Quantum coherence tomography of light-controlled superconductivity

The coupling between superconductors and oscillation cycles of light pulses, i.e., lightwave engineering, is an emerging control concept for superconducting quantum electronics. Although progress has been made towards terahertz-driven superconductivity and supercurrents, the interactions able to drive non-equilibrium pairing are still poorly understood, partially due to the lack of measurements of high-order correlation functions. In particular, the sensing of exotic collective modes that would uniquely characterize light-driven superconducting coherence, in a way analogous to the Meissner effect, is very challenging but much needed. Here we report the discovery of parametrically driven superconductivity by light-induced order-parameter collective oscillations in iron-based superconductors. The time-periodic relative phase dynamics between the coupled electron and hole bands drives the transition to a distinct parametric superconducting state out-of-equalibrium. This light-induced emergent coherence is characterized by a unique phase–amplitude collective mode with Floquet-like sidebands at twice the Higgs frequency. We measure non-perturbative, high-order correlations of this parametrically driven superconductivity by separating the terahertz-frequency multidimensional coherent spectra into pump–probe, Higgs mode and bi-Higgs frequency sideband peaks. We find that the higher-order bi-Higgs sidebands dominate above the critical field, which indicates the breakdown of susceptibility perturbative expansion in this parametric quantum matter. Multidimensional coherent spectroscopy measurements in iron-based superconductors demonstrate how the coupling between a superconductor and strong light pulses can drive the transition into a non-equilibrium superconducting state with distinct collective modes.

of lightwave-controlled superconductivity via parametric time-periodic driving of the strongly-coupled bands in iron-based superconductors (FeSCs) by a unique phase-amplitude collective mode assisted by broken-symmetry THz supercurrents.We are able to measure non-perturbative, high-order correlations in this strongly-driven superconductivity by separating the THz multidimensional coherent spectra (THz-MDCS) into conventional pump-probe, Higgs collective mode, and pronounced bi-Higgs frequency sideband peaks with highly nonlinear field dependence.We attribute the drastic transition in the coherent spectra to parametric excitation of time-dependent pseudospin canting states modulated by a phase-amplitude collective mode that manifests as a strongly nonlinear shift from ω Higgs to 2ω Higgs .Remarkably, the latter higher-order sidebands dominate over the lower-order pump-probe and Higgs mode peaks above critical field, which indicates the breakdown of the susceptibility perturbative expansion in the parametrically-driven SC state.
Correlation tomography by THz-MDCS provides opportunities for sensing of parametric quantum matter and non-equilibrium SC pairing that even processes finite center-of-mass momentum, with implications for THz supercurrent acceleration to extend gigahertz quantum circuits.
Alternating "electromagnetic" bias, in contrast to DC bias, has shown promise to enable dynamical functionalities by terahertz (THz) modulation and control of topological-/supercurrents and quantum order parameters during timescales faster than a cycle of lightwave oscillations [1][2][3][4][5][6][7][8][9].THz-lightwave-accelerated SC and topological currents [8][9][10][11][12][13][14][15][16][17] have revealed exotic quantum dynamics, e. g., harmonic modes [9,10,18] and gapless quantum fluid states [19] forbidden by equilibrium SC pairing symmetry, or light-induced Weyl and Dirac nodes [1,14].However, high-order correlation characteristics far exceeding the known two-photon light coupling to superconductors are hidden in conventional spectroscopy signals and perturbative responses, where a mixture of multiple excitation pathways contribute to the same low-order responses [20,21].A compelling solution to sensing SC coherence of lightdriven states is to be able to identify unambiguously their collective modes [8,[22][23][24][25][26][27].The dominant collective excitations of the equilibrium SC phase range from amplitude fluctuations (Higgs mode) to fluctuations of interband phase differences (Leggett mode) of the SC order parameter.Although amplitude modes have been observed close to equilibrium when external DC [17,28,29] and AC [9,18] fields break inversion symmetry (IS), here we show that the phase coherent dynamics of the SC order parameters can parametrically drive quantum states, yet-to-be-observed, accessed by strong THz coherent two-pulse excitations.These states are characterized by distinct phase-amplitude collective modes arising from strong light-induced couplings between the amplitude and phase channels.
THz frequency, multi-dimensional coherent nonlinear spectroscopy (THz-MDCS) ) [11,[30][31][32][33][34][35] represents a correlation tomography tool to distinguish between different many-body response functions and light-induced collective modes in superconductors under strong twopulse THz excitation.Unlike for THz-MDCS studies of semiconductors [11,31,32,36], magnets [33], and molecular crystals [34], Fig. 1a illustrates three distinct features of our scheme in superconductors, not explored so far.First, our approach is based on measuring the phase of the supercurrent coherent nonlinear emission, in addition to the amplitude, by using phaseresolved coherent measurements with two intense phase-locked THz pulses of similar field strengths that has not been applied on superconductors.Taking advantage of both the real time and the relative phase of the two THz fields, we separate in two-dimensional (2D) frequency space spectral peaks generated by light-induced correlations and collective mode interactions as a function of gate time t at a fixed delay time between the two pulses, τ = 6.5 ps, under THz driving fields of 229 kV/cm at temperature of 5 K. e, Temporal dynamics of the E NL (t, τ ) amplitude decay below (red diamond, 5 K) and above (black cross, 40 K) T c , as a function of pulse-pair time delay τ under THz driving fields of 333 kV/cm.The correlated nonlinear signal E NL (t, τ )(red) decays over timescales much longer than the pulse duration.from the conventional pump-probe, four-wave-mixing, and high-harmonic-generation signals [9,23].This 2D separation of spectral peaks arising from high-order nonlinear processes achieves a "super" resolution of many-body interactions and collective modes in highly nonperturbative states, which is not possible with the conventional spectroscopy one-dimensional measurements of prior works [9,19,23].Second, as a result of lightwave condensate acceleration by the effective local field inside a thin-film SC induced by two sub-gap THz pulses and electromagnetic propagation effects, the Cooper pairs (k, −k) of the equilibrium BCS state experience a highly nonlinear center-of-mass momentum, p S (t), i. e., SC pairing with finite center-of-mass momentum (Fig. 1a).Precisely, the state persists well after the two strong pulses and results in (k + p S (t)/2, −k + p S (t)/2) Cooper pairing, due to dynamical symmetry breaking of the centrosymmetric pairing states [9].Third, the driven quantum state with current-flow ∝ p S (t) controllable by two-pulse interference can host distinct collective modes that provide parametric excitation of the time-dependent quantum states (Fig. 1a), whose nonlinear interactions determine the THz-MDCS spectral profile [37].
In this Article, we reveal a superconducting state parametrically driven by time-periodic light-induced dynamics of the order parameter phase in a Ba(Fe 1−x Co x ) 2 As 2 superconductor.Such parametric driving becomes important when the phase dynamics is amplified by a phaseamplitude collective mode that develops with increasing THz pulse-pair driving.The change in the character of the SC state manifests itself via the drastic changes in the THz-MDCS spectra observed with increased two-pulse driving.In particular, we observe a transition in THz-MDCS, from wave-mixing peaks centered at the Higgs amplitude mode frequency ω H to bi-Higgs frequency sidebands at 2ω H . Remarkably, the higher-order 2ω H peaks dominate over the lower order single-Higgs peaks and conventional pump-probe spectra above critical THz field strength.We attribute the non-perturbative sidebands at bi-Higgs frequencies to the parametrically driven SC state by the collective temporal oscillations in the phases of the s ± -symmetry order parameter, consistent with our quantum kinetic simulations.
We measured optimally Co-doped BaFe 2 As 2 (Ba-122) epitaxial thin film (60 nm) with T c ∼23 K and lower SC gap 2∆ 1 ∼6.8 meV (Methods section 1.1).We used THz-MDCS to measure the responses to two phase-locked, nearly single-cycle THz pulses A and B of similar field strength (Fig. 1b), with central frequency ω 0 ∼4 meV (black arrow, Figs. , was recorded as a function of both the gate time t (Fig. 1d) and the delay time τ between the two pulses A and B (Fig. 1e).We note three points.First, as demonstrated by E NL (t, τ ) shown in Fig. 1d (pink cross), measured at fixed delay τ = 6.5 ps, the electric field in the time domain allows for simultaneous amplitude-/phase-resolved detection of the coherent nonlinear responses induced by the pulse-pair and has negligible contributions from the individual pulses.This is achieved by subtracting the individual responses, E A (t) and E B (t, τ ) (red and blue solid lines), from the full signal obtained in response to both phase-locked pulses, E AB (t, τ ) (black solid line).Second, the E NL (t, τ ) in Fig. 1e vanishes above the SC transition temperature T c , as seen by comparing the 5 K (red diamond) and 40 K traces (black cross).Third, the THz-MDCS signals persist even when the two pulses do not overlap in time, e. g., at τ = 6.5 ps (Figs.1d-1e).The long-lived correlated signal E NL (t, τ ) indicates that the two sub-gap laser excitations, centered below 2∆ 1 (Fig. 1c), have generated robust supercurrent-carrying macroscopic states persisting well after the pulse.for relatively weak (Fig. 2a), intermediate (Fig. 2b), and strong (Fig. 2c) driving fields.The E NL (t, τ ) dynamics reveals that pronounced coherent temporal oscillations last much longer than the temporal overlap between the two driving pulses (Fig. 1b).These long-lived coherent sponding THz 2D coherent spectra E NL (ω t , ω τ ) at 5 K for the above three pump electric fields, respectively.g-i, The normalized E NL (ω t , ω τ ) spectra are plotted for the same pump fields to highlight the pump field-dependent evolution of the correlation peaks along the 2D frequency vector space.Peaks marked by the dashed lines are located at frequencies associated with the Higgs (green) mode and bi-Higgs frequency sideband (red and blue) consistent with the theory.-2f).We observe multiple distinguishing and well-defined resonances with unique lineshapes that drastically change with increasing field strength.These E NL (ω t , ω τ ) spectra differ strongly from the conventional ones measured, e. g., in semiconductors [11,31,32].In the latter uncorrelated systems, peaks are observable at multiples of the THz driving pulse frequency ω 0 ∼4 meV (magenta dashed line), as expected in the case of a rigid excitation energy bandgap.The observed peaks in FeSCs are much narrower than the excitation pulse width ∆ω (Fig. 1c).This result implies that E NL (t, τ ) oscillates with the frequencies of SC collective mode excitations that lie within the ∆ω of the few-cycle driving pulses, with the width of the THz-MDCS spectral peaks determined by the SC mode damping and not by ∆ω of the driving pulses.For such high fields, the two strongest THz-MDCS peaks are roughly located at (2.3, 0) meV and (2.3, −2.3) meV, while two weaker peaks become detectable at higher frequencies, at (6.2, 0) meV and (6.2, −6.2) meV (Fig. 2i).These high field peaks should be distinguished from the low field ones at similar frequencies (Fig. 2g), as the latter have red-shifted with increasing field due to the SC gap reduction.The evolution of the well-defined MDCS spectral peaks reflect the emergence of different collective modes with increasing driving field, which characterize the transition to different non-equilibrium SC states.
We use three principles to classify the observed peaks in (ω t , ω τ ) space.First, we introduce frequency vectors characterizing the two pulses A and B, ω A = (ω 0 ± ∆ω, 0) and ω B = (ω 0 ± ∆ω, −ω 0 ∓ ∆ω), which are centered around ω 0 ∼4 meV (black arrow, Fig. 1c).The corresponding "time vectors" t = (t, τ ) allow us to represent the driving electric fields of Fig. 1b in the form E A (t ) sin(ω A t ) and E B (t ) sin(ω B t ), respectively.Second, the non-equilibrium SC state driven by the above pulse-pair is characterized by a quenched asymptotic value of the time-evolved SC order parameter, which defines the Higgs frequencies ω H,i = 2∆ ∞,i , where i = 1 (i = 2) denotes the hole (electron) pocket of the FeAs bandstructure.The above Higgs mode frequencies decrease from their equilibrium values of 2∆ 0,i with increasing field, which leads to a redshift of the THz-MDCS spectral features observed in Figs.2g-2i.
To identify which nonlinear process generates each peak measured in Figs.2g-2i, we use our quantum kinetic simulations (Methods section 1.3) and the above three principles.Lightwave propagation inside a SC thin film geometry determines the effective driving field E(t) = E THz (t) − µ 0 c 2n J(t), which is obtained from Maxwell's equations [37] and differs from the applied field E THz (t) (n is the refractive index).This effective field drives the nonlinear supercurrent J(t), described self-consistently by solving the gauge-invariant SC Bloch equations [23,35,37] (Methods section 1.3) for a 3-pocket SC model with strong electron-hole pocket interaction U far exceeding the intra-band pairing interaction.Using the above results, we simulate directly the E NL (t, τ ) temporal dynamics measured in the experiment (Fig. 2j as an example) and then obtain the E NL (ω t , ω τ ) spectra (Figs.2k-2n).These simulations are fully consistent with the observed drastic change in the THz-MDCS spectra, where non-perturbative spectral peaks emerging with increasing field, as shown in Figs.2g-2i, are indicative of a transition to light-driven SC states with different, emergent collective modes.We elaborate the above quantum state transition by using three different excitation regimes, marked in Figs.2o (red arrows) as field-strength dependence of the interband phase difference δθ(ω) peak: I, perturbative regime; II, the non-perturbative state with dominant Higgs amplitude mode; regime III, the parametrically-driven SC state determined by phase-amplitude collective mode.We first examine the perturbative susceptibility regime I, where the Higgs frequency ω H,1 remains close to its equilibrium value, 2∆ 1 ∼ 6.8 meV, similar to the "rigid" excitation energy gap in semiconductors.The simulated THz-MDCS spectrum (Fig. 2k) then shows several peaks (Table 1, Methods) splitting along the ω τ vertical axis, at ω t = ω 0 (dashed magenta line) and ω t = ω H,1 (dashed green line).The conventional pump-probe signals are observed at (ω 0 , −ω 0 ) and (ω 0 , 0) in Fig. 2k, generated by the familiar third-order processes ω A − ω A + ω B and ω B − ω B + ω A , respectively.Four-wave mixing signals are also observed at (ω 0 , ω 0 ) and (ω 0 , −2ω 0 ), generated by the third-order processes 2ω A − ω B and 2ω B − ω A .However, the perturbative behavior in this regime are inconsistent with the dominance of higher-order peaks (Fig. 2g) for the much stronger fields used in the experiment to achieve the necessary signal-to-noise ratio.
By increasing the field strength (Figs.2l-2n), the calculated signals along the ω τ vertical axis and at (ω 0 , −ω 0 ), (ω 0 , 0) diminish.Only peaks along (ω t , 0) and (ω t , −ω t ) are then predicted by our calculation, consistent with the experiment in Figs.2g-2i.For the lower field strength of 250 kV/cm (Fig. 2l), our calculated THz-MDCS spectrum shows two weak peaks at ω t ∼ 2 meV (black dashed line) and two strong broken-IS peaks at ω t = ω H,1 ∼ 6 meV (green dashed line), similar to the experimental THz-MDCS peaks shown in Fig. 2g.The weak peaks at ω t ∼ 2 meV (black dashed line) arise from high-order difference-frequency Raman processes (PP, Table 2 in Methods), which generate pump-probe signals at ω t = ω H,1 − ω 0 , as observed in Figs.2d and 2g.The strong peaks at the Higgs frequency ω t = ω H,1 ∼6 meV (green dashed line) dominate for intermediate fields up to ∼400 kV/cm (regime II, Fig. 2o), but vanish if we neglect the electromagnetic propagation effects as discussed later.The BCS ground state evolves into a finite-momentum-pairing SC state, which is determined by the condensate and persisting well after the pulse.Higgs frequency peaks then arise from ninth-order IS breaking nonlinear processes generated by the coupling between the Higgs mode and the lightwave accelerated supercurrent J(t) (IS Higgs, Table 2 in Methods).The superior resolution achieved for sensing the collective modes using THz-MDCS with 2D coherent excitation is far more than a static IS symmetry breaking scheme using a DC current (Supplementary Fig. 9, Note 7).
For an even higher field strength of 350 kV/cm (Fig. 2m) and of 700 kV/cm (Fig. 2n), the THz-MDCS spectra change above the excitation threshold where the order parameter phase dynamics becomes significant, as shown in Fig. 2o (regime III).As discussed in Methods (section 1.4), time-periodic phase deviations from their equilibrium values, with frequency ω H,1 , parametrically drive pseudo-spin time-dependent canting (Supplementary Fig. 4d), which coincides with a highly nonlinear field-dependence of p S (Supplementary Fig. 4c).In this regime III, new dominant THz-MDCS peaks emerge at ω t = 2ω H,1 − ω 0 (blue dashed line), referred as to bi-Higgs frequency sideband, while satellite peaks are also observed at ω t = 2ω   for a field strength of 250 kV/cm between (i) the full calculation that includes electromagnetic propagation and interference effects leading to slowly decaying p S (t) after the pulse (Fig. 4a), and (ii) a calculation where these effects are switched off, in which case p S (t) oscillates during the THz pulse and vanishes afterwards (Fig. 4b).In the latter case, the ω t = ω H,1 peak vanishes in Fig. 4b (green dashed line), and the THz-MDCS spectrum is dominated by broad pump-probe (PP) peaks at ω t = ω 0 ∼4 meV, similar to Fig. 2k.This result suggests that the peaks at ω H,1 dominating the PP peaks in nonlinear regime II, as in Fig. 2g (data) and Fig. 4a (theory), provide coherent sensing of non-perturbative Higgs collective modes underpinning the finite-momentum-pairing SC phase (Supplementary Note 7) different from the ground state in By comparing Figs.4c and.4d, we see that the signals at frequencies ω t = 2ω H,1 − ω 0 (blue dashed line) and ω t = 2ω H,1 − 2ω 0 (red dashed line) are absent when the order parameter phase can be approximated by its equilibrium value.We also compare the full result with a calculation without interband Coulomb interaction between the electron hole pockets (Fig. 4e), which diminishes the bi-Higgs frequency signals.If the inter-band Coulomb coupling exceeds the intra-band pairing interaction, the Leggett mode phase oscillations lie well within the quasiparticle continuum (regime I), so they are overdamped (Fig. 2o).Above critical THz driving (regime III), however, the THz-modulated superfluid density of strongly-Coulomb-coupled electron and hole pockets (Methods section 1.5 and Supplementary Note 4) enhances the nonlinear coupling between the phase-amplitude collective mode (ω H,1 ) and quasi-particle excitations (ω H,1 ) in the parametrically-driven SC state that processes supercurrent J(t)-oscillations at 2ω H,1 visualized as the unique bi-Higgs sidebands in the THz-MDCS spectra (Fig. 2i).

aFig. 1 .
Fig. 1.Terahertz (THz) multidimensional coherent spectroscopy of lightwave accelerated non-equilibrium superfluid states in FeSCs.a, Schematics of lightwave supercurrent generation, coherent control and detection of the parametrically-driven SC state via two phaselocked THz pulses in our experiment.b, Temporal waveforms of the nearly single-cycle THz pulse-pair used in the experiment (red and blue lines), and c, spectra of the used pulses, centered at ω 0 = 4 meV.d, Temporal dynamics of the measured coherent nonlinear transmission E NL (t, τ )(pink) = E AB (t, τ )(black) − E A (t)(red) − E B (t, τ )(blue) as a function of gate time t at a fixed delay time between the two pulses, τ = 6.5 ps, under THz driving fields of 229 kV/cm at temperature of 5 K. e, Temporal dynamics of the E NL (t, τ ) amplitude decay below (red diamond, 5 K) and above (black cross, 40 K) T c , as a function of pulse-pair time delay τ under THz driving fields of 333 kV/cm.The correlated nonlinear signal E NL (t, τ )(red) decays over timescales much longer than the pulse duration.
1c) and broadband frequency width of ∆ω ∼6 meV (purple dashed line, Fig. 1c) (Methods section 1.2).Representative time scans of these THz-MDCS experiments driven by laser fields E THz,A,B =229 kV/cm, are shown in Fig. 1d.The measured nonlinear differential emission cor-

Figure 2
Figure 2 compares the 2D THz temporal profile of the coherent nonlinear signal E NL (t, τ )

Fig. 2 .
Fig.2.Drastic changes of correlations and collective modes revealed in the driving electric field dependence of THz-MDCS.a-c, Two-dimensional (2D) false-colour plot of the measured coherent nonlinear transmission E NL (t, τ ) of FeSCs superconductors at 5 K induced by THz pump electric fields of (a) 229 kV/cm, (b) 333 kV/cm and (c) 475 kV/cm.d-f , The corresponding THz 2D coherent spectra E NL (ω t , ω τ ) at 5 K for the above three pump electric fields, respectively.g-i, The normalized E NL (ω t , ω τ ) spectra are plotted for the same pump fields to highlight the pump field-dependent evolution of the correlation peaks along the 2D frequency vector space.Peaks marked by the dashed lines are located at frequencies associated with the Higgs (green) mode and bi-Higgs frequency sideband (red and blue) consistent with the theory.j, An example of calculated E NL (t, τ ) as a function of gate time t and delay time τ for 250 kV/cm pump field.k-n, 2D Fourier transform of E NL (t, τ ).As predicted by the theory, dashed black (blue) lines indicate pump-probe ω t = ω H,1 − ω 0 (bi-Higgs frequency sideband ω t = 2ω H,1 − ω 0 ) while IS-breaking signals at Higgs ω t = ω H,1 (bi-Higgs frequency sideband ω t = 2ω H,1 −2ω 0 ) are marked by vertical dashed green (red) line; pump-probe peaks at ω t = ω 0 are indicated by vertical dashed magenta lines.o, Field-strength dependence of the dominant peak in the spectrum of the interband phase difference δθ(ω).p, Field-strength dependence of the bi-Higgs frequency sidebands at 2ω H,1 − ω 0 follows the δθ(ω) behavior in (o), which identifies the importance of light-induced time-periodic phase dynamics at the ω H,1 frequency in driving a non-equilibrium SC state.Three excitation regimes are marked (main text).
The normalized E NL (ω t , ω τ ) experimental spectra shown in Figs.2g-2i visualize nonlin-ear couplings of SC collective mode resonances and their field-dependences.For the weaker pump field of E 0 = 229 kV/cm in Fig.2g, the THz-MDCS spectrum shows four dominant peaks.Intriguingly, the two strongest peaks are located at the higher frequencies, roughly (6, 0) meV and (6, −6) meV, with the weaker peaks at the lower frequencies, slightly below (2, 0) meV and (2, −2) meV.This observation is in strong contrast to the expectation from conventional harmonic generation that high-order nonlinear signals should be weaker than lowerorder ones.Such a reversal of coherent nonlinear signal strengths indicates a breakdown of a susceptibility perturbative expansion around the SC equilibrium state.For the intermediate field of E 0 = 333 kV/cm (Fig.2h), the THz-MDCS spectrum shows several peaks close to each other (red and green lines), centered at new frequencies ∼ (5, 0) meV and (5, −5) meV which exhibit the similar non-perturbative behavior with dominant high order THz-MDCS spectral peaks.The spectral profile changes again with increasing THz driving: four peaks are observable in the THz-MDCS spectrum for the highest studied pump field of E 0 = 475 kV/cm.

Fig. 3 .
Fig. 3. Temperature dependence of THz-MDCS signals.a, THz-MDCS spectra E NL (ω t , ω τ ) at 16 K for pump electric field 333 kV/cm.b, Temporal profiles of two-pulse THz coherent signals E NL (t, τ ) at various temperatures from 5 K to 22 K for a peak THz pump electric field of E pump = 229 kV/cm and τ = 6.5 ps.Traces are offset for clarity.c, The corresponding Fourier spectra of the coherent dynamics in (b).d-e, A 2D false-color plot of THz coherent signals (e) as a function of temperature and frequency ω t with (d) integrated spectral weight at various temperatures.Dashed gray line indicates the SC transition temperature.

,1 − 2ω 0 (
red dashed line).The spectral position of these emergent sideband peaks in 2D frequency space indicates that they arise from a second harmonic of the Higgs frequency, 2ω H,1 .Figure 2p demonstrates a threshold nonlinear behavior of these bi-Higgs frequency sideband peak strengths, which coincides with the development of strong phase dynamics as seen in Fig. 2o.These theoretical prediction are fully consistent with our experimental observations in Figs.2h and 2i.For the intermediate field in Fig. 2m, the THz-MDCS peaks at ω t = 2ω H,1 − 2ω 0 ∼4 meV (red dashed line) and ω t = ω H,1 ∼6 meV (green dashed line) are close to each other, so they merge into a single broad resonance around (5, 0) meV and (5, −5) meV in the calculated 2D spectrum.This result agrees with the measured broad, overlapping THz-MDCS peaks ∼5meV in Fig.2h, while the calculated ω t = 2ω H,1 − ω 0 peak (blue line) is not visible experimentally due to the substrate absorption.For the highest studied field strength (Fig.2n), the calculated THz-MDCS signals are dominated by the bi-Higgs frequency nonlinear sidebands at ω t = 2ω H,1 − ω 0 ∼ 6.0 meV and ω t = 2ω H,1 − 2ω 0 ∼ 2.0 meV.Both sidebands peaks now fall into the substrate transparency region and are clearly resolved in Fig.2i.The emergence of these new MDCS peaks in Regime III will be discussed later as manifestations of a phase-amplitude collective mode that parametrically drive pseudo-spin canting with respect to the s ± -symmetry equilibrium directions.

Figure 3
Figure3demonstrates the strong temperature dependence and redshift of the observed peaks as we appoach T c .The THz-MDCS spectrum E NL (ω t , ω τ ) at temperature 16 K is shown in

Figure 4
Figure4offers more insight into the physical mechanism behind the observed transition in the THz-MDCS spectra with increasing field.First, we compare the spectra E NL (ω t , ω τ )
j, An example of calculated E NL (t, τ ) as a function of gate time t and delay time τ for 250 kV/cm pump field.k-n, 2D Fourier transform of E NL (t, τ ).As predicted by the theory, dashed black (blue) lines indicate pump-probe ω t = ω H,1 − ω 0 (bi-Higgs frequency sideband ω t = 2ω H,1 − ω 0 ) while IS-breaking signals at Higgs ω t = ω H,1 (bi-Higgs frequency sideband ω t = 2ω H,1 −2ω 0 ) are marked by vertical dashed green (red) line; pump-probe peaks at ω t = ω 0 are indicated by vertical dashed magenta lines.o, Field-strength dependence of the dominant peak in the spectrum of the interband phase difference δθ(ω).p, Field-strength dependence of the bi-Higgs frequency sidebands at 2ω H,1 − ω 0 follows the δθ(ω) behavior in (o), which identifies the importance of light-induced time-periodic phase dynamics at the ω H,1 frequency in driving a non-equilibrium SC state.Three excitation regimes are marked (main text).responses generate sharp THz-MDCS spectral peaks visible up to ∼8 meV below substrate absorption (Methods).These spectra were obtained by Fourier transform of E NL (t, τ ) with respect to both t (frequency ω t ) and τ (frequency ω τ ) (Figs. 2d