## Main

Non-equilibrium orders in complex materials include photo-induced ferroelectricity1,2, magnetic polarization in antiferromagnets3 and transient superconductivity in the normal state of cuprates and organic conductors4,5,6,7,8,9. Among these, much work has been dedicated to alkali-doped fullerides of the A3C60 family (Fig. 1a), which exhibit tunable high-temperature superconductivity at equilibrium10,11,12,13,14,15 (Fig. 1b). The dynamical manipulation of superconductivity in these materials6,7 has been demonstrated by using optical pulses at mid-infrared frequencies that are tuned in the vicinity of local vibrational resonances of the C60 molecules.

The evidence reported so far is summarized in Fig. 1c,d. K3C60 powders were held at a base temperature of $$T = 100\,{\mathrm{K}} \gg T_{\mathrm{c}} = 20\,{\mathrm{K}}$$ and irradiated with 100-fs-long, 7.3-µm-wavelength ($$\hbar {\omega}\approx 170\,{\mathrm{meV}}$$, where ħ is Planck’s constant and ω is angular frequency) pulses at a fluence of 3 mJ cm−2, which yielded a short-lived transient state with large changes in the terahertz optical properties. The transient optical response was probed with phase-sensitive, time-domain terahertz spectroscopy, which directly yields the real and imaginary part of the reflectance and hence can be used to retrieve the complex optical conductivity without the need of Kramers–Kronig transformations (Supplementary Section 4 and refs. 6,7). The transient low-frequency optical conductivity induced by femtosecond excitation was almost indistinguishable from that of the equilibrium superconducting state measured in the same material at $$T \ll T_{\mathrm{c}} = 20\,{\mathrm{K}}$$ (compare to Fig. 1c). Such ‘superconducting-like’ optical properties consist of a perfect low-energy reflectance (R = 1), a vanishingly small real part of the optical conductivity σ1(ω) for all photon energies lower than the energy gap 2Δ and an imaginary conductivity σ2(ω) that diverges toward low frequencies as 1/ω, which itself is indicative of a large zero-frequency conductivity.

These observations have generated interest, as they may make it possible to achieve photo-induced superconducting states4,5,6,7,9,16,17 at or in the vicinity of room temperature. However, all of the experiments reported have indicated the presence of states that disappear immediately after optical excitation (see optical properties measured at 5 ps time delay in Fig. 1d). These short lifetimes would prevent most applications and have even raised controversy over the data interpretation itself18,19.

In this Article, we explore the possibility of longer-lived superconductivity under a sustained optical drive. First, we modified the experimental set-up used for the experiments shown in Fig. 1 and lengthened the 7.3 µm wavelength pump pulses by making them propagate in a dispersive CaF2 rod (Fig. 2a). These chirped pump pulses had a duration τp ≈ 1 ps, which enabled a sixfold increase in the pulse energy density (up to 18 mJ cm−2) without a corresponding increase in the peak electric field. This longer pulse duration allowed us to conduct these experiments at pulse energy densities that would damage the sample at femtosecond pulse durations, and hence it enabled the exploration of a new regime of excitation.

The pump-induced changes in the low-frequency reflectivity and complex optical conductivity (Fig. 2b) were measured on the same sample and at the same temperature T = 100 K as for the experiments of Fig. 1. Representative reflectivity spectra R(ω) and complex optical conductivity, σ1(ω)+2(ω), measured for a 1 ps pump-pulse duration and 18 mJ cm−2 fluence are reported for time delays of –5 ps, 10 ps, 300 ps and 12 ns. These plots indicate a similar response to that shown in Fig. 1d (perfect reflectivity (R = 1), gapped σ1(ω) and a divergent σ2(ω)), although with a far longer relaxation time of at least 300 ps.

A more comprehensive exploration of sustained optical driving, beyond the limited pulse-width tunability of the set-up shown in Fig. 2, is shown in Figs. 3, 4 and 5. These measurements made use of a newly developed optical device, based on a CO2-gas laser that was optically synchronized to a femtosecond Ti:Al2O3 laser, and that delivered 10.6 µm wavelength pulses with durations that could be tuned between 5 ps and 1.3 ns (Fig. 3a).

Near-infrared femtosecond pulses from a Ti:Al2O3 laser were converted to a wavelength of 10.6 µm with an optical parametric amplifier and were used to seed a CO2 laser oscillator20. The oscillator emitted trains of nanosecond-long pulses, out of which the most intense was selected by a Pockels cell and amplified to an energy of 10 mJ in a second, multi-pass CO2 laser amplifier. The duration of these amplified pulses was then tuned as shown in Fig. 3a. The ‘front’ and the ‘back’ of the 1-ns-long CO2 pulses were ‘sliced’ using a pair of photoexcited semiconductor wafers as plasma mirrors21,22,23. As the wafers were set at Brewster’s angle and their bandgap was much larger than the 117 meV photon energy of the CO2 laser, these were almost perfectly transparent when unexcited. Pairs of femtosecond optical pulses struck each wafer at adjustable time delays, making these reflective owing to the injection of dense electron hole plasmas. Pump pulses with a duration tunable from 5 ps to 1.3 ns were generated in this way and used to pump the K3C60 sample (Supplementary Section 3), which was probed with the same time-domain terahertz reflectivity probe used for the experiments of Figs. 1 and 2.

Figure 3b displays snapshots of the transient optical properties (R(ω), σ1(ω), σ2(ω)) measured for K3C60 at T = 100 K before photoexcitation and 100 ps and 1 ns after photoexcitation with a 300-ps-long pulse centred at a wavelength of 10.6 µm. As these measurements were obtained for a repetition rate of 18 Hz, the signal-to-noise ratio is reduced compared with that of the cases shown in Figs. 1 and 2, which were measured at 500 Hz.

The transient optical spectra measured in these conditions showed the same superconducting-like features as reported in Figs. 1d and 2b for all pump-pulse durations up to 1 ns after excitation. Note that the 10.6 µm wavelength radiated by the CO2 laser is different from the 7.3 µm wavelength used in the experiments of Figs. 1 and 2. However, excitation with femtosecond optical pulses at this wavelength had previously been shown to induce the same transient optical signatures generated with 7.3 µm wavelength excitation, although with a lower efficiency6.

The time evolution of the terahertz optical properties is shown in Fig. 3c. The top panel shows the average reflectivity R(ω) in the region where σ1(ω) exhibited a gap (2–10 meV). The lower panel shows the average value of the corresponding real part of the optical conductivity σ1(ω), which reached zero after optical excitation, reflecting full gapping. Both quantities remained unchanged after excitation for all time delays measured up to 1 ns. Extended measurements indicate a lifetime of the light-induced superconducting state of τd ≈ 10 ns (Supplementary Section 11).

From the optical spectra, we also extracted an estimate of the ‘zero-frequency resistivity’ $$\rho _0 = 1/\lim _{\omega \to 0}\sigma _1\left( \omega \right)$$, which is based on a Drude–Lorentz fit to the transient optical properties (Supplementary Section 6). This fitting procedure yielded a vanishingly small ρ0(t) for all time delays after excitation (Fig. 4a).

Estimates of ρ0 from optical measurements were complemented with direct electrical measurements. The K3C60 pellets were incorporated into lithographically patterned microstrip transmission lines. Their resistance was tracked at different times after excitation by transmitting a 1 ns voltage probe pulse that yielded time-resolved two-terminal resistance measurements (Supplementary Sections 7 and 8). The contributions due to contact resistance were normalized by performing equilibrium four-terminal measurements that were subtracted from the time-resolved resistance measurements (Supplementary Section 8). Figure 4b shows the time evolution of the two-terminal resistance of a K3C60 pellet measured at T = 100 K upon photoexcitation in similar conditions to those of the optical experiments reported in Fig. 3. As seen in the resistivity extrapolated from the optical measurements (compare to Fig. 4a), upon excitation the resistance drops to a value that is compatible with zero and recovers on the same timescale of tens of nanoseconds that was extracted from the fitted results of Fig. 4a (Supplementary Section 11).

The electrical probe experiments were repeated by varying the excitation pulse duration and fluence. Figure 5a shows exemplary pump-pulse duration dependencies of the measured sample photoresistivity for excitation fluences of 1.5 mJ cm−2, 10 mJ cm−2 and 25 mJ cm−2. The photoresistivity was mostly independent of the pump-pulse duration and depended only on the total energy of the excitation pulse. This is also underscored by the data shown in Fig. 5b, which illustrates the dependence of the sample resistance on the excitation fluence at a constant pulse duration. A long-lived state featuring zero resistance was observed for all excitation fluences in excess of 20 mJ cm−2.

These data provide evidence for a metastable state of K3C60 with a very large positive photoconductivity. The observation of such a large and positive photoconductivity would be highly unconventional for a metal, which generally exhibits photoconductivities of less than 1% (refs. 24,25,26) and which are often negative27, especially when excitation is performed in the mid-infrared and far away from inter-band transitions. Rather, the combined observation of a vanishingly small resistance with the transient terahertz optical spectra shown in Fig. 3 substantiates the assignment of a metastable superconducting state.

Note that the photo-induced high-temperature state survives far longer than the drive pulse, and hence exhibits intrinsic rigidity at timescales when the coherence is no longer supplied by the external drive. In search for a mechanism behind this long lifetime, we applied a phenomenological time-dependent Ginzburg–Landau model to capture the dynamics of the superconducting order parameter after laser excitation. In this model, we did not address the microscopic mechanism for the formation of superconducting order, but posited its emergence under optical excitation for a base temperature far in excess of the equilibrium transition temperature Tc. As argued in ref. 28, the normal state of unconventional superconductors that are susceptible to being ordered with light may be that of a phase-incoherent bosonic fluid, in which superconducting fluctuations are present already. Hence, we consider a hypothetical situation in which phase synchronization is established by a light field, and we study how one such order is lost after the driving pulse has been turned off.

In our time-dependent Ginzburg–Landau simulations, a local superconducting order parameter $$\psi _m\left( t \right) = \left| {\psi _m} \right|e^{i\varphi _m}$$ was assigned on each lattice site. The average of these wave functions through the whole lattice $$\left\langle {\psi _m} \right\rangle$$ described the macroscopic properties of the system (Supplementary Section 13). The relaxation of a superconducting state $$\left( {\left| {\left\langle {\psi _m} \right\rangle } \right| > 0} \right)$$ to the equilibrium metallic one $$\left( {\left| {\left\langle {\psi _m} \right\rangle } \right| = 0} \right)$$ can happen either by a fast decrease of the amplitudes of the local order parameter |ψm| (that is, by annihilation of Cooper pairs) or by randomization of the order-parameter phase φm. By construction, the latter dominates in a phase-incoherent superconductor, in which phase fluctuations are considerably larger than the amplitude ones. In this case, the local free energy surface shows a minimum, even above Tc, at a finite local order-parameter amplitude |ψm| and suppresses amplitude fluctuations (Fig. 6a). We found that, whereas the relaxation to the non-superconducting ground state occurs by thermally driven diffusion of the local phases, the synchronized state can survive considerably longer than that observed in many cases that consider amplitude relaxation only29,30.

Figure 6b displays the time dependence of the integrated spectral weight loss over the optical gap $${\Delta}\sigma _1 = {\int} {\left[ {\sigma _1^{{\mathrm{equil}}}\left( \omega \right) - \sigma _1^{{\mathrm{trans}}}\left( \omega \right)} \right]{\rm{d}}\omega }$$, a quantity that is proportional to the superfluid density in a superconductor. Here, $$\sigma _1^{{\mathrm{trans}}}\left( \omega \right)$$ and $$\sigma _1^{{\mathrm{equil}}}\left( \omega \right)$$ are the real parts of the optical conductivities measured in the photo-induced and equilibrium state, respectively. This quantity can be compared to the normalized amplitude of the averaged order parameter |ψm| that is extracted from the time-dependent Ginzburg–Landau simulations (Supplementary Section 13), yielding a relaxation time of ~12 ns. Figure 6c–e complements this observation and shows the time evolution of the local complex order parameters ψm that were extracted from the same simulations 0 ns, 1 ns and 25 ns after excitation. These snapshots show directly how in this model the local order parameters ψm evolve by randomizing their phase φm around a ring at constant |ψm| until the metallic state at $$\left| {\left\langle {\psi _m} \right\rangle } \right| = 0$$ is reached.

In this context, questions remain about the microscopic origin of the metastable light-induced superconducting state. From the observation that the amplitude of the effect is dependent only on the integrated pulse area, it becomes clear that, at least in the long-pulse regime, the previously proposed non-linear phonon mechanism6, which was suggested to generate a displaced crystal structure, is most likely not correct. If it were correct, one would expect a response that depends on some power of the electric field, rather than on only the total energy of the pulse. That said, mechanisms based on the parametric coupling of the light field to the electronic properties, either to amplify a superconducting order parameter31,32,33 or to cool selected degrees of freedom34,35, are not necessarily inconsistent with these observations.

Our findings prompt comparison with previously measured responses that were observed under sustained driving, such as the microwave enhancement of conventional superconductivity36,37. One evident difference with these measurements is that the temperature scale observed here is far larger than that reported in the microwave-enhancement case. Furthermore, in the case of microwave-enhanced Bardeen–Cooper–Schrieffer superconductivity, the effect was observed only for excitation below the superconducting gap, which was interpreted to be a result of quasi-particle redistribution and of renormalization of the parameters entering the Bardeen–Cooper–Schrieffer equations38. In the microwave experiments, complementary measurements, in which excitation was tuned immediately above the gap, yielded a reduction of the superconducting order. Here, our experiments are conducted in a different regime; that is, at pump-photon energies that are at least one order of magnitude larger than the low-temperature equilibrium superconducting gap, and in a regime in which the primary coupling of the mid-infrared radiation is not with the condensate, but with other high-energy excitations, such as molecular vibrations or collective electronic modes. Hence, it is unlikely that the mechanism invoked for microwave enhancement can explain these observations.

Other experimental reports have documented a sustained, metastable enhancement of superconductivity in oxygen-deficient YBa2Cu3O7−δ samples after exposure to radiation with frequencies ranging from the ultraviolet to the near-infrared39,40,41. However, in all of these cases, the superconducting transition temperature of the irradiated superconductor never exceeded the equilibrium transition temperature at optimal doping. These observations were either interpreted as a result of photodoping towards a more metallic state favouring superconductivity, or involved annealing of oxygen-deficient samples. By contrast, the data reported here show enhanced superconducting properties well above the highest equilibrium transition temperature observed in any one of the A3C60 compounds13, by excitation with photon energies away from inter-band transitions. Therefore, none of the previously invoked mechanisms offers a plausible explanation for our findings.

The discovery of a metastable light-induced state with clear signatures of superconductivity holds compelling promise in the quest to extend lifetimes even further. New lasers that are capable of generating longer pulses or suitably designed trains of pulses could be developed to sustain the coherence of this state. Extended lifetimes will also open up the possibility of studying these effects with other low-frequency probes, which range from measurements of magnetic susceptibility to scattering and transport methods. Recent theoretical42 and experimental43 reports suggest that the superconducting order parameter can be influenced also by the electromagnetic environment of an optical cavity. Indeed, observations made in alkali-doped fullerides43 could be expanded on by combining cavity settings with external driving44 as a means to reduce the required excitation and hence dissipation and heating.

## Methods

All methods can be found in the Supplementary Information.