Experimental scheme and sample characterization

A 550 μm thick free standing perovskite-poly(methylmethacrylate) (PMMA) thin film was made by embedding μm size CH 3 NH 3 PbI 3 crystals (Fig. 1b) in the polymer matrix, as shown in SEM image of the CH 3 NH 3 PbI 3 /PMMA film (Fig. 1c). Room temperature X-ray diffraction measurements of the film and pure CH 3 NH 3 PbI 3 powder corroborate the inclusion of the perovskites into the PMMA matrix (Fig. 1d). We also performed comprehensive temperature-dependent absorption (Fig. 1e), photoluminescence and X-ray diffraction measurements of our samples down to T=6 K (Supplementary Figs 1–3), which are consistent with prior studies of high-quality perovskite materials12,13,18. In addition, there is no evidence for the presence of any tetragonal impurities at low temperature, for example, 8 K at which most of our experiments are performed. Note the absorption curves in Fig. 1e display very big light scattering as manifested by the large background above 820 nm, which is below the perovskite band gap and little absorption is expected. We attribute this to the high inhomogeneity of perovskite crystals dispersed in PMMA matrix, as confirmed in ref. 29 (See Supplementary Note 4 for details).

We perform optical-pump and THz-probe spectroscopy, which is driven by a 1 kHz Ti:Sapphire regenerative amplifier with 790 nm central wavelength and 40 fs pulse duration22,23,30,31. One part of the output is used to pump the sample either directly at the fundamental wavelength of 790 nm or at 399 nm after going through a β-barium-borate (BBO) crystal. The pump spectra are shown in Fig. 1e together with the linear absorption spectra of the CH 3 NH 3 PbI 3 /PMMA film. The other part of the output is used to generate and detect phase-locked THz fields in time-domain via optical rectification and electro-optic sampling of two 0.2 mm thick ZnTe crystals, respectively. This increases by an order of magnitude of the signal-to-noise ratio at the important, high frequency regions in the range of 10–15 meV comparing to 1 mm thick ZnTe emitter/detector. The THz electrical fields exhibit broadband spectral width from 2 to 15 meV used as a probe. To obtain time-resolved complex THz conductivity of the sample, we record the THz electrical fields in time-domain transmitted through a clear aperture E air (t), static sample (t) and its pump-induced change Δ (t) at a fixed pump-probe delay Δt (Fig. 2a). Note there is negligible pump-induced change for a pure PMMA control sample ΔE PMMA (t).

Figure 2: The direct observation of excitonic Rydberg states in CH 3 NH 3 PbI 3 . (a) THz fields transmitted (raw data) through a clear aperture (grey), the sample without pump (black) and its pump-induced change (red) under 790 nm wavelength and 550 μJ cm−2 fluence, at T=8 K and pump probe delay Δt=60 ps. There is negligible pump-induced change for pure PMMA (blue). (b,c) Ultrafast THz spectra Δσ 1 (ω) and Δɛ 1 (ω) under the same pumping conditions as (a) for various temperatures. This demonstrates the distinct resonant, internal quantum transitions, the 1s→2p and 1s→3p marked as A 1 and A 2 , which are clearly different from the phonon bleachings in the CH 3 NH 3 PbI 3 (B j=1,..,4 ) and PMMA (P k=1,2 ) (Supplementary Fig. 4 and Supplementary Note 4). Shown together are the complete (black lines) and partial (blue lines, without excitonic contribution) model calculations by equation (1). (d) A 3D view of the temperature-dependent Δσ 1 (ω) spectra further confirms the fine details of the phonon bleaching modes. Full size image

Correlated THz resonances from excitonic Rydberg states

Figure 2a allows us to simultaneously obtain the sub-ps, frequency-dependent complex conductivity Δσ 1 (ω) (Fig. 2b) and dielectric function Δɛ 1 (ω) (Fig. 2c), associated with dissipative and inductive responses of the sample31 (also see Methods and Supplementary Note 3). At high temperature above the structural phase transition at T S =160 K, the 295 K trace exhibits two strong photoinduced bleaching features, that is, negative conductivity (Δσ 1 <0), at 4.2 and 10 meV. These arise from pump-induced softening of phonon modes in CH 3 NH 3 PbI 3 at B 1 ∼4.2 meV, B 2 ∼8 meV, and in PMMA at P 1 ∼10 meV (see Supplementary Fig. 4 and Supplementary Note 4). The last two merge into a broad bleaching mode. Below T S two new bleaching modes become progressively pronounced, as marked in the 8 K trace at B 3 ∼3 meV and B 4 ∼6.2 meV, fully consistent with prior observations13,24,25. However, remarkable photoinduced absorptive resonances (Δσ 1 >0), previously unobserved, appear at frequencies above 10 meV, but only below T S . Most intriguingly, the pronounced absorptive resonances, centred at A 1 ∼10.1 meV and A 2 ∼12.1 meV, completely prevail over the bleaching behaviours, as seen in the 8 K trace (Fig. 2b). Figure 2d further shows the resonant absorption significantly increases with decreasing temperature, which cannot be attributed to the phonon bleaching. Moreover, the high energy spectral shapes in Δσ 1 and Δɛ 1 are characterized by the correlated, dissipative and inductive features of two well-defined A 1 and A 2 resonances. These point to the internal transitions between Rydberg states, that is, 1s→2p and 1s→3p. We emphasize that the experimental ratio of the oscillator frequencies A 1 /A 2 =0.835 is in perfect agreement with the theoretical value ω 1s→2p /ω 1s→3p =(3/4)/(8/9)=0.844 derived from the quantized bound states E n =−E 1s /n2, where . μ r , ɛ r and R H are the relative effective mass, relative permittivity and Rydberg constant, respectively. This yields exciton binding energy E 1s =13.5 meV. In addition, the disappearance of the resonances across T S indicates a sudden drop in E 1s below ∼2 meV, our low-frequency resolution, consistent with the tetragonal-to-orthorhombic transition27,28.

To put our observation of resonant quantum transitions of excitonic Rydberg states on a solid footing, we fit the experimentally-determined, sub-ps THz response function by a model that consists of the 1s→np (n=2,3) excitonic resonances, the ‘local’ resonant phonon bleaching and Drude–Smith responses from disorder-/backscattering-induced transport of unbound carriers (see Methods). The experimentally-determined, sub-ps THz response functions can be fitted by a model that assumes co-existence of excitons, phonons and e-h plasma (see Supplementary Note 4):

The first two terms describe the 1s→np (n=2,3) excitonic resonances which are proportional to the effective transition strength , where f is the intra-excitonic oscillator strength, and the plasma frequency . N X is the exciton density. This way, is proportional to the population difference between the two Rydberg states involved in the transitions (Fig. 1a), that is, ΔN 1s,np =N 1s −N np (ref. 22). Next, the ‘local’ resonant bleaching features are well represented by the third term in equation (1), the sum of the phonon contributions in the CH 3 NH 3 PbI 3 (B j=1,..,4 ) and PMMA (P k=1,2 ) (see Supplementary Fig. 4 and Supplementary Note 4). Lastly, the non-resonant component is described by the fourth Drude–Smith term in equation (1), which describes disorder-/backscattering-induced transport of unbound carriers of density N eh 32,33. The last term is justified from the suppressed Δσ 1 (ω) (Fig. 2b) and rapidly increased Δɛ 1 (ω) (Fig. 2c) as ω→0, in contrast to the conventional Drude transport of unbound e-h carriers. The Drude-Smith term is proportional to .

Both Δσ 1 (ω) and Δɛ 1 (ω) can be consistently fitted very well over the entire spectral, up to 14 meV, and temperature ranges (black solid lines in Fig. 2b,c). This is remarkable considering that the fit are strongly restrained by the requirement of simultaneously describing both responses over a broad spectral range, and by the distinctly different spectral shapes of the excitons and charge carriers. Therefore, by fitting the experimentally obtained THz response functions with the theoretical model, we are able to calculate the densities of excitons ΔN 1s,np and unbound charge carriers N eh , as shown in the discussion below.

Excitonic formation pathways characterized by THz responses

To further investigate the buildup of the internal quantum transitions, Fig. 3a highlights the complete characterization of full response functions from three electronic contributions at 8 K after removing the phonon contributions, that is, photo-generated excitons 1s→2p (dashed red lines), 1s→3p (dashed green lines) and from unbound e-h carriers (dashed blue lines). This has not been possible in prior measurements in the perovskites. Such spectra are plotted in Fig. 3b for various time delays Δt under 790 nm excitation at 550 μJ cm−2, T=8 K, that is, for pump tuned slightly below the lowest bound, 1s exciton (pump #1, Fig. 1e). For such below-resonance pumping we expect to mainly generate electronic coherence at early times with minimum heating since only absorption appears at the high energy tail of the pump spectrum. Remarkably, the 1s→2p and 1s→3p quantum transitions in Δσ 1 (ω) exhibit a delayed rise persisting up to ∼17 ps. In strong contrast, excitation of the higher energy, e-h continuum under 399 nm at 120 μJ cm−2, T=8 K (pump #2, Fig. 1e) gives rise to much longer buildup times of 10s of ps (Fig. 3c). Such continuum, e-h excitations are unbound and thus expected to lose electronic coherence quasi-instantaneously following the fs pulse, which leads to a hot electron distribution. An excitonic 1s population can then be formed only when the continuum of phonons are scattered to cool the hot state. Such a process is slowed down by the requirement of many scattering events and hot-phonon effects34.

Figure 3: Ultrafast THz Snapshots of formation pathways of excitonic Rydberg states. (a) The THz response functions (red dots) measured under 790 nm wavelength and 550 μJ cm−2 fluence, at T=8 K and pump probe delay Δt=60 ps. The fit (black lines), based on the analytical model of equation (1), is the sum of 1s→2p (dashed red lines), 1s→3p (dashed green lines) and unbound e-h carriers (dashed blue lines). (b,c) Photoinduced conductivity changes Δσ 1 (ω) at several pump-probe delays and T=8 K after excitation at 790 nm (550 μJ cm−2 fluence) and 399 nm (120 μJ cm−2 fluence), respectively. The shaded circles are the effective transition strength , extracted from photoinduced internal quantum transitions of excitons. The pump fluences are chosen to induce approximately equivalent for both excitations which allows to underpin their distinctly different rise times. Full size image

Time-dependent densities of excitons and unbound carriers

Access to both the 2p and higher-lying, 3p, dark bound states has been very scarce in materials21,31, which allows us to quantitatively analyse distribution functions and cooling curves of excitons in the perovskite system to corroborate these observations. The distinct spectral shapes in Fig. 3b,c allow us to faithfully extract the effective transition strength (see Supplementary Fig. 5 and Supplementary Note 4) and, thereby, densities of excitons (Fig. 4a,b) and unbound carriers (Fig. 4c). For the below-resonance excitation at 790 nm and 550 μJ cm−2, Fig. 4a presents the time-evolution of the extracted exciton population differences ΔN 1s,2p =N 1s −N 2p and ΔN 1s,3p =N 1s −N 3p . They both show the same delayed buildup in time, which yields a nearly time-independent ratio ∼98% (red squares in the inset of Fig. 4d). This result indicates that the incoherent exciton population after fs photoexcitation is mostly distributed at the 1s state, that is, ΔN 1s,2p ≈ΔN 1s,3p ≈N 1s . This is, again, in contrast to pump excitation of continuum states at 399 nm and 120 μJ cm−2, where the ΔN 1s,2p and ΔN 1s,3p now exhibit substantially longer rise times on 10s of ps time scale and display different temporal dependence (Fig. 4b) as compared to the 790 nm excitation (Fig. 4a). This results in a time-dependent population ratio varying by >10% over 60 ps (blue dots in the inset of Fig. 4d). This allows us to extract the cooling curve of the hot state as shown in Fig. 4d, that is, the thermalized, transient electronic temperature T* as a function of time (see Supplementary Note 4). T* is ∼270 K after the 399 nm excitation and gradually reaches lattice temperature of 8 K on a 10s of ps time scale (blue dots), while T* is already close to 8 K after the 790 nm excitation (red squares).

Figure 4: The time-evolution of the exciton distribution distinguishing various processes. (a,b) Exciton population difference ΔN 1s,2p (red diamonds), ΔN 1s,3p (blue dots) as a function of time delay in a logarithmic scale at 8 K for 790 nm (550 μJ cm−2) and 399 nm (120 μJ cm−2) excitation, respectively. The detailed THz time scan with 50 fs resolution is shown together in a. The error bars of carrier densities indicate the uncertainty from theoretical fitting of the experimental results. (c) The time-dependent density N eh of unbound carriers for two pumping conditions. (d) Time-dependent ratios of the population difference ΔN 1s,2p /ΔN 1s,3p are shown in the inset, which allows to extract the cooling curves for two pumping conditions. (e) The photoinduced THz transmission under 790 nm and 550 μJ cm−2 pumping for various temperatures. A two-step rise, ∼1.0±0.03 ps (grey) and ∼11.2±1.06 ps (cyan) is separated for the 8 K trace. Full size image

The distinct rise times of the relatively cold 1s exciton populations associated with the below-resonance pumping underpin a formation pathway different from the cooling of high-energy carriers. The time-dependent exciton density faithfully follows the detailed THz time scan with 50 fs resolution (black line, Fig. 4a) under 790 nm pumping, which displays a two-step rise with characteristic times ∼1.0±0.03 ps and ∼11.2±1.06 ps at 8 K (blue dots, Fig. 4e). We attribute this two-step exciton buildup to (1) the loss of exciton coherence and (2) the resonant scattering of 1s exciton with finite momentum distribution with discrete THz phonon modes B 1 –B 4 in CH 3 NH 3 PbI 3 . Initially the formation of incoherent 1s exciton populations occurs on a timescale determined by the quantum dephasing time that characterizes the optical coherence-to-population conversion arising from multiple scattering contributions and disorder effects. The linewidths of the observed resonant internal THz transitions indicate optical polarization dephasing times of the order of 1 ps, which is consistent with the measured τfast. Subsequently, the presence of discrete phonon states, with energies on the order of 2–8 meV comparable to the exciton kinetic energies, are unique, which make the 1s exciton populations with finite momenta relax to the K=0 state on a timescale of τslow. This is much faster than conventional semiconductors. For example, the near-resonant photoexcitation in GaAs relaxes through acoustic phonon scattering which slows thermalization by orders of magnitude because of the absence of THz LO phonons2 (see Discussion and Supplementary Note 5). In addition, the delayed formation observed becomes faster with increasing temperature, for example, 8 and 100 K versus 160 K in Fig. 4e, consistent with thermally-induced dephasing and phonon scattering.