Abstract
The Higgs mechanism, i.e., spontaneous symmetry breaking of the quantum vacuum, is a crossdisciplinary principle, universal for understanding dark energy, antimatter and quantum materials, from superconductivity to magnetism. Unlike oneband superconductors (SCs), a conceptually distinct Higgs amplitude mode can arise in multiband, unconventional superconductors via strong interband Coulomb interaction, but is yet to be accessed. Here we discover such hybrid Higgs mode and demonstrate its quantum control by light in ironbased hightemperature SCs. Using terahertz (THz) twopulse coherent spectroscopy, we observe a tunable amplitude mode coherent oscillation of the complex order parameter from coupled lower and upper bands. The nonlinear dependence of the hybrid Higgs mode on the THz driving fields is distinct from any known SC results: we observe a large reversible modulation of resonance strength, yet with a persisting mode frequency. Together with quantum kinetic modeling of a hybrid Higgs mechanism, distinct from chargedensity fluctuations and without invoking phonons or disorder, our result provides compelling evidence for a lightcontrolled coupling between the electron and hole amplitude modes assisted by strong interband quantum entanglement. Such lightcontrol of Higgs hybridization can be extended to probe manybody entanglement and hidden symmetries in other complex systems.
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Introduction
Amplitude modes and their competition with chargedensity fluctuations are currently intensely studied in oneband superconductors (SCs). In multiband, unconventional SCs, a hybrid Higgs amplitude mode, controllable by terahertz (THz) laser pulses, can arise via strong interband Coulomb interaction. A recent prominent example to explore such collective mode is seen in ironarsenide based superconductors (FeSCs). Phase coherence between multiple SC condensates in different strongly interacting bands is well established in FeSCs. As illustrated in Fig. 1a, a dominant Coulomb coupling between the h and elike Fermi sea pockets, unlike in other SCs, is manifested by, e.g., s± pairing symmetry^{1,2}, spindensity wave resonant peaks, and nesting wave vectors (black arrows)^{3,4,5,6}. Despite the extensive studies, experimental evidence for Higgs amplitude coherent excitations in FeSCs has not been reported yet, despite recent progress in nonequilibrium superconductivity and collective modes^{7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22}.
The condensates in different bands of Ba(Fe_{1−x}Co_{x})_{2}As_{2} studied here, shown in Fig. 1a, are coupled by the strong interband e–h interaction U (blue double arrow vector), which is about one order of magnitude stronger than the intraband interaction V (gray and red double arrows)^{23}. For U ≫ V, the formation channels of collective modes are distinct from oneband SC^{7,8,9,24} and multiband MgB_{2} with dominant intraband interaction, V ≫ U^{25,26}. For the latter, only Leggett modes are observed thus far^{10}. In contrast, for U ≫ V, one expects Higgs amplitude modes arising from the condensates in all Coulombcoupled bands, i.e., in the h pocket at the Γpoint (gray circle, mode frequency ω_{H,1}), and in the two e pockets at (0, π) and (π, 0) (red ellipses, ω_{H,2}). A singlecycle THz oscillating field (red pulse) can act like a quantum quench, with impulsive nonadiabatic driving of the Mexicanhatlike quantum fields (dark green) and, yet, with minimum heating of other degrees of freedoms. Consequently, the multiband condensates are forced out of the free energy minima, since they cannot follow the quench adiabatically. Most intriguingly, such coherent nonlinear driving not only excites amplitude mode oscillations in the different Fermi sea pockets, but also transiently modifies their coupling, assisted by the strong interband interaction U. Such coherent transient coupling can be regarded as nonlinear amplitude mode hybridization with a timedependent phase coherence. In this way, THz laser fields can manipulate hybrid Higgs emerging collective modes in FeSCs, assisted by the strong interband interaction.
Here we present evidence of hybrid Higgs modes that are excited and controlled by THzfielddriven interband quantum entanglement in a multiband SC, optimally doped Ba(Fe_{1−x}Co_{x})_{2}As_{2}, using two phaselocked nearsinglecycle THz laser fields. We reveal a striking nonlinear THz field dependence of coherent amplitude mode oscillations: quick increase to maximum spectral weight (SW) with negligible mode frequency shift, followed by a huge SW reduction by more than 50%, yet with robust mode frequency position, with less than 10% redshift. These distinguishing features of the observed collective mode are different from any oneband and conventional SC results and predictions so far. Instead, they are consistent with coherent coupling between the e and hlike amplitude modes . To support this scenario, we perform quantum kinetic modeling of a hybrid Higgs mechanism without invoking extra disorder or phonons. This simulation identifies the key role of the interband interaction U for coherently coupling two amplitude modes and for controlling the SW of the lower Higgs mode observed in the experiment.
Results
The amplitude coherent oscillations detected by THz twopulse coherent spectroscopy
The equilibrium complex conductivity spectra, i.e., real and imaginary parts, σ_{1}(ω) and σ_{2}(ω), of our epitaxial Ba(Fe_{1−x}Co_{x})_{2}As_{2} (x = 0.08) film^{27} (Methods) measure the lowfrequency quasiparticle (QP) electrodynamics and condensate coherence, respectively (Fig. 1b, c)^{28}. The normal state (black circles) displays Drudelike behavior, while the QP spectral weight in σ_{1}(ω) is depleted in the SC state due to SC gap openings, seen, e.g., in the 4.2 K trace (red circles). The lowest SC gap value 2Δ_{1} ~ 6.8 meV obtained is in agreement with the literature values 6.2–7 meV^{29,30} (Methods). Such σ_{1}(ω) spectral weight depletion is accompanied by an increase of condensate fraction n_{s}/n_{0} (inset, Fig. 1c), extracted from a diverging 1/ω condensate inductive response, marked by blue arrow, e.g., in the 4.2 K lineshape of σ_{2}(ω) (Fig. 1c). Note that the superfluid density n_{s} vanishes above T_{c} ~ 23 K (inset Fig. 1c).
We characterize the THz quantum quench coherent dynamics directly in the time domain^{31,32,33} (Methods) by measuring the responses to two phaselocked THz pulses as differential field transmission of the weak THz probe field ΔE/E_{0} (blue circles, Fig. 1d) for THz pump field, E_{pump} = 56 kV/cm and as a function of pump–probe time delay Δt_{pp}. The central pump energy ℏω_{pump} = 5.4 meV (gray shade, Fig. 1e) is chosen slightly below the 2Δ_{1} gap. Intriguingly, the ΔE/E_{0} dynamics reveals a pronounced coherent oscillation, superimposed on the overall amplitude change, which persists much longer than the THz photoexcitation (pink shade). This mode is excited by the quadratic coupling of the pump vector potential, \({{\bf{A}}}^{2}(t)\propto {E}_{{\rm{pump}}}^{2}/{\omega }^{2}\) due to the SC equilibrium symmetry^{7}. Such coherent responses yield information within the general framework of coherent nonlinear spectroscopy^{34} (Methods). The origin of the observed coherent ΔE/E_{0} oscillation is better illustrated by its Fourier transformation (FT), shown in Fig. 1e. The FT spectrum of the coherent nonlinear signals (blue solid line) displays a pronounced resonance at 6.9 meV, indicative of 2Δ_{1} coherent amplitude mode oscillations. This FT spectrum strongly differs from the spectra of both THz pump E_{pump}(ω) centered at ω_{pump} ~ 5.4 meV (gray shade) and second harmonic, Anderson pseudospin (APS) precession at 2ω_{pump} from \({E}_{{\rm{pump}}}^{2}(\omega )\) (pink shade). The broadband spectrum of the fewcycle pump pulse used in the experiment overlaps with the mode resonances such that the ΔE/E_{0} oscillates with the collective mode frequencies^{9}. After the oscillation, timedependent complex conductivity spectra, σ_{1}(ω, Δt_{pp}) and σ_{2}(ω, Δt_{pp}), can be measured (Supplementary Figs. 5–6). They show that ΔE/E_{0} closely follows the pumpinduced change in condensate density, Δn_{s}/n_{0}^{13}. The THz excitation at E_{pump} = 56 kV/cm only reduces n_{s} slightly, Δn_{s}/n_{0} ~ ΔE/E_{0} ~ −3% at Δt_{pp} = 5 ps. Furthermore, the pumpinduced peak amplitude (blue diamond), marked by the red dashed line in Fig. 1d, diminishes above T_{c} (inset). These evidence indicate that the measured coherent oscillations reflect the emergence of a hybrid Higgs multiband collective mode between two Coulombcoupled lower and higher modes, ω_{H,1} and ω_{H,2}.
Figure 2 reveals a strong temperature dependence of the Higgs mode oscillations. The coherent dynamics of ΔE/E_{0} is shown in Fig. 2a for temperatures 4.2–30 K. Approaching T_{c} from below, the coherent oscillations quickly diminish, as seen by comparing the 4.2 K (black line) and 16 K (gray) traces versus 22 K (cyan) and 24 K (pink) traces. Figure 2b shows the temperaturedependent Fourier spectra of ΔE/E_{0}, in the range 4–12 meV. Figure 2c plots the integrated spectral weight SW_{5→14meV} of the amplitude mode (Fig. 2b). The strong temperature dependence correlates the mode with SC coherence. Importantly, while the mode frequency is only slightly redshifted, by less than 10% before full SW depletion close to T_{c}, SW is strongly suppressed, by ~55% at 16 K (T/T_{c} ~ 0.7). Such a spectral weight reduction in FeSCs is much larger than expected in oneband BCS superconductors or for U = 0 shown later. We also note that the temperature dependence of Higgs oscillations, observed in FeSCs here, has not been measured experimentally in conventional BCS systems, and could represent a key signature of quantum quench dynamics of unconventional SCs. The observed behavior is consistent with our simulations of the hybrid Higgs mode in multiband SCs with dominant interband U, shown later.
Nonlinear THz field dependence of hybrid Higgs mode in FeSCs
Figure 3 presents distinguishing experimental evidence for the hybrid Higgs mode in FeSCs, which is different from oneband SCs – a highly nonlinear THz electric field dependence of coherent 2Δ_{1} oscillations that manifests as a huge SW change, yet with persisting mode frequency, i.e., with only very small redshift. Figure 3a shows the detailed pumpfluence dependence of ΔE/E_{0} as a function of time delay, presented as a falsecolor plot at T = 4.2 K for up to E_{pump} ~ 600 kV/cm. It is clearly seen that amplitude mode oscillations depend nonlinearly on the E_{pump} field strength. 1 + ΔE/E_{0} (red solid line) at Δt_{pp} = 5 ps is shown in Fig. 3b. This, together with the measured ~1/ω divergence in σ_{2}(ω, Δt_{pp}), allows the determination of the condensate fraction n_{s}(E_{pump})/n_{0} (blue circles) in the driven state (Supplementary Fig. 6). There shows three different excitation regimes, marked by black dash lines in Fig. 3a–b: (1) in regime #1, the condensate quench is minimal, e.g., n_{s}/n_{0} ≥ 97% below the field E_{#1} = 56 kV/cm; (2) Regime #2 displays partial SC quench, where n_{s}/n_{0} is still significant, e.g., condensate fraction ≈60% at E_{#2} = 276 kV/cm; (3) A saturation regime #3 is observed ~E_{#3} = 600 kV/cm, which leads to a slowly changing n_{s}/n_{0} approaching a saturation ~25%. The saturation is expected since below gap THz pump is used, especially ℏω_{pump} ≪ 2Δ_{2} ~ 15–19 meV at the elike pockets^{29,30}.
Quantum quenching of the singleband BCS pairing interaction has been well established to induce Higgs oscillations with amplitude scaling as 1/\(\sqrt{{\Delta }_{\text{SC},\infty }}\), determined by the longtime asymptotic nonthermal order parameter Δ_{SC,∞}. The latter decreases with pump field^{35,36,37}. Both model and experimental results establish that the Higgs mode amplitude increases with THz pumping until full depletion of the condensate, concurrent with a continuous Higgs resonance redshift to zero^{11,35,36,37}. In contrast to this expected behavior for conventional SCs, Fig. 3a and the Fourier spectra of the coherent oscillations, Fig. 3c, d, show a distinct, nonmonotonic pumpfield dependence of the Higgs mode amplitude that is unique here. Specifically, the Fourier spectra exhibit a clear resonance at low pump fluences, which coincides with the frequency of the lower Higgs mode ω_{H,1}. This resonance grows quickly up to a field of E_{pump} = 173 kV/cm (Fig. 3c), saturates up to 276 kV/cm (Fig. 3d) and then exhibits a significant reduction in pump regime #3, e.g., by more than 50% at 629 kV/cm (black line, Fig. 3d). Therefore, the coherent oscillations in Fig. 3a quickly increase in pump regime #1 and saturate in regime #2, prior to any significant mode resonance redshift (blue dashed line, Fig. 3c). Above this relatively low field regime, the oscillation amplitude starts to decrease at 276 kV/cm, even though there is still more than 60% of condensate, marked in Fig. 3b: the driven state is still far from full SC depletion. This striking SW reduction is also seen in the integrated spectral weight analysis, SW_{5→14meV}, summarized in Fig. 3e. Most intriguingly, while there is a large reduction of the Higgs mode SW ~ 50% at 629 kV/cm (regime #3), the resonance peak position is nearly persistent, with ≤10% redshift (blue dashed lines, Fig. 3d). These observations of the hybrid Higgs mode differ from any known behavior in oneband SCs, but are consistent with expectations from lightinduced nonlinear coherent coupling of Higgs modes in multiband SCs by a dominant interband interaction U.
The distinct mode amplitude and position variation with pump field extracted from the coherent oscillations clearly show the transition from SW growth and saturation to reduction, marked by the black dashed line at E_{pump} = 276 kV/cm (Fig. 3e). The saturation and reduction of SW in the amplitude oscillation, yet with a persisting mode frequency, cannot be explained by any known mechanism. This can arise from the coupling of the two amplitude modes ω_{H,1} and ω_{H,2} expected in iron pnictides due to the strong interpocket interaction U. The coherent coupling process can be controlled and detected nonlinearly by THz twopulse coherent spectroscopy, as clearly shown in Fig. 3e. We argue that (I) At low driving fields, ω_{H,1} dominates the hybrid collective mode due to less damping than ω_{H,2} arising from the asymmetry between the electron and hole pockets; (II) For higher fields, SW of ω_{H,1} mode decreases due to the coherent transfer to ω_{H,2} mode expected for the strong interband interaction in iron pnictides. Moreover, it is critical to note that the THz driving is of highly nonthermal nature, which is distinctly different from that obtained by temperature tuning in Fig. 2. Specifically, Fig. 3f, g compare the hybrid Higgs mode spectra for similar condensate faction n_{s}/n_{0}, i.e., ≈60% (f) and 25% (g), induced by tuning either the temperature (red shade) or THz pump (black line). The mode amplitude is clearly much stronger in the THzdriven coherent states than in the temperature tuned ones.
Quantum kinetic calculation of the THzdriven hybrid Higgs dynamics
To put the above hybrid Higgs mode findings on a rigorous footing, we calculate the THz coherent nonlinear spectra (Methods) by extending the gaugeinvariant density matrix equations of motion theory of ref. ^{38}, as outlined in the Supplementary Note 6. Using the results of these calculations, we propose a physical mechanism that explains the distinct differences of the Higgs mode resonance in the fourwave mixing spectra between the strong and weak interband interaction limits. For this, we calculate the APS and quantum transport nonlinearities^{38} driven by two intense phaselocked THz Efield pulses for an effective 3pocket BCS model of FeSCs^{39}. This model includes both intraband and interband pairing interactions, as well as asymmetry between electron and hole pockets. We thus calculate the nonlinear differential field transmission ΔE/E_{0} for two phaselocked THz pulses, which allows for a direct comparison of our theory with the experiment (Supplementary Note 6).
The inset of Fig. 4a presents the calculated ΔE/E_{0} (black line), shown together with \({E}_{{\rm{pump}}}^{2}(t)\) of the applied experimental pump pulse (pink shade). The calculated Higgs mode spectra, Fig. 4a (regime I) and Fig. 4b (regime II), are dominated by a resonance close to 6.8 meV for low pump fluences, which corresponds to ω_{H,1}. This resonance grows up to pump fields E_{THz} ≈ 320 kV/cm for the parameters used here, with minimal redshift and without any significant SW at ω_{H,2} (Fig. 4a). Interestingly, for higher fields (Fig. 4b), we obtain both a redshift and a decrease of the oscillation amplitude. In this regime II, SW emerges close to 15.0 meV, outside of our experimental bandwidth, in the frequency regime of the ω_{H,2} Higgs mode. The latter mode is strongly suppressed due to damping induced via electron–hole asymmetry (Supplementary Note 6). Specifically, the ellipticity of the e pockets increases the DOS along the pump field direction and thus increases the damping of mainly the ω_{H,2} resonance, which leads to a transfer of oscillator strength to the continuum. This damping has a much smaller influence on the ω_{H,1} resonance that arises largely from the hole pockets. Most importantly, the strong interband coupling expected in FeSCs leads to a decrease in the ω_{H,1} resonance amplitude, with SW reduction accompanied by a persisting mode frequency. This behavior of the multiband model with strong U is clearly seen in the raw experimental data. Note that, while in regime I we observe an increase in the mode amplitude without any significant redshift, in regime II, the decrease in ω_{H,1} resonance is accompanied by a small redshift. This behavior of the hybrid Higgs mode contradicts the oneband behavior, recovered by setting U = 0, and is in excellent agreement with our experimental observations in the FeSC system (Fig. 3c–e).
To scrutinize further the critical role of the strong interband interaction U, we show the fluence dependence of the coherent Higgs SW close to ω_{H,1} in Fig. 4c. SW_{0→14meV} differs markedly between the calculation with strong U ≠ 0 (blue circles) and that without interpocket interaction U = 0 (red circles), which resembles the oneband BCS quench results. Importantly, the Higgs mode SW_{0→14meV} for strong U grows at low pump fluences (regime I), followed by a saturation and then decrease at elevated E_{pump} (regime II), consistent with the experiment. Meanwhile, Fig. 4d demonstrates that the resonance frequency remains constant in regime I, despite the strong increase of SW, and then redshifts in regime II, yet by much less than in the oneband system (compare U ≠ 0 (blue circles) vs. U = 0 (red circles)). Without interpocket U (U = 0, red circles in Fig. 4c–d), the SW of Higgs mode ω_{H,1} grows monotonically up to a quench of roughly 90%. A further increase of the pump field leads to a complete quench of the order parameter Δ_{1} and a decrease of the SW of Higgs mode ω_{H,1} to zero, due to transition from a damped oscillating Higgs phase to an exponential decay. Based on the calculations in Fig. 4c–d, the decrease of the spectral weight with interband coupling appears at a Δ_{1} quench close to 15%, while without interband coupling the decrease of spectral weight is only observable close to the complete quench (~90%) of the SC order parameter Δ_{1}. We conclude from this that the spectral weight decrease at the lower Higgs resonance with low redshift is a direct consequence of the strong coupling between the electron and hole pockets due to large U. This E_{pump} dependence is the hallmark signature of the Higgs mode in FeSCs and is fully consistent with the Higgs mode behaviors observed experimentally.
Finally, the temperature dependence of the hybrid Higgs mode predicted by our model is shown in Fig. 4e, f for U = 0 (red circles) and U ≠ 0 (blue circles). With interband coupling, the SW is strongly suppressed, by about 60% up to a temperature of 0.6T_{c}, while at the same time the mode frequency is only slightly redshifted, by about 15%, before a full spectral weight depletion is observable towards T_{c}. The strong suppression results from transfer of SW from mode ω_{H,1} to the higher mode ω_{H,2} with increasing temperature, since the higher SC gap Δ_{2} experiences stronger excitation by the applied pump E^{2} with growing T. These simulations are in agreement with the hybrid Higgs behavior in Fig. 2 and differ from oneband superconductors showing comparable change of both SW and position of the Higgs mode with increasing temperature (red circles, Fig. 4e, f). Moreover, our calculation without lightinduced changes in the collective effects (only chargedensity fluctuations) produces a significantly smaller ΔE/E_{0} signal in the nonperturbative excitation regime (Supplementary Fig. 11). Therefore, we conclude that the hybrid Higgs mode dominates over chargedensity fluctuations in twopulse coherent nonlinear signals in FeSCs, due to the different effects of the strong interband U and multipocket bandstructure on QPs and on Higgs collective modes.
In summary, we provide distinguishing features for hybrid Higgs modes and thier coherent excitations in multiband FeSCs, which differ significantly from any previously observed collective mode in other superconducting materials: 2Δ_{SC} amplitude oscillations displaying a robust mode resonance frequency position despite a large change of its spectral weight, more than 50%, on the THz electric field. This unusual nonlinear quantum behavior provides evidence for Higgs hybridization from the interband quantum entanglement in FeSCs. Such discovery and lightcontrol of the hybrid Higgs mode in multiband unconventional SCs inspire future research of quantum tomography of manybody correlated states in complex systems.
Methods
Sample preparation
We measure optimally Codoped BaFe_{2}As_{2} epitaxial singlecrystal thin films^{27} which are discussed in Supplementary Figs. 1 and 2. They are 60 nm thick, grown on 40 nm thick SrTiO_{3} buffered (001)oriented (La;Sr)(Al;Ta)O_{3} (LSAT) singlecrystal substrates. The sample exhibits a SC transition at T_{c} ~ 23 K (Supplementary Fig. 2). 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/sec). The Codoped Ba122 target was prepared by solidstate reaction with a nominal composition of Ba/Fe/Co/As = 1:1.84:0.16:2.2. 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 %) optimal Codoping. The 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} (Supplementary Note 1). The epitaxial and crystalline quality of the optimally doped BaFe_{2}As_{2} thin films was measured by fourcircle Xray diffraction (XRD) shown in Supplementary Fig. 1 for the outofplane θ–2θ scans of the films. As an example, the full width half maximum (FWHM) of the (004) reflection rocking curve of the films is as narrow as 0.7^{∘}, which indicates highquality epitaxial thin films. Furthermore, we have also performed chemical, structural, and electrical characterizations of epitaxial Codoped BaFe_{2}As_{2} (Ba122) superconducting thin films. We determined the chemical composition of Ba122 thin films by wavelength dispersive spectroscopy (WDS) analyses. 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 %) optimal Codoping. We measured temperaturedependent electrical resistivity for superconducting transitions by fourpoint method (Supplementary Fig. 2). 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, which are the highest and narrowest values for Ba122 thin films. In our prior papers, one can also check a zerofieldcooled magnetization T_{c} and it clearly shows a diamagnetic signal by superconducting quantum interference device (SQUID) magnetometer measurements.
Properties of the thin film Codoped BaFe_{2}As_{2} samples used
The lowest SC gap values 2Δ_{1} ~ 6.8 meV were obtained in our THz conductivity data using a MattisBardeen approach similar to that used in^{28,29,30}. Specifically, Supplementary Fig. 7 plots together the THz conductivity of our singlecrystal film sample and prior FTIR conductivity of singlecrystal samples at the similar doping^{29}. They show an excellent agreement which indicate similar superconducting energy gaps of both samples. The two superconducting energy gaps obtained from the FTIR data are 2Δ_{1} ~ 6.2 meV that agrees with our 2Δ_{1} value, and 2Δ_{2} ~ 14.8–19 meV^{29,30}. The further analysis of THz conductivity spectra of our sample also allows the determination of other key equilibrium electrodynamics and transport parameters, consistent with the prior literature^{3,6}. First, we can obtain the plasma frequency ω_{p}, obtained by fitting the normal state conductivity spectra with the Drude model, which gives the plasma frequency to be ~194 THz (~804 meV) and the scattering rate ~2.5 THz (~10 meV). These measurements are in excellent agreement with bulk singlecrystal samples at optimal doping^{29}: a normal state Drude plasma frequency of 972 meV, scattering rate 15 meV was obtained there. Second, the London penetration depth λ_{L} = 378 nm, corresponding to a condensed fraction of ~50%. These are again consistent with the prior measurements of high quality bulk samples, which show ~50% of the free carriers participating in superfluidity below T_{c} and a penetration depth of 300 nm.
Twopulse THz coherent nonlinear spectroscopy
Our THz pump–THz probe setup, illustrated in Supplementary Figs. 3 and 4, can be understood within the general framework of THz 2D spectroscopy. The experiment is driven by a 1 kHz fs laser amplifier^{40} and performed in the collinear geometry with two pulses E_{A} and E_{B} with wave vectors \({\overrightarrow{k}}_{A}\)=\({\overrightarrow{k}}_{B}\)=\({\overrightarrow{k}}_{{\rm{NL}}}\). Measuring the electric fields in time domain through electrooptic sampling (EOS) by a third pulse (red shade) allows for phaseresolved detection of the sample response. In general, such a two pulse experiment can provide a number of nonlinear (NL) responses as shown in the Supplementary Table 1. The signals arise from the third order (χ^{(3)}) nonlinear pump–probe responses of the superconducting state, which are separated from the linear response background. Two main contributions relevant here are measured along the same phase matching direction \({\overrightarrow{k}}_{{\rm{NL}}}\): (1) pump–probe (PP) signals that access condensate quench and recovery; (2) Fourwave mixing (FWM) signals that access amplitude channel coherence and/or density fluctuations. For the sake of consistency, we follow our prior publications^{13} to label the time delays Δt_{pp} and t_{gate} corresponding to pulse E_{A} and E_{B} in Supplementary Fig. 3, which can be varied independently. For the PP contribution, a polarization response \(\widetilde{P}(\omega ,\Delta {t}_{\text{pp}})\) is measured and used to obtain the time and frequencyresolved response functions by performing deconvolution along the t_{gate} axis. For the FWM contribution, one usually measures it in terms of the interpulse delay τ = t_{gate} − Δt_{pp}^{34} as ΔE_{FWM}(τ, t_{gate}) = E_{AB}(t_{g} − Δt_{pp}, t_{gate}) − E_{A}(Δt_{pp}) − E_{B}(t_{gate}) where E_{AB} means E_{A} and E_{B} are both present. In our experiment, two optical choppers synchronized by the f = 1 kHz laser repetition rate modulate the pump and probe THz beams at f/2 (500 Hz) and f/4 (250 Hz), respectively. The measured data is then divided into 4 channels CH0 to CH3 as shown in the schematic of the timing sequence in Supplementary Fig. 4. After the data acquisition in each channel, we obtain different backgroundfree nonlinear signals as ΔE = E_{AB} − E_{A} − E_{B} = E_{CH0} + E_{CH1} − E_{CH2} − E_{CH3}. We provide more details in supplementary.
Timeresolved complex THz conductivity spectra
Unlike for the Fourier spectra of coherent FWM signals of main interest here, THz PP signals (Supplementary Table 1 and Supplementary Eq. (1), Supplementary Note 2) directly access the timedependent complex conductivity spectra, i.e., real and imaginary parts, σ_{1}(ω, Δt_{PP}), σ_{2}(ω, Δt_{PP}), shown in Supplementary Fig. 5 for a fixed time delay Δt_{PP} = 5 ps. PP signal measurements extend the conventional, equilibrium complex conductivity σ_{1}(ω) and σ_{2}(ω) measurement (black and gray circles, Supplementary Fig. 5) to characterize the nonequilibrium postTHzquench superconducting states. σ_{1}(ω, Δt_{PP}), σ_{2}(ω, Δt_{PP}) are shown for Efield = 44 kV/cm (red circles) and Efield = 88 kV/cm (blue circles) in Supplementary Fig. 5. The static conductivity can be fitted with Drudelike behavior of quasiparticles in the normal state (black circles) and condensate gaps of 2Δ = 6.8 meV in the SC state (red circles) (discussed later in Supplementary Fig. 7). The PP conductivity spectra identify the minimal quenching of condensate density of few percent for tens of kV/cm Efield used in observing coherent oscillations. Timedependence of superfluid density can be directly obtained by the diverging σ_{2}(ω, Δt_{PP}) ∝ n_{s}/ω in Supplementary Fig. 5. Furthermore, we also plot Supplementary Fig. 6a from three representative linecuts in Fig. 3a (main text) and then compare this with the condensate fraction (Supplementary Fig. 6b) extracted from the usual diverging response in σ_{2}(ω) (Supplementary Fig. 6c). It is clear that, after the oscillation, ΔE/E_{0} closely follows the pumpinduced change in condensate density, Δn_{s}/n_{0}. For example, for weak THz excitation E_{pump} = 56 kV/cm and below T_{c}, the THz pump pulse only reduces n_{s} slightly, e.g., ΔE/E_{0} ∝ Δn_{s}/n_{0} ~ −3%.
Gaugeinvariant theory and simulations of THz coherent nonlinear spectroscopy in FeSCs
We start from the microscopic spatialdependent Boguliobov–de Gennes Hamiltonian for multiband superconductors
Here the Fermionic field operators \({\psi }_{\alpha ,\nu }^{\dagger }({\bf{x}})\) and ψ_{α,ν}(x) create and annihilate an electron with spin α in pocket ν; ξ_{ν}(p + eA(x, t)) corresponds to the dispersion of the pocket with momentum operator p = −i∇_{x} (ℏ = 1), vector potential A(x, t), and electron charge −e; μ is the chemical potential while ϕ(x, t) denotes the scalar potential.
The SC complex order parameter components arising from the condensates in the different Fermi sea pockets are
while the Fock energy is given by
and ensures charge conservation in the SC system. Here g_{λ,ν} is the effective inter (λ ≠ ν) and intra (λ = ν) electron–electron interaction in the BCS theory developed in previous works.
Hamiltonian (1) is gaugeinvariant under the gauge transformation
when vector potential, scalar potential, and different order parameter component phases transform as
with the field operator for band ν in Nambu space \({\Psi }_{\nu }({\bf{x}})={({\psi }_{\uparrow ,\nu }({\bf{x}}),{\psi }_{\downarrow ,\nu }^{\dagger }({\bf{x}}))}^{T}\) and Pauli spin matrix \({\tau }_{3}=\left(\begin{array}{ll}1&0\\ 0&1\end{array}\right)\). The conventional density matrix describing band ν, \({\rho }^{(\nu )}({\bf{x}},{\bf{x}}^{\prime} )=\langle {\Psi }_{\nu }{({\bf{x}})}^{\dagger }{\Psi }_{\nu }({\bf{x}}^{\prime} )\rangle\), depends on the choice of the gauge. To simplify the gauge transformation of the density matrix, we define centerofmass and relative coordinates \({\bf{R}}=({\bf{x}}+{\bf{x}}^{\prime} )/2\) and \({\bf{r}}={\bf{x}}{\bf{x}}^{\prime}\) and introduce a new density matrix,
where \({\rho }^{(\nu )}({\bf{r}},{\bf{R}})=\left.\left\langle {\Psi }_{\nu }^{\dagger }\left({\bf{R}}+\frac{{\bf{r}}}{2}\right){\Psi }_{\nu }\left({\bf{R}}\frac{{\bf{r}}}{2}\right)\right)\right\rangle\). This new density matrix \({\tilde{\rho }}^{(\nu )}({\bf{r}},{\bf{R}})\) transforms as
under the gauge transformation (4), where the transformed phase Λ(R) only depends on the centerofmass coordinate and not on both coordinates R and r as in the original density matrix. This property simplifies the gaugeinvariant description of the photoexcited nonequilibrium SC dynamics.
The equation of motion for \({\tilde{\rho }}^{(\nu )}({\bf{r}},{\bf{R}})\) is now derived by using the Heisenberg equation of motion technique. To simplify the equations of motion, we Fourier transform the obtained exact results with respect to the relative coordinate r and then apply a gradient expansion which is valid for SC systems with condensate centerofmass spatial fluctuations smoother than the spatial dependence of Cooper pair relative motion. To simplify the problem further, we eliminate the phase of the order parameter \({\Delta }_{{\nu }_{0}}({\bf{R}})\) by applying the gauge transformation
After assuming a homogeneous SC system and homogeneous excitation conditions by neglecting Rdependence, we obtain the gaugeinvariant Bloch equations for multiband superconductors that were solved numerically here
Here we introduced the gaugeinvariant superfluid momentum
and effective chemical potential
The Leggett mode corresponds to oscillations of the phase difference
while the Higgs mode is defined by the amplitude oscillation of the multicomponent complex SC order parameter. The latter is expressed in terms of the gaugeinvariant density matrix (6). To lowest order in the gradient expansion,
There are three mechanisms contributing to the driving of the Higgs modes. First, quantum transport contributions ∝ E in the equations of motion (9) lead to an acceleration of the Cooper pair condensate by the pump electric field (Lightwave Quantum Electronics)
which is neglected in the Anderson pseudospin model. This results in SC order parameter nonlinearities that are of odd order in the electric field. The condensate acceleration breaks equilibriuminversion symmetry of the SC system and can lead to dc supercurrent generation when lightwave propagation effects are included. Second, the minimal coupling, ξ(k − p_{S}/2) + ξ(k + p_{S}/2), known from the Anderson pseudospin model drives evenorder nonlinearities of the SC order parameter and depends on the band dispersion nonparabolicity. Third, the induced condensate momentum leads to a displacement of populations and coherences in momentum space by p_{S}/2 in the equations of motion which is also neglected in the Anderson pseudospin model. While the linear coupling to the superconductor via the quantum transport terms dominate the driving of the Higgs modes in the perturbative excitation regime, the excitation of the hybrid Higgs mode in the nonperturbative regime is dominated by the quadratic \({{\bf{p}}}_{{\rm{S}}}^{2}\)coupling known from the Anderson pseudospin model^{38}.
In our calculations we solve the gaugeinvariant optical Bloch equations (9) for a 3pocket model with a hole (h) pocket centered at the Γpoint and two electron (e) pockets located at (π, 0) and (0, π). We include the inter e–h pocket interactions (U = g_{e,h} = g_{h,e}) as well as intrapocket interactions (V_{λ} = g_{λ,λ}) while inter e–e pocket interactions are neglected for simplicity. The dominance of interband coupling U between e–h pockets over intraband in Febased SCs is taken into account by using an interbandtointraband interaction ratio of r = U/V = 10. The pockets are modeled using the square lattice nearestneighbor tightbinding dispersion \({\xi }_{\nu }({\bf{k}})=2\ [{J}_{\nu ,x}\cos ({k}_{x})+{J}_{\nu ,y}\cos ({k}_{y})]+{\mu }_{\nu }\) with hopping parameter J_{ν,i} and bandoffset μ_{ν}. We choose a circular hole pocket with J_{1,x} = J_{1,y} = 10.0 meV and μ_{1} = −37.5 meV. We introduce the known particle–hole asymmetry between electron and hole pockets in our system by considering elliptical electron pockets with J_{2,x} = J_{3,y} = −10.0 meV, J_{2,y} = J_{3,x} = −50.0 meV, and μ_{2} = μ_{3} = 57.5 meV. The latter can lead to coexistence of superconductivity and spindensity wave by changing the doping level. For the doping levels considered here, such asymmetry in our calculation strongly suppresses the ω_{H,2} mode in the spectra of coherent ΔE/E dynamics presented in Fig. 4a–b in the main text as discussed in more detail below. We assume s± pairing symmetry with equilibrium SC order parameters Δ_{1} = 3.4 meV for the hole pocket and Δ_{2} = Δ_{3} = 9.7 meV for the electron pockets. To directly model our phasecoherent nonlinear pump–probe spectroscopy experiments analogous to 2D THz phasecoherent nonlinear spectroscopy in semiconductors, we calculate the nonlinear differential transmission, ΔE, and not just the order parameter dynamics. The nonlinear differential transmission ΔE/E_{0} is obtained by computing the transmitted Efield of both pump and probe pulse, E_{pp}(t, τ), as a function of gate time t and pump–probe delay τ, as well as the transmitted electric field resulting from the probe pulse, E_{probe}(t), and the pump pulse, E_{pump}(t, τ) separately, following the experimental protocols discussed above. Here, the calculated transmitted Efield is given by
where E_{THz}(t) is the applied THz electric field, n is the refractive index of the SC system, and
is the current expressed in terms of the gaugeinvariant density matrix (9). This result is obtained by solving Maxwell’s equations in a thin film geometry^{38}. We then calculate the nonlinear differential transmission which is defined by ΔE = E_{pp}(t, τ) − E_{pump}(t, τ) − E_{probe}(t) for the collinear pump–probe geometry used in the experiment. All the presented theoretical results in manuscript Fig. 4 and the supplementary are based on ΔE calculated as above, i.e., the signal comes from interaction of the excitations by the two phasecoherent pulses and vanishes for independent excitations. In particular, in the inset of manuscript Fig. 4a, we show ΔE/E_{0} as a function of pump–probe delay for a fixed gate time t, where E_{0} is the peak electric field strength of the applied pump Efield. The spectra of ΔE/E_{0} for different pump fluences are plotted in manuscript Fig. 4a, b, while the spectral weights of the Higgs mode resonances presented in Fig. 4c, e and the inset of Fig. 4d are extracted from ΔE/E_{0} spectra.
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.
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Acknowledgements
This work was supported by National Science Foundation 1905981 (THz spectroscopy and data analysis). The work at UWMadison (synthesis and characterizations of epitaxial thin films) was supported by the US Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES), under award number DEFG0206ER46327. Theory work at the University of Alabama, Birmingham was supported by the US Department of Energy under contract # DESC0019137 (M.M and I.E.P) and was made possible in part by a grant for high performance computing resources and technical support from the Alabama Supercomputer Authority (quantum kinetic calculations). L.L. was supported by the Ames Laboratory, the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division under contract No. DEAC0207CH11358 (technical assistance).
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C.V., L.L., and X.Y. performed the THz spectroscopy measurements with J.W.’s supervision. J.H.K, C.S., and C.B.E. grew the samples and performed structural characterizations and transport measurements. M.M. and I.E.P. developed the theory for the hybrid Higgs mode and performed calculations. Y.G.C and E.E.H made Ba122 target for epitaxial thin films. J.W. and C.V. analyzed the THz data with the help of L.L., D.C., C.H., R.J.H.K., and Z.L. The paper is written by J.W., M.M., and I.E.P. with discussions from all authors. J.W. conceived and supervised the project.
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Vaswani, C., Kang, J.H., Mootz, M. et al. Light quantum control of persisting Higgs modes in ironbased superconductors. Nat Commun 12, 258 (2021). https://doi.org/10.1038/s41467020203506
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DOI: https://doi.org/10.1038/s41467020203506
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