Recent experiments with strong THz fields in both conventional and unconventional superconductors have clearly evidenced a marked third-harmonic generation below the superconducting temperature T c . Its interpretation challenged substantial theoretical work aimed at establishing the relative efficiency of quasiparticle excitations and collective modes in triggering such a resonant response. Here we compute the nonlinear current by implementing a time-dependent Bogoljubov-de Gennes approach, with the twofold aim to account nonperturbatively for the effect of local disorder, and to include the contribution of all collective modes, i.e., superconducting amplitude (Higgs) and phase fluctuations, and charge fluctuations. We show that, in agreement with previous work, already at small disorder the quasiparticle response is dominated by paramagnetic effects. We further demonstrate that paramagnetic processes mediate also the response of all collective modes, with a substantial contribution of charge/phase fluctuations. These processes, which have been overlooked so far, turn out to dominate the third-order current at strong disorder. In addition, we show that disorder strongly influences the polarization dependence of the nonlinear response, with a marked difference between the clean and the disordered case. Our results are particularly relevant for recent experiments in cuprates, whose band structure is in a first approximation reproduced by our lattice model.

Time-resolved spectroscopies using intense THz pulses appear as a promising tool to address collective electronic excitations in condensed matter. In particular recent experiments showed the possibility to selectively excite collective modes emerging across a phase transition, as it is the case for superconducting and charge-density-wave (CDW) systems. One possible signature of these excitations is the emergence of coherent oscillations of the differential probe field in pump-probe protocols. While the analogy with the case of phonon modes suggests that the basic underlying mechanism should be a sum-frequency stimulated Raman process, a general theoretical scheme able to describe the experiments and to define the relevant optical quantity is still lacking. Here we provide this scheme by showing that coherent oscillations as a function of the pump-probe time delay can be linked to the convolution in the frequency domain between the squared pump field and a Raman-like non-linear optical kernel. This approach is applied and discussed in few paradigmatic examples: ordinary phonons in an insulator, and collective charge and Higgs fluctuations across a superconducting and a CDW transition. Our results not only account very well for the existing experimental data in a wide variety of systems, but they also offer an useful perspective to design future experiments in emerging materials.In the last decade, a significant advance in the investigation of complex systems has been gained thanks to the huge experimental progress in time-resolved spectroscopic techniques 1,2 . On very general grounds, the basic idea behind any pump-probe protocol is to first excite the system with a short and very intense electromagnetic pulse (pump), and then to monitor its relaxation towards equilibrium by using a secondary, weak pulse (probe) applied with a finite time delay with respect to the pump. This general protocol can then be implemented in several different ways, according to the nature of the spectroscopic measurement (angle-resolved photoemission, optical reflection or transmission, etc.) or to the wavelength of the pump/probe fields. However, in all cases one has to face with two phenomena which mark the difference with respect to ordinary equilibrium spectroscopies: (i) the use of an intense pulse triggers in general non-linear optical processes; (ii) the subsequent relaxation encodes by definition non-equilibrium phenomena on time scales which depend on the characteristics of the experiment and of the system under investigation. Due in part to these innovative aspects, many pump-probe protocols still lack a general theoretical framework able to connect the measured quantities to the material properties. More specifically, while Kubo linear-response theory 3 represents nowadays the standard theoretical tool needed to compute the optical response of any system to a weak external perturbation, an analogous protocol for timeresolved spectroscopies has not been established yet.The present work aims at filling in part this knowl- * mat...

The hallmark of superconductivity is the rigidity of the quantum-mechanical phase of electrons, responsible for superfluid behavior and Meissner effect. The strength of the phase stiffness is set by the Josephson coupling, which is strongly anisotropic in layered cuprates. So far, THz light pulses have been used to achieve non-linear control of the out-of-plane Josephson plasma mode, whose frequency lies in the THz range. However, the high-energy in-plane plasma mode has been considered insensitive to THz pumping. Here, we show that THz driving of both low-frequency and high-frequency plasma waves is possible via a general two-plasmon excitation mechanism. The anisotropy of the Josephson couplings leads to markedly different thermal effects for the out-of-plane and in-plane response, linking in both cases the emergence of non-linear photonics across Tc to the superfluid stiffness. Our results show that THz light pulses represent a preferential knob to selectively drive phase excitations in unconventional superconductors.

Raman experiments on bulk FeSe revealed that the low-frequency part of the B 1g Raman response R B1g ðΩÞ, which probes nematic fluctuations, rapidly decreases below the nematic transition at T n ∼ 85 K. Such behavior is expected when a gap opens up and at a first glance is inconsistent with the fact that FeSe remains a metal below T n . We argue that the drop of R B1g ðΩÞ can be ascribed to the fact that the nematic order drastically changes the orbital content of low-energy excitations near hole and electron pockets, making them nearly mono-orbital. In this situation, the B 1g Raman response gets reduced by the same vertex corrections that enforce charge conservation in the symmetric Raman channel. The reduction holds at low frequencies and gives rise to gaplike behavior of R B1g ðΩÞ. We also show that the enhancement of the B 1g Raman response near T n is consistent with the sign change of the nematic order parameter between hole and electron pockets.

Recent experiments with strong THz fields in unconventional cuprates superconductors have clearly evidenced an increase of the non-linear optical response below the superconducting critical temperature Tc. As in the case...