Abstract
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 terahertzdriven superconductivity and supercurrents, the interactions able to drive nonequilibrium pairing are still poorly understood, partially due to the lack of measurements of highorder correlation functions. In particular, the sensing of exotic collective modes that would uniquely characterize lightdriven 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 lightinduced orderparameter collective oscillations in ironbased superconductors. The timeperiodic relative phase dynamics between the coupled electron and hole bands drives the transition to a distinct parametric superconducting state outofequalibrium. This lightinduced emergent coherence is characterized by a unique phase–amplitude collective mode with Floquetlike sidebands at twice the Higgs frequency. We measure nonperturbative, highorder correlations of this parametrically driven superconductivity by separating the terahertzfrequency multidimensional coherent spectra into pump–probe, Higgs mode and biHiggs frequency sideband peaks. We find that the higherorder biHiggs sidebands dominate above the critical field, which indicates the breakdown of susceptibility perturbative expansion in this parametric quantum matter.
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Main
Alternating ‘electromagnetic’ bias, in contrast to d.c. bias, is emerging as a universal control concept to enable dynamical functionalities by terahertz (THz) modulation^{1,2,3,4,5,6,7,8,9,10}. THzlightwaveaccelerated superconducting (SC) and topological currents^{8,9,11,12,13,14,15,16,17,18} have revealed exotic quantum dynamics, for example, high harmonics^{9,11,19} and gapless quantum fluid states^{20}, and lightinduced Weyl and Dirac nodes^{1,15}. However, highorder correlation characteristics far exceeding the known twophoton light coupling to superconductors are hidden in conventional singleparticle spectroscopies and perturbative responses, where a mixture of multiple excitation pathways contribute to the same loworder responses^{21,22}. A compelling solution to sensing lightinduced SC coherence far from equilibrium is to be able to identify their correlations and collective modes^{8,23,24,25,26,27,28}. The dominant collective excitations of the equilibrium SC phase range from amplitude oscillations (Higgs mode) to oscillations of interband phase differences (Leggett mode) of the SC order parameter. Although amplitude modes have been observed close to equilibrium when external a.c.^{9,19} and d.c.^{18,29,30} fields break inversion symmetry (IS), the orderparameter phase–amplitude coherent oscillations have never been observed in spite of their fascinating opportunity to parametrically drive quantum phases.
Figure 1a illustrates the parametrically driven SC state, which is characterized by distinct phase–amplitude collective modes arising from strong lightinduced couplings between the amplitude and phase channels in ironbased superconductors (FeSCs). The ground state of FeSCs is known to have s_{±}, rather than Bardeen–Cooper–Schrieffer (BCS), pairing symmetry, which is determined by strong coupling between the electron (e) and hole (h) bands. The state can be viewed as the relative orientation of correlated Anderson pseudospins located at different momentum points k (ref. ^{9}). Up and down pseudospins correspond to filled and empty (k, −k) Cooperpair states, whereas canted spins are the superposition of these up and down pseudospins. In FeSCs, the pseudospins are antiparallelly oriented between the e−h bands with orderparameter phase difference of π (red and blue arrows, Fig. 1a (top left)) (the ‘THzMDCS of Anderson pseudospin canting states’ section). The THzdriven dynamics causes timedependent deviations in this s_{±} relative phase and leads to the precession of the correlated pseudospins, which, in turn, parametrically drives pseudospin canting (Fig. 1a, top right) from the equilibrium antiparallel configuration. The latter results from the different dynamics of the SC phase in each band, which leads to different e−h pseudospin rotations. Consequently, the strong lightinduced coupling between the pseudospins (Δρ) in each band and the relative phase between them (Δθ) (Fig. 1a) can lead to phase–amplitude collective oscillations and emergent nonlinear sidebands, absent for either d.c. currents or weak field driving.
Terahertzfrequency multidimensional coherent nonlinear spectroscopy (THzMDCS)^{12,31,32,33,34,35,36} represents a correlation tomography and control tool to distinguish between different manybody response functions and collective modes. Unlike for THzMDCS studies of semiconductors^{12,32,33,37}, magnets^{34} and molecular crystals^{35}, Fig. 1a illustrates three distinct features of our THzMDCS scheme applied for the first time here on superconductors, to the best of our knowledge. 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 phaselocked THz pulses of similar field strengths. Taking advantage of both realtime and relative phase of the two THz fields, we separate in twodimensional (2D) frequencyspace spectral peaks generated by lightinduced correlations and collective mode interactions from the conventional pump–probe (PP), fourwavemixing (FWM) and highharmonicgeneration signals^{9,24}. This 2D separation of spectral peaks arising from highorder nonlinear processes achieves a ‘super’ resolution of higherorder interactions and collective modes in highly nonperturbative states. This capability is not possible with traditional singleparticle or PP spectroscopies^{9,20,24}. Second, as a result of lightwave condensate acceleration by the effective local field inside a thinfilm SC induced by THz pulse pairs and electromagnetic propagation effects^{9}, the Cooper pairs (k, –k) of the equilibrium BCS state experience SC pairing with finite centreofmass momentum p_{S}(t) (Fig. 1a). Precisely, this phase persists well after the two strong pulses and exhibits (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 finitemomentumpairing quantum state with supercurrent flow that is directly proportional to p_{S}(t) controllable by twopulse interference can host distinct collective modes that parametrically drive the timedependent pseudospin oscillators. This process triggers the phase–amplitude dynamics illustrated in Fig. 1a, whose nonlinear interactions determine the THzMDCS spectral profile^{36}.
In this Article, we reveal a parametrically driven SC state by timeperiodic lightinduced dynamics of the order parameter phase in a Ba(Fe_{1−x}Co_{x})_{2}As_{2} superconductor. Such an effect parametrically drives timedependent Anderson pseudospin canting and precession from the antiparallel equilibrium orientation, consistent with our quantum kinetic simulations. Such parametric driving becomes important when the phase dynamics is amplified by a unique phase–amplitude collective mode that develops with increasing THz pulsepair driving and gives rise to the drastic nonlinear shift from ω_{Higgs} to 2ω_{Higgs} Floquet sideband peaks.
THzMDCS of FeSCs
We measured Codoped BaFe_{2}As_{2} (Ba122) epitaxial thin film (60 nm) at the optimal doping with T_{c} ≈ 23 K and lower SC gap of 2Δ_{1} ≈ 6.8 meV (the ‘Sample preparation and quality’ section). We used THzMDCS to measure the responses to two phaselocked, nearly singlecycle THz pulses A and B of similar field strength (Fig. 1b), with a central frequency of ω_{0} ≈ 4 meV (Fig. 1c, black arrow) and broadband frequency width of Δω ≈ 6 meV (Fig. 1c, purple dashed line). Representative time scans of these THzMDCS experiments driven by laser fields E_{THz,A,B} = 229 kV cm^{−1} are shown in Fig. 1d. The measured nonlinear differential emission correlated signal, E_{NL}(t, τ) = E_{AB}(t, τ) − E_{A}(t) − E_{B}(t, τ), was recorded as a function of both gate time t (Fig. 1d) and delay time τ = t_{B} − t_{A} between the two pulses A and B (Fig. 1e). We note three points. First, as demonstrated by E_{NL}(t, τ) (Fig. 1d, pink cross), measured at a fixed delay of −τ = 6.5 ps, the electric field in the time domain allows for simultaneous amplitude/phaseresolved 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) (Fig. 1b, red and blue solid lines, respectively), from the full signal obtained in response to both phaselocked pulses, namely, E_{AB}(t, τ) (Fig. 1d, black solid line). Second, E_{NL}(t, τ) in Fig. 1e vanishes above T_{c}, as seen by comparing the 5 K (red diamond) and 40 K (black cross) traces. Third, the THzMDCS signals persist even when the two pulses do not overlap in time, for example, at −τ = 6.5 ps (Fig. 1d,e). The longlived correlated signal E_{NL}(t, τ) indicates that the two subgap laser excitations, centred below 2Δ_{1} (Fig. 1c), have generated robust supercurrentcarrying macroscopic states persisting well after the pulse.
Figure 2 compares the 2D THz temporal profile of the coherent nonlinear signal E_{NL}(t, τ) 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). One can introduce frequency vectors characterizing the two pulses A and B, namely, ω_{A} = (ω_{0} ± Δω, 0) and ω_{B} = (ω_{0} ± Δω, −ω_{0} ∓ Δω), respectively, which are centred around ω_{0} ≈ 4 meV (Fig. 1c, black arrow). Following these notations, the observed longlived coherent responses generate sharp THzMDCS spectral peaks visible up to ~8 meV below substrate absorption (the ‘Sample preparation and quality’ section). These spectra were obtained by the Fourier transform of E_{NL}(t, τ) with respect to both t (frequency ω_{t}) and τ (frequency ω_{τ}) (Fig. 2d–f). We observe multiple distinguishing and welldefined resonances with unique lineshapes that drastically change with increasing field strength. These E_{NL}(ω_{t}, ω_{τ}) spectra strongly differ from the conventional ones measured, for example, in semiconductors^{12,32,33}, where peaks are observable at multiples of the THz driving pulse frequency ω_{0} ≈ 4 meV (Fig. 2d–f, 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 Δω (Fig. 1c, purple dashed line). This result implies that E_{NL}(t, τ) oscillates with the frequencies of SC collective mode excitations that lie within the Δω value of the fewcycle driving pulses. The width of the THzMDCS spectral peaks is determined by the SC mode damping and not by Δω of the driving pulses.
The normalized E_{NL}(ω_{t}, ω_{τ}) experimental spectra (Fig. 2g–i) visualize the nonlinear couplings of SC collective mode resonances and their field dependences. For the weaker pump field of E_{0} = 229 kV cm^{−1} (Fig. 2g), E_{NL}(ω_{t}, ω_{τ}) shows four dominant peaks. Intriguingly, the two strongest peaks, roughly (6, 0) and (6, − 6) meV, are located at higher frequencies, whereas the weaker peaks, slightly below (2, 0) and (2, −2) meV, are located at lower frequencies. This observation is in strong contrast to the expectation from conventional harmonic generation that highorder nonlinear signals should be weaker than lowerorder ones. For the intermediate field of E_{0} = 333 kV cm^{−1} (Fig. 2h), E_{NL}(ω_{t}, ω_{τ}) shows several peaks close to each other (red and green lines), centred at new frequencies of ~(5, 0) and (5, −5) meV, which exhibit clear nonperturbative behaviour as the dominant highorder THzMDCS spectral peaks over loworder ones. This observation indicates the breakdown of the susceptibility perturbative expansion that is valid around the SC equilibrium state. The spectral profile changes again with increasing THz driving: four peaks are observable in the THzMDCS spectrum for the highest studied pump field of E_{0} = 475 kV cm^{−1}. The two strongest THzMDCS peaks are roughly located at (2.3, 0) and (2.3, −2.3) meV, whereas two weaker peaks become detectable at (6.2, 0) and (6.2, −6.2) meV (Fig. 2i). These highfield peaks should be distinguished from the lowfield ones at similar frequencies (Fig. 2g), as the latter have redshifted with increasing field due to the SC gap reduction. The evolution of THzMDCS spectral peaks reflects the emergence of different collective modes with increasing driving field, which characterizes the transition to different nonequilibrium SC states.
Lightinduced drastic changes in collective modes
We first introduce some basic principles to classify the observed peaks in (ω_{t}, ω_{τ}) space. First, the nonequilibrium SC state driven by the THz pulse pair is characterized by a quenched asymptotic value of the timeevolved SC order parameter, which defines the Higgs frequencies ω_{H,i} = 2Δ_{∞,i}, where i = 1 (i = 2) denotes the h (e) pocket of the FeAs band structure. The above Higgs mode frequencies decrease from their equilibrium values of 2Δ_{0,i} with increasing field, which leads to a redshift in the THzMDCS spectral features (Fig. 2g–i). Note that we only probe the lower Higgs mode, ω_{H,1} ≈ 6.8 meV, whereas the higher Higgs frequency, ω_{H,2} ≈ 19 meV, lies outside the measured spectral range (Fig. 1c). Second, the THz pulses drive the Anderson pseudospin oscillators^{9,36} at different momenta k (the ‘Pseudospin canting driven parametrically by phase oscillations’ section). The pseudospin dynamics is dominated by frequencies ~ω_{H,i;A} = (ω_{H,i}, 0) and ~ω_{H,i;B} = (ω_{H,i}, − ω_{H,i}), that is, fielddependent Higgs and quasiparticle pair excitations, or ~2ω_{A,B}, that is, quasiparticle excitations driven at the laser frequency.
To identify which nonlinear process generates each peak measured in Fig. 2g–i, we use our quantum kinetic simulations (the ‘Gaugeinvariant theory and simulations of THzMDCS signals’ section) and the above three principles. Lightwave propagation inside an SC thinfilm geometry determines the effective driving field \(E(t)={E}_{{{{\rm{THz}}}}}(t)\frac{{\mu }_{0}c}{2n}J(t)\), which is obtained from Maxwell’s equations^{38} and differs from the applied field E_{THz}(t) (n is the refractive index and c is the speed of light). This effective field drives the nonlinear supercurrent J(t), described selfconsistently by solving the gaugeinvariant SC Bloch equations^{24,36,38} (the ‘Gaugeinvariant theory and simulations of THzMDCS signals’ section) for a threepocket SC model with strong e–h pocket interaction U far exceeding the intraband pairing interaction. We directly simulate the E_{NL}(t, τ) temporal dynamics measured in the experiment (Fig. 3a) and then obtain the E_{NL}(ω_{t}, ω_{τ}) spectra (Fig. 3b–e). These simulations are fully consistent with the observed drastic change in the THzMDCS spectra, where nonperturbative spectral peaks emerging with increasing field (Fig. 2g–i) are indicative of a transition to lightdriven SC states with different, emergent collective modes.
We elaborate the above quantum state transitions by identifying three different excitation regimes. They are marked in Fig. 3f (black dashed lines) and distinguished by the fieldstrength dependence of the interband phase difference ∆θ(ω) peak: regime (i), the perturbative susceptibility regime; regime (ii), the nonperturbative state with dominant Higgs amplitude mode; regime (iii), the parametrically driven SC state determined by phase–amplitude collective mode. We first examine regime (i), where the Higgs frequency ω_{H,1} remains close to its equilibrium value, 2Δ_{1} ≈ 6.8 meV, similar to the ‘rigid’ excitation energy gap in semiconductors. The simulated THzMDCS spectrum (Fig. 3b) then shows several peaks (Table 1 and 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 PP signals are observed at (ω_{0}, −ω_{0}) and (ω_{0}, 0) (Fig. 3b), generated by the familiar thirdorder processes ω_{A} − ω_{A} + ω_{B} and ω_{B} − ω_{B} + ω_{A}, respectively. FWM signals are also observed at (ω_{0}, ω_{0}) and (ω_{0}, −2ω_{0}), generated by the thirdorder processes 2ω_{A} − ω_{B} and 2ω_{B} − ω_{A}. However, the perturbative behaviour in this regime is inconsistent with the dominance of higherorder peaks (Fig. 2g) for the stronger fields used in the experiment (to achieve the necessary signaltonoise ratio).
By increasing the field strength (Fig. 3c–e), the calculated signals along the ω_{τ} vertical axis and at (ω_{0}, −ω_{0}) and (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 shown in Fig. 2g–i. For the lower field strength of 250 kV cm^{−1} (Fig. 3c), our calculated E_{NL}(ω_{t}, ω_{τ}) 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 THzMDCS peaks (Fig. 2g). The weak peaks at ω_{t} ≈ 2 meV (black dashed line) arise from the highorder differencefrequency Raman processes (Methods and Table 2, PP), which generate PP signals at ω_{t} = ω_{H,1} − ω_{0} (Fig. 2g). The strong peaks at the Higgs frequency ω_{t} = ω_{H,1} ≈ 6 meV (Fig. 2g,h, green dashed line) dominate for intermediate fields up to ~400 kV cm^{−1} (Fig. 3f, regime (ii)). However, they vanish if we neglect the electromagnetic propagation effects, as discussed later. The BCS ground state evolves into a finitemomentumpairing SC state, which is determined by the condensate momentum p_{S} generated by nonlinear processes (Supplementary Fig. 4c and Supplementary Note 4). This condensate momentum persists well after the pulse. Higgs frequency peaks then arise from ninthorder ISbreaking nonlinear processes generated by the coupling between the Higgs mode and lightwaveaccelerated supercurrent J(t) (Methods and Table 2, IS Higgs). The superior resolution achieved for sensing the collective modes by using THzMDCS with 2D coherent excitation is far more than that achieved by using a static ISbreaking scheme using a direct current (Supplementary Fig. 9 and Supplementary Note 7).
For even higher field strengths of 350 kV cm^{−1} (Fig. 3d) and 700 kV cm^{−1} (Fig. 3e), the THzMDCS spectra change above the excitation threshold where the orderparameter phase dynamics becomes substantial (Fig. 3f, regime (iii)). In this regime, new dominant THzMDCS peaks emerge at ω_{t} = 2ω_{H,1} − ω_{0} (Fig. 3d,e, blue dashed line), referred to as biHiggs frequency sideband. Satellite peaks are also observed at ω_{t} = 2ω_{H,1} − 2ω_{0} (Fig. 3d,e, red dashed line). Figure 3g demonstrates a threshold nonlinear behaviour of these biHiggs frequency sideband peak strengths, which coincides with the development of strong phase dynamics (Fig. 3f). These theoretical predictions are fully consistent with our experimental observations (Fig. 2h,i). For the intermediate field (Fig. 3d), the THzMDCS 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. As a result, they merge into a single broad resonance around (5, 0) and (5, −5) meV, which agrees with the measured broad, overlapping THzMDCS peaks of ~5 meV (Fig. 2h). The calculated ω_{t} = 2ω_{H,1} − ω_{0} peak (Fig. 3d, blue line) is not experimentally visible due to substrate absorption. For the highest studied field strength (Fig. 3e), the calculated THzMDCS signals are dominated by the biHiggs frequency nonlinear sidebands at ω_{t} = 2ω_{H,1} − ω_{0} ≈ 6 meV and ω_{t} = 2ω_{H,1} − 2ω_{0} ≈ 2 meV (ω_{H,1} redshifted to ~5 meV with the highest driving field used). Both sidebands peaks now fall into the substrate transparency region and are clearly resolved in Fig. 2i. The emergence of these new THzMDCS peaks in regime (iii) is a direct manifestation of the phasedriven Anderson pseudospin canting (Fig. 1a), as further discussed later.
Figure 4 demonstrates the strong temperature dependence and redshift in the observed peaks as we appoach T_{c}. The THzMDCS spectrum E_{NL}(ω_{t}, ω_{τ}) at a temperature of 16 K is shown in Fig. 4a for the intermediate field of E_{0} = 333 kV cm^{−1}. It is compared in Fig. 2h with the spectrum at T = 5 K for the same excitation. The broken IS signals observed at the Higgs mode frequency ω_{t} = ω_{H,1} redshift with increasing temperature, from (5.0, 0) and (5.0, −5.0) meV peaks at 5 K to broad peaks slightly below (2.5, −2.5) and (2.5, 0) meV at 16 K (Fig. 4a, green line). This redshift arises from the thermal quench of the SC order parameter 2Δ_{1} with increasing temperature. However, unlike for the case of THz coherent control of the order parameter (Fig. 2h), a thermal quench does not produce any obvious biHiggs frequency THzMDCS peaks, expected at ~1 meV, which is indicative of the coherent origin of the latter. Figure 4b,c shows the temperature dependence of the measured differential coherent emission E_{NL}(t, τ) and corresponding E_{NL}(ω_{t}, τ) at a fixed pulse separation of −τ = 6.5 ps. By comparing the 5 K (black line) and 22 K (grey) traces (Fig. 4b,c), it is evident that when approaching T_{c} from below, the coherent nonlinear emissions quickly diminish and redshift. Finally, Fig. 4d,e shows a detailed plot of E_{NL}(ω_{t}, τ) up to 100 K. The integrated spectral weight (Fig. 4d) correlates with the SC transition at T_{c} (grey dashed line).
Phase–amplitude mode and parametric driving
Figure 5 offers more insights into the physical mechanism behind the observed transition in the THzMDCS spectra with increasing field. First, we compare E_{NL}(ω_{t}, ω_{τ}) for a field strength of 250 kV cm^{−1} between (1) the full calculation that includes electromagnetic propagation and interference effects leading to slowly decaying p_{S}(t) after the pulse (Fig. 5a) and (2) a calculation without electromagnetic propagation effects where p_{S}(t) oscillates during the THz pulse and vanishes afterwards (Fig. 5b). The ω_{t} = ω_{H,1} peaks vanish in Fig. 5b (green dashed line) and E_{NL}(ω_{t}, ω_{τ}) is dominated by broad PP peaks at ω_{t} = ω_{0} ≈ 4 meV. This result suggests that the peaks at ω_{H,1}, which dominate the PP peaks in nonlinear regime (ii) (Fig. 2g (experiment) and Fig. 5a (theory)), provide the coherent sensing of nonperturbative Higgs collective modes underpinning the finitemomentumpairing SC phase (Supplementary Note 7).
Next, we turn to the transition from Higgs to dominant biHiggs signals at ω_{t} = 2ω_{H,1} − ω_{0}. We associate this transition with the development of a timedependent pseudospin canting from the equilibrium antiparallel pseudospin directions, which is parametrically driven by amplified relative phase dynamics at frequency ω_{H,1} (the ‘Pseudospin canting driven parametrically by phase oscillations’ and ‘Phase–amplitude collective mode and biHiggs frequency sidebands’ sections). If the interband Coulomb coupling exceeds the intraband pairing interaction, the Leggettmode phase oscillations lie well within the quasiparticle continuum (regime (i) for weak THz fields; Fig. 3f), so they are overdamped. Above the critical THz driving (Fig. 3f, regime (iii)), however, the THzmodulated superfluid density of strongly Coulombcoupled e–h pockets (the ‘Phase–amplitude collective mode and biHiggs frequency sidebands’ section and Supplementary Note 4) enhances the nonlinear coupling of the orderparameter phase and amplitude oscillations. This leads to phase oscillations at the same Higgs frequency ω_{H,1} as the amplitude oscillations, which we refer as to the phase–amplitude collective mode. The latter interacts with quasiparticle excitations at an energy of ~ω_{H,1}, which amplifies the THzMDCS sideband peaks at frequencies ~2ω_{H,1} (Fig. 3e). This amplification is at the expense of the Higgs mode peak at ω_{H,1}, which dominates in regime (ii) (Fig. 3f).
To further corroborate the transition from Higgs amplitude to phase–amplitude collective mode, we compare (Fig. 5c,d) the THzMDCS spectra obtained from the full calculation for 700 kV cm^{−1} driving with those obtained by turning off the pseudospin canting around the s_{±} equilibrium state driven by the ω_{H,1} timeperiodic phase oscillations (Supplementary Figs. 4d and 6 and Supplementary Note 4). Our formulation of the gaugeinvariant SC Bloch equations in terms of two coupled pseudospin nonlinear oscillators (the ‘Pseudospin canting driven parametrically by phase oscillations’ section) shows that nonadiabatic pseudospin canting is parametrically driven with the timedependent strength of \(\sim \vert{{{\varDelta }}}_{1}\vert^{2}\sin (2\Delta \theta )\Delta{\rho }_{1,2}\), where ∆ρ_{1} and ∆ρ_{2} are the THzdriven deviations of the x and y Anderson pseudospin components from equilibrium, respectively. Such phasedependent contribution to the nonlinear response is amplified by the strong THz modulation of the superfluid density characterized by lightinduced changes in Δ_{1}^{2}. Consequently, the threshold nonlinear field dependence of this coupling (Supplementary Fig. 4d) leads to the strong field dependence of the ~2ω_{H,1} sideband (Fig. 3g). By comparing Fig. 5c,d, 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 e and h pockets (Fig. 5e), which again diminishes the biHiggs frequency signals.
Conclusion
We demonstrate parametrically driven superconductivity enabled by lightinduced phase–amplitude coupling and by timeperiodic relative phase dynamics. Higgs coherence tomography of the lightwaveinduced highorder correlations and entanglement in parametric quantum matter is of direct interest to quantum information, sensing and superconducting electronics.
Note added after publication: We could have cited ref. ^{39}, as it contributes a potentially helpful item in the existing literature. We thank Dr Matteo Puviani for bringing this to our attention.
Methods
Sample preparation and quality
We measure optimally Codoped BaFe_{2}As_{2} epitaxial singlecrystal thin films^{40} (Supplementary Note 8 and Supplementary Figs. 10–12). They are 60 nm thick, grown on 40nmthick SrTiO_{3} buffered (001)oriented (La;Sr)(Al;Ta)O_{3} singlecrystal substrates. The sample exhibits an SC transition at T_{c} ≈ 23 K (Supplementary Fig. 11). THz spectra are visible up to ~8 meV below substrate absorption (Supplementary Fig. 12). The base pressure is below 3 × 10^{−5} Pa and the films were synthesized by pulsed laser deposition with a KrF (248 nm) ultraviolet excimer laser in a vacuum of 3 × 10^{−4} Pa at 730 °C (growth rate, 2.4 nm s^{–1}). The Codoped Ba122 target was prepared by a solidstate reaction with a nominal composition of Ba/Fe/Co/As = 1.00:1.84:0.16:2.20. The chemical composition of the thin film is found to be Ba(Fe_{0.92},Co_{0.08})_{2}As_{1.8}, which is close to the stoichiometry of Ba122 with 8% (atomic percentage) optimal Co doping. The pulsed laser deposition (PLD) targets were made in the same way using the same nominal composition of Ba(Fe_{0.92},Co_{0.08})_{2}As_{2.2}.
The epitaxial and crystalline quality of the sample were confirmed by fourcircle Xray diffraction, complex THz conductivity and other extensive chemical, structural and electrical characterizations (Supplementary Figs. 10–12). Equilibrium lowfrequency electrodynamics measurements show that the superfluid density n_{s} vanishes above T_{c} ≈ 23 K and that the lower SC gap is ~6.2–7.0 meV, in agreement with the values quoted in the literature^{41,42}. We measured temperaturedependent electrical resistivity for SC transitions by the fourpoint method (Supplementary Fig. 11). Onset T_{c} and T_{c} at zero resistivity are as high as 23.4 and 22.0 K, respectively, and ΔT_{c} is as narrow as 1.4 K. These are the highest and narrowest values for Ba122 thin films. In our prior papers, we also checked the zerofieldcooled magnetization T_{c} and clearly showed a diamagnetic signal by superconducting quantum interference device magnetometer measurements.
THzMDCS of Anderson pseudospin canting states
Following the early work of Anderson^{43}, an SC state can be viewed in terms of an Nspin/pseudospin state, with one 1/2 spin at each momentum point interacting with all the rest. The spin texture resulting from this longrange spin interaction, determined by the relative orientation of the different correlated spins located at different momentum points, describes the properties of the SC state. Unlike in oneband SCs with BCS order parameter, the properties of the multiband iron pnictide superconductors studied here are determined by the strong coupling between the e and h bands and the corresponding pseudospins. The ground state is known to have s_{±} order parameter symmetry. This means that, in equilibrium, the Anderson pseudospins are antiparallelly oriented between the e and h bands (Fig. 1a). This antiparallel pseudospin orientation between different bands reflects a phase difference of π between the e and h components of the order parameter. Collective excitations of the SC state may be described as magnonlike collective excitations of the Anderson pseudospins, whereas quasiparticle excitations correspond to flipping a single spin. THz excitation leads to precession of the correlated pseudospins around their equilibrium positions. The new lightdriven dynamics proposed here causes timedependent deviations in the orderparameter relative phase between the e and h bands, which, in turn, parametrically drive Anderson pseudospin timedependent canting from the equilibrium antiparallel orientation corresponding to the e–h phase difference of π. Such canting from the antiparallel pseudospin orientation between the e and h bands results from the different dynamics of the SC phase in each band, which leads to different e–h pseudospin rotations.
Close to equilibrium, timedependent oscillations of the e–h relative phase that drives pseudospin canting results in the Leggettphase collective modes. These collective modes are additional to the Higgs amplitude modes. However, the Leggett linear response modes are damped in iron pnictides, as their energy lies within the quasiparticle excitation continuum, higher than the frequency of the lower Higgs amplitude mode that coincides with twice the SC gap 2Δ_{1}. Our numerical results presented in the main text show that strong lightinduced timedependent nonlinear coupling of the orderparameter phase and amplitude oscillations leads to phase oscillations at the same Higgs frequency as the amplitude oscillations. We refer to such phase–amplitude simultaneous oscillations at the same frequency of twice the SC energy gap as the phase–amplitude collective mode of the highly driven nonequilibrium state. This phase–amplitude collective mode replaces the Higgs amplitude and Leggettphase collective modes, which describe the perturbative loworder responses to the THz driving electric field. Since the relative phase of the orderparameter e–h components determines the antiparallel pseudospin configuration that defines the SC equilibrium state of iron pnictides, the development of relative phase oscillations around π at the same frequency as the amplitude Higgs oscillations nonadiabatically drives an ultrafast spin canting from the equilibrium antiparallel pseudospin configuration. Such phasedriven Anderson pseudospin canting (Fig. 1a) corresponds to a timedependent change in the relative pseudospin orientation between the e and h bands. This ultrafast canting from the antiparallel orientation oscillates at the Higgs rather than Leggett frequency above the THz excitation threshold. It is in addition to the collective pseudospin precession within each individual band, which describes the hybridHiggs lightinduced collective mode of iron pnictides introduced in our previous work^{24}. The relative phase oscillations around the equilibrium value of π are long lived when their frequency shifts to the Higgs mode frequency of twice the SC energy gap, which is below the quasiparticle continuum leading to the damping of the Leggett phase mode. Therefore, this frequency shift, induced by the strong lightinduced coherent coupling of the orderparameter phase and amplitude oscillations above the threshold field, parametrically drives a nonequilibrium state characterized by timedependent pseudospin canting from the antiparallel equilibrium configuration. The nonlinear coupling between the pseudospin and phase oscillations at the Higgs frequency results in THzMDCS sidebands at twice the Higgs frequency, which is confirmed by our numerical results in the main text.
To create and characterize the Anderson pseudospin canting states, the FeAs SC film is excited with two broadband THz pulses of similar amplitudes, with a centre frequency of ~1.0 THz (4.1 meV) and broadband frequency width of Δω ≈ 1.5 THz (Fig. 1c). The measured nonlinear coherent differential transmission E_{NL}(t, τ) = E_{AB}(t, τ) − E_{A}(t) − E_{B}(t, τ) is plotted as a function of gate time t and the delay time between the two pulses A and B, that is, τ. The timeresolved coherent nonlinear dynamics is then explored by varying the interpulse delay τ between the two THz pulses. Measuring the electric fields in the time domain through electrooptic sampling by a third pulse allows the phaseresolved detection of the sample response as a function of gate time t. The signals arise from third and higherorder nonlinear PP responses of the SC state, which are separated from the linear response background to obtain an enhanced resolution. Details of our THz setup can be found elsewhere^{9,20,44,45}.
Gaugeinvariant theory and simulations of THzMDCS signals
Details are presented in Supplementary Notes 1 and 2. In thinfilm SCs, electromagnetic propagation effects combined with strong SC nonlinearity leads to an effective driving electric field and Cooperpair centreofmass momentum that persist well beyond the duration of the laser pulse. To model both effects in a gaugeinvariant way, we use the Boguliobov–de Gennes Hamiltonian^{46,47}
where the fermionic field operators \({\psi }_{\alpha ,\nu }^{{\dagger} }({{{\bf{x}}}})\) create an electron characterized by spin index α and band index ν; ξ_{ν}(p + eA(x, t)) is the band dispersion, with momentum operator p = −i∇_{x}, vector potential A(x, t) and electron charge −e; μ denotes the chemical potential; and ϕ(x, t) is the scalar potential. The complex SC order parameter component in band ν is given by
The Hartree and Fock energy contributions are
and
respectively, where \({n}_{\sigma ,\nu }({{{\bf{x}}}})=\langle {\psi }_{\sigma ,\nu }^{{\dagger} }({{{\bf{x}}}}){\psi }_{\sigma ,\nu }({{{\bf{x}}}})\rangle\). Here V(x – x′) is the longranged Coulomb potential, with Fourier transform V_{q} = e^{2}/(ε_{0}q^{2}), which pushes the ingap Nambu–Goldstone mode up to the plasma frequency according to the Anderson–Higgs mechanism^{43}. The Fock energy \({\mu }_{{{{\rm{F}}}}}^{\alpha ,\nu }({{{\bf{x}}}})\) ensures charge conservation. Also, g_{λ,ν} describes the effective interband (λ ≠ ν) and intraband (λ = ν) pairing interactions.
The multiband Hamiltonian (1) is gauge invariant under the general gauge transformation^{48}
when the corresponding vector potential, scalar potential and SC order parameter phases transform as
Here we introduced the field operator in Nambu space, \({{{\varPsi }}}_{\nu }({{{\bf{x}}}})={({\psi }_{\uparrow ,\nu }^{{\dagger} }({{{\bf{x}}}}),{\psi }_{\downarrow ,\nu }({{{\bf{x}}}}))}^{\mathrm{T}}\), and the Pauli spin matrix \({\sigma }_{3}=\left(\begin{array}{ll}1&0\\ 0&1\end{array}\right)\). However, the density matrix \({\rho }^{(\nu )}({{{\bf{x}}}},{{{\bf{x}}}}^{\prime} )=\langle {\hat{\rho }}^{(\nu )}({{{\bf{x}}}},{{{\bf{x}}}}^{\prime} )\rangle =\langle {{{\varPsi }}}_{\nu }({{{\bf{x}}}}){{{\varPsi }}}_{\nu }^{{\dagger} }({{{\bf{x}}}}^{\prime} )\rangle\) depends on the specific choice of the gauge. To obtain gaugeinvariant SC equations of motion, we introduce centreofmass and relative coordinates R = (x + x′)/2 and r = x – x′ and define the transformed density matrix^{38}
where \({\rho }^{(\nu )}({{{\bf{r}}}},{{{\bf{R}}}})=\langle {{{\varPsi }}}_{\nu }({{{\bf{R}}}}+\frac{{{{\bf{r}}}}}{2}){{{\varPsi }}}_{\nu }^{{\dagger} }\left.({{{\bf{R}}}}\frac{{{{\bf{r}}}}}{2})\right)\rangle\). By applying the gauge transformation in equation (5), the density matrix \({\tilde{\rho }}^{(\nu )}({{{\bf{r}}}},{{{\bf{R}}}})\) transforms as^{38}
After applying a Fourier transformation with respect to the relative coordinate r, we perform an additional gauge transformation,
to eliminate phase \({\theta }_{{\nu }_{0}}\) of the SC order parameter for reference band ν_{0}. The equations of motion then depend on the phase difference, \(\delta {\theta }_{\nu }={\theta }_{{\nu }_{0}}{\theta }_{\nu }\), of the order parameter components between different bands ν ≠ ν_{0}. The latter relative phases also determine the equilibrium symmetry of the multiband SC order parameter (for example, s_{++} or s_{±}).
Assuming that the multiband SC system is only weakly spatially dependent^{36,38}, we express the gaugeinvariant density matrix in terms of Anderson pseudospin components at each wavevector k:
Here σ_{n}, n = 1,…3, are the Pauli spin matrices, σ_{0} is the unit matrix and \({\tilde{\rho }}_{n}^{(\nu )}({{{\bf{k}}}})\) are the pseudospin components of band ν. We then straightforwardly derive gaugeinvariant SC Bloch equations of pseudospins, thus generalizing the data from another work^{38} to the multiband case:
where k_{±} = k ± p_{S}/2. The above equations were solved numerically coupled to Maxwell’s equations to compare with the experiment. Three different sources drive lightinduced pseudospin motion in the above equations: condensate centreofmass momentum p_{S}, effective chemical potential μ_{eff} and orderparameter phase difference between different bands ν, namely, \(\delta {\theta }_{\nu }={\theta }_{{\nu }_{0}}{\theta }_{\nu }\). The coupling of the laser field leads to the timedependent band energy
This time dependence comes from the lightinduced condensate momentum p_{S}(t), the effective chemical potential μ_{eff} and the Fock energy
The spin↑ and spin↓ electron populations determine the phasespace filling contributions:
Compared with the conventional pseudospin models used in the literature before, the gaugeinvariant SC Bloch equations (equation (11)) include quantum transport terms proportional to the laser electric field, such as \(e\,{{{\bf{E}}}}\cdot {\nabla }_{{{{\bf{k}}}}}{\tilde{\rho }}_{3}^{(\nu )}({{{\bf{k}}}})\), which also lead to ±p_{S}(t)/2 kspace displacements of the coherences and populations in the finitemomentumpairing SC state. Dynamically induced IS breaking leads to the coupling between \({\tilde{\rho }}_{0}({{{\bf{k}}}})\) and \({\tilde{\rho }}_{3}({{{\bf{k}}}})\) described by equation (11). As already discussed in other work^{9,19,20,36,38}, by also including the lightwave electromagnetic propagation effects inside the SC system, described by Maxwell’s equations, the above IS breaking persists after the driving pulse.
We obtain the experimentally measured signals by calculating the gaugeinvariant supercurrent^{38}
with pseudospin component \({\tilde{\rho }}_{0}^{(\lambda )}({{{\bf{k}}}})\). The nonlinear differential transmission measured in the experiment is obtained in terms of J(t) by calculating the transmitted electric field after solving Maxwell’s equations. For a superconducting thinfilm geometry, we obtain the effective driving field^{38}
where E_{THz}(t) is the applied THz electric field, c is the speed of light and n is the refractive index of the SC system. THz lightwave propagation inside the SC thin film is included in our calculation by selfconsistently solving equation (16) and the gaugeinvariant SC Bloch equations (equation (11))^{38}. Using the above results, we calculated the nonlinear differential transmissioncorrelated signal measured in the THzMDCS experiment, which is given by
for the collinear twopulse geometry used in the experiment (Fig. 1a). E_{AB}(t, τ) is the transmitted electric field induced by both pulses A and B, which depends on both gate time t and delay time between the two pulses, τ. E_{A}(t) and E_{B}(t, τ) are the transmitted electric fields resulting from separate driving by pulse A and pulse B. The THzMDCS spectra are obtained by the Fourier transform of E_{NL}(t, τ) with respect to both t (frequency ω_{t}) and τ (frequency ω_{τ}). To analyse the spectra, we introduce ‘time vectors’ t′ = (t, τ) and ‘frequency vectors’ (ω_{t}, ω_{τ}), such that the electric fields used in the calculations can be written as E_{A}(t)sin(ω_{A}t′) and E_{B}(t)sin(ω_{B}t′). In these calculations, we assume Gaussian envelope functions E_{A,B}(t′). The corresponding frequency vectors of the two pulses A and B are ω_{A} = (ω_{0}, 0) and ω_{B} = (ω_{0}, −ω_{0}), where ω_{0} is the central frequency of the pulses.
We solve the gaugeinvariant optical Bloch equations (equation (11)) for a threepocket model with an h pocket centred at the Γpoint and the two e pockets located at (π, 0) and (0, π). We include the intere–h pocket interactions (g_{e,h} = g_{h,e}) as well as intrapocket interactions (V_{λ} = g_{λ,λ}) and neglect the intere–e pocket interactions for simplicity. We use an interbandtointraband interaction ratio of U = g_{e,h}/V_{λ} = 3 to model the dominance of interband coupling between the e–h pockets over the intraband interaction in FeSCs. The pockets are described by using the square lattice nearestneighbour tightbinding dispersion ξ_{ν}(k) = –2[J_{ν}_{,x}cos(k_{x}) + J_{ν,y}cos(k_{y})] + µ_{ν} with hopping parameter J_{ν,i} and band offset μ_{ν}. We choose a circular h pocket with J_{1,x} = J_{1,y} = 25.0 meV and μ_{1} = −15.0 meV. We introduce the known particle–hole asymmetry between the e and h pockets in our system^{49,50,51} by considering elliptical e pockets with J_{2,x} = J_{3,y} = −25.0 meV, J_{2,y} = J_{3,x} = −80.0 meV and μ_{2} = μ_{3} = 15.0 meV. Such asymmetry strongly suppresses the higher Higgs mode in the spectra of E_{NL} in our calculation, as discussed elsewhere^{24}. We assume s_{±} pairing symmetry with equilibrium SC order parameters Δ_{1} = 3.4 meV for the h pockets and Δ_{2} = Δ_{3} = 9.7 meV for the e pockets. The multiband SC system is excited with two equal broadband pulses with centre frequency ω_{0} = 1 THz.
Pseudospin canting driven parametrically by phase oscillations
To identify the physical origin of the THzMDCS peaks, we derive nonlinear oscillator equations of motion from the full gaugeinvariant equations of motion (equation (11)). Details are discussed in Supplementary Note 3. First, we express the density matrix \({\tilde{\rho }}^{(\nu )}({{{\bf{k}}}})\) describing the nonequilibrium SC state as
where \({\tilde{\rho }}^{(\nu ),{{{\rm{0}}}}}({{{\bf{k}}}})\) is the density matrix of the equilibrium (stationary) state and \({{\Delta }}{\tilde{\rho }}^{(\nu )}({{{\bf{k}}}})\) is the nonequilibrium change induced by the strong driving fields. We consider s_{±} symmetry in the SC ground state, as in the studied iron pnictide system. Then, \(\delta {\theta }_{\nu }^{0}=0,\,\uppi\) defines the equilibrium pseudospin orientations in the different bands, whereas \({\tilde{\rho }}_{2}^{(\nu ),{{{\rm{0}}}}}({{{\bf{k}}}})=0\) and \({\tilde{\rho }}_{1}^{(\nu ),{{{\rm{0}}}}}({{{\bf{k}}}})\ne 0\), that is, the pseudospins point along the x axis and are antiparallel between the e and h bands. By taking the second time derivative of equation (11), we obtain the deviations of the x and y pseudospin components from equilibrium, namely, \({{\Delta }}{\tilde{\rho }}_{1}^{(\nu )}({{{\bf{k}}}})\) and \({{\Delta }}{\tilde{\rho }}_{2}^{(\nu )}({{{\bf{k}}}})\), respectively, in terms of equations of motion for two nonlinearly coupled oscillators:
The above coupled oscillator equations of motion describe lightinduced pseudospin canting parametrically driven by longlived timedependent phase oscillations \({{\Delta }}{\theta }_{\nu }=\delta {\theta }_{\nu }\delta {\theta }_{\nu }^{0}\). Also, \(\delta {{{\varDelta }}}_{\nu }^{\prime}={{{\varDelta }}}_{\nu }^{\prime}{{{\varDelta }}}_{\nu ,0}^{\prime}\) and \(\delta {{{\varDelta }}}_{\nu }^{^{\prime\prime} }={{{\varDelta }}}_{\nu }^{^{\prime\prime} }\) describe the lightinduced orderparameter collective dynamics, where we introduced the real and imaginary parts of the complexvalued order parameters
The first terms on the righthand side of equation (19), \({S}_{\nu }^{(1,2)}({{{\bf{k}}}})\), describe pseudospin driving by sum and differencefrequency Raman and quantum transport processes, previously discussed in another work^{36}, modified here by Δθ_{ν} ≠ 0 (Supplementary Note 3). The second term on the righthand side of equation (19), proportional to the lightinduced order parameter deviations from equilibrium \(\delta {{{\varDelta }}}_{\nu }^{\prime}\) and \(\delta {{{\varDelta }}}_{\nu }^{^{\prime\prime} }\), describes the collective modes arising from the nonperturbative coupling of the different k pseudospins.
The main new effect here comes from the parametric driving of the nonlinear oscillator equations of motion (equation (19)) by the timedependent orderparameter relative phase Δθ_{ν}(t). This parametric driving results in nonadiabatic canting of the pseudospins from their equilibrium directions defined by \(\delta {\theta }_{\nu }^{0}=0,\uppi\). It originates from the dependence of the lefthand side of equation (19) on Δθ_{ν}(t), which is enhanced by the phase–amplitude collective mode of the driven nonequilibrium SC state, discussed in the next section. By expanding the nonlinearcoupledoscillator equations of motion to the lowest order in driving Δθ_{ν}(t) and p_{S}(t), we show that pseudospin canting from the equilibrium direction, \({{\Delta }}{\tilde{\rho }}_{2}^{(\nu )}({{{\bf{k}}}})\ne 0\), is described by the timedependent coupling to
where ∂_{t}E_{ν}(k) ≈ e (E(t) ⋅ ∇_{k})(p_{S} ⋅ ∇_{k}) ξ(k) is approximated by expanding the band dispersions in powers of the centreofmass momentum p_{S}. Equation (21) drives lightinduced pseudospin canting determined by the competition between condensate momentum and orderparameter relative phase dynamics oscillating at ~ω_{H,1}. In particular, ∂_{t}E_{ν}(k) drives pseudospin canting via differencefrequency Raman processes ω_{A,B} − ω_{A,B} ≈ 0 and sumfrequency Raman processes ω_{A,B} + ω_{A,B} ≈ 2ω_{0} > ω_{H,1}. On the other hand, the time dependence of the interband phase difference is dominated by strong oscillations close to the Higgs frequency ω_{H,1}, rather than the Leggett mode frequency well within the quasiparticle continuum, when a phase–amplitude collective mode develops above the critical field (Supplementary Fig. 1).
When the quasiparticle excitations are resonantly driven by the pulse E^{2} spectrum, as is the case for the broad pulses used here, the \(\Delta{\tilde{\rho }}_{1}^{(\nu )}({{{\bf{k}}}})\) spectra are dominated by a momentumdependent peak centred at the quasiparticle excitation energy,
determined by the quenched orderparameter asymptotic value Δ_{1,∞}. The dominant contribution of the pseudospin oscillations that determine the time dependence of \({{\Delta }}{\tilde{\rho }}_{1}^{(\nu )}({{{\bf{k}}}})\) comes from quasiparticle excitations close to the excitation energy minimum, which is located close to the Higgs mode energy ω_{H,1} (Supplementary Fig. 2). As a result, \({{\Delta }}{\tilde{\rho }}_{1}^{(\nu )}({{{\bf{k}}}})\) mainly oscillates close to ω_{H,1}. We thus obtain, through the nonlinear coupling (equation (21)), THzMDCS sidebands centred at the sum of the frequencies of Δθ_{ν}(t) oscillations (phase–amplitude collective mode frequency close to ω_{H,1}) and \({{\Delta }}{{{{\rm{\tilde{\rho} }}}}}_{1}^{(\nu )}\) oscillations (quasiparticle excitations close to ω_{H,1}). Above critical driving, this nonlinear coupling is amplified by the lightinduced quench of the superfluid density (Supplementary Fig. 4d), as discussed in more detail in Supplementary Notes 3 and 4.
Phase–amplitude collective mode and biHiggs frequency sidebands
To clarify the role of the phase–amplitude collective mode in enhancing the parametric pseudospin driving, we have studied the field dependence of the orderparameter amplitude spectrum Δ_{1}(ω) and the spectrum of the relative phase Δθ(ω) for strong Coulomb interband coupling as in FeSCs studied here (Supplementary Fig. 4a,b). At low fields, Δ_{1}(ω) is dominated by a peak at ~ω_{H,1}, whereas Δθ(ω) shows a peak located within the quasiparticle continuum, corresponding to the Leggett mode. This lowfield result reproduces previous collective mode results obtained by using susceptibility expansions. With an increasing driving field, nonlinear coupling between the phase and amplitude moves the relative phase mode towards ω_{H,1} by creating phase–amplitude collective modes. The emergence of a strong Δθ(ω) peak at the Higgs frequency ω_{H,1} results from a lightinduced phase–amplitude collective mode at ω_{H,1}. This collective mode displays strong phase oscillations at ω_{H,1}, which allows the resonant parametric driving of the coupled nonlinear harmonic oscillators (equation (19)). The resulting timedependent pseudospin canting leads to the sideband signals at twice the Higgs energy ω_{H,1} (Supplementary Fig. 4e).
To clarify the transition from the Higgs collective mode to coupled phase–amplitude mode with energy ω_{H,1}, we compare the field dependence of the persisting superfluid momentum p_{S} (Supplementary Fig. 4c), which breaks the IS and characterizes the strength of the Higgs mode signals at ω_{t} = ω_{H,1}, and the maximum of the Δ_{1}^{2}sin2Δθ spectrum (Supplementary Fig. 4d), which drives pseudospin canting in response to the timedependent changes in the relative phase. Here p_{S} arises from the coupling between SC nonlinearity and electromagnetic propagation effects, which determines the effective driving field (equation (16)) dependent on supercurrent J(t). At low driving fields, the increase in p_{S} is proportional to \({E}_{0}^{3}\), since in this regime, it is generated to the lowest order by thirdorder nonlinear processes when lightwave propagation effects are included^{38}. This initial excitation regime is, however, followed by another excitation regime, where in a twoband SC, the quench of the SC gap is only slightly modified as the driving field increases (Supplementary Fig. 4a). This behaviour is unlike the oneband case^{19,38} and results from the strong interband coupling between the e and h pockets leading to the formation of a hybridHiggs collective mode^{24}. In this nonperturbative excitation regime, the contribution of Higgs collective effects to the nonlinear response dominates over quasiparticle excitations, which results in the different nonlinear increases in p_{S} compared with the initial regime (Supplementary Fig. 4c). Above 600 kV cm^{−1} driving, a further increase in the driving field leads to a complete quench of the SC gap (results are only shown up to an order parameter quench of 25%; Supplementary Fig. 4), which results in a stronger nonlinear increase in p_{S} compared with the one in the initial excitation regime (Supplementary Fig. 4c). This behaviour is in agreement with the results in oneband superconductors discussed elsewhere^{19,38}. Compared with the increasing superfluid momentum p_{S}, the maximum of the Δ_{1}^{2}sin2Δθ spectrum (Supplementary Fig. 4d) remains near zero in the perturbative excitation regime. In this susceptibility regime, parametric timeperiodic driving of pseudospin canting by the phase dynamics is negligible, and we recover previously obtained results without any biHiggs frequency sidebands. However, above the critical laser field, Supplementary Fig. 4d shows a (twostep) nonlinear increase in Δ_{1}^{2}sin2Δθ, up to 400 kV cm^{−1} excitation. In this regime, the coupled phase–amplitude mode emerges when the relative phase mode gets close to ω_{H,1} (Supplementary Fig. 4b). A further increase in E_{0} leads to a strong increase in the maximum of the Δ_{1}^{2}sin2Δθ spectrum, which coincides with the emergence of the nonperturbative biHiggs frequency sidebands in the THzMDCS spectra (Supplementary Fig. 4e). In this highexcitation nonlinear regime, the SC order parameter is quenched, which leads to a stronger increase in the relative phase oscillation amplitude, enhanced by 1/Δ_{1}. Due to the strong nonlinear increase above the driving field threshold (Supplementary Fig. 4d), the pseudospin canting driven by Δ_{1}^{2}sin2Δθ dominates over that due to the increase in p_{S} (Supplementary Fig. 4c). This is in contrast to the behaviour at lower fields, where p_{S} increases whereas Δ_{1}^{2}sin2Δθ remains small. As a result, the THzMDCS signals generated by the phase–amplitude mode at ω_{t} = 2ω_{H,1} − ω_{0} and ω_{t} = 2ω_{H,1} − 2ω_{0} dominate over the nonlinear signals at ω_{t} = ω_{H,1} in the Higgs collective mode.
Nonlinear processes contributing to the THzMDCS spectra
In this section, we summarize all the nonlinear processes that contribute to the THzMDCS spectra (Figs. 2 and 3 and Supplementary Notes 5–7). First, we list the conventional PP and FWM signals already known from THzMDCS spectroscopy experiments on semiconductors. Such peaks significantly contribute to the THzMDCS spectra here only in the perturbative excitation regime (Fig. 3b) and result from the nonlinear processes summarized in Table 1.
With increasing field strength, new highorder correlated wavemixing signals emerge, which dominate over the above conventional PP and FWM signals. In particular, lightwave propagation inside the SC system leads to dynamical IS breaking persisting after the pulses. As a result, new wavemixing signals (IS wave mixing) emerge at ω_{t} = ω_{H,1}, which are generated by nonlinear processes involving amplitude Higgs mode excitation. BiHiggs frequency and ISbreaking biHiggs frequency (IS biHiggs) sideband peaks emerge with increasing field strength and exceed the Higgs signals at elevated E_{0}. In addition, differencefrequency Raman process assisted by quasiparticle excitations leads to highorder correlated PP signals. All the highorder correlated wavemixing spectral peaks and corresponding nonlinear processes are summarized in Table 2.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Code availability
All computer codes are available from the corresponding author upon reasonable request.
Change history
07 June 2024
A Correction to this paper has been published: https://doi.org/10.1038/s4156702402576z
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Acknowledgements
THz spectroscopy work was designed and supported by National Science Foundation 1905981 (M.M. and J.W.). Data processing and analysis (L.L.) was supported by the Ames National Laboratory, the US Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES), Materials Science and Engineering Division, under contract no. DEAC0207CH11358. THz instrument (J.W.) was supported by the W.M. Keck Foundation (initial design and commission) and by the US DOE, Office of Science, National Quantum Information Science Research Centers, Superconducting Quantum Materials and Systems Center (SQMS), under contract no. DEAC0207CH11359 (for improved cryogenic operation and for benchmarking of coherent spectroscopy). The work at University of WisconsinMadison (J.H.K. and C.B.E) was supported by the US DOE, Office of Science, Basic Energy Sciences (BES), the Materials Sciences and Engineering (MSE) Division, under award no. DEFG0206ER46327 (synthesis of pnictide thin films and characterizations of epitaxial thin films). Work by K.E. and J.W.L. was funded by the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF9065. Modelling work at the University of Alabama, Birmingham (I.E.P.), was supported by the US DOE under contract no. DESC0019137 and was made possible in part by a grant for highperformance computing resources and technical support from the Alabama Supercomputer Authority.
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J.W., M.M. and I.E.P. designed the project. L.L., with help of C.H. and C.V., processed and analysed the raw data. I.E.P., M.M. and J.W. developed the physical picture with discussions from all the authors and M.M. performed the calculations. J.H.K., K.E., J.W.L. and C.B.E. grew the samples and performed the crystalline quality and transport characterizations. Y.G.C. and E.E.H. prepared the Ba122 target for the epitaxial thin films. The paper is written by J.W., M.M. and I.E.P. with help of all the authors. J.W. coordinated the project.
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Luo, L., Mootz, M., Kang, J.H. et al. Quantum coherence tomography of lightcontrolled superconductivity. Nat. Phys. 19, 201–209 (2023). https://doi.org/10.1038/s41567022018271
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DOI: https://doi.org/10.1038/s41567022018271
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