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We present a novel theoretical approach for computing and analyzing two-dimensional spectroscopy of bosonic collective excitations in disordered many-body systems. Specifically, we employ the Keldysh formalism to derive the nonlinear response and obtain two-dimensional spectroscopy maps with particular emphasis on the rephasing sector, which allows to disentangle different sources of broadening. Our many-body approach successfully distinguishes elastic and inelastic scattering mechanisms contributing to the excitation linewidth. Additionally, using a non-perturbative conserving approach, we demonstrate that the echo peak exhibits a universal asymmetric shape in the sole presence of static disorder, a feature that remains robust against quantum fluctuations. This is in stark contrast to the standard theory based on isolated two-level systems, which fails to account for the dispersive nature of excitations and the interactions between different momentum components., Comment: 14 + 4 pages, 9 + 2 figures

Two-dimensional terahertz spectroscopy (2DTS), a terahertz analogue of nuclear magnetic resonance, is a new technique poised to address many open questions in complex condensed matter systems. The conventional theoretical framework used ubiquitously for interpreting multidimensional spectra of discrete quantum level systems is, however, insufficient for the continua of collective excitations in strongly correlated materials. Here, we develop a theory for 2DTS of a model collective excitation, the Josephson plasma resonance in layered superconductors. Starting from a mean-field approach at temperatures well below the superconducting phase transition, we obtain expressions for the multidimensional nonlinear responses that are amenable to intuition derived from the conventional single-mode scenario. We then consider temperatures near the superconducting critical temperature $T_c$, where dynamics beyond mean-field become important and conventional intuition fails. As fluctuations proliferate near $T_c$, the dominant contribution to nonlinear response comes from an optical parametric drive of counter-propagating Josephson plasmons, which gives rise to 2D spectra that are qualitatively different from the mean-field predictions. As such, and in contrast to one-dimensional spectroscopy techniques, such as third harmonic generation, 2DTS can be used to directly probe thermally excited finite-momentum plasmons and their interactions. Our theory provides a clear interpretation of recent 2DTS measurements on cuprates, and we discuss implications beyond the present context of Josephson plasmons.

Inhomogeneities play a crucial role in determining the properties of quantum materials. Yet methods that can measure these inhomogeneities are few, and apply to only a fraction of the relevant microscopic phenomena. For example, the electronic properties of cuprate materials are known to be inhomogeneous over nanometer length scales, although questions remain about how such disorder influences supercurrents and their dynamics. Here, two-dimensional terahertz spectroscopy is used to study interlayer superconducting tunneling in near-optimally-doped La1.83Sr0.17CuO4. We isolate a 2 THz Josephson echo signal with which we disentangle intrinsic lifetime broadening from extrinsic inhomogeneous broadening. We find that the Josephson plasmons are only weakly inhomogeneously broadened, with an inhomogeneous linewidth that is three times smaller than their intrinsic lifetime broadening. This extrinsic broadening remains constant up to 0.7Tc, above which it is overcome by the thermally-increased lifetime broadening. Crucially, the effects of disorder on the Josephson plasma resonance are nearly two orders of magnitude smaller than the in-plane variations in the superconducting gap in this compound, which have been previously documented using Scanning Tunnelling Microscopy (STM) measurements. Hence, even in the presence of significant disorder in the superfluid density, the finite frequency interlayer charge fluctuations exhibit dramatically reduced inhomogeneous broadening. We present a model that relates disorder in the superfluid density to the observed lifetimes.

Planar oxygen nuclear magnetic resonance (NMR) relaxation and shift data from all cuprate superconductors available in the literature are analyzed. They reveal a temperature independent pseudogap at the Fermi surface, which increases with decreasing doping in family specific ways, i.e., for some materials the pseudogap is substantial at optimal doping while for others it is nearly closed at optimal doping. The states above the pseudogap, or in its absence are similar for all cuprates and doping levels, and Fermi liquid-like. If the pseudogap is assumed exponential it can be as large as about 1500 K for the most underdoped systems, relating it to the exchange coupling. The pseudogap can vary substantially throughout a material, being the cause of cuprate inhomogeneity in terms of charge and spin, and consequences for the NMR analyses are discussed. This pseudogap appears to be in agreement with the specific heat data measured for the YBaCuO family of materials, long ago. Nuclear relaxation and shift show deviations from this scenario near $T_{\mathrm{c}}$, possibly due to other in-gap states., Comment: 26 pages, 9 figures

Very recently, there has been significant progress with establishing a common phenomenology of the superconducting cuprates in terms of nuclear magnetic resonance (NMR) shift and relaxation. Different from the old interpretation, it was shown that the shifts demand two coupled spin components with different temperature dependencies. One spin component couples isotropically to the planar Cu nucleus and is likely to reside at planar O, while the other, anisotropic component has its origin in the planar copper $3d(x^2-y^2)$ orbital. Nuclear relaxation, on the other hand, was found to be rather ubiquitous and Fermi liquid-like for planar Cu, i.e., it is independent of doping and material, apart from the sudden drop at the superconducting transition temperature, $T_{\rm c}$. However, there is a doping and material dependent anisotropy that is independent on temperature, above and below $T_{\rm c}$. Here we present a slightly different analysis of the shifts that fits all planar Cu shift data. In addition we are able to derive a simple model that explains nuclear relaxation based on these two spin components. In particular, the only outlier so far, \lsco, can be understood, as well. While this concerns predominantly planar Cu, it is argued that the two component model should fit all cuprate shift and relaxation data., Comment: 7 pages, 5 figures

Nuclear magnetic resonance (NMR) is a fundamental bulk probe that provides key information about the electronic properties of materials. Very recently, the analysis of all available planar copper shift as well as relaxation data proved that while the shifts cannot be understood in terms of a single temperature dependent spin component, relaxation can be explained with one dominating Fermi liquid-like component, without enhanced electronic spin fluctuations. For the shifts, a doping dependent isotropic term, as well as doping independent anisotropic term became obvious. Here we focus on planar $^{17}$O NMR shifts and quadrupole splittings. Surprisingly, we find that they demand, independently, a similar two-component scenario and confirm most of the previous conclusions concerning the properties of the spin components, in particular that a negative spin polarization survives in the superconducting state. This should have consequences for the pairing scenario., Comment: 9 pages, 7 figures

Nuclear relaxation is an important thermodynamic probe of electronic excitations, in particular in conducting and superconducting systems. Here, an empirical phenomenology based on all available literature data for planar Cu in hole-doped cuprates is developed. It is found that most of the seemingly different relaxation rates among the systems are due to a temperature independent anisotropy that affects the mostly measured $1/T_{1\parallel}$, the rate with an external magnetic field along the crystal $c$-axis, while $1/T_{1\perp}$ is largely independent on doping and material above the critical temperature of superconductivity ($T_c$). This includes very strongly overdoped systems that show Fermi liquid behavior and obey the Korringa law. Below $T_c$ the relaxation rates are similar, as well, if plotted against the reduced temperature $T/T_c$. Thus, planar Cu nuclear relaxation is governed by a simple, dominant mechanism that couples the nuclei with varying anisotropy to a rather ubiquitous bath of electronic excitations that appear Fermi liquid-like irrespective of doping and family. In particular, there is no significant enhancement of the relaxation due to electronic spin fluctuations, different from earlier conclusions. Only the La$_{2-x}$Sr$_x$CuO$_4$ family appears to be an outlier as additional relaxation is present, however, the anisotropy remains temperature independent. Also systems with very low doping levels, for which there is a lack of data, may behave differently., Comment: 7 pages, 7 figures

Nuclear magnetic resonance (NMR) provides local, bulk information about the electronic properties of materials, and it has been influential for theory of high-temperature superconducting cuprates. Importantly, NMR found early that nuclear relaxation is much faster than what one expects from coupling to fermionic excitations above the critical temperature for superconductivity ($T_{\rm c}$), i.e. what one estimates from the Knight shift with the Korringa law. As a consequence, special electronic spin fluctuations have been invoked. Here, based on literature relaxation data it is shown that the electronic excitations, to which the nuclei couple with a material and doping dependent anisotropy, are rather ubiquitous and Fermi liquid-like. A suppressed NMR spin shift rather than an enhanced relaxation leads to the failure of the Korringa law for most materials. Shift and relaxation below $T_{\rm c}$ support the view of suppressed shifts, as well. A simple model of two coupled electronic spin components, one with $3d(x^2-y^2)$ orbital symmetry and the other with an isotropic $s$-like interaction can explain the data. The coupling between the two components is found to be negative, and it must be related to the pseudogap behavior of the cuprates. We can also explain the negative shift conundrum and the long-standing orbital shift discrepancy for NMR in the cuprates., Comment: 10 pages, 7 figures