Nonlinear optical properties of materials are useful both for their practical applications, but also as a characterization tool for symmetry in novel systems. However, the nonlinear optical response of a material is often not as one of its fundamental properties, along with band structure, linear conductivity, etc. This has particularly been the case with the recent interest in “quantum materials”: materials such as high-Tc superconductors, topological materials, and exotic magnets. Weyl semimetals are a class of materials where band-touching points have divergent Berry curvature, and are host to a number of unique physical phenomena such as the chiral anomaly and Fermi arc surface states. From the nonlinear optics perspective, Weyl semimetals are interesting since inversion-symmetry breaking Weyl semimetals have been experimentally realized, and breaking of inversion-symmetry is crucial for second order nonlinear optical phenomena.

In this dissertation, I present results from a number of studies of the Weyl semimetal TaAs and related compounds using nonlinear optical techniques. As detailed here, TaAs was discovered to have the largest measured nonlinear optical susceptibility of any material, and further, this susceptibility was found to be caused by a resonance in the spectrum of nonlinear optical conductivity. TaAs was also measured to have a photogalvanic effect that has qualitatively different characteristics depending on the direction of measurement. These nonlinear optical properties of the TaAs family of materials may lead to interesting applications in near-field nonlinear optics, and optical pulse shaping below the diffraction limit.

In recent years, the study of cuprate superconductors has been dominated by the investigation of normal state properties. Of particular interest is the nature of interactions between superconductivity and other incipient orders which emerge above the superconducting transition temperature, T_c. The discovery of charge density wave (CDW) correlations in YBa₂Cu₃O_6+x (YBCO) and HgBa₂CuO_4+d (Hg-1201) has established that some form of charge order is ubiquitous in the cuprates. In this work, we explore the non-equilibrium dynamics of systems which sit near the boundary between superconductivity and competing orders.

Ultrafast pump-probe spectroscopy is ideally suited to the study of competing order. Exciting the sample with an optical pulse perturbs the system from equilibrium, altering the balance between the co-existing orders. The return to equilibrium is then monitored by a time-delayed probe pulse, revealing multiple decay processes as well as collective excitations. We first apply this technique to Hg-1201, conducting a detailed study of the phase diagram. At temperatures near T_c, the pump pulse induces a non-equilibrium quasiparticle population. At T_c we observe a doping-dependent peak in the relaxation time of these quasiparticles which we associate with a divergence in the coherence time of the fluctuating CDW. Using heterodyne probing in the transient grating geometry, we are able to disentangle the transient reflectivity components associated with superconductivity and the pseudogap, domonstrating competition across the phase diagram. We also discuss the observation of a sharp transition in the nature of the pseudogap signal at ∼ 11% doping.

In YBCO, we explore the temperature and doping dependence of coherent oscillations excited by the pump pulse. We associate these oscillations with the excitation of the CDW amplitude mode, and model their temperature dependence within the framework of a Landau model of competing orders.

We conclude with an investigation of pseudogap dynamics in the electron doped compound Nd_2-xCe_xCuO_4+d as a function of temperature and doping. Near optimal doping, we observe the impulsive excitation of a critically damped mode, with time-temperature scaling consistent with quantum-critical fluctuations. This mode competes with superconductivity in a dynamical fashion, such that the suppression of this mode below T_c can be lifted via photo-evaporation of the superconducting condensate.