coupling in combination with nanoscale quantum architectures 23 .
HPP IR s-SNOM also lays the foundation for adapting other state-of-the-art ultrafast spectroscopy to nanoimaging. For example, by implementing two pump pulses with controlled time delay, ultrafast non-degenerate two-dimensional nanospectroscopy 4,66 can be realized, probing coherence and population transfer among different modes on the nanoscale. Our implementation with low-repetition excitation is particularly beneficial to potentially implement nonlinear spectroscopy with infrared 67 and THz 68 excitations at the nanoscale, which would require a strong pump fluence that is only attainable in amplifier laser sources. Further, in combination with interferometric heterodyne detection, adiabatic plasmonic nano-focused electronic four-wave mixing 27,28 would provide two-dimensional electronic nano-spectroscopy to characterize local electron or exciton dynamics in, e.g., two-dimensional materials 69 .
HPP s-SNOM can thus resolve the full spatiotemporal-spectral evolution of key elementary excitations which define the properties of a wide range of functional materials. With the pump modulation and heterodyne detection, HPP s-SNOM bridges the prolific success and plethora of modalities of far-field ultrafast spectroscopy to ultrafast nano-spectroscopy and -imaging to probe coupling and dynamics at the nanoscale.

Data availability
The data generated in this study have been deposited in the Open Science Framework (OSF) at https://osf.io/nkyta/.