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One of the most intensely studied scenarios of high-temperature superconductivity (HTS) postulates pairing by exchange of magnetic excitations. Indeed, such excitations have been observed up to optimal doping in the cuprates. In the heavily overdoped regime, neutron scattering measurements indicate that magnetic excitations have effectively disappeared, and this has been argued to cause the demise of HTS with overdoping. Here we use resonant inelastic X-ray scattering, which is sensitive to complementary parts of reciprocal space, to measure the evolution of the magnetic excitations in La(2-x)Sr(x)CuO4 across the entire phase diagram, from a strongly correlated insulator (x = 0) to a non-superconducting metal (x = 0.40). For x = 0, well-defined magnon excitations are observed. These magnons broaden with doping, but they persist with a similar dispersion and comparable intensity all the way to the non-superconducting, heavily overdoped metallic phase. The destruction of HTS with overdoping is therefore caused neither by the general disappearance nor by the overall softening of magnetic excitations. Other factors, such as the redistribution of spectral weight, must be considered.

Cuprate materials hosting high-temperature superconductivity (HTS) also exhibit various forms of charge and spin ordering whose significance is not fully understood. So far, static charge-density waves (CDWs) have been detected by diffraction probes only at particular doping levels or in an applied external field . However, dynamic CDWs may also be present more broadly and their detection, characterization and relationship with HTS remain open problems. Here we present a method based on ultrafast spectroscopy to detect the presence and measure the lifetimes of CDW fluctuations in cuprates. In an underdoped La(1.9)Sr(0.1)CuO4 film (T(c) = 26 K), we observe collective excitations of CDW that persist up to 100 K. This dynamic CDW fluctuates with a characteristic lifetime of 2 ps at T = 5 K that decreases to 0.5 ps at T = 100 K. In contrast, in an optimally doped La(1.84)Sr(0.16)CuO4 film (T(c) = 38.5 K), we detect no signatures of fluctuating CDWs at any temperature, favouring the competition scenario. This work forges a path for studying fluctuating order parameters in various superconductors and other materials.