Terahertz High-Harmonics

Introduction

The generation of harmonics, i.e. integer multiples of a fundamental frequency, is an elementary and heavily used process in modern nonlinear optics. Its main use is the generation of photons with higher energy for technical use, e.g. in extreme UV photoemission spectroscopy. But harmonic generation can also reveal  information about the nonlinear medium, such as fundamental electronic properties and their timescales, crystal structures, symmetries and phases. From an application point of view, THz harmonic generation can be a viable way to bridge the gap between electronic and optical technologies.

As optical harmonic generation is a highly nonlinear process, extremely intense driving fields are typically necessary as a driver, which are only available in the form of ultra-short pulsed amplified laser systems. This is one of the main reasons why harmonic generation in the terahertz regime has been largely unexplored until recently. The required electric field strengths of several tens to hundreds of kV/cm, narrow bandwidth and also high repetition rates in the kHz to MHz range are extremely challenging to produce in the THz range. The accelerator-based TELBE THz source at HZDR offers all of these parameters in the range between 0.2 THz and 2.5 THz, making it an ideal platform for exploring THz harmonic generation.

As a result, TELBE has been pioneering THz harmonic generation as a general tool for exploring nonlinear light matter interaction in a variety of systems, most notably (i) Dirac materials, (ii) superconductors, and (iii) correlated oxides.

In THz harmonic generation, an incoming multicycle THz pulse drives quickly alternating currents in the sample. The nonlinear response of the sample manifests itself in the emission of THz radiation at multiples of the incident frequency, as schematically shown in the figure above. The microscopic mechanisms leading to the emission of harmonics are manifold and depend directly on the response of the carriers in the sample to intense electric field transients, as described in detail below.

Dirac materials

Dirac materials are an advanced class of systems characterized by low-energy electronic excitations that mimic relativistic particles, moving effectively without mass. A closer look at the electronic bandstructure reveals that the valence and conduction bands touch at a single point known as the Dirac point, forming a linear dispersion relation E(k). This linear energy spectrum forces charge carriers to behave like massless Dirac fermions, which travel at a constant, high velocity. Consequently, these materials exhibit extraordinary electronic mobility, high thermal conductivity, and unique quantum phenomena like the anomalous quantum Hall effect. 

Upon excitation with intense THz pulses, the charge carriers at the Fermi level near the Dirac point are therefore accelerated instantly by the electric field of the THz pulse, leading to a strong deformation of the distribution of electrons in the bandstructure. This is equivalent to a heating of the electronic system by thousands of Kelvin, all within less than a picosecond. In the hot state, the optical properties of the graphene layer become drastically different: The THz absorption drops from roughly 13% to 10% - all within than the duration of the oscillation cycle of the THz electric field. As a consequence, the THz field that passes through the sample is strongly modulated. This modulation manifests itself in the emission of odd integer harmonics, i.e. THz pulses with 3, 5, ... times the initial frequency.

Graphene metamaterials

The nonlinear response of graphene is mainly determined by its carrier concentration next to the Fermi level. Therefore, shifting the Fermi level by intentional doping or by electronic gating has a significant impact on the harmonic generation efficiency. One approach is the variation of the concentration of quasi-free carriers at the Fermi edge by electrical gating. In these gate-tunable samples we have demonstrated a power conversion efficiency modulation by two orders of magnitude - just by tuning the gate voltage over a range of about 1 V [2].

Another approach is the addition of a photonic structure which is designed to locally enhance the incident THz electric field in the graphene layer. This approach is of particular technical relevance since the incident field strength does not have to be extremely high (i.e. ⪆100 kV/cm) to drive nonlinear currents in the material. Using gold stripes with few micrometer width, a field enhancement by a factor of five can be obtained. This leads to an increase in the third order susceptibility χ⁽³⁾ by a factor of 40 in the low field regime [3]. As a result, the obtained field conversion efficiency for third harmonic generation reaches 1% for moderade incident field strength of 30 kV/cm. However, at higher field strengths graphene shows strong saturation effects due to the effective overheating of the electronic system [4]. As the electronic heat is dissipated on picosecond timescales, the sample cannot relax back to its initial state before the next field cycle of the incident THz pulse starts. To overcome these saturation effects that limit the applicability of THz harmonic generators based on graphene for extremely high incident field (towards 1 MV/cm), materials with faster cooling times of the electronic system are needed.

Topological insulators

Topological insulators (TIs), such as bismuth telluride, exhibit the same linear dispersion relation E(k) as graphene at their topological surface states. At the same time, THz pump, optical probe experiments have shown that the relaxation time scales of hot carriers in these Dirac bands is significantly shorter – few hundred femtoseconds instead of picoseconds in graphene. This faster cooling is a consequence of the Coulomb scattering between hot carriers in the Dirac surface states and bulk states. As a result, no saturation is observed in THz harmonic generation in with high incident field strengths [5]. Still, the effective nonlinear susceptibilities are smaller compared to those of graphene. The combination of TIs with field enhancing photonic structures however is capable of harnessing this non-saturable behavior: We observed the current record field conversion efficiencies of ~8% in a bismuth selenide metamaterial [6].

Related publications

• [1] H. A. Hafez, S. Kovalev et al., Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac fermions. Nature 561, 507-511 (2018). (press release)
• S. Kovalev, S., et al. Non-perturbative terahertz high-harmonic generation in the three-dimensional Dirac semimetal Cd₃As₂. Nat Commun 11, 2451 (2020). (press release)
• [2] S. Kovalev et al., Electrical tunability of terahertz nonlinearity in graphene. Sci. Adv. 7, eabf9809 (2021). (press release)
• [3] J.-C. Deinert et al., Grating-Graphene Metamaterial as a Platform for Terahertz Nonlinear Photonics. ACS nano 15, 1145-1154 (2020). (press release)
• [4] H. A. Hafez, S. Kovalev et al., Terahertz Nonlinear Optics of Graphene: From Saturable Absorption to High-Harmonics Generation. Adv. Optical Mater. 8, 1900771 (2020).
• [5] S. Kovalev, S., KJ Tielrooij, J.-C. Deinert, et al., Terahertz signatures of ultrafast Dirac fermion relaxation at the surface of topological insulators. npj Quantum Mater. 6, 84 (2021).
• [6] KJ Tielrooij, A. Principi, D.S. Reig et al., Milliwatt terahertz harmonic generation from topological insulator metamaterials. Light Sci Appl 11, 315 (2022). (press release)

Superconductors – Higgs spectroscopy

In superconductors, the Anderson-Higgs mechanism gives mass to the gauge field (the photon), resulting in the Meissner effect. Because the superconducting state breaks the U(1) gauge symmetry, the collective excitations of the Cooper-pair condensate split into two modes: a massless Goldstone phase mode and a massive Higgs (amplitude) mode. Because the Higgs mode is a scalar amplitude oscillation, it does not couple directly (linearly) to electromagnetic fields. Instead, it can be probed via nonlinear optical processes, primarily Third-Harmonic Generation (THG). 

How the Higgs Mechanism Drives Harmonic Generation

When a superconductor is driven by an intense alternating electromagnetic field (typically using Terahertz (THz) pulses), the incident light acts as a nonlinear Raman-like drive.

• Nonlinear Drive: An incident electric field with frequency ω couples quadratically to the superconducting order parameter, effectively driving the superconducting gap Δ with a frequency of 2ω.
• Higgs Resonance: When twice the driving frequency matches the energy gap of the Higgs mode (i.e., 2ω = 2Δ), a pronounced resonance occurs. This causes the order parameter to oscillate strongly at the Higgs frequency.
• Third-Harmonic Generation (THG): These resonant oscillations of the superconducting gap mix with the incident electric field, resulting in the emission of radiation at the third harmonic 3ω.

At TELBE a series of THG experiments on different types of superconductors were carried out. The aim was to disentangle the different collective degrees of freedom in the low THz range and to establish Higgs probing as an ultrafast analogon to the Meissner effect. Cuprate superconductors, such as LSCO, exhibited a strong THG signal, which interestingly persists even at temperatures more than 1.5 times the critical temperature. 

Related publications

• Chu, H., Kim, MJ., Katsumi, K. et al. Phase-resolved Higgs response in superconducting cuprates. Nat Commun 11, 1793 (2020). (press release)
• Kovalev, S., Dong, T. et al. Band-selective third-harmonic generation in superconducting MgB2: Possible evidence for the Higgs amplitude mode in the dirty
  limit. Phys. Rev. B 104, L140505 (2021).
• Chu, H., Kovalev, S., Wang, Z.X. et al. Fano interference between collective modes in cuprate high-T_c superconductors. Nat Commun 14, 1343 (2023). 
• Feng, L. et al., Dynamical interplay between superconductivity and charge density waves: A nonlinear terahertz study of coherently driven 2⁢𝐻−NbSe₂. Phys. Rev. B 108, L100504 (2023).
• Kim, M.-J. et al., Tracing the dynamics of superconducting order via transient terahertz third-harmonic generation. Sci. Adv. 10, eadi7598 (2024).

Correlated systems

Terahertz (THz) harmonic generation in correlated electron systems, such as rare-earth nickelates (RNiO₃), is an emergent field at the intersection of ultrafast photonics and solid-state physics. It serves as a highly sensitive, non-invasive probe of nonlinear dynamics, strongly coupled degrees of freedom, and phase transitions in quantum materials.
When subjected to intense THz electric fields, strongly correlated systems – driven by Coulomb repulsion (U) and Hund's coupling – exhibit significant nonlinear light-matter interactions. These interactions produce measurable higher harmonics of the fundamental driving frequency (most commonly the third harmonic, THG).

Physical Mechanisms in Rare-Earth Nickelates (RNiO₃)

Rare-earth nickelates (RNiO₃) are prototype materials for studying the Mott insulator-metal transition (IMT). The THz nonlinear response is distinct across the three thermodynamic phases of these negative charge-transfer insulators:

• Antiferromagnetic (AFM) Insulating Phase: At low temperatures, the THz third-harmon (THG) amplitude increases upon cooling. Here, the harmonics are predominantly driven by trong spin-charge and orbital-charge couplings.
• Paramagnetic (PM) Insulating Phase: In the intermediate PM insulating phase, harmonic generation intensity is suppressed as temperature drops. This decrease is linked to the reduction of available charge carriers as the Mott gap opens.
• Paramagnetic (PM) Metallic Phase: At high temperatures, he THz harmonic signal increases again. In this metallic state, nonlinearities are dominated by intraband currents from renormalized quasiparticles with frequency-dependent scattering rates.

Experimental and Theoretical Approaches

• Driving Fields: By pumping nickelates and other correlated materials (like heavy quasiparticle ruthenates such as CaRuO₃) with intense, multi-cycle THz pulses, researchers can track quantum critical kinetics without overwhelming the system with heat.
• Tunable Parameters: The THz harmonic yield is highly tunable. By adjusting the Ni-O-Ni bond angle (via rare-earth ionic radius), epitaxial strain from the substrate, or film thickness, researchers can sharply control both the metal-insulator transition temperature (\(T_{MI}\)) and the corresponding harmonic generation efficiency.
• Theoretical Modeling: Harmonic generation across these phases is typically modeled using nonequilibrium many-body techniques, including single- and two-band Hubbard models and time-dependent Boltzmann transport frameworks.

Related publications

• Prajapati, G. L., et al., Terahertz harmonic generation across the Mott insulator-metal transition. Phys. Rev. B 113, 045118 (2026). 
• Prajapati, G. L., et al., Evolution of terahertz third harmonic response across rare-earth nickelate phase-diagram. arXiv:2606.09622 [cond-mat.str-el]
• Azab, a. et al., Higher harmonics in Mott-Hubbard insulators as sensors. arXiv:2603.05254 [cond-mat.str-el]