Terahertz-light quantum tuning of a metastable emergent phase hidden by superconductivity

Abstract

‘Sudden’ quantum quench and prethermalization have become a cross-cutting theme for discovering emergent states of matter1,2,3,4. Yet this remains challenging in electron matter5,6,7,8,9, especially superconductors10,11,12,13,14. The grand question of what is hidden underneath superconductivity (SC)15 appears universal, but poorly understood. Here we reveal a long-lived gapless quantum phase of prethermalized quasiparticles (QPs) after a single-cycle terahertz (THz) quench of a Nb3Sn SC gap. Its conductivity spectra is characterized by a sharp coherent peak and a vanishing scattering rate that decreases almost linearly towards zero frequency, which is most pronounced around the full depletion of the condensate and absent for a high-frequency pump. Above a critical pump threshold, such a QP phase with coherent transport and memory persists as an unusual prethermalization plateau, without relaxation to normal and SC thermal states for an order of magnitude longer than the QP recombination and thermalization times. Switching to this metastable ‘quantum QP fluid’ signals non-thermal quench of coupled SC and charge-density-wave (CDW)-like orders and hints quantum control beneath the SC.

Main

Exotic states in correlated materials have been discovered by traditional tuning methods, such as chemical substitution, applied pressure or magnetic fields. These methods correspond to slow changes of parameters g in the Hamiltonian H(g) and mostly access states of thermodynamic equilibrium, as illustrated in Fig. 1a. The availability of single-cycle intense THz pulses (red arrow in Fig. 1a) opens fascinating possibilities for the non-thermal and non-adiabatic modification of Hamiltonians, in analogy to parameter quenches3,4 in ultracold atoms. During postquench prethermalization time evolution, the system can reach persisting plateau states that are far from equilibrium1. Such states are inaccessible by conventional tuning or high-frequency optical pumping methods. In the latter case, the induced heating and dissipation couple excited QPs with, for example, thermal baths of hot phonons, which smear out the quantum effects. When applied to superconductors with competing orders, a THz quench of the dominant SC phase without heating other degrees of freedom allows us to discover exotic prethermalized quantum phases and pre-empted ‘hidden’ orders beneath SC (Fig. 1a). This has not yet been demonstrated, despite recent progress in THz-driven dynamics, for example, collective modes11,12 excited by a weak quench and light-enhanced SC in cuprates13,14. A strong THz quench field, one order of magnitude higher than prior studies10,11,12, is desired to drive phase transitions16,17, but remains scarce so far. Prior studies also indicate photoinduced, spatially inhomogeneous SC/metallic phases due, in part, to dirty limit samples10,18.

Fig. 1: Single-cycle THz quantum quench and phase transition in a Nb3Sn A15 superconductor.
figure1

a, Schematic of an out-of-equilibrium quantum-tuning scheme via non-thermal quenching of the SC order \(\left|{\psi }_{{\rm{SC}}}\right\rangle\) to discover a hidden phase, marked as \(\left|{\psi }_{{\rm{Plateau}}}\right\rangle\). b, A typical single-cycle THz quench electric field in the time domain. c, Quench-field spectrum (black) with a central frequency well within the 2ΔSC gap shown by σ1(ω) at 4.1 K. σ1(ω) at zero frequency is marked by a red arrow and is proportional to the superfluid density ns. d,e, The complex conductivity is shown as σ2(ω) (d) and σ1(ω) (e). Insets: n s (d) and 2D false-colour plot of the static transmission spectrum overlaid by the extracted 2ΔSC gap (e) at different temperatures. f, A 2D false-colour plot of a THz pump-induced change under a peak pump field of ETHz = 120 kV cm–1 in a THz probe electric field, ΔE(tgatetpp). The normalized temporal profile of ΔE/E0, measured at tgate = −0.08 ps (inset, red line), closely follows the dynamic superfluid density change Δns/ns.

Here we present evidence of a quantum-quench phase transition to a gapless, prethermalized QP phase \(\left|{\psi }_{{\rm{Plateau}}}\right\rangle\) in a Nb3Sn superconductor. Subpicosecond single-cycle THz fields up to 620 kV cm–1 (Fig. 1b) non-adiabatically excite QPs without excessive heating, with central frequency ħω = 4 meV (Fig. 1c) slightly below twice the QP excitation gap, 2ΔSC ≈ 5.1 meV at 4.1 K. Such non-thermal depletion of the SC condensate is implemented on timescales comparable to SC gap fluctuation times of ħ/2ΔSC ≈ 0.8 ps in a nearly single-crystal Nb3Sn film. A sample of 20 nm thickness on a 1 mm Al2O3(10–12) substrate exhibited a SC transition at Tc ≈ 16 K and a competing martensitic transition around T M  ≈ 47 K (Supplementary Information). The latter has been attributed to optical phonon condensation (‘dimerization’ of Nb atoms)19, possibly driven by a Van Hove singularity-like electronic density of states and by strong electron–phonon interaction20,21,22,23,24,25,26. The equilibrium frequency-dependent complex conductivities σ1(ω,T) and σ2(ω,T) measure the dissipative and inductive responses, respectively, shown in Fig. 1d,e. Specifically, the equilibrium 4.1 K line shape exhibits a large 1/ω SC response in σ2 and zero conductivity in σ1 below 2ΔSC. These features change in the normal state, which displays a Drude-like behaviour in which σ2(ω) gradually decreases towards low frequency and σ1(ω) shows no gap (16 K traces).

To characterize the non-equilibrium postquench states we measured the complex conductivity (σ1(ωtpp), σ2(ωtpp)) as a function of the pump–probe delay Δtpp (Fig. 1f). To obtain this, we first constructed a two-time THz pump and THz probe map of the induced THz probe field transmission through the excited sample, ΔE(tgatetpp), by scanning both gating pulse delay tgate and Δtpp (Methods). The top panel of Fig. 1f shows an example of raw data at 4.1 K for ETHz = 120 kV cm–1. The time-dependent dielectric response functions were then retrieved at each Δtpp via Fourier transformation along the tgate axis (Methods and Supplementary Information). The low-frequency spectra Δσ2(ω) = Δnse2/ reflect the pump-induced change in the superfluid density Δns/ns (Fig. 1f). The peak of the differential THz transmission ΔE/E0 at tgate = −80 fs (Fig. 1f, inset) closely follows the above-obtained Δns/ns, in both dynamics and magnitude. This indicates that the measured ΔE/E0 originates from condensate quench and recovery.

Figure 2 demonstrates two regimes of postquench states obtained by tuning, via ETHz, the ‘distance’ from the equilibrium SC state \(\left|{\psi }_{{\rm{SC}}}\right\rangle\). At T = 4.1 K and Δtpp = 12 ps, the red arrows in Fig. 2a mark two regimes: (1) a partial quench regime at fields E1E3 = 47, 120 and 155 kV cm–1, respectively, in which −Δns/ns ΔE/E0 shows a suppression of ns and decays on a subnanosecond timescale (Fig. 1f); (2) a strong quench regime at E4E6 = 256, 446 and 620 kV cm–1, in which the non-linear saturation of the SC condensate density to a minimum is accompanied by a long-lived prethermalization plateau that lasts for ~10 ns timescales (discussed further in Fig. 3a). Quantum quenching of the Bardeen–Cooper–Schrieffer (BCS) pairing interaction has been predicted to induce a non-thermal transition to a gapless coherent state with exotic correlations3,4. Although it differs from the normal state27, the two have indistinguishable THz conductivities in dirty-limit superconductors28. Here we observed a new upturn feature in both the non-equilibrium σ1(ω) and σ2(ω) at low frequencies that differs from the equilibrium (Fig. 2b). First, a non-equilibrium σ1(ω) (upper panel) of postquench states (filled circles in Fig. 2b) displays a remarkable coherent QP transport peak, which manifests as a sharp increase towards zero frequency and a 10% larger integrated spectral weight near EF (1–10 meV) as compared to the normal state (grey diamonds in Fig. 2b). Such a coherent peak in σ1 is already observed in the partial-quench regime, in which it can arise from condensate coherence factors. Most intriguingly, it even persists in regime 2 above the non-linear saturation of Δns/ns to its minimum or zero value, for example, at E4E6 (blue, purple and red circles, respectively). The persisting σ1 peak in a gapless state could arise, for example, from SC fluctuations. Second, the lower panel in Fig. 2b shows a ~1/ω divergence in σ2(ωtpp) (filled circles) that, for E1E3 in regime 1, nearly coincides with that of the equilibrium states at various temperatures (grey diamonds), except a low- energy kink of 2–3 meV that could be correlated to the sharp upturn in the non-equilibrium σ1(ω) line shape. As the Δns/ns reduction saturates at E4 = 256 kV cm–1, the 1/ω dependence in σ2 (filled circles in Fig. 2b) is suppressed. The remaining upturn differs from the residual condensate line shape at any temperature, for example, 14–16 K close to Tc in the E4 panel. The σ2 suppression in the deep non-linear saturation regime, E5E6, is consistent with the strong SC quench, and a possible residual upturn shows up below ~1 meV outside our spectral window, which is expected to correlate with the persisting σ1 peak. More importantly, as shown in Fig. 2c, the coherent σ1 peak disappears for high-frequency pump pulses tuned at 1.55 eV (at a threshold fluence of 4 μJ cm–2 (black lines)), unlike in prior studies9,29, or when quenching above Tc (T = 18 K and 620 kV cm–1 (red lines). In addition, the observed non-equilibrium line shape is clearly distinct from an inhomogeneous mixed SC/normal state (Supplementary Fig. 2) and beyond the predictions of the standard BCS quench model3,4,28. These distinct features could arise from order-parameter fluctuations in a gapless QP quantum phase, tuned by THz pulses out-of-equilibrium, that emerges after non-thermal ns depletion to its minimum (or zero) value (Fig. 2a) and evolves as a long-lived prethermalization plateau, as discussed below.

Fig. 2: The distinct spectral features of the gapless quantum state differ from both normal metallic states and thermal behaviours.
figure2

a, Non-linear pump-field dependence of a peak–peak probe electric field transmission change ΔE/E0 for fields up to 620 kV cm–1. b, THz response functions, expressed as σ1 and σ2, of the postquench states (filled circles) at the various pump electric fields marked in a, which correspond to partial (E1E3) and strong (E4E6) SC order quench regimes. Shown together are the equilibrium responses σ1 and σ2 at various temperatures from 6 K to 18 K (grey diamonds). The 1/ω dependence is suppressed as Δns/ns saturates to a minimum in regime 2; for example, the E4 line shape is distinctly different from the residual condensate in equilibrium traces plotted together (14–16 K). c, The postquench state conductivities at an initial T = 18 K above Tc for E6 pumping (red line) and at T = 4.1 K below Tc, but for optical pumping at 1.55 eV (black line). d, Frequency-dependent scattering rate 1/τ(ω) for the postquench states pumped by E2 and E5 compared to the normal state result that converges to 1/τimp (grey circles). e, 1/τ(ω) for the equilibrium SC (4.1 K) and normal (18 K) states as marked.

Fig. 3: The persisting prethermalized plateau state with non-thermal characteristics and a long-lived memory.
figure3

a, The temporal dynamics of ΔE/E0 at different quench fields ETHz shows a clear transition between two different decay profiles, marked as τfast and τslow at the threshold field Eth. Inset: THz response σ2 as a function of time delay in a 2D false colour plot on the right, with representative traces for time delays Δtpp = −10 ps, 12 ps, 500 ps and 1 ns at ETHz = 120 kV cm–1 at 4.1 K on the left. b, The characteristic timescales extracted from our measurements, which range from order-parameter coherence, QP decay and thermalization to postquench prethermalization above the threshold. c, σ2 of the postquench state for various time delays, Δtpp = −10 ps, 12 ps, 500 ps and 1 ns at ETHz = 620 kV cm–1 at 4.1 K. Inset: ΔE/E0 dynamics. d, The same spectral–temporal characteristics as in c, but for the normal state at 18 K. e, A comparison of 1/τ(ω) that corresponds to c and d at the given time delays. Shown together is the final SC state after relaxation (grey squares). a.u., arbitrary units.

The sharp peak in the conductivity spectra in Fig. 2b indicates a narrow linewidth proportional to the effective scattering rate 1/τ(ω), that is, the imaginary part of the self-energy Σ2(ω). A standard extended Drude analysis using the self-energy (Methods) produces a frequency-dependent scattering rate 1/τ(ω), which is consistent with the existence of residual fluctuations after the THz quench that is different from equilibrium. As shown in Fig. 2d, 1/τ(ω) decreases towards zero almost linearly with ω at ω → 0, at the expense of enhanced scattering above 2ΔSC, for example, seen for both a strong (E5) and a partial (E2) SC quench. In comparison, Fig. 2e shows that, in equilibrium, 1/τ(ω) at T > Tc (18 K, grey circles) remains fairly constant and converges to the elastic impurity scattering 1/τimp ≈ 7.7 meV ≈ 2Δ SC , whereas for T < Tc, the 1/τ(ω) spectra clearly reveal zero scattering below 2ΔSC. Following the THz quantum quench, vanishing 1/τ(ω) at ω → 0 is significantly lower than the 1/τimp. Therefore, these differences from the normal state indicate that the THz light-tuning scheme reveals the emergence of a gapless quantum QP fluid.

Figure 3 reveals a photoexcitation threshold for relaxation slowdown, that is, a prethermalized plateau temporal behaviour above Eth = 256 kV cm–1 that corroborates the transition to the persisting QP quantum phase \(\left|{\psi }_{{\rm{Plateau}}}\right\rangle\) at E4E6. This is clearly seen in the detailed pump-fluence dependence of ΔE/E0 as a function of time delay, shown in Fig. 3a on a logarithmic scale at T = 4.1 K. At a low quench field, ETHz < Eth, we observe a smooth picosecond condensate recovery with a relaxation time τfast ≈ 0.6 ns. This is typical for SC recovery due to QP decay, as seen, for example, in the 27 kV cm–1 and 120 kV cm–1 traces in Fig. 3a. This is corroborated by the detailed time evolution of Δσ2 up to 1,000 ps (Fig. 3a, inset) shown for E2. In contrast, at high quench fields, ETHz > Eth, in which Δns/ns saturates to its minimum (or zero) value (Fig. 2a), a much longer many-nanosecond quasi-steady temporal regime emerges and dominates the out-of-equilibrium dynamics. This is witnessed, for example, by an order of magnitude longer relaxation time τslow ≈ 7 ns in the 445 kV cm–1 (red line) and 620 kV cm–1 (black line) traces in Fig. 3a. The 1/ω divergence in the postquench, non-equilibrium σ2(ω) (Fig. 3c) has already changed markedly at Δtpp = 12 ps as compared to Δtpp = −10 ps, and persists at very long times, for example, 500 ps and 1,000 ps, with very little recovery to the initial SC state (inset). These gapless quantum states still exhibit the coherent transport (Fig. 2b) with vanishing scattering (Fig. 3e), distinct from the normal states. In addition, the dynamics is markedly faster above Tc, which again indicates the distinct difference between the driven quantum-phase evolution and normal state thermalization. For a normal state quench at 620 kV cm–1, σ2(ω) has mostly recovered to its ground state behaviour in the 1,000 ps trace in Fig. 3d, indicative of a short ΔE/E0 decay constant of ~0.6 ns (inset). This energy relaxation τth is comparable to the thermalization time observed for the optical pump (Fig. 2c). Figure 3b summarizes the characteristic timescales extracted and distinguishes their different physical origins from the initial coherent oscillations (green) to the SC recovery (black) and QP thermalization (blue), which exhibit much shorter times than the lifetime of the prethermalized plateau state (red).

To highlight the correlation physics in the gapless \(\left|{\psi }_{{\rm{Plateau}}}\right\rangle\), we compare in Fig. 3e the 1/τ(ω) at T = 4.1 K versus T = 18 K for various time delays Δtpp = 12 ps, 500 ps and 1,000 ps at E6 = 620 kV cm–1. Interestingly, besides their different ω dependence discussed above, the 1/τ(ω) spectra of these postquench states exhibit a sharp isosbestic point at a frequency of ωc ≈ 2ΔSC. This frequency marks the crossing from a suppressed to an enhanced scattering and exhibits very little shift for a wide range of time delays measured up to 1,000 ps (vertical dashed line in Fig. 3e). The formation of an isosbestic point as a function of time delay represents a hallmark for correlated state build-up and signifies a memory of hidden correlation gaps that manifests as a transient spectral weight transfer from higher to lower frequencies. For example, the 1,000 ps 1/τ(ω) spectra retain memory of a nearly linear frequency dependence and the emergent correlation gap with no apparent relaxation to either the normal (grey circles in Fig. 3e) or the initial SC state behaviour (grey rectangles in the bottom panel of Fig. 3e). Such very weak time dependence of spectral redistribution in the prethermalization \(\left|{\psi }_{{\rm{Plateau}}}\right\rangle\) differs strongly from the quench experiment at 18 K, which leads to a largely frequency-independent 1/τ(ω) (grey circles in Fig. 3e), always greater than 1/τimp, and to a full relaxation before 1,000 ps.

The observation of an additional spectral weight near EF in the hidden \(\left|{\psi }_{{\rm{Plateau}}}\right\rangle\) as compared to the normal state (Fig. 2b) indicates that the equilibrium condensate nsδ(ω) at ω = 0 did not come exclusively from the spectral weight (W) at ω ≤ 2ΔSC but also involved a larger competing gap ΔW > ΔSC. Bilbro and McMillan20 introduced a mean-field model to predict the coexistence of a CDW-like order and SC in the ground state of Nb3Sn, which is driven by strong electron–phonon coupling. This coexistence results in a correlation energy gap that varies between different regions in k-space20. Here we use this model to demonstrate the emergence of additional gapless excitations after the decrease of ΔW(t) that accompanies the THz quench of ΔSC(t). A σ1 peak at low frequencies is then formed by transferring the spectral weight from higher frequencies (Supplementary Information). Even a relatively small (~10%) laser-induced decrease of ΔW(t) in a low T transient state with ΔSC = 0 leads to a coherent peak in σ1(ω) for a suppressed scattering rate 1/τ(ω) (Fig. 4a–c) and a partially gapped Fermi surface with regions of gapless excitations (shaded area in Fig. 4e,f). For example, for ΔW(t) = 0.8ΔW,0 (Fig. 4c) and 0.9ΔW,0 (Fig. 4b), σ1 develops a sharp peak at low frequencies (Fig. 4e,f) that is absent for the equilibrium ΔW,0 ≈ 40 meV (ref. 22) (Fig. 4d). The emergence of such a peak when (ΔW(t) < ΔW,0, ΔSC(t) = 0) in a transient state with suppressed scattering is consistent with our experimental observation (Fig. 2b). This calculation suggests that our observed additional spectral weight and low-frequency coherent peak could arise from a THz-quench-induced decrease in a competing order parameter and/or the partial k-dependent Fermi surface that gap together with a suppressed 1/τ(ω).

Fig. 4: Predictions of the theoretical model for the hidden gapless quantum state with extraordinary conductivity.
figure4

af, Simulation of the conductivity (ac) and Fermi surface (df) with the reduced gap ΔW as discussed in the main text. The red and dashed black lines illustrate the electron and hole pockets. Blue-shaded areas highlight the regions of gapless excitations. gi, Free–energy density for the model Hamiltonian (Supplementary Information) as a function of the CDW-like order parameter ΔW. ΔSC = ΔSC,4K for the equilibrium state below Tc (g); ΔSC,18K = 0 describes the thermal normal state that shows an increase in the equilibrium ΔW (dashed red line) (h); ΔSC = 0 and T(t) < Tc describes a prethermalized gapless state that follows the THz quench of the SC gap with minimal heating that cannot be realized in equilibrium (i).

A THz ultrafast quench of a low energy ΔSC with a simultaneous decrease in high energy ΔW can arise from coherent/non-thermal quench dynamics (green line in Fig. 3b) during, for example, THz excitation of a short duration. Over timescales longer than that from relaxation due to inelastic scattering, the above ultrafast k-dependent change of the electronic correlation gap creates a non-thermal ‘initial condition’ that can access different regions of the free energy landscape30. To illustrate how a long-lived prethermalized state may emerge and become trapped for a long time after the coherent temporal regime (Fig. 3b), a free-energy calculation based on the mean–field model of Bilbro and McMillan20 (Methods), is shown in Fig. 4g–i. Compares the dependence on ΔW for three cases: (1) ΔSC = ΔSC,4K (Fig. 4g), as in the equilibrium state below Tc. The free-energy minimum is then located at ΔW = ΔW,0 and corresponds to an equilibrium homogeneous phase characterized by a two-component k-dependent order parameter20. (2) ΔSC,18K = 0 (Fig. 4h) describes the thermal normal state that shows an increase in its equilibrium ΔW (dashed red line) due to the reduction of the competing SC order20. We expect such a landscape to follow the quenching of ΔSC via high-energy optical pumping (Fig. 2c). (3) ΔSC = 0 with T(t) < Tc (Fig. 4i), achieved via non-thermal THz light-induced dynamics, changes the free-energy landscape from that of the thermal normal state (Fig. 4h). Within the model of Bilbro and McMillan20 (Methods), a sharp local minimum then develops at Δ W  ≈ 0 (blue arrow in Fig. 4i), which, as seen in Fig. 4c, is accompanied by a sharp peak in σ1(ω → 0) for the suppressed scattering 1/τ(ω). Additional couplings to acoustic phonons19,23,24 and the details of the band structure influence such metastable states with reduced lattice distortion/dimerization26. The THz tuning ‘suddenly’ brings the system to an excited state during laser pulses with ΔSC = 0 and initiates a coherent/anharmonic motion of ΔW(t) that can access to such free-energy local minima for sufficient pump fluence. The system then becomes trapped there due to damping of the coherent motion prior to returning to the global minimum. This is analogous to the quantum femtosecond magnetism discovered in the manganites6 that was shown to non-adiabatically generate ferromagnetic correlation during laser pulses and subsequently drive a magnetic phase transition. Such a simultaneously suppressed ΔW and SC order parameters, going beyond the thermodynamic self-consistency limitation20, leads to our observation of the metastable gapless QP quantum phase above the quench threshold. THz-quantum-quench prethermalization of the competing orders discovered may be extended to access hidden density-wave phases and quantum criticality under the SC dome in high Tc materials31.

Methods

Sample preparation

A nearly single-crystal Nb3Sn film 20 nm thick with a critical temperature of Tc ≈ 16 K was grown by magnetron sputtering on a 1 mm Al2O3(100) substrate. It was grown by the co-sputtering of Nb and Sn at high temperatures. Nb and Sn targets were current regulated at 0.33 A and 0.1 A, respectively, in 3 mtorr of Ar, and positioned 15.5 cm from the substrate. The base pressure was 9.4 × 10−8 mtorr. The 10 × 10 × 1 mm R-plane sapphire substrate was exposed to a SiC-coated graphite heating element above, and exposed to the Nb and Sn flux from below. The deposited material was heated directly by the radiation, which largely passed through the sapphire. The film temperature during the growth was estimated at 1,020 °C by measuring the film temperature on thick films grown under similar conditions. Temperatures were measured with an optical pyrometer.

Two-time THz pump and THz probe spectroscopy of complex conductivity

We characterize the non-equilibrium postquench states by measuring the complex conductivity (σ1(ωtpp), σ2(ωt pp )) as a function of Δtpp, as illustrated in Supplementary Fig. 1). This scheme is implemented by using three pulses: THz pump \({E}_{{\rm{THz}}}^{^{\prime} }\left({t}_{{\rm{pu}}}\right)\), THz probe \({E}_{{\rm{THz}}}^{{\prime \prime} }\left({t}_{{\rm{pr}}}\right)\) and optical gating at time tgate (Supplementary Fig. 1). The three pulses were split from a 1 kHz Ti:sapphire regenerative amplifier (35 fs, 800 nm centre wavelength). Subsequently, (1) the single-cycle THz pump field with a peak amplitude of up to \({E}_{{\rm{THz}}}^{^{\prime} }\) = 620 kV cm–1 was generated by the tilted-pulse-front phase-matching method through a 1.3% MgO doped LiNbO3 crystal (red line in Supplementary Fig. 1); (2) the weak THz probe field (blue line) was generated by optical rectification and detected by free-space electro-optic sampling through two 1 mm thick \(\left\langle 110\right\rangle\) ZnTe emitter/detector crystals; (3) the THz polarization responses in the time domain were directly measured by the third optical gating pulse at time tgate.

To measure the static THz conductivity responses without the pump, the first-order dielectric response of the sample, induced by an oscillatory probe field \({E}_{{\rm{THz}}}^{\prime \prime }\left({t}_{{\rm{pr}}}\right)\), was detected at time tgate:

$$\begin{array}{l}\widetilde{P}\left({t}_{{\rm{gate}}}\right)=\underset{-\infty }{\overset{{t}_{{\rm{gate}}}}{\int }}{\rm{d}}{t}_{{\rm{pr}}}{E}_{{\rm{THz}}}^{\prime \prime }\left({t}_{{\rm{pr}}}\right){\widetilde{\chi }}^{(1)}\left({t}_{{\rm{gate}}}-{t}_{{\rm{pr}}}\right)\end{array}$$
(1)

Through Fourier transformation, the THz response functions in equilibrium are then readily obtained as:

$$\begin{array}{l}\widetilde{\sigma }(\omega )=-i\omega {\widetilde{\chi }}^{(1)}(\omega )=\frac{\widetilde{P}(\omega )}{{E}_{{\rm{THz}}}^{\prime \prime }(\omega )}\end{array}$$
(2)

The polarization response of a transient state that is ‘suddenly’ driven away from equilibrium by the pump pulse is naturally described by the third-order non-linear susceptibility \({\widetilde{\chi }}^{(3)}\). This is the case because the interaction involves three electric fields, one from the probe and two from the pump. Importantly, the polarization \(\widetilde{P}\) at time tgate can no longer be expressed as a function of a single time variable, as the pump field gives rise to an additional change in the system as \(\widetilde{P}\) propagates in time. Therefore, the non-equilibrium responses can only be described properly by a two-time response function:

$$\begin{array}{l}\widetilde{P}\left({t}_{{\rm{gate}}},{t}_{{\rm{pu}}}\right)=\underset{-\infty }{\overset{{t}_{{\rm{gate}}}}{\int }}{\rm{d}}{t}_{{\rm{pr}}}\underset{-\infty }{\overset{{t}_{{\rm{gate}}}-{t}_{{\rm{pu}}}}{\int }}{E}_{{\rm{THz}}}^{\prime \prime }\left({t}_{{\rm{pr}}}\right){E}_{{\rm{THz}}}^{^{\prime} 2}(t){\widetilde{\chi }}^{(3)}\left({t}_{{\rm{gate}}}-{t}_{{\rm{pr}}},t\right){\rm{d}}t\end{array}$$
(3)

In analogy to the definition of equation (1), one can then rewrite equation (3) in a more transparent way as:

$$\begin{array}{l}\widetilde{P}\left({t}_{{\rm{gate}}},{t}_{{\rm{pu}}}\right)=\underset{-\infty }{\overset{{t}_{{\rm{gate}}}}{\int }}{\rm{d}}{t}_{{\rm{pr}}}{E}_{{\rm{THz}}}^{\prime \prime }\left({t}_{{\rm{pr}}}\right){\widetilde{\chi }}^{(1)}\left({t}_{{\rm{gate}}}-{t}_{{\rm{pr}}},{t}_{{\rm{gate}}}-{t}_{{\rm{pu}}}\right){\rm{d}}t\end{array}$$
(4)

by defining a new response function \({\widetilde{\chi }}^{(1)}\left({t}_{{\rm{gate}}}-{t}_{{\rm{pr}}},{t}_{{\rm{gate}}}-{t}_{{\rm{pu}}}\right)\) for describing the out-of-equilibrium responses:

$$\begin{array}{l}{\widetilde{\chi }}^{(1)}\left({t}_{{\rm{gate}}}-{t}_{{\rm{pr}}},{t}_{{\rm{gate}}}-{t}_{{\rm{pu}}}\right)=\underset{-\infty }{\overset{{t}_{{\rm{gate}}}-{t}_{{\rm{pu}}}}{\int }}{E}_{{\rm{THz}}}^{^{\prime} 2}(t){\widetilde{\chi }}^{(3)}\left({t}_{{\rm{gate}}}-{t}_{{\rm{pr}}},t\right){\rm{d}}t\end{array}$$
(5)

A simple deconvolution of equation (4) along the tgate axis fails because both arguments of \({\widetilde{\chi }}^{(1)}\left({t}_{{\rm{gate}}}-{t}_{{\rm{pr}}},{t}_{{\rm{gate}}}-{t}_{{\rm{pu}}}\right)\) exhibit a dependence on tgate, which makes it impossible to retrieve the response function as in equation (2). Such a retrieval problem can be circumvented at a fixed time delay Δtpp = tgate − tpu, referred to as the pump–probe delay in the text. In this way, the polarization response of the out-of-equilibrium state can be described as:

$$\begin{array}{l}\widetilde{P}\left({t}_{{\rm{gate}}},{\rm{\Delta }}{t}_{{\rm{pp}}}\right)=\underset{-\infty }{\overset{{t}_{{\rm{gate}}}}{\int }}{\rm{d}}{t}_{{\rm{pr}}}{E}_{{\rm{THz}}}^{\prime \prime }\left({t}_{{\rm{pr}}}\right){\widetilde{\chi }}^{(1)}\left({t}_{{\rm{gate}}}-{t}_{{\rm{pr}}},{\rm{\Delta }}{t}_{{\rm{pp}}}\right)\end{array}$$
(6)

Such a two-time polarization response is directly measured in our experiment via the emitted electrical field, shown in Fig. 1f as an example, which allows us to further obtain simultaneously the time- and frequency-resolved response functions by performing deconvolution along the tgate axis:

$$\begin{array}{l}\widetilde{\sigma }\left(\omega ,{\rm{\Delta }}{t}_{{\rm{pp}}}\right)=-i\omega {\widetilde{\chi }}^{(1)}\left(\omega ,{\rm{\Delta }}{t}_{{\rm{pp}}}\right)=\frac{\widetilde{P}\left(\omega ,{\rm{\Delta }}{t}_{{\rm{pp}}}\right)}{{E}_{{\rm{THz}}}^{\prime \prime }(\omega )}\end{array}$$
(7)

Experimentally, we determine the complex transmission \({\widetilde{t}}_{{\rm{\exp }}}\) by comparing the transmitted THz electric fields through the sample and reference with Fourier transformation and Fresnel equations:

$$\begin{array}{l}{\widetilde{t}}_{{\rm{\exp }}}=\frac{\frac{{E}_{{\rm{out}}}^{{\rm{ref}}}}{{E}_{{\rm{in}}}}}{\frac{{E}_{{\rm{out}}}^{\rm{s}}}{{E}_{{\rm{in}}}}}=\frac{1+{n}_{{\rm{sub}}}}{\left(1+{n}_{{\rm{sub}}}\right){\rm{\cos }}{\beta }_{{\rm{s}}}-\left({n}_{{\rm{s}}}+\frac{{n}_{{\rm{sub}}}}{{n}_{{\rm{s}}}}\right)i\,{\rm{\sin }}{\beta }_{{\rm{s}}}}\times {{\rm{e}}}^{-i\frac{2{\rm{\pi }}}{{\lambda }_{0}}({{{d}}}_{{\rm{s}}})}\end{array}$$
(8)

where nsub and ns are the optical indices of substrate and sample, λ0 is wavelength, ds is the thickness of sample and \({\beta }_{{\rm{s}}}=\frac{2{\rm{\pi }}}{{\lambda }_{0}}{n}_{{\rm{s}}}{d}_{{\rm{s}}}\). This allows to extract the complex conductivity \(\widetilde{\sigma }=i\omega /4{\rm{\pi }}(1-\widetilde{\epsilon })\). The extracted real and imaginary parts σ1(ωtpp) and σ2(ωtpp) are presented in Figs. 2 and 3 via a data-extraction procedure (Supplementary Section 2). In this way we access the electronic correlation and fluctuations in the postquantum-quench states.

Spot sizes of the THz pump and probe are 1.2–1.5 mm and 0.8 mm, respectively (Supplementary Section 2 gives further details). We further minimize the size mismatch of the pump and probe by placing a pinhole 2 mm in diameter before the sample. In this way, the pump–probe overlap of our set-up is determined by this hard aperture that is comparable to the pump beam size, which leverages the effect of a larger probe size at a low frequency. In this way, we further guaranteed a uniform pump illumination of the whole probe detection area. The signals are closely monitored and analysed by three boxcar integrators to record both the pump and probe with on and off conditions, shot by shot, as well as with chopper phase that are synchronized with the laser repetition rate.

Frequency-dependent electric transport

The optical self-energy has been identified to be more effective than \(\widetilde{\sigma }(\omega )\) itself for underpinning the exact nature of correlated electronic states. This self-energy can be regarded as a frequency–dependent memory function in analogy to the way in which we analyse many-body interactions in the Green’s function approach. More information is given in Supplementary Section 5. Specifically, we calculate the complex optical self-energy Σ(ω,T) in terms of both the frequency-dependent momentum scattering rate 1/τ(ω) and the electron–phonon mass renormalization 1 + λ(ω), for example, which 1/τ(ω) is obtained as follows:

$$\frac{1}{\tau (\omega )}=\frac{{\omega }_{{\rm{p}}}^{2}}{4{\rm{\pi }}}\frac{{\sigma }_{1}}{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}$$
(9)

Here the plasma frequency ωp is obtained by fitting the normal-state conductivity spectra with the Drude model, which gives the plasma frequency as ~6.66 eV. The superfluid density in Fig. 1d is obtained by sum rule:

$$\begin{array}{l}\underset{0}{\overset{+\infty }{\int }}\left({\sigma }_{1}^{n}(\omega )-{\sigma }_{1}^{s}(\omega )\right){\rm{d}}\omega =\frac{{\rm{\uppi }}}{2}\frac{{n}_{{\rm{s}}}{e}^{2}}{m}\end{array}$$
(10)

The density ratio ns/n obtained agrees well with the extrapolated \({\sigma }_{2}^{s}(\omega )\) in the limit of ω → 0 in the superconducting state:

$$\begin{array}{l}\frac{{n}_{{\rm{s}}}(T)}{n}={\left(\frac{{\omega }_{{\rm{p}}}^{2}}{4\pi }\right)}^{-1}\mathop{{\rm{lim}}}\limits_{\,\omega \to 0}\omega {\sigma }_{2}\end{array}$$
(11)

Effective medium theory calculations

The observed prethermalized quantum state is well distinguished from thermally mixed states (phase separation) as demonstrated in Supplementary Fig. 2a,b. Specifically, we perform an effective medium calculation of the dielectric response of a spatially inhomogeneous state from phase-separated patches that consist of SC and normal states. Assuming that the non-equilibrium state is a result of spatial inhomogeneity, the optical response should satisfy:

$$\begin{array}{l}f\frac{{\sigma }_{{\rm{N}}}(\omega )-{\sigma }_{{\rm{eff}}}(\omega )}{k{\sigma }_{{\rm{N}}}(\omega )+(1-k){\sigma }_{{\rm{eff}}}(\omega )}+(1-f)\frac{{\sigma }_{{\rm{S}}}(\omega )-{\sigma }_{{\rm{eff}}}(\omega )}{k{\sigma }_{{\rm{S}}}(\omega )+(1-k){\sigma }_{{\rm{eff}}}(\omega )}=0\end{array}$$
(12)

where σN(ω) and σS(ω) are the static conductivities of the normal state (18 K) and the SC state (4 K). The coefficient f describes the volume fraction of the normal state in a superconductor and k is the depolarization factor determined by the shape of the hot spots, which we assume to be 0.5.

Theoretical model of the gapless conducting state

To interpret the qualitative features of our experimental results, we use an extension of the mean–field one-dimensional (1D) electron model introduced by Bilbro and McMillan20 to a 3D anisotropic model with two electronic bands coupled by an optical phonon, as proposed to explain equilibrium experiments in our material19,23,24. As in Bilbro and McMillan20, we assume for simplicity that the Fermi surface consists of two k-regions: region 1 corresponds to momenta in which the coupling of the two bands with a condensed optical phonon or other CDW-like order is strong. This small k region favours the emergence of a CDW-like order coexisting with a SC order, which enhances the electronic correlation energy gap in this part of the Fermi surface. Also, we assume that an only SC order is possible in the rest of the Fermi surface, referred to as region 2.

Our model Hamiltonian has the form:

$$H={H}_{0}+{H}_{{\rm{SC}}}+{H}_{{\rm{ph}}}+{H}_{{\rm{el}}-{\rm{ph}}}+\frac{{{\rm{\Delta }}}_{{\rm{SC}}}^{2}}{{V}_{{\rm{SC}}}}+{H}_{{\rm{light}}}(t)$$
(13)

In equation (13), the band Hamiltonian is:

$${H}_{0}=\sum _{{\bf{k}},\sigma ,n}{\xi }_{{\bf{k}},n}{c}_{{\bf{k}},n,\sigma }^{\dagger }{c}_{{\bf{k}},n,\sigma }$$
(14)

where the operator \({c}_{{\bf{k}},n,\sigma }^{\dagger }\) creates an electron with crystal momentum ħk and spin σ in two bands, n = − and +, close to the Fermi surface, with dispersions:

$${\xi }_{{\bf{k}},-}=-{\xi }_{-,0}+\sum _{j}\frac{{\hslash }^{2}{k}_{j}^{2}}{2{m}_{j,1}}-\mu ,{\xi }_{{\bf{k}},+}={\xi }_{+,0}-\sum _{j}\frac{{\hslash }^{2}{k}_{j}^{2}}{2{m}_{j,2}}-\mu$$
(15)

where the chemical potential is μ, band offsets are ξn,0 and the effective masses are mj,n (j = x,y, z)23,24,26. The SC pairing interaction is given by:

$${H}_{{\rm{SC}}}=-\sum _{{\bf{k}},n}\left[{{\it{\Delta }}}_{{\rm{SC}}}{c}_{{\bf{k}},n,\uparrow }^{\dagger }{c}_{-{\bf{k}},n,\downarrow }^{\dagger }+{\rm{h.c.}}\right]$$
(16)

where the SC order parameter is:

$${{\rm{\Delta }}}_{{\rm{SC}}}={V}_{{\rm{SC}}}\sum _{{\bf{k}},n}\left\langle {c}_{-{\bf{k}},n,\downarrow }{c}_{{\bf{k}},n,\uparrow }\right\rangle$$
(17)

where VSC describes the strength of the interaction, and h.c. is the hermitian conjugate. The k-sum includes the set W of wavevectors k in both regions 1 and 2 with \(\left|{\xi }_{{\bf{k}}n}\right|\le \hslash {\omega }_{{\rm{D}}}\), where ωD is the cut-off frequency32. The above two electronic bands are coupled to an additional CDW–like order, which can arise from the condensation of a bosonic excitation, such as an optical phonon with momentum q ≈ 0, coupled to the interband electronic excitations19,20. The corresponding free-phonon Hamiltonian is:

$${H}_{{\rm{ph}}}=\hslash {\omega }_{{\rm{ph}}}{b}^{\dagger }b,$$
(18)

where b is the bosonic annihilation operator and ħωph is the phonon energy. The coupling of this boson to the two electronic bands is described by23,24:

$${H}_{{\rm{el}}-{\rm{ph}}}=-g_{\mathrm{ph}}\left(b+{b}^{\dagger }\right)\sum _{{\bf{k}},\sigma }\left({c}_{{\bf{k}},-,\sigma }^{\dagger }{c}_{{\bf{k}},+,\sigma }+{\rm{h}}.{\rm{c}}.\right)$$
(19)

where gph describes the strength of the electron–phonon interaction. Here, the k sum includes only the wavevectors of region 1, in which the electron–boson coupling is strong. In the very early coherent temporal regime, to be discussed elsewhere, we must also include the coupling of the THz laser A field that drives the time dependence of both CDW–like and SC order parameters. The coupling used in Papenkort et al.32 and Schnyder et al.33 has the form:

$${H}_{{\rm{light}}}(t)=\sum _{{\bf{k}},{\bf{q}},n,\sigma }\left(-{{\bf{j}}}_{n}\left({\bf{k}}+\frac{{\bf{q}}}{2}\right)\cdot {{\bf{A}}}_{{\bf{q}}}+\frac{{e}^{2}}{2{m}_{n}}\sum _{{\bf{q}}^{\prime} }{{\bf{A}}}_{{\bf{q}}-{\bf{q}}^{\prime} }\cdot {{\bf{A}}}_{{\bf{q}}}\right){c}_{{\bf{k}}+{\bf{q}},n,\sigma }^{\dagger }{c}_{{\bf{k}},n,\sigma }$$
(20)

with current density is:

$${{\bf{j}}}_{n}({\bf{k}})=-\frac{\left|e\right|}{\hslash }\frac{\partial {\xi }_{{\bf{k}},n}}{\partial {\bf{k}}}$$
(21)

and the THz pump pulse described by the transverse vector potential pulse is

$${{\bf{A}}}_{{\bf{q}}}(t)={{\bf{A}}}_{0}{{\rm{e}}}^{-{\left(\frac{2\sqrt{{\rm{ln}}2t}}{{\tau }_{{\rm{p}}}}\right)}^{2}}\left({\delta }_{{\bf{q}},{{\bf{q}}}_{{\rm{p}}}}{{\rm{e}}}^{-i{\omega }_{p}t}+{\delta }_{{\bf{q}},-{{\bf{q}}}_{{\rm{p}}}}{{\rm{e}}}^{i{\omega }_{{\rm{p}}}t}\right)$$
(22)

with finite duration τp, amplitude A0, photon energy ħωp and photon wavevector qp.

Following Porer et al.8, we adopt a mean-field approximation to describe the boson condensation and coherent phonon light-induced dynamics: b = \(\langle b \rangle + B\). Equation (13) then yields:

$$\begin{array}{l}{H}_{{\rm{ph}}}^{MF}+{H}_{{\rm{el}}-{\rm{ph}}}^{MF}=\frac{{{\rm{\Delta }}}_{{\rm{W}}}^{2}}{{V}_{{\rm{W}}}}\\ +\hslash {\omega }_{{\rm{ph}}}{B}^{\dagger }B+\hslash {\omega }_{{\rm{ph}}}\left[\left\langle b\right\rangle {B}^{\dagger }+{\left\langle b\right\rangle }^{\star }B\right]\\ -g_{\mathrm{ph}}\sum _{{\bf{k}},\sigma }\left(\left\langle {c}_{{\bf{k}},-,\sigma }^{\dagger }{c}_{{\bf{k}},+,\sigma }\right\rangle +{\rm{c}}.{\rm{c}}.\right)\left(B+{B}^{\dagger }\right)\\ -{{\rm{\Delta }}}_{{\rm{W}}}(t)\sum _{{\bf{k}},\sigma }\left({c}_{{\bf{k}},-,\sigma }^{\dagger }{c}_{{\bf{k}},+,\sigma }+{\rm{h}}.{\rm{c}}.\right)\end{array}$$
(23)

where

$${{\rm{\Delta}}}_{{\rm{W}}}(t)=g_{\mathrm{ph}}\left(\left\langle b\right\rangle +{\left\langle b\right\rangle }^{* }\right)$$
(24)

is the CDW order parameter, VW = 4gph2/(ħωph), and c.c. is the complex conujugate. The condensed boson amplitude \(\left\langle b\right\rangle\) is modified from its ground state value due to a non-thermal interband electronic coherence and populations induced by the THz light coupling, which can result in a transient lattice configuration after the damping of coherent CDW order parameter amplitude oscillations:

$$\frac{{{\rm{d}}}^{2}{{\rm{\Delta }}}_{{\rm{W}}}}{{\rm{d}}{t}^{2}}+{\omega }_{{\rm{ph}}}^{2}{{\rm{\Delta }}}_{{\rm{W}}}+2\Gamma {\omega }_{{\rm{ph}}}\frac{{\rm{d}}{{\rm{\Delta }}}_{{\rm{W}}}}{{\rm{d}}t}=\frac{{V}_{{\rm{W}}}{\omega }_{{\rm{ph}}}^{2}}{2}\sum _{{\bf{k}},\sigma }\left(\left\langle {c}_{{\bf{k}},-,\sigma }^{\dagger }{c}_{{\bf{k}},+,\sigma }\right\rangle +{\rm{c}}.{\rm{c}}.\right)$$
(25)

where we added the damping rate Γ. Following damping of the coherent oscillations, the quasi-equilibrium CDW displacement is given by the steady-state solution:

$${{\rm{\Delta }}}_{{\rm{W}}}^{{\rm{eq}}}=\frac{{V}_{{\rm{W}}}}{2}\sum _{{\bf{k}},\sigma }\left({\left\langle {c}_{{\bf{k}},-,\sigma }^{\dagger }{c}_{{\bf{k}},+,\sigma }\right\rangle }_{{\rm{eq}}}+{\rm{c}}.{\rm{c}}.\right)$$
(26)

where the above interband electronic coherence is modified from its ground state value due to the time evolution of the electronic system driven by the THz ultrafast excitation. This coherent/non-thermal time evolution, to be discussed elsewhere, is described by density matrix equations of motion32, obtained here for the mean-field electronic Hamiltonian:

$${H}_{{\rm{el}}}(t)={H}_{0}+{H}_{{\rm{SC}}}-{{\rm{\Delta }}}_{{\rm{W}}}(t)\sum _{{\bf{k}},\sigma }\left({c}_{{\bf{k}},-,\sigma }^{\dagger }{c}_{{\bf{k}},+,\sigma }+{\rm{h}}.{\rm{c}}.\right)+{H}_{{\rm{light}}}(t)$$
(27)

The non-linear response to the THz pulses is treated similar to the ultrafast coherent response of semiconductors and leads to strong changes in both ΔSC and ΔW from their equilibrium values. We can diagonalize the above Hamiltonian Hel for given ΔSC and ΔW by using a Bogoliubov canonical transformation20 to obtain a basis of quasiparticle excitations32 that depend on both SC and CDW coherence. These QP excitation energies are given by:

$$\begin{array}{l}{E}_{\alpha ,1,{\bf{k}}}^{2}={{\it{\Delta }}}_{{\rm{SC}}}^{2}+{{\it{\Delta }}}_{{\rm{W}}}^{2}+\frac{1}{2}\left[{\xi }_{{\bf{k}},-}^{2}+{\xi }_{{\bf{k}},+}^{2}\pm \left({\xi }_{{\bf{k}},-}+{\xi }_{{\bf{k}},+}\right)\sqrt{4{{\it{\Delta }}}_{{\rm{W}}}^{2}+{\left({\xi }_{{\bf{k}},-}-{\xi }_{{\bf{k}},+}\right)}^{2}}\right]\\ {E}_{\alpha ,2,{\bf{k}}}^{2}={{\it{\Delta }}}_{{\rm{SC}}}^{2}+\frac{1}{2}\left[{\xi }_{{\bf{k}},-}^{2}+{\xi }_{{\bf{k}},+}^{2}\pm \left({\xi }_{{\bf{k}},-}^{2}-{\xi }_{{\bf{k}},+}^{2}\right)\right]\end{array}$$
(28)

in regions 1 and 2, respectively. The two quasiparticle branches at a given k are described by a = ±.

After damping of the coherent/non-thermal order parameter motion, the system reaches a quasi-equilibrium state determined by the adiabatic free energy f(ΔSC,ΔW,T) = \(\left\langle H\right\rangle -TS\). Approximating, for simplicity, the quasiparticle populations after relaxation by a Fermi–Dirac distribution, we obtain, after some algebra:

$$f\left({{\it{\Delta }}}_{{\rm{SC}}},{{\it{\Delta }}}_{{\rm{W}}},T\right)=\frac{{{\it{\Delta }}}_{{\rm{SC}}}^{2}}{{V}_{{\rm{SC}}}}+\frac{{{\it{\Delta }}}_{{\rm{W}}}^{2}}{{V}_{{\rm{W}}}}-2{k}_{{\rm{B}}}T\sum _{{\bf{k}},a,l}{\rm{log}}\left[2{\rm{\cosh }}\left(\frac{{E}_{a,l,{\bf{k}}}}{2{k}_{{\rm{B}}}T}\right)\right]$$
(29)

The equilibrium values of the order parameters ΔSC and ΔW are determined by the global minimum of the above free energy for a given temperature: ∂f(ΔSC,ΔW)/∂ΔSC = ∂f(ΔSC,ΔW)/∂ΔW = 0 gives the temperature-dependent energy-gap equations:

$$\begin{array}{l}{{\it{\Delta }}}_{{\rm{SC}}}={V}_{{\rm{SC}}}\sum _{{\bf{k}},a,l}\frac{{{\it{\Delta }}}_{{\rm{SC}}}}{{E}_{a,l,{\bf{k}}}}{\rm{\tanh }}\left(\frac{{E}_{a,l,{\bf{k}}}}{2{k}_{{\rm{B}}}T}\right)\\ {{\it{\Delta }}}_{{\rm{W}}}={V}_{{\rm{W}}}\sum _{{\bf{k}},a}\frac{{{\it{\Delta }}}_{{\rm{W}}}\left({E}_{a,1,{\bf{k}}}^{2}-{{\it{\Delta }}}_{{\rm{SC}}}^{2}+{\xi }_{{\bf{k}},-}{\xi }_{{\bf{k}},+}-{{\it{\Delta }}}_{{\rm{W}}}^{2}\right)}{{E}_{a,1,{\bf{k}}}\left({E}_{a,1,{\bf{k}}}^{2}-{E}_{\bar{a},1,{\bf{k}}}^{2}\right)}{\rm{\tanh }}\left(\frac{{E}_{a,1,{\bf{k}}}}{2{k}_{{\rm{B}}}T}\right)\end{array}$$
(30)

where \(\bar{a}=-a\). ΔW is determined by k summation in region 1 and ΔSC depends on both k regions. At equilibrium, the two order parameters above can coexist and their relative values are restricted by the self-consistent solution of the coupled equations (30) for a given temperature T. With increasing T, ΔSC decreases as ΔW increases. Such a trend is consistent with the experimental observations for optical pumps, but contradicts the behaviour in the case of a high-field THz quench, in which both ΔSC and ΔW decrease from equilibrium. A time-dependent density matrix calculation of the coherent ultrafast dynamics using the above Hamiltonian (discussed elsewhere) shows that, for high THz pump fluences, ΔSC → 0 immediately after the short pump pulse, that is, well before the significant thermal changes that would be dictated by the above self-consistent coupled equations. At the same time, ΔW > ΔSC also decreases from its equilibrium value, together with ΔSC, as a result of non-thermal populations and coupled dynamics (to be discussed elsewhere). By ultrafast quenching Δ SC  = 0, the system can be trapped at a local minimum of the free energy as function of ΔW, accessed via large-amplitude coherent dynamics.

To compute the conductivity for ΔW ≠ 0 and ΔSC = 0, we followed the steps used to derive the conductivity of a magnetically ordered SDW phase presented in Fernandes et al.34. We thus obtain for the real part of the conductivity:

$${\sigma }_{1}(\omega )=2{e}^{2}{v}_{{\rm{F}}}^{2}\left(\frac{{\tau }^{-1}}{{\omega }^{2}+{\tau }^{-2}}\right)\sum _{a=1}^{2}\sum _{{\bf{k}},l}\frac{h\left({E}_{a,l,{\bf{k}}},\omega \right)}{\left({E}_{a,l,{\bf{k}}}-{E}_{\bar{a},l,{\bf{k}}}\right)}\frac{\left(2\omega +{E}_{a,l,{\bf{k}}}-{E}_{\bar{a},l,{\bf{k}}}\right)}{{\left(\omega +{E}_{a,l,{\bf{k}}}-{E}_{\bar{a},l,{\bf{k}}}\right)}^{2}+{\tau }^{-2}}$$
(31)

with hole-band velocity vF, Fermi–Dirac distribution nF, scattering rate τ and

$$h\left({E}_{a,l,{\bf{k}}},\omega \right)=\left[\frac{{n}_{{\rm{F}}}\left({E}_{a,l,{\bf{k}}}\right)-{n}_{{\rm{F}}}\left({E}_{a,l,{\bf{k}}}+\omega \right)}{\omega }\right]\left(-\frac{2{{\it{\Delta }}}_{{\rm{W}}}^{2}}{\bar{m}}+\frac{{C}_{-,l,{\bf{k}}}^{(a)}}{{\bar{m}}^{2}}+{C}_{+,l,{\bf{k}}}^{(a)}\right)$$
(32)

where

$${C}_{n,l,{\bf{k}}}^{(a)}=\left({E}_{a,l,{\bf{k}}}-{\xi }_{{\bf{k}},n}\right)\left({E}_{a,l,{\bf{k}}}+\omega -{\xi }_{{\bf{k}},n}\right)$$
(33)

and

$$\bar{m}={m}_{x,2}{\rm{/}}{m}_{x,1}$$
(34)

Finally, besides that part of the pulse above the SC gap excites non-equilibrium QP populations, the portion below the gap drives time-dependent inductive currents. The two effects coexist in a strongly time-dependent system here, which can significantly increase the total non-linearity and modify the quantum-quench results. However, this complete theory and simulation, beyond the scope of this paper, is not crucial for the present picture of the observed metastable state during unusually long ~10 ns timescales and will be published elsewhere.

Data availability

The data sets generated during and/or analysed during the current study are available from the corresponding author (J.W.) on reasonable request.

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Acknowledgements

Work at Iowa State University was supported by the Army Research Office under award W911NF-15-1-0135 (THz quantum quench spectroscopy). Work at the University of Wisconsin was supported by funding from the DOE Office of Basic Energy Sciences under award number DE-FG02-06ER46327 (structural and electrical characterizations) and DOE Grant no. DE-SC100387-020 (sample growth). Work at the University of Alabama at Birmingham was supported by start-up funds. The THz instrument was supported in part by the M. W. Keck Foundation (J.W.).

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X.Y. and C.V. performed the THz pump–probe spectroscopy measurements and collected the data. C.S., J.H.K. and C.B.E. grew the samples and performed structural and electrical characterizations. M.M. and I.E.P. developed the theory for the hidden phase calculations. J.W., X.Y., C.V. and L.L. analysed the results with the help of P.P.O. and P.G. The paper was written by J.W. and I.E.P., with discussions from all the authors. J.W. conceived and supervised the project.

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Correspondence to J. Wang.

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Supplementary Information, 21 pages, Supplementary Figures 1–3, Supplementary References 1–35

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Yang, X., Vaswani, C., Sundahl, C. et al. Terahertz-light quantum tuning of a metastable emergent phase hidden by superconductivity. Nature Mater 17, 586–591 (2018). https://doi.org/10.1038/s41563-018-0096-3

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