Abstract
‘Sudden’ quantum quench and prethermalization have become a crosscutting theme for discovering emergent states of matter^{1,2,3,4}. Yet this remains challenging in electron matter^{5,6,7,8,9}, especially superconductors^{10,11,12,13,14}. The grand question of what is hidden underneath superconductivity (SC)^{15} appears universal, but poorly understood. Here we reveal a longlived gapless quantum phase of prethermalized quasiparticles (QPs) after a singlecycle terahertz (THz) quench of a Nb_{3}Sn 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 highfrequency 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 nonthermal quench of coupled SC and chargedensitywave (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 singlecycle intense THz pulses (red arrow in Fig. 1a) opens fascinating possibilities for the nonthermal and nonadiabatic modification of Hamiltonians, in analogy to parameter quenches^{3,4} in ultracold atoms. During postquench prethermalization time evolution, the system can reach persisting plateau states that are far from equilibrium^{1}. Such states are inaccessible by conventional tuning or highfrequency 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 preempted ‘hidden’ orders beneath SC (Fig. 1a). This has not yet been demonstrated, despite recent progress in THzdriven dynamics, for example, collective modes^{11,12} excited by a weak quench and lightenhanced SC in cuprates^{13,14}. A strong THz quench field, one order of magnitude higher than prior studies^{10,11,12}, is desired to drive phase transitions^{16,17}, but remains scarce so far. Prior studies also indicate photoinduced, spatially inhomogeneous SC/metallic phases due, in part, to dirty limit samples^{10,18}.
Here we present evidence of a quantumquench phase transition to a gapless, prethermalized QP phase \(\left{\psi }_{{\rm{Plateau}}}\right\rangle\) in a Nb_{3}Sn superconductor. Subpicosecond singlecycle THz fields up to 620 kV cm^{–1} (Fig. 1b) nonadiabatically excite QPs without excessive heating, with central frequency ħω = 4 meV (Fig. 1c) slightly below twice the QP excitation gap, 2Δ_{SC} ≈ 5.1 meV at 4.1 K. Such nonthermal depletion of the SC condensate is implemented on timescales comparable to SC gap fluctuation times of ħ/2Δ_{SC} ≈ 0.8 ps in a nearly singlecrystal Nb_{3}Sn film. A sample of 20 nm thickness on a 1 mm Al_{2}O_{3}(10–12) substrate exhibited a SC transition at T_{c} ≈ 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 singularitylike electronic density of states and by strong electron–phonon interaction^{20,21,22,23,24,25,26}. The equilibrium frequencydependent complex conductivities σ_{1}(ω,T) and σ_{2}(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 σ_{2} and zero conductivity in σ_{1} below 2Δ_{SC}. These features change in the normal state, which displays a Drudelike behaviour in which σ_{2}(ω) gradually decreases towards low frequency and σ_{1}(ω) shows no gap (16 K traces).
To characterize the nonequilibrium postquench states we measured the complex conductivity (σ_{1}(ω,Δt_{pp}), σ_{2}(ω,Δt_{pp})) as a function of the pump–probe delay Δt_{pp} (Fig. 1f). To obtain this, we first constructed a twotime THz pump and THz probe map of the induced THz probe field transmission through the excited sample, ΔE(t_{gate},Δt_{pp}), by scanning both gating pulse delay t_{gate} and Δt_{pp} (Methods). The top panel of Fig. 1f shows an example of raw data at 4.1 K for E_{THz} = 120 kV cm^{–1}. The timedependent dielectric response functions were then retrieved at each Δt_{pp} via Fourier transformation along the t_{gate} axis (Methods and Supplementary Information). The lowfrequency spectra Δσ_{2}(ω) = Δn_{s}e^{2}/mω reflect the pumpinduced change in the superfluid density Δn_{s}/n_{s} (Fig. 1f). The peak of the differential THz transmission ΔE/E_{0} at t_{gate} = −80 fs (Fig. 1f, inset) closely follows the aboveobtained Δn_{s}/n_{s}, in both dynamics and magnitude. This indicates that the measured ΔE/E_{0} originates from condensate quench and recovery.
Figure 2 demonstrates two regimes of postquench states obtained by tuning, via E_{THz}, the ‘distance’ from the equilibrium SC state \(\left{\psi }_{{\rm{SC}}}\right\rangle\). At T = 4.1 K and Δt_{pp} = 12 ps, the red arrows in Fig. 2a mark two regimes: (1) a partial quench regime at fields E_{1}–E_{3} = 47, 120 and 155 kV cm^{–1}, respectively, in which −Δn_{s}/n_{s} ∝ ΔE/E_{0} shows a suppression of n_{s} and decays on a subnanosecond timescale (Fig. 1f); (2) a strong quench regime at E_{4}–E_{6} = 256, 446 and 620 kV cm^{–1}, in which the nonlinear saturation of the SC condensate density to a minimum is accompanied by a longlived 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 nonthermal transition to a gapless coherent state with exotic correlations^{3,4}. Although it differs from the normal state^{27}, the two have indistinguishable THz conductivities in dirtylimit superconductors^{28}. Here we observed a new upturn feature in both the nonequilibrium σ_{1}(ω) and σ_{2}(ω) at low frequencies that differs from the equilibrium (Fig. 2b). First, a nonequilibrium σ_{1}(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 E_{F} (1–10 meV) as compared to the normal state (grey diamonds in Fig. 2b). Such a coherent peak in σ_{1} is already observed in the partialquench regime, in which it can arise from condensate coherence factors. Most intriguingly, it even persists in regime 2 above the nonlinear saturation of Δn_{s}/n_{s} to its minimum or zero value, for example, at E_{4}–E_{6} (blue, purple and red circles, respectively). The persisting σ_{1} 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 σ_{2}(ω,Δt_{pp}) (filled circles) that, for E_{1}–E_{3} 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 nonequilibrium σ_{1}(ω) line shape. As the Δn_{s}/n_{s} reduction saturates at E_{4} = 256 kV cm^{–1}, the 1/ω dependence in σ_{2} (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 T_{c} in the E_{4} panel. The σ_{2} suppression in the deep nonlinear saturation regime, E_{5}–E_{6}, 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 σ_{1} peak. More importantly, as shown in Fig. 2c, the coherent σ_{1} peak disappears for highfrequency pump pulses tuned at 1.55 eV (at a threshold fluence of 4 μJ cm^{–2} (black lines)), unlike in prior studies^{9}^{,29}, or when quenching above T_{c} (T = 18 K and 620 kV cm^{–1} (red lines). In addition, the observed nonequilibrium line shape is clearly distinct from an inhomogeneous mixed SC/normal state (Supplementary Fig. 2) and beyond the predictions of the standard BCS quench model^{3,4,28}. These distinct features could arise from orderparameter fluctuations in a gapless QP quantum phase, tuned by THz pulses outofequilibrium, that emerges after nonthermal n_{s} depletion to its minimum (or zero) value (Fig. 2a) and evolves as a longlived prethermalization plateau, as discussed below.
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 selfenergy Σ_{2}(ω). A standard extended Drude analysis using the selfenergy (Methods) produces a frequencydependent 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 2Δ_{SC}, for example, seen for both a strong (E_{5}) and a partial (E_{2}) SC quench. In comparison, Fig. 2e shows that, in equilibrium, 1/τ(ω) at T > T_{c} (18 K, grey circles) remains fairly constant and converges to the elastic impurity scattering 1/τ_{imp} ≈ 7.7 meV ≈ 2Δ_{ SC }, whereas for T < T_{c}, the 1/τ(ω) spectra clearly reveal zero scattering below 2Δ_{SC}. Following the THz quantum quench, vanishing 1/τ(ω) at ω → 0 is significantly lower than the 1/τ_{imp}. Therefore, these differences from the normal state indicate that the THz lighttuning 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 E_{th} = 256 kV cm^{–1} that corroborates the transition to the persisting QP quantum phase \(\left{\psi }_{{\rm{Plateau}}}\right\rangle\) at E_{4}–E_{6}. This is clearly seen in the detailed pumpfluence dependence of ΔE/E_{0} as a function of time delay, shown in Fig. 3a on a logarithmic scale at T = 4.1 K. At a low quench field, E_{THz} < E_{th}, we observe a smooth picosecond condensate recovery with a relaxation time τ^{fast} ≈ 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 Δσ_{2} up to 1,000 ps (Fig. 3a, inset) shown for E_{2}. In contrast, at high quench fields, E_{THz} > E_{th}, in which Δn_{s}/n_{s} saturates to its minimum (or zero) value (Fig. 2a), a much longer manynanosecond quasisteady temporal regime emerges and dominates the outofequilibrium dynamics. This is witnessed, for example, by an order of magnitude longer relaxation time τ^{slow} ≈ 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, nonequilibrium σ_{2}(ω) (Fig. 3c) has already changed markedly at Δt_{pp} = 12 ps as compared to Δt_{pp} = −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 T_{c}, which again indicates the distinct difference between the driven quantumphase evolution and normal state thermalization. For a normal state quench at 620 kV cm^{–1}, σ_{2}(ω) has mostly recovered to its ground state behaviour in the 1,000 ps trace in Fig. 3d, indicative of a short ΔE/E_{0} decay constant of ~0.6 ns (inset). This energy relaxation τ^{th} 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 Δt_{pp} = 12 ps, 500 ps and 1,000 ps at E_{6} = 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 ω_{c} ≈ 2Δ_{SC}. 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 buildup 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 frequencyindependent 1/τ(ω) (grey circles in Fig. 3e), always greater than 1/τ_{imp}, and to a full relaxation before 1,000 ps.
The observation of an additional spectral weight near E_{F} in the hidden \(\left{\psi }_{{\rm{Plateau}}}\right\rangle\) as compared to the normal state (Fig. 2b) indicates that the equilibrium condensate n_{s}δ(ω) at ω = 0 did not come exclusively from the spectral weight (W) at ω ≤ 2Δ_{SC} but also involved a larger competing gap Δ_{W} > Δ_{SC}. Bilbro and McMillan^{20} introduced a meanfield model to predict the coexistence of a CDWlike order and SC in the ground state of Nb_{3}Sn, which is driven by strong electron–phonon coupling. This coexistence results in a correlation energy gap that varies between different regions in kspace^{20}. Here we use this model to demonstrate the emergence of additional gapless excitations after the decrease of Δ_{W}(t) that accompanies the THz quench of Δ_{SC}(t). A σ_{1} peak at low frequencies is then formed by transferring the spectral weight from higher frequencies (Supplementary Information). Even a relatively small (~10%) laserinduced decrease of Δ_{W}(t) in a low T transient state with Δ_{SC} = 0 leads to a coherent peak in σ_{1}(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 Δ_{W}(t) = 0.8Δ_{W,0} (Fig. 4c) and 0.9Δ_{W,0} (Fig. 4b), σ_{1} develops a sharp peak at low frequencies (Fig. 4e,f) that is absent for the equilibrium Δ_{W,0} ≈ 40 meV (ref. ^{22}) (Fig. 4d). The emergence of such a peak when (Δ_{W}(t) < Δ_{W,0}, Δ_{SC}(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 lowfrequency coherent peak could arise from a THzquenchinduced decrease in a competing order parameter and/or the partial kdependent Fermi surface that gap together with a suppressed 1/τ(ω).
A THz ultrafast quench of a low energy Δ_{SC} with a simultaneous decrease in high energy Δ_{W} can arise from coherent/nonthermal 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 kdependent change of the electronic correlation gap creates a nonthermal ‘initial condition’ that can access different regions of the free energy landscape^{30}. To illustrate how a longlived prethermalized state may emerge and become trapped for a long time after the coherent temporal regime (Fig. 3b), a freeenergy calculation based on the mean–field model of Bilbro and McMillan^{20} (Methods), is shown in Fig. 4g–i. Compares the dependence on Δ_{W} for three cases: (1) Δ_{SC} = Δ_{SC,4K} (Fig. 4g), as in the equilibrium state below T_{c}. The freeenergy minimum is then located at Δ_{W} = Δ_{W,0} and corresponds to an equilibrium homogeneous phase characterized by a twocomponent kdependent order parameter^{20}. (2) Δ_{SC,18K} = 0 (Fig. 4h) describes the thermal normal state that shows an increase in its equilibrium Δ_{W} (dashed red line) due to the reduction of the competing SC order^{20}. We expect such a landscape to follow the quenching of Δ_{SC} via highenergy optical pumping (Fig. 2c). (3) Δ_{SC} = 0 with T(t) < T_{c} (Fig. 4i), achieved via nonthermal THz lightinduced dynamics, changes the freeenergy landscape from that of the thermal normal state (Fig. 4h). Within the model of Bilbro and McMillan^{20} (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 σ_{1}(ω → 0) for the suppressed scattering 1/τ(ω). Additional couplings to acoustic phonons^{19,23,24} and the details of the band structure influence such metastable states with reduced lattice distortion/dimerization^{26}. The THz tuning ‘suddenly’ brings the system to an excited state during laser pulses with Δ_{SC} = 0 and initiates a coherent/anharmonic motion of Δ_{W}(t) that can access to such freeenergy 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 manganites^{6} that was shown to nonadiabatically generate ferromagnetic correlation during laser pulses and subsequently drive a magnetic phase transition. Such a simultaneously suppressed Δ_{W} and SC order parameters, going beyond the thermodynamic selfconsistency limitation^{20}, leads to our observation of the metastable gapless QP quantum phase above the quench threshold. THzquantumquench prethermalization of the competing orders discovered may be extended to access hidden densitywave phases and quantum criticality under the SC dome in high T_{c} materials^{31}.
Methods
Sample preparation
A nearly singlecrystal Nb_{3}Sn film 20 nm thick with a critical temperature of T_{c} ≈ 16 K was grown by magnetron sputtering on a 1 mm Al_{2}O_{3}(100) substrate. It was grown by the cosputtering 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 Rplane sapphire substrate was exposed to a SiCcoated 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.
Twotime THz pump and THz probe spectroscopy of complex conductivity
We characterize the nonequilibrium postquench states by measuring the complex conductivity (σ_{1}(ω,Δt_{pp}), σ_{2}(ω,Δt_{ pp })) as a function of Δt_{pp}, 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 t_{gate} (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 singlecycle THz pump field with a peak amplitude of up to \({E}_{{\rm{THz}}}^{^{\prime} }\) = 620 kV cm^{–1} was generated by the tiltedpulsefront phasematching method through a 1.3% MgO doped LiNbO_{3} crystal (red line in Supplementary Fig. 1); (2) the weak THz probe field (blue line) was generated by optical rectification and detected by freespace electrooptic 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 t_{gate}.
To measure the static THz conductivity responses without the pump, the firstorder 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 t_{gate}:
Through Fourier transformation, the THz response functions in equilibrium are then readily obtained as:
The polarization response of a transient state that is ‘suddenly’ driven away from equilibrium by the pump pulse is naturally described by the thirdorder nonlinear 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 t_{gate} 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 nonequilibrium responses can only be described properly by a twotime response function:
In analogy to the definition of equation (1), one can then rewrite equation (3) in a more transparent way as:
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 outofequilibrium responses:
A simple deconvolution of equation (4) along the t_{gate} 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 t_{gate}, 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 Δt_{pp} = t_{gate} − t_{pu}, referred to as the pump–probe delay in the text. In this way, the polarization response of the outofequilibrium state can be described as:
Such a twotime 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 frequencyresolved response functions by performing deconvolution along the t_{gate} axis:
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:
where n_{sub} and n_{s} are the optical indices of substrate and sample, λ_{0} is wavelength, d_{s} 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 σ_{1}(ω,Δt_{pp}) and σ_{2}(ω,Δt_{pp}) are presented in Figs. 2 and 3 via a dataextraction procedure (Supplementary Section 2). In this way we access the electronic correlation and fluctuations in the postquantumquench 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 setup 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.
Frequencydependent electric transport
The optical selfenergy has been identified to be more effective than \(\widetilde{\sigma }(\omega )\) itself for underpinning the exact nature of correlated electronic states. This selfenergy can be regarded as a frequency–dependent memory function in analogy to the way in which we analyse manybody interactions in the Green’s function approach. More information is given in Supplementary Section 5. Specifically, we calculate the complex optical selfenergy Σ(ω,T) in terms of both the frequencydependent 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:
Here the plasma frequency ω_{p} is obtained by fitting the normalstate 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:
The density ratio n_{s}/n obtained agrees well with the extrapolated \({\sigma }_{2}^{s}(\omega )\) in the limit of ω → 0 in the superconducting state:
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 phaseseparated patches that consist of SC and normal states. Assuming that the nonequilibrium state is a result of spatial inhomogeneity, the optical response should satisfy:
where σ_{N}(ω) and σ_{S}(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 onedimensional (1D) electron model introduced by Bilbro and McMillan^{20} to a 3D anisotropic model with two electronic bands coupled by an optical phonon, as proposed to explain equilibrium experiments in our material^{19,23,24}. As in Bilbro and McMillan^{20}, we assume for simplicity that the Fermi surface consists of two kregions: region 1 corresponds to momenta in which the coupling of the two bands with a condensed optical phonon or other CDWlike order is strong. This small k region favours the emergence of a CDWlike 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:
In equation (13), the band Hamiltonian is:
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:
where the chemical potential is μ, band offsets are ξ_{n,0} and the effective masses are m_{j,n} (j = x,y, z)^{23,24,26}. The SC pairing interaction is given by:
where the SC order parameter is:
where V_{SC} describes the strength of the interaction, and h.c. is the hermitian conjugate. The ksum 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 ω_{D} is the cutoff frequency^{32}. 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 excitations^{19,20}. The corresponding freephonon Hamiltonian is:
where b is the bosonic annihilation operator and ħω_{ph} is the phonon energy. The coupling of this boson to the two electronic bands is described by^{23,24}:
where g_{ph} 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:
with current density is:
and the THz pump pulse described by the transverse vector potential pulse is
with finite duration τ_{p}, amplitude A_{0}, photon energy ħω_{p} and photon wavevector q_{p}.
Following Porer et al.^{8}, we adopt a meanfield approximation to describe the boson condensation and coherent phonon lightinduced dynamics: b = \(\langle b \rangle + B\). Equation (13) then yields:
where
is the CDW order parameter, V_{W} = 4g_{ph}^{2}/(ħω_{ph}), 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 nonthermal 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:
where we added the damping rate Γ. Following damping of the coherent oscillations, the quasiequilibrium CDW displacement is given by the steadystate solution:
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/nonthermal time evolution, to be discussed elsewhere, is described by density matrix equations of motion^{32}, obtained here for the meanfield electronic Hamiltonian:
The nonlinear response to the THz pulses is treated similar to the ultrafast coherent response of semiconductors and leads to strong changes in both Δ_{SC} and Δ_{W} from their equilibrium values. We can diagonalize the above Hamiltonian H_{el} for given Δ_{SC} and Δ_{W} by using a Bogoliubov canonical transformation^{20} to obtain a basis of quasiparticle excitations^{32} that depend on both SC and CDW coherence. These QP excitation energies are given by:
in regions 1 and 2, respectively. The two quasiparticle branches at a given k are described by a = ±.
After damping of the coherent/nonthermal order parameter motion, the system reaches a quasiequilibrium state determined by the adiabatic free energy f(Δ_{SC},Δ_{W},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:
The equilibrium values of the order parameters Δ_{SC} and Δ_{W} are determined by the global minimum of the above free energy for a given temperature: ∂f(Δ_{SC},Δ_{W})/∂Δ_{SC} = ∂f(Δ_{SC},Δ_{W})/∂Δ_{W} = 0 gives the temperaturedependent energygap equations:
where \(\bar{a}=a\). Δ_{W} is determined by k summation in region 1 and Δ_{SC} depends on both k regions. At equilibrium, the two order parameters above can coexist and their relative values are restricted by the selfconsistent solution of the coupled equations (30) for a given temperature T. With increasing T, Δ_{SC} decreases as Δ_{W} increases. Such a trend is consistent with the experimental observations for optical pumps, but contradicts the behaviour in the case of a highfield THz quench, in which both Δ_{SC} and Δ_{W} decrease from equilibrium. A timedependent density matrix calculation of the coherent ultrafast dynamics using the above Hamiltonian (discussed elsewhere) shows that, for high THz pump fluences, Δ_{SC} → 0 immediately after the short pump pulse, that is, well before the significant thermal changes that would be dictated by the above selfconsistent coupled equations. At the same time, Δ_{W} > Δ_{SC} also decreases from its equilibrium value, together with Δ_{SC}, as a result of nonthermal 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 Δ_{W}, accessed via largeamplitude coherent dynamics.
To compute the conductivity for Δ_{W} ≠ 0 and Δ_{SC} = 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:
with holeband velocity v_{F}, Fermi–Dirac distribution n_{F}, scattering rate τ and
where
and
Finally, besides that part of the pulse above the SC gap excites nonequilibrium QP populations, the portion below the gap drives timedependent inductive currents. The two effects coexist in a strongly timedependent system here, which can significantly increase the total nonlinearity and modify the quantumquench 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.
References
 1.
Aoki, H. et al. Nonequilibrium dynamical meanfield theory and its applications. Rev. Mod. Phys. 86, 779–837 (2014).
 2.
Langen, T. et al. Experimental observation of a generalized Gibbs ensemble. Science 348, 207–211 (2015).
 3.
Barankov, R. A. & Levitov, L. S. Synchronization in the BCS pairing dynamics as a critical phenomenon. Phys. Rev. Lett. 96, 230403 (2006).
 4.
Yuzbashyan, E. A. & Dzero, M. Dynamical vanishing of the order parameter in a fermionic condensate. Phys. Rev. Let. 96, 230404 (2006).
 5.
Torchinsky, D. H. et al. Fluctuating chargedensity waves in a cuprate superconductor. Nat. Mater. 12, 387–391 (2013).
 6.
Li, T. et al. Femtosecond switching of magnetism via strongly correlated spin charge quantum excitations. Nature 496, 69–73 (2013).
 7.
Patz, A. et al. Ultrafast observation of critical nematic fluctuations and giant magnetoelastic coupling in iron pnictides. Nat. Commun. 5, 3229 (2014).
 8.
Porer, M. et al. Nonthermal separation of electronic and structural orders in a persisting charge density wave. Nat. Mater. 13, 857–861 (2014).
 9.
Zhang, J. et al. Cooperative photoinduced metastable phase control in strained manganite films. Nat. Mater. 15, 956–960 (2016).
 10.
Matsunaga, R. & Shimano, R. Nonequilibrium BCS state dynamics induced by intense terahertz pulses in a superconducting NbN film. Phys. Rev. Lett. 109, 187002 (2012).
 11.
Matsunaga, R. et al. Higgs amplitude mode in the BCS superconductors Nb _{1−x} Ti _{ x } N induced by terahertz pulse excitation. Phys. Rev. Lett. 111, 057002 (2013).
 12.
Matsunaga, R. et al. Lightinduced collective pseudospin precession resonating with Higgs mode in a superconductor. Science 345, 1145–1149 (2014).
 13.
Dienst, A. et al. Bidirectional ultrafast electricfield gating of interlayer charge transport in a cuprate superconductor. Nat. Photon. 5, 485–488 (2011).
 14.
Fausti, D. et al. Lightinduced superconductivity in a stripeordered cuprate. Science 331, 189–192 (2011).
 15.
Broun, D. M. What lies beneath the dome? Nat. Phys. 4, 170–172 (2008).
 16.
Kampfrath, T., Tanaka, K. & Nelson, K. A. Resonant and nonresonant control over matter and light by intense terahertz transients. Nat. Photon. 7, 680–690 (2013).
 17.
Liu, M. et al. Terahertzfieldinduced insulatortometal transition in vanadium dioxide metamaterial. Nature 487, 345–348 (2012).
 18.
Beck, M. et al. Transient increase of the energy gap of superconducting NbN thin films excited by resonant narrowband terahertz pulses. Phys. Rev. Lett. 110, 267003 (2013).
 19.
Shirane, G. & Axe, J. D. Neutron scattering study of the latticedynamical phase transition in Nb_{3}Sn. Phys. Rev. B 4, 2957–2963 (1971).
 20.
Bilbro, G. & McMillan, W. L. Theoretical model of superconductivity and the martensitic transformation in A15 compounds. Phys. Rev. B 14, 1887–1892 (1976).
 21.
Markiewicz, R. S. A survey of the Van Hove scenario for high Tc superconductivity with special emphasis on pseudogaps and striped phases. J. Phys. Chem. Solids 58, 1179–1310 (1997).
 22.
Escudero, R. & Morale, F. Point contact spectroscopy of Nb_{3}Sn crystals: evidence of a CDW gap related to the martensitic transition. Solid State Commun. 150, 715–719 (2010).
 23.
Bhatt, R. N. Microscopic theory of the martensitic transition in A15 compounds based on a threedimensional band structure. Phys. Rev. B 16, 1915–1932 (1977).
 24.
Bhatt, R. N. & Lee, P. A. Theory of coherence length and phonon softening in A15 compounds. Phys. Rev. B 16, 4288–4301 (1977).
 25.
Weber, W. & Mattheiss, L. F. Electronic structure of tetragonal Nb_{3}Sn. Phys. Rev. B 25, 2270–2284 (1982).
 26.
Sadigh, B. & Ozolins, V. Structural instability and electronic excitations in Nb_{3}Sn. Phys. Rev. B 57, 2793–2800 (1997).
 27.
Owen, C. S. & Scalapino, D. J. Superconducting state under the influence of external dynamic pair breaking. Phys. Rev. Lett. 28, 1559–1561 (1972).
 28.
Chou, Y. Z. et al. Twisting Anderson pseudospins with light: quench dynamics in terahertzpumped BCS superconductors. Phys. Rev. B 95, 104507 (2017).
 29.
Stojchevska, L. et al. Ultrafast switching to a stable hidden topologically protected quantum state in an electronic crystal. Science 344, 177–180 (2014).
 30.
Lingos, P. C., Wang, J. & Perakis, I. E. Manipulating femtosecond spinorbit torques with laser pulse sequences to control magnetic memory states and ringing. Phys. Rev. B 91, 195203 (2015).
 31.
Yang, X. et al. Nonequilibrium pair breaking in Ba(Fe_{1−x}Co_{ x })_{2}As_{2} superconductors: evidence for formation of photoinduced excitonic spindensitywave state. Preprint at https://arxiv.org/abs/1804.04987 (2018).
 32.
Papenkort, T., Axt, V. M. & Kuhn, T. Coherent dynamics and pumpprobe spectra of BCS superconductors. Phys. Rev. B 76, 224522 (2007).
 33.
Schnyder, A. P., Manske, D. & Avella, A. Resonant generation of coherent phonons in a superconductor by ultrafast optical pump pulses. Phys. Rev. B 84, 214513 (2011).
 34.
Fernandes, R. M. & Schmalian, J. Transfer of optical spectral weight in magnetically ordered superconductors. Phys. Rev. B 82, 014520 (2010).
Acknowledgements
Work at Iowa State University was supported by the Army Research Office under award W911NF1510135 (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 DEFG0206ER46327 (structural and electrical characterizations) and DOE Grant no. DESC100387020 (sample growth). Work at the University of Alabama at Birmingham was supported by startup funds. The THz instrument was supported in part by the M. W. Keck Foundation (J.W.).
Author information
Affiliations
Contributions
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.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Supplementary Information
Supplementary Information, 21 pages, Supplementary Figures 1–3, Supplementary References 1–35
Rights and permissions
About this article
Cite this article
Yang, X., Vaswani, C., Sundahl, C. et al. Terahertzlight quantum tuning of a metastable emergent phase hidden by superconductivity. Nature Mater 17, 586–591 (2018). https://doi.org/10.1038/s4156301800963
Received:
Accepted:
Published:
Issue Date:
Further reading

LightDriven Raman Coherence as a Nonthermal Route to Ultrafast Topology Switching in a Dirac Semimetal
Physical Review X (2020)

Ultrafast Control of Excitonic Rashba Fine Structure by Phonon Coherence in the Metal Halide Perovskite CH3NH3PbI3
Physical Review Letters (2020)

Terahertz SecondHarmonic Generation from Lightwave Acceleration of SymmetryBreaking Nonlinear Supercurrents
Physical Review Letters (2020)

Coherent bandedge oscillations and dynamic longitudinaloptical phonon mode splitting as evidence for polarons in perovskites
Physical Review B (2020)

Charge density wave modulation in superconducting BaPbO3/BaBiO3 superlattices
Physical Review B (2020)