Nonlinear light–Higgs coupling in superconductors beyond BCS: Effects of the retarded phonon-mediated interaction

Abstract

We study the contribution of the Higgs amplitude mode on the nonlinear optical response of superconductors beyond the BCS approximation by taking into account the retardation effect in the phonon-mediated attractive interaction. To evaluate the vertex correction in nonlinear optical susceptibilities that contains the effect of collective modes, we propose an efficient scheme which we call the "dotted DMFT" based on the nonequilibrium dynamical mean-field theory (nonequilibrium DMFT) to go around the difficulty of solving the Bethe-Salpeter equation and analytical continuation. The vertex correction is represented by the derivative of the self-energy with respect to the external driving field, which is self-consistently determined by the differentiated ("dotted") DMFT equations. We apply the method to the Holstein model, a prototypical electron-phonon-coupled system, to calculate the susceptibility for the third-harmonic generation including the vertex correction. The results show that, in sharp contrast to the BCS theory, the Higgs mode can contribute to the third-harmonic generation for general polarization of the laser field with an order of magnitude comparable to the contribution from the pair breaking or charge density fluctuations. The physical origin is traced back to the nonlinear resonant light-Higgs coupling, which has been absent in the BCS approximation.

Theory of coherent-oscillations generation in terahertz pump-probe spectroscopy: From phonons to electronic collective modes

“…While in a generic lattice model the Higgs is always excited by a light pulse [23][24][25][26][27][28] , its contribution to the differential probe field (49) is crucially determined by its Raman visibility. We then show that in general the experimental observation of THz-induced coherent oscillations at twice the gap value 5 or at twice the pump frequency 6 could be well reproduced by considering only density-like fluctuations, which certainly dominate the kernel K(ω) in the clean limit, as it has been widely discussed within the context of the third-harmonic generation 32,[34][35][36][37][38][39] . More recently, it has been pointed out 38,39 that in the presence of disorder the hierarchy of the two contributions can be reverted, making the Higgs contribution to the non-linear kernel K(ω) the dominant one.…”

“…(1) the differential field δE probe (t pp ) shows marked oscillations when entering in the SC state [4][5][6][7][8] . At the same time, below the SC critical temperature T c a marked THG has been observed 6,11,31 , which has been also ascribed to SC collective modes 6,8,[32][33][34][35][36][37][38][39] . For the case of conventional superconductors, like thin films of NbN, both pump-probe protocols and THG measurements show that the relevant frequency is twice the value of the SC gap ∆.…”

“…In other words, even if the gauge field is able to excite the Higgs mode as soon as χ A 2 ∆ is non zero, the contribution of the Higgs fluctuations to the nonlinear kernel K(ω), and then to the physically accessible quantity δE probe (t pp ), is negligible. The situation can change, and eventually even be reversed, when strong interactions 20,35 are considered, even if this also implies a significant smearing of the resonance itself, challenging the understanding of the experiments. Recently, it has been also suggested that disorder can play a crucial role, by activating paramagnetic-like processes which are absent without disorder 9,38,39 .…”

“…While the frequency of the SC oscillations is well captured by the parabolic-band approximation (35) for the non-linear kernel, their long-time decay depends crucially on the band-structure details. This can be again understood by relying on the general equation (8) for the transmitted probe field.…”

Theory of Time-Resolved Optical Conductivity of Superconductors: Comparing Two Methods for Its Evaluation

“…In addition to a reduction in the order parameter, the system exhibits Higgs (or Anderson-Higgs) oscillations, which are an oscillatory decay in the relaxation of the excited population of the Cooper pairs in superconducting condensates subject to perturbation by ultrafast pump fields-these were previously discussed based on similar calculations using the non-equilibrium Keldysh formalism [5,19,24,29]. Higgs oscillations arise here due to a time-dependence of the superconducting order parameter and we observe them in the time-dependent conductivity σ(ω, t delay ) as time-dependent oscillations of the gap edge and minimum around the phonon energy.…”