Phase-resolved Higgs response in superconducting cuprates

Field theories with broken U(1) symmetry host two collective modes: a phase (Goldstone) mode and an amplitude (Higgs) mode. In superconductors, the amplitude mode of the order parameter—analogous to the Higgs boson—has been theoretically anticipated and more recently detected, but is challenging to observe because it lacks direct electric or magnetic dipole coupling. Ultrafast terahertz techniques can excite the Higgs mode either as free oscillations at 2Δ following a quench or as a driven response at 2ω via nonlinear coupling to the vector potential A(ω). The latter process leads to third-harmonic generation (THG) that is resonantly enhanced near 2ω = 2Δ. While free and driven Higgs oscillations have been observed in s‑wave superconductors, d‑wave cuprates pose additional complexity due to gap anisotropy and low-energy quasiparticles that strongly damp and dephase the amplitude mode. This study applies phase-resolved, multicycle, carrier-envelope phase stable THz excitation (0.7 THz, up to ~50 kV/cm) from the TELBE source to probe the driven Higgs response in optimally doped LSCO (T_c = 45 K), DyBCO (~90 K), YBCO (88 K), and overdoped BSCCO (65 K) thin films. The goals are to identify signatures of the Higgs contribution to THG, determine whether coupling to other collective modes affects the response, and test for finite pairing amplitude above T_c.

Prior work established the Higgs (amplitude) mode in superconductors and its excitation via terahertz pulses. Theory predicts THG enhancement when the driven Higgs oscillation at 2ω resonates with 2Δ and that the amplitude mode can reveal superconducting gap symmetry, multiplicity, and coupled collective modes. Experiments in s‑wave superconductors (e.g., NbN) have shown polarization-resolved THG dominated by the Higgs mode, and optical studies in the d‑wave cuprate Bi2Sr2CaCu2O8+x reported an A^2 response of the condensate to intense THz pulses, consistent with an isotropic A1g Higgs component. Competing or additional nonlinear mechanisms have been discussed: the nonlinear Meissner effect (NME), often associated with Josephson-like phase dynamics, which typically yields a narrow peak in third-order response around T_c; and charge density fluctuations (CDF), expected to produce anisotropic nonlinear optical responses in both s- and d‑wave superconductors. Theoretical works also consider coupling among Higgs, Leggett, phonon, and other order-parameter collective modes, suggesting complex dynamics in unconventional superconductors.

Samples: Thin films of LSCO(OP45, 80 nm on LSAO), DyBCO(OP90, 70 nm on LSAT), YBCO(OP88, 200 nm on NGO) grown by MBE or PLD at the Max Planck Institute; BSCCO(OD65, 160 nm on MgO) grown by sputtering at Laboratoire de Physique des Solides. T_c values determined by mutual inductance (LSCO, DyBCO, YBCO) or SQUID magnetometry (BSCCO). THz excitation and detection: Multicycle, CEP-stable THz pulses at 0.7 THz (photon energy ~3 meV) from the TELBE superradiant undulator source with peak fields up to ~50 kV/cm. Electro-optic sampling (2 mm ZnTe, 100 fs, 800 nm gate). Timing jitter ~20 fs with pulse-resolved synchronization. Residual FH and generated TH (2.1 THz) isolated using FFT filtering; external 2.1 THz (or 1.9 THz) bandpass filters used in certain measurements to suppress FH and to analyze fluence dependence. Measurement protocols:

• Temperature sweeps through T_c to measure FH and TH waveforms and extract TH intensity (I_TH) and relative phase Φ_TH with respect to FH (phase extracted via fitting of filtered waveforms; two-sigma uncertainties reported).
• Polarization dependence of THG versus in-plane angle θ relative to Cu-O bonds to separate isotropic A1g Higgs response from anisotropic CDF contributions.
• Terahertz pump–optical probe (TPOP): 0.7 THz pump and 80 fs optical probe reflectivity to detect A(ω)^2-driven condensate response manifesting as 1.4 THz modulation while the pump is present.
• Fluence dependence: Test perturbative regime via scaling I_TH ∝ I_FH^3. Analysis and modeling: Assess temperature dependence of I_TH and normalized I_TH/I_FH^3 to test for resonance-like behavior at 2ω = 2Δ(T). Use a driven coupled harmonic oscillators toy model (Higgs mode coupled to another underdamped collective mode with temperature-dependent resonance energies Δ(T) and δΔ(T)) to account for phase behavior (anti-resonance and negative phase jump) and amplitude minima, including screening effects in thin films (Fresnel analysis excludes linear origin of phase jump).

• Strong THG below T_c in all cuprate films studied; FH transmission decreases monotonically with cooling, whereas TH grows and exhibits non-monotonic temperature dependence.
• Polarization dependence: THG dominated by an isotropic response consistent with A1g Higgs symmetry; any anisotropic (e.g., CDF) contribution is subdominant but possibly finite.
• Terahertz pump–optical probe shows 1.4 THz modulation in ΔR only while the THz pump is on, consistent with an |A|^2 condensate response.
• Fluence dependence: Perturbative scaling I_TH ∝ I_FH^3 verified.
• Temperature dependence of I_TH: • LSCO(OP45): Main I_TH peak near ~0.67 T_c and a smaller peak near ~0.9 T_c. • DyBCO(OP90): Similar main peak near ~0.6 T_c. • YBCO(OP88): Sharp peak near ~0.97 T_c and a hump around T_c. • BSCCO(OD65): I_TH increases continuously upon cooling. The main peaks arise from competition between increasing nonlinear response and increasing screening, not from a sharp 2ω = 2Δ(T) resonance (expected immediately below T_c for 0.7 THz given Δ ≥ 20 meV in optimally doped cuprates and strong damping).
• Phase-resolved result: A universal, abrupt, negative phase jump of nearly π in the relative TH phase Φ_TH at T_φ < T_c is observed in LSCO(OP45), DyBCO(OP90), and YBCO(OP88). The sign and sharpness of the jump are inconsistent with a simple heavily damped Higgs resonance.
• Modeling: A driven coupled-oscillators model (Higgs coupled to another underdamped collective mode of comparable energy) reproduces a negative phase jump across an anti-resonance and an amplitude minimum; suggests the coupled mode is underdamped and has temperature-dependent energy and/or coupling.
• Above-T_c response: (I_TH/I_FH^3)^(1/4) shows order-parameter-like behavior below T_c and remains finite up to T ≳ 1.5 T_c, indicating either preformed Cooper pairs without global phase coherence, field-induced coherence above T_c, and/or roles of pseudogap/CDW-related orders.
• Experimental parameters: Driving frequency 0.7 THz (~3 meV), fields up to ~50 kV/cm; T_c values LSCO 45 K, DyBCO ~90 K, YBCO 88 K, BSCCO 65 K.

The central question is whether the THz-driven THG response in cuprate superconductors reveals the dynamics of the Higgs (amplitude) mode and its coupling to other collective degrees of freedom. The observed isotropic polarization dependence and |A|^2-driven condensate response support a significant A1g Higgs contribution to THG. However, the universal, abrupt, negative jump of Φ_TH at T_φ < T_c across different families is not compatible with a simple, strongly damped Higgs resonance at 2ω = 2Δ(T). Instead, it is naturally explained by an anti-resonance arising from coupling between the Higgs mode and another underdamped collective excitation of comparable energy. Candidate coupled modes include CDW-related charge fluctuations, paramagnons (spin resonance), phonon-coupled amplitude modes, or other superconducting collective modes (e.g., Bardasis–Schrieffer or anisotropic Higgs components). The persistence of finite THG above T_c further suggests a non-vanishing pairing amplitude without long-range phase coherence, consistent with superconducting fluctuations observed by other probes, though strong-field-induced coherence and pseudogap/CDW contributions are alternative explanations. Thus, phase-resolved THG provides sensitive access to coupling among collective modes and to pairing correlations above T_c in d‑wave cuprates.

This work establishes phase-resolved terahertz THG as a probe of the superconducting Higgs response in cuprate thin films and reveals universal phase anomalies indicative of coupling to an additional underdamped collective mode. Key contributions include: (i) observation of strong, predominantly isotropic THG below T_c consistent with a driven A1g Higgs response; (ii) discovery of a universal, negative Φ_TH jump near T_φ < T_c across several cuprates, best explained by an anti-resonance from mode coupling; and (iii) detection of finite THG well above T_c, pointing to pairing amplitude persisting to T ≳ 1.5 T_c. These findings motivate comprehensive quantum-mechanical treatments of driven collective-mode dynamics in d‑wave superconductors and systematic experiments varying doping, magnetic field, and driving frequency to identify the coupled mode and disentangle contributions from superconducting, charge, spin, and phononic excitations. The approach also opens avenues for probing nonequilibrium superconductivity and gap evolution near T_c.

• Source attribution: While isotropic polarization dependence and |A|^2 response support a Higgs contribution, THG cannot uniquely exclude other nonlinear mechanisms (e.g., charge density fluctuations, nonlinear Meissner effect, or additional nonlinearities); anisotropic components may contribute.
• Modeling: The coupled-oscillators analysis is a classical toy model; a full quantum mechanical treatment is required for microscopic interpretation and quantitative parameter extraction.
• Identification of coupled mode: The precise nature of the additional underdamped mode remains unresolved; distinguishing among CDW fluctuations, paramagnons, phonon-coupled modes, or other superconducting collective modes requires further measurements (doping, magnetic field, frequency sweeps).
• Temperature resolution and screening: Peaks and dips in I_TH(T) are influenced by competition with temperature-dependent screening in thin films; sharp features may be narrowed beyond the experimental temperature step size in some compounds.
• Above-T_c interpretation: Finite THG above T_c may arise from preformed pairs, field-induced coherence, or pseudogap/CDW effects; present data do not uniquely differentiate these scenarios.