Giant THz photoconductivity and possible non-equilibrium superconductivity in metallic K₃C₆₀

Abstract

The non-equilibrium control of emergent phenomena in solids is an important research frontier, encompassing effects like the optical enhancement of superconductivity ¹ . Recently, nonlinear excitation ^{2 , 3} of certain phonons in bilayer cuprates was shown to induce superconducting-like optical properties at temperatures far above T_c ^4,5,6. This effect was accompanied by the disruption of competing charge-density-wave correlations^7,8, which explained some but not all of the experimental results. Here, we report a similar phenomenon in a very different compound. By exciting metallic K₃C₆₀ with mid-infrared optical pulses, we induce a large increase in carrier mobility, accompanied by the opening of a gap in the optical conductivity. Strikingly, these same signatures are observed at equilibrium when cooling metallic K₃C₆₀ below the superconducting transition temperature (T_c = 20 K). Although optical techniques alone cannot unequivocally identify non-equilibrium high-temperature superconductivity, we propose this scenario as a possible explanation of our results.

A₃C₆₀ molecular solids crystallize often in face-centred cubic structures (Fig. 1a) in which each C₆₀^3- ion contributes three half-filled molecular orbitals to form narrow bands⁹. These electronic states give rise to superconductivity at equilibrium^10,11, mediated by a combination of electronic correlations ^{12 , 13} and local molecular vibrations^14,15,16.

Fig.1. Structure and equilibrium optical properties of K₃C₆₀.

a, Face centered cubic (fcc) unit cell of K₃C₆₀²⁶. Blue bonds link the C atoms on each C₆₀ molecule. K atoms are represented as red spheres. b, C₆₀ molecular distortion (red) along the T_1u(4) vibrational mode coordinates. The equilibrium structure is displayed in blue. c-e, Equilibrium reflectivity and complex optical conductivity of K₃C₆₀ measured at T = 25 K (red) and T = 10 K (blue).

In the experiments reported here, femtosecond optical pulses were used to excite K₃C₆₀ powders (see Methods section and Extended Data Fig. 1 for sample characterization). The excitation pulses were tuned to wavelengths between 6 μm and 15 μm (200 - 80 meV photon energy), a spectral region where both the metallic plasma and local molecular vibrations could be excited. The resulting changes in THz-frequency reflectivity and optical conductivity were measured with a second probe pulse, a quasi-single-cycle THz-field transient, detected after reflection from the photo-excited material (see Methods section and Extended Data Figs. 2, 3). These transient changes were determined at different pump-probe time delays and normalized to the absolute equilibrium optical properties measured on the same sample (Extended Data Fig. 4).

Fig. 1c-1e display two representative equilibrium reflectivity spectra measured above and below the superconducting transition temperature T_c = 20 K, along with the corresponding optical conductivities extracted by Kramers-Kronig transformations. Similar to what reported for K₃C₆₀ single crystals^17,18,19, in the metallic state (T = 25 K) the optical conductivity displayed a narrow Drude peak and a polaronic band²⁰ centred at 55 meV. When cooling below T_c, a saturated reflectivity at R=1, an optical gap in σ₁(ω) and a divergent σ₂(ω) emerged from the metal.

This equilibrium superconducting response is to be compared with that measured in metallic high-temperature K₃C₆₀ after excitation at 7 μm wavelength (180 meV photon energy), tuned to the T_1u(4) vibration (see Fig. 2 for measurements at 1 ps time delay). For all temperatures between T_c = 20 K and T = 100 K, photo-excitation induced transient changes in the optical properties very similar to those observed when cooling at equilibrium (see also Extended Data Fig. 5). At 1 ps pump-probe time delay, we observed a saturated reflectivity at R=1 (Fig. 2a and 2d), gapped σ₁(ω) (Fig. 2b and 2e), and divergent σ₂(ω) (Fig. 2c and 2f).

Fig. 2. Transient optical response of photo-excited K₃C₆₀ at T = 25 K and T = 100 K.

Reflectivity and complex optical conductivity of K₃C₆₀ at equilibrium (red) and 1 ps after photo-excitation (blue) at a pump fluence of 1.1 mJ/cm², measured at base temperatures T = 25 K (a-c) and T = 100 K (d-f). Fits to the data are displayed as dashed lines.

Furthermore, all transient optical spectra could be fitted with the same function used for the low-temperature superconductor at equilibrium. An extension of the Mattis-Bardeen model was used, applicable to superconductors of arbitrary purity and taking into account both an optical gap (superconducting component) and a fluid of normal quasi-particles (Drude component) ²¹. The non-equilibrium fits evidenced an 11-meV gap in the non-equilibrium σ₁(ω), nearly twice as large as the 6-meV superconducting gap at equilibrium.

Figure 3 reports similar measurements taken at higher base temperatures, for which the effect progressively disappeared. Only a partial reduction in σ₁(ω) and a less pronounced divergence in σ₂(ω) were observed. For these higher temperatures the Drude component alone was sufficient to fit the transient spectra, which did not exhibit a clear gap. However, note that the σ₁(ω) Drude peak in the light-induced phase (blue curves in Fig. 3b and 3e) was always narrower than for the equilibrium metal (red curves in Fig. 3b and 3e), underscoring an optical enhancement in carrier mobility for all temperatures.

Fig. 3. Transient optical response of photo-excited K₃C₆₀ at T = 200 K and T = 300 K.

Reflectivity and complex optical conductivity of K₃C₆₀ at equilibrium (red) and 1 ps after photo-excitation (blue) with a pump fluence of 1.1 mJ/cm², measured at base temperatures T = 200 K (a-c) and T = 300 K (d-f). Fits to the data are displayed as dashed lines.

In Fig. 4, we plot the transient integrated loss in spectral weight (0.75 - 2.5 THz, i.e., 3.1 – 10.3 meV) as a function of base temperature (Fig. 4a), pump-probe time delay (Fig. 4b), excitation fluence (Fig. 4c) and wavelength (Fig. 4d). The regions in which the optical properties required the full Mattis-Bardeen fit are shaded in blue, whereas those regions in which the fit yielded a gapless, Drude conductivity are shaded in white.

Fig. 4. Scaling of the σ₁(ω) gap with experimental parameters.

Photo-induced reduction in σ₁(ω) spectral weight, integrated between 0.75 – 2.5 THz (circles in all panels). The lost spectral weight is plotted as a function of base temperature (a), pump-probe time delay (b), pump fluence (c), and pump wavelength (d). Error bars represent maximum uncertainties estimated from different sets of measurements. The regions shaded in blue indicate the parameter ranges for which the transient response was fitted with a model for a superconductor. The red curve in b is the pump pulse profile (cross-correlation signal), while the dashed line is a double exponential fit (τ₁ ≃ 1 ps, τ₂ ≃ 10 ps). Dashed vertical lines in d are the frequencies of the four T_1u vibrational modes. The experiments were not possible for pump wavelengths between 6 and 3 μm due to absorption in the diamond window that contained the K₃C₆₀ powders. White and blue dots are data taken at fluence values of 0.4 and 1.1 mJ/cm², respectively. Data in b-d were measured at T = 25 K.

Figure 4a displays a crossover between a light-induced state with a Mattis-Bardeen gap (blue), observed up to T = 100 K, and a high-mobility metal (white) observed at higher temperatures.

Similarly, in the time dependence (Fig. 4b) a spectral weight reduction was observed immediately after the excitation pulse (red curve), followed by a gap opening only at later time delays (blue shading). The gapped state disappeared after approximately 2 ps, and the rest of the decay involved relaxation of a high-mobility metal (white shading) with an exponential time constant of ~10 ps (see Extended Data Fig. 6 for representative spectra as a function of pump-probe delay).

The pump fluence dependence of Fig. 4c reveals a similar crossover. Metallic conductivities (white) were observed for low excitation levels (F ≲ 0.8 mJ/cm²), whereas fully gapped states (blue) emerged only at higher fluences (F ≳ 0.8 mJ/cm²). Finally, when tuning the pump (Fig. 4d) to short wavelengths (2 μm, 620 meV photon energy) no response was found. On the contrary, a strong response was measured when the pump was made resonant to the T_1u(4) and T_1u(3) modes, which correspond to stretching and compressing of the hexagons and pentagons of the fullerene molecule (see Fig. 1b). A weaker response was still observed for wavelengths extending toward the T_1u(2) and T_1u(1) resonances.

One possible interpretation for the data above posits that excitation of molecular vibrations can coherently stimulate a transition from the high-temperature metallic state into a non-equilibrium superconducting phase.

To linear order, T_1u vibrations do not couple to the t_1u electronic states that cross the Fermi level, because T_1u ≠ t_1u × t_1u. However, for the strong optical fields used here (~1.5 MV/cm), and from the polarizability of the T_1u molecular vibrations, we estimate large oscillatory distortions of the C-C bonds amounting to several percents of their equilibrium bond lengths. Hence, the large amplitude excitation is expected to extend beyond linear response and to deform the structure of the solid along other, anharmonically coupled coordinates. To lowest order, these couplings are described by q_T1u²Q terms in the nonlinear lattice Hamiltonian^2,22,23 , where q_T1u is the directly driven mode coordinate, and Q is the coordinate of any distortion contained in the irreducible representation of T_1u × T_1u.

As discussed in the Methods section, excitation of the T_1u molecular mode is expected to drive a sizeable distortion along normal mode coordinates of H_g symmetry. Because H_g modes are thought to assist pairing at equilibrium⁹, it is possible that a quasi-static H_g molecular deformation might favour stronger superconductivity, for example by causing an increase in the electron-phonon coupling (see Extended Data Fig. 7).

Excitation of T_1u vibrations is also expected to modulate local electronic correlations. For a molecular mode of coordinate q_IR with classical vibrational coordinate q_IR = C sin(Ω_IR τ), where C is a proportionality constant, Ω_IR is the driving frequency and τ the time delay, we expect the charge density on each molecule to “slosh” backwards and forwards²⁴. Deformation of the molecular orbital causes a modification of the Coulomb repulsion U_a for the t_1u orbital (a = {x,y,z}), which can be written as U_a(τ) = U_a + Δ U_a (1 − cos (2 Ω_IR τ)). This highlights two effects: a modulation of the onsite Coulomb integral at twice the frequency of the driven molecular mode, and an average change in correlation energy, ΔU_a, which have been observed in a different organic compound²⁵.

Although a complete quantitative analysis requires a dynamical quantum chemistry calculation, an order-of-magnitude estimate for this effect can be extracted from frozen atomic motions for the T_1u(4) mode. This estimate predicts a change in electronic repulsion ΔU_a as high as ~10% of the equilibrium correlation energy for the orbitals orthogonal to the vibrational polarization (see Methods section and Extended Data Fig. 8). Such large asymmetric increase in U_a for only one orbital unbalances the occupancy of the three t_1u orbitals. This effect may interplay with the dynamical Jahn-Teller coupling known to contribute to superconductivity near equilibrium⁹.

However, the interpretation proposed above is not unique. Firstly, some other transient non-superconducting states would also be consistent with the optical properties revealed by our experiments. A large reduction in carrier scattering rate without condensation may be sufficient to explain these results. The measured gap in the transient conductivity would then simply emerge from the equilibrium polaronic band at 55 meV as the broad Drude absorption narrows. However, this scenario is difficult to reconcile with the estimated increase, rather than decrease, of the carrier effective mass (see Methods section). Alternatively, one may also consider a sliding, non-commensurate charge density wave, which would be promoted by the enhanced electron-phonon coupling.

Secondly, the mid-infrared pulses used to excite the sample couple not only to the vibrations but also to the metallic plasma. The measured photo-conductivity signal exhibits a peak in the region of the molecular vibrations (Fig. 4d), but this peak appears broadened compared to the mode linewidths. This might be indication that also a modulation of the local charge density plays a role in driving the photo-induced state.

Other observations appear important but are not well understood. The light-induced high-mobility phase is only induced for temperatures above, but not below T_c = 20 K. Unlike in the case of light-enhanced superconductivity in the cuprates, in which excitation of a phonon below T_c caused a stiffening of the Josephson Plasma Resonance⁴, in the present case we did not observe any widening of the equilibrium superconducting gap at base temperatures below T_c. Rather, we measured very little perturbation at the same pump-probe delay (1 ps) for which the response above T_c was maximum (Extended Data Fig. 9), followed by a filling in of the gap by hot quasi-particles at later time delays (Extended Data Fig. 10). This suggests that the equilibrium superconductor and the high-mobility transient state compete with one another. One may rationalize this observation by hypothesizing that equilibrium superconductivity depletes states near the Fermi level, which then cannot participate in the formation of the non-equilibrium state when stimulated by the light.

In the balance, although a definitive assignment should rest on further experimentation and theoretical analysis, we favour an interpretation based on stimulated superconductivity. However, the data above is indicative of striking emergent physics away of equilibrium regardless of the specific mechanism, opening many new opportunities and challenges for both theory and experiments.

Methods

Sample growth and characterization

Stoichiometric amounts of finely ground C₆₀ powder and potassium metal were sealed in a cylindrical vessel and closed in a Pyrex vial under vacuum (~10⁻⁶ Torr). The potassium was kept separated from the fullerene powder during the thermal treatment, therefore only potassium vapors came in contact with C₆₀. The two reagents were heated at 523 K for 72 h and then at 623 K for 28 h. The vessel was then opened and the recovered black powder was reground and pelletized. Afterwards, the pellets were further annealed at 623 K for 5 days. All described operations were performed in inert atmosphere (vacuum or Ar glove box with <0.1 ppm O₂ and H₂O). The final product, K₃C₆₀, was characterized by laboratory powder X-ray diffraction and SQUID magnetometry (Extended Data Fig. 1).

Equilibrium optical properties and fitting models

The K₃C₆₀ equilibrium optical response was measured on compacted powder pellets (100 – 400 nm average grain size) sealed in a custom sample holder and pressed against a diamond window, ensuring an optically flat interface.

Broadband infrared reflectivity measurements were carried at the SISSI beamline (Elettra Synchrotron Facility, Trieste)²⁷. The sample was mounted on a cryostat coupled to a Bruker Vertex70 interferometer through a Hyperion microscope. The sample temperature was varied between 300 K and 25 K. The spectra were referenced against the reflectivity of a gold mirror placed into the holder at the sample position.

Data below 3 meV have been extrapolated with the Hagen-Rubens formula, while at high frequency the recalculated sample-diamond reflectivity from single crystal data was used¹⁸. The complex optical conductivity was determined through a Kramers-Kronig algorithm for samples in contact with a transparent window²⁸.

All spectra at T > T_c are reported in Extended Data Fig. 4a. The low energy (≲ 200 meV) part of the complex conductivity was fitted at all T > T_c with a Drude term and a Lorentz oscillator, which reproduced the polaronic absorption centered at ~55 meV¹⁷ (a representative fit to the 25 K data is also displayed). The sample reflectivity in the superconducting state was determined instead as follows. We first measured the reflected electric field at different T < T_c using THz time domain spectroscopy. Each spectrum was then referenced against the reflected field at 25 K, for which the broadband reflectivity was known from the synchrotron measurements.

In Extended Data Fig. 4b the reflectivity measured at different T < T_c is shown. The data were extrapolated below 3 meV using the Zimmermann model²¹, a generalization of the Mattis-Bardeen optical conductivity for BCS superconductors with arbitrary purity. The optical conductivity was determined through the same Kramers-Kronig procedure described above. The real and imaginary conductivities are also reported in Extended Data Fig. 4b.

The gap opening in σ₁(ω) and the 1/ω divergence in σ₂(ω) observed upon cooling below T_c are indicative of the superconducting transition. Through fits to the data performed with the Zimmermann model²¹ the optical gap 2Δ could be determined as a function of temperature. Both its zero temperature value 2Δ(T=0) ≃ 6 meV and temperature dependence 2Δ(T) - closely following the mean field prediction - were found to be in full agreement with previously published data on K₃C₆₀ single crystals¹⁸, as shown in the inset of Extended Data Fig. 4b.

In Extended Data Fig. 4c-d we show a direct comparison of the equilibrium optical properties (below and above T_c) measured on our K₃C₆₀ powders with those reported for single crystals by L. Degiorgi et al.¹⁸

Firstly, we observe that the polaronic absorption at ~55 meV maintains the very same shape and central frequency. Secondly, the effect that the superconducting transition has on the low-frequency optical properties in our K₃C₆₀ powder is identical to that measured in the single crystals, which exhibit the same T_c ≃ 20 K and the same optical gap 2Δ(T=0) ≃ 6 meV.

The only difference between the single crystal data and the present measurements is in the carrier density and scattering rate. As is well known for this class of materials¹⁹, in powdered samples both these quantities can be different from those of a single crystal. From Drude-Lorentz fits to the data, we could estimate a reduction of a factor of ~4 in carrier density n (the Drude plasma frequency ${\omega }_p\propto \sqrt{n}$ reduces from ${\omega }_p^{\mathit{crystal}}$≃ 410 meV to ${\omega }_p^{\mathit{powder}}$≃ 175 meV) and of a factor of ~4 in the scattering rate (from Γ^crystal≃ 18 meV to Γ^powder≃ 4 meV).

Also the spectral weight ${\Omega }_{\mathit{pol}}^2$ of the 55-meV polaronic band is a factor of ~4 higher in the single crystal (${\Omega }_{\mathit{pol}}^{\mathit{crystal}}$≃ 1100 meV against ${\Omega }_{\mathit{pol}}^{\mathit{powder}}$≃ 540 meV). This implies that the quantity ${\omega }_p^2∕{\Omega }_{\mathit{pol}}^2$ extracted for our sample is very close to that reported in Ref. 17, thus indicating that the relative ratio between mobile and localized carriers is the same in powders and single crystals.

Determination of the transient optical properties

THz-frequency probe pulses, generated by optical rectification of 800 nm pulses in a ZnTe crystal, were used to measure the transient response of the sample after excitation. These probe pulses were detected after reflection from the sample-diamond interface in a second ZnTe crystal, yielding pump-induced reflectivity changes for frequencies between 0.75 and 2.5 THz (3.1 - 10.3 meV) and with a pump-probe time-delay resolution of approximately 300 fs.

The pump-induced change in the THz electric field $\Delta E_R(t,\tau )=E_R^{\mathit{pumped}}(t,\tau )-E_R\left(t\right)$ was acquired at each time delay τ by filtering the electro-optic sampling signal with a lock-in amplifier, triggered by modulation of the mid-infrared pump with a mechanical chopper. This measurement yielded “pump on” minus “pump off” reflected electric field.

The differential electric field ΔE_R (t, τ) and the stationary reflected electric field E_R (t) were independently Fourier transformed to obtain the complex-valued, frequency dependent $\Delta {\tilde{E}}_R(\omega ,\tau )$ and ${\tilde{E}}_R\left(\omega \right)$.

Importantly, because the pump-induced changes were large (~2%), the same measurement was repeated by directly recording ${\tilde{E}}_R^{\mathit{pumped}}(\omega ,\tau )$ and ${\tilde{E}}_R\left(\omega \right)$ without chopping the pump, and then calculating $\Delta {\tilde{E}}_R(\omega ,\tau )={\tilde{E}}_R^{\mathit{pumped}}(\omega ,\tau )-{\tilde{E}}_R\left(\omega \right)$. This method does not require calibration of the absolute phase of the lock-in amplifier, and avoids phase errors in estimating the optical properties. The two methods yielded identical results.

In Extended Data Fig. 2, E_R (t) and ΔE_R (t, τ > 0) are displayed both below and above T_c with the corresponding normalized reflectivity changes. For all measured T > T_c and for any τ > 0, all ΔE_R (t, τ) and E_R (t) have the same phase, indicating that above T_c one always finds an increase in reflectivity. On the other hand, all measurements below T_c exhibit an opposite phase, i.e., reflectivity is reduced upon excitation.

The complex reflection coefficient of the photo-excited material, $\tilde{\mathrm{r}}(\omega ,\tau )$, was determined using the relation

$$\frac{\Delta {\tilde{E}}_R(\omega ,\tau )}{{\tilde{E}}_R\left(\omega \right)}=\frac{\tilde{\mathrm{r}}(\omega ,\tau )-{\tilde{\mathrm{r}}}_0\left(\omega \right)}{{\tilde{\mathrm{r}}}_0\left(\omega \right)}$$

To calculate these ratios, the stationary reflection coefficient ${\tilde{\mathrm{r}}}_0\left(\omega \right)$ was extracted at all temperatures from the equilibrium optical properties, determined independently (at the same temperature and in the same sample holder) with synchrotron-based infrared spectroscopy.

These “raw” light-induced changes, which indicate increase and decrease of the reflectivity above and below T_c, respectively, were reprocessed to take into account the penetration depth mismatch between THz probe (L~700-800 nm) and mid-infrared pump (d~200 nm) (Extended Data Fig. 3a). Importantly, this renormalization only affects the size of the response, whereas the qualitative change in optical properties is independent on it and on the specific model chosen.

As in, for example, Ref. 1, this mismatch can be taken into account most simply by modeling the response of the system as that of a homogeneously photo-excited layer with the unperturbed bulk beneath it (Extended Data Fig. 3b). However, this model applies well only in case of large pump-probe penetration depth mismatches (d/L~10²-10³).

A more precise method consists in treating the excited surface as a stack of thin layers with a homogeneous refractive index and describing the excitation profile by an exponential decay⁴ (see Extended Data Fig. 3c). By calculating the coupled Fresnel equations of such multi-layer system, the refractive index at the surface, $\tilde{n}(\omega ,\tau )$, can be retrieved, and from this the complex conductivity for a volume that is homogeneously transformed,

$$\tilde{\sigma }(\omega ,\tau )=\frac{\omega }{4\pi i}[\tilde{n}{(\omega ,\tau )}^2-{\epsilon }_{\infty }].$$

In Extended Data Fig. 3d-f we compare the transient optical properties obtained with the exponential decay model (same curves of Fig. 2a-2c in main text) with those obtained with the single-layer description (having used the same pump penetration depth d = 220 nm). Both treatments yield very similar results, although we consider the exponential model more accurate.

Note that the extracted optical properties from pump-probe data can sometimes be deformed at the earliest time delays by perturbed free induction decay²⁹. This issue was raised recently in discussing transient THz probing of superconductors³⁰ . However, this effect occurs only when coherence is transiently induced on a timescale far shorter than momentum relaxation time and response time of the detector³¹. Here the excitation of the transient high-mobility state is achieved with 300 fs mid-infrared pulses, the risetime of the signal is 1 ps, and the relaxation occurs within 2 – 10 ps. Therefore these deformations are negligible³².

Comparing transient conductivities below and above T_c

Transient optical properties above and below T_c measured at the same time delay after photo-excitation are shown in Extended Data Fig. 9 (τ = 1 ps, same above-T_c data as in Fig. 2a-c in main text).

Importantly, although the equilibrium properties of the unperturbed solid are very different (normal and superconducting state, red and blue curves in Extended Data Fig. 9a-c and d-f, respectively), the two light-induced states are quite similar to one another (light blue curves). In both cases, the state at τ = 1 ps after excitation is gapped.

As shown in Extended Data Fig. 2, these transient properties descend in one case (T > T_c) from a light-induced increase in reflectivity (which saturates to R ≃ 1 at ω < 2Δ), while in the other case (T < T_c), from a decrease in reflectivity (partial gap filling).

The relaxation dynamics of these two states towards equilibrium is shown in Extended Data Figs. 6, 10. Above T_c, the gapped state measured at τ = 1 ps rapidly relaxes into a state with a gapless optical conductivity. The metallic ground state is then fully recovered within ~5 ps. Below T_c, after the early time response of Extended Data Fig. 9, at τ = 3 ps the conductivity gap becomes filled. The superconducting ground state is then recovered over relatively long time scales (> 20 ps).

Uncertainties in determining the transient optical properties

We examine here different sources of uncertainty and their propagation on the transient optical properties. The uncertainty in the absolute value of the measured equilibrium reflectivity is of the order of ±1%. Extended Data Fig. 5a shows as colored bands the propagated error bars in the equilibrium and transient optical properties for a ±1% and a larger ±2.5% uncertainty in R(ω). In both cases the response remains gapped.

We next examine uncertainty in the Fresnel phase coefficient β, used in the Kramers-Kronig transformations. This has been set to β = 217 meV to precisely match the central frequency of the polaronic band (55 meV) with that reported in previous optical studies^17,18 on the same compound. In Extended Data Fig. 5b we show how a ±10% uncertainty in β would affect the equilibrium and transient optical response.

Another possible source of error resides in the pump penetration depth value used to retrieve the transient reflectivities and optical conductivities. This value was set to d = 220 nm for all analyzed data. In Extended Data Fig. 5c we show how a ±25% change in d would affect the transient optical properties.

Finally, in Extended Data Fig. 5d we show the effect of a different choice of functional form for the decay of $\tilde{n}(\omega ,z)$ in the multi-layer model. The exponential decay used for all data analyzed in the paper is compared with a single layer (already introduced above) and with a Gaussian-like decay, all having the same d value.

In all these cases the impact of the different sources of error on the calculated reflectivities and optical conductivities is moderate and the qualitative behavior of the spectra is not affected.

Estimate of the carbon atom displacement

The peak electric field of the mid-infrared pulses can be estimated from

$$E=\sqrt{\frac{F}{2{\epsilon }_{0}c\Delta t}}$$

where c is the speed of light, ε₀ is the vacuum permittivity and Δt is the pulse duration. The fluence F typically used in these experiments is 1 mJ/cm² that, for a pulse duration of approximately 300 fs, corresponds to a peak electric field E = 800 kV/cm.

From previous optical measurements³³, it is known that the real part of the optical conductivity on the pumped T_1u(4) phonon is σ₁[T_1u(4)]=120 Ω⁻¹cm⁻¹ , while its central frequency is ω₀ = 168.62 meV. The pump-induced polarization P is then given by:

$$P=\frac{{\sigma }_1\left({\omega }_0\right)}{{\omega }_0}E=2.35\cdot {10}^{-6}{}^{\mathrm{C}}∕_{{\mathrm{cm}}^2}$$

The polarization arises due to a light-induced dipole moment P = n · δ · Z_eff e, where n is the number of dipoles per unit volume, Z_eff e is the effective charge defined as in Ref. 34 and δ is the atomic displacement. The Z_eff determined for this vibration is 0.147. Under the assumption that all the incident photons are contributing to the excitation, one can estimate from the polarizability a displacement δ = 0.24 Å, i.e. approximately 17% of the C-C bond length.

Mode coupling and electronic structure calculations

Electronic structure, lattice dynamics, and nonlinear phonon couplings after excitation of the T_1u(4) phonon were obtained using density functional theory (DFT) calculations with plane-wave basis set and projector augmented wave pseudopotentials (VASP software package)³⁵. These calculations were performed within the local density approximation (LDA) with a 900 eV cut-off for the plane-wave basis set. A 6×6×6 k-point grid and a 0.1 eV Gaussian smearing were used for the Brillouin-zone integration during the self-consistency. Lattice parameters and atomic positions were obtained by energy minimization, ignoring the C₆₀ orientational disorder. The calculated lattice constant is 13.888 Å, in good agreement with the experimental value of 14.240 Å.

Phonon frequencies and eigenvectors were determined using the frozen-phonon approach (PHONOPY software package)³⁶. After the normal modes were identified, total energy curves as a function of the Raman mode amplitude Q were calculated while simultaneously keeping the amplitude of the pumped T_1u(4) mode finite. A shift of the Raman mode energy minimum for finite amplitude of the T_1u(4) vibration allowed to identify a cubic q_T1u² Q coupling between the pumped infrared mode and the Raman vibration (see Extended Data Fig. 7a).

Electron-phonon calculations were performed using DFT-LDA (Quantum ESPRESSO software package)³⁷ with 42 and 420 Ry cutoffs for plane wave basisset and charge-density expansions, respectively. A 6×6×6 k-point grid and a 0.020 Ry Gaussian smearing were used for Brillouin-zone integration during the self-consistency.

The majority of Raman modes couple to the T_1u(4) mode. In Extended Data Fig. 7a we show the shift in the H_g(1) mode minimum when the amplitude of the T_1u(4) mode is 2.0 Å √amu. Such a shift implies that the structure is displaced along the H_g(1) coordinate while the T_1u(4) mode is pumped. We also report the band structure, electronic density of states (Extended Data Fig. 7b), and electron-phonon coupling (Extended Data Fig. 7c) for a structure displaced along the H_g(1) coordinate with an amplitude of 1.5 Å √amu. We find that such displacement slightly lifts the degeneracy of the t_1u states. However, these bands remain part of a single manifold, crossing each other at various points along the high-symmetry path of the Brillouin zone.

We also find a modest change in density of states, which is caused by the shift of a van Hove singularity near Γ as the bands are moved by the H_g(1) mode displacement. Furthermore, the bandwidth of the t_1u manifold steadily increases as a function of H_g(1) mode amplitude. This is reasonable since the H_g(1) displacements elongate the C₆₀ molecule and reduces the intermolecular distances, thus increasing the intermolecular hopping.

The H_g(1) displaced structure develops strong coupling between electrons and intermolecular phonon modes (Extended Data Fig. 7c). At equilibrium the three intermolecular modes with 5 meV energy do not couple to the t_1u electrons, while under displacement along the H_g(1) coordinate two of these modes exhibit very large electron-phonon coupling.

The electron phonon coupling, obtained by integrating the Eliashberg function α²F(ω), is 0.52 at equilibrium and 1.44 for the H_g(1) displaced structure. Energy-resolved changes in the electron–phonon coupling function, evaluated as Δα²F(ω)=α²F(ω)H_g(1)-α²F(ω)_equilibrium, are reported in Extended Data Fig. 7d. The increased coupling at low energy and a strong reshaping at high frequency (>130 meV) are apparent.

Time-dependent onsite correlation energies

We estimated the magnitude of changes to the inter- and intra-orbital Coulomb repulsion of the t_1u orbitals on a single C₆₀ molecule when the T_1u(4) vibrational mode is driven. The relevant C₆₀ molecular orbitals are described by pπ orbitals projecting radially outwards at each C atom. A tight-binding Hückel model captures the hybridization of these orbitals in the C₆₀ cage. Hopping amplitudes are assumed to vary with the distance d between C atoms as

$$V\left(d\right)=\frac{\Lambda }{d^2},$$ (1)

where Λ = 5.63 eV Ų. We take the C-C and C=C bond lengths as d_s = 1.45 Å and d_D = 1.37 Å, corresponding to V_S = 2.69 eV and V_D = 3.00 eV hopping integrals for the pentagonal-hexagonal and hexagonal-hexagonal edges of the C₆₀ cage. The x, y and z t_1u orbitals are the degenerate 31^st-33^rd single electron eigenstates of the Hückel model and the lowest unoccupied molecular orbitals of free C₆₀³⁸. In Extended Data Fig. 8a, the wavefunction (chosen to be real) for each t_1u orbital is plotted over the bucky-ball with the size of the spheres denoting the magnitude and the color denoting the sign.

C atom motions corresponding to the T_1u(4) vibrational mode were taken from a complete nearest-neighbor force field model calculation for C₆₀, optimized to fit the optically accessible frequencies of the molecule³⁹. We applied a “frozen-phonon” approach, which assumes the Born-Oppenheimer approximation to be strictly obeyed and the molecular orbitals adiabatically following the atomic distortions. Although this model is not appropriate for high-frequency vibrations and low-energy electronic properties, we posit that this approach can provide the order of magnitude of the response. The t_1u orbitals were then computed for T_1u(4) distorted configurations of C₆₀ using the Hückel model with hopping integrals varied according to Eq. (1). Snapshots of the driven t_1u(Z) orbital are shown in Extended Data Fig. 8b.

The Hubbard U’s on each C₆₀ site are estimated from

$$U=F^0=\int d^3{r}_1\int d^3{r}_2\frac{\rho \left({\mathit{r}}_1\right)\rho \left({\mathit{r}}_2\right)}{4\pi {ϵ}_{\text{eff}}{ϵ}_0\mid {\mathit{r}}_1-{\mathit{r}}_2\mid }$$ (2)

where ρ(r) is the charge density corresponding to a t_1u orbital.

This bare U will be reduced due to screening from valence electrons, intramolecular correlations and metallic background in a solid. However, since we are concerned with relative changes in U this renormalization is not crucial. We estimated F⁰ by assuming point charges on each C atom and calculating their mutual interactions as

$$F^0=\sum _{i\ne j}\frac{p_i{p}_j{e}^2}{4\pi {ϵ}_{\text{eff}}{ϵ}_0\mid {\mathit{r}}_1-{\mathit{r}}_2\mid }+\sum _i{p}_i^2{F}_C^0$$ (3)

where p_i is the probability given by a t_1u orbital of an electron being on the ith C atom, and $F_C^0\sim 12$ eV is the atomic on-site repulsion⁴⁰. The variation of U with the vibrational driving was then obtained using the p_i’s of the driven t_1u orbitals and is shown, together with the single-particle energies of the orbitals over a vibration period, in Extended Data Fig. 8c-d. The amplitude of the vibrational driving A denotes the maximum displacement of any C atom. For A = 5 pm, both quantities in the y, z t_1u orbitals vary by nearly 10% at twice the frequency of the vibration.

Since the vibration is polarized along the x axis the y, z orbitals remain degenerate and possess identical U’s. The modulation in y, z is significantly larger than that in x because these orbitals have the equatorial distribution running across the vibrational axis, causing their charge distribution to compress at the poles of the x axis (Extended Data Fig. 8b).

The modulation magnitudes as a function of vibrational driving amplitude are shown in Extended Data Fig. 8e-f. An amplitude A = 5 pm is consistent with the strong vibrational driving considered in this work.

Extended Data

Extended Data Figure 1. K₃C₆₀ sample characterization.

a, Powder X-ray diffraction of K₃C₆₀ (black circles) fitted with a single fcc phase Rietveld refinement (red). b, Temperature dependent magnetic susceptibility (FCC: Field cooled cooling. ZFC: Zero field cooling). The extracted superconducting critical temperature is T_c = 19.8 K.

Extended Data Figure 2. “Raw” electric field transients below and above T_c.

a-d, Stationary electric field reflected at the sample-diamond interface and pump-induced changes in the same quantity, measured at τ = 3 ps in a-b and at τ = 1 ps in c-d. Data are shown both below and above T_c. e-f, Corresponding frequency-dependent differential changes in reflectivity, calculated as Fourier transform magnitude ratios of the quantities in a-d.

Extended Data Figure 3. Models for penetration depth mismatch.

a, Schematics of pump-probe penetration depth mismatch. Single-layer model (b) and multi-layer model with exponential decay (c) (see Methods section) used to calculate the pump-induced changes in the complex refractive index $\tilde{n}\left(\omega \right)$. d-f, Reflectivity and complex optical conductivity of K₃C₆₀ at τ = 1 ps pump-probe delay and T = 25 K, extracted using the single-layer model (light blue) and the multi-layer model with exponential decay (dark-blue).

Extended Data Figure 4. Equilibrium optical properties.

a, Reflectivity, real and imaginary part of the optical conductivity of K₃C₆₀ displayed at different temperatures above T_c. Dashed lines are fits to the 25 K data performed with a Drude-Lorentz model. b, Same quantities displayed at different T < T_c. In the inset, the temperature dependent optical gap (full circles) is compared with previously published data on K₃C₆₀ single crystals (open circles, from Ref. 18). c, Optical properties of K₃C₆₀ compressed powders from a and b shown at representative temperatures below and above T_c. The R(ω) has been recalculated at the sample-vacuum interface. d, Same quantities as in c, measured on single crystals by L. Degiorgi et al.¹⁸.

Extended Data Figure 5. Uncertainties in determining the transient optical properties.

Reflectivity and complex optical conductivity of K₃C₆₀ at equilibrium (red) and 1 ps after photo-excitation (blue) at T = 25 K. Error bars, displayed as colored bands, have been propagated from (a) a ±1% and ±2.5% uncertainty in the equilibrium R(ω); (b) a ±10% uncertainty in the equilibrium Fresnel phase coefficient β (see Methods); (c) a ±25% change in the pump penetration depth d = 220 nm. In d we analyze the effect of different functional forms for modeling the pump-probe penetration depth mismatch (see Methods): Single-layer model, multi-layer model with exponential and Gaussian-like decay, all with the same pump penetration depth d = 220 nm.

Extended Data Figure 6. Relaxation dynamics at T>T_c.

Reflectivity and complex optical conductivity of K₃C₆₀ at equilibrium (gray) and after photo-excitation (red) at T = 25 K. Data have been measured with a pump fluence of ~1 mJ/cm² and are shown at selected pump-probe time delays: 1.5 ps, 5.5 ps, and 21 ps.

Extended Data Figure 7. Mode coupling and electronic structure calculations.

a, Calculated total energy curves as a function of H_g(1) mode amplitude when the amplitude of the T_1u(4) mode is 0.0 (blue) and 2.0 Å √amu (red). b, Calculated band structure and electronic density of states of K₃C₆₀. c, Calculated electron-phonon coupling function α²F(ω). In b-c, blue lines are for the equilibrium structure and red lines are for the structure displaced along the H_g(1) coordinate with an amplitude of 1.5 Å √amu. d, Differential changes in the electron-phonon coupling function, evaluated from the curves in c.

Extended Data Figure 8. Changes in time of single particle energy and Coulomb repulsion.

a, Depiction of the x, y, z t_1u orbital wavefunctions of C₆₀ according to the Hückel model (see Methods). b, Snap-shots of the calculated t_1u(z) orbital at various points in the T_1u(4) vibration polarized along the x axis. Color and sizes follow those of a. c, Changes in the single-particle energies of the t_1u orbitals over one period of the T_1u(4) vibration with an amplitude A = 5 pm. d, Relative changes in the intra-orbital Coulomb repulsions. e, Maximum relative changes in the single-particle energies of the t_1u orbitals as a function the driving amplitude A. f, Relative changes in the intra-orbital Coulomb repulsions under the same driving conditions.

Extended Data Figure 9. Response at early time delays below and above T_c.

Reflectivity and complex optical conductivity of K₃C₆₀ at equilibrium and 1 ps after photo-excitation, measured at T > T_c (a-c) and T < T_c (d-f). Data were taken using pump fluences of ~1 mJ/cm² in a-c and ~0.5 mJ/cm² in d-f.

Extended Data Figure 10. Relaxation dynamics at T<T_c.

Reflectivity and complex optical conductivity of K₃C₆₀ at equilibrium (gray) and after photo-excitation (red) at T = 10 K. Data have been measured with a pump fluence of ~0.5 mJ/cm² and are shown at selected pump-probe time delays: 3 ps, 11 ps, and 21 ps.