Ultrafast Two-Dimensional Spectroscopy Uncovers Ubiquitous Electron-Paramagnon Coupling in Cuprate Superconductors

Abstract

The coupling between electronic excitations and collective bosonic modes is fundamental to the emergence of high-temperature superconductivity in cuprates. Despite extensive effort, conventional equilibrium and pump-probe optical spectroscopies still struggle to disentangle couplings to different bosonic modes when their energy scales overlap. Here we overcome this limitation using ultrafast two-dimensional electronic spectroscopy (2DES), which correlates coherent excitation and detection photon energies with femtosecond time resolution. Applied to optimally doped Bi$_2$Sr$_2$Ca$_{0.92}$Y$_{0.08}$Cu$_2$O$_{8+δ}$, 2DES reveals a pronounced off-diagonal resonance arising from the ultrafast generation of non-thermal bosons with energy $\hbarΩ_\mathbf{q}\simeq200$ meV. By comparing the measured spectra with a theoretical framework that explicitly includes the interaction between charge-transfer and magnetic excitations, we identify these bosons as paramagnons with momenta centered near $(π/2,π/2)$ and extending toward $(0,π)$ and $(π,0)$. The resonance persists across a large range of temperatures and doping concentrations, demonstrating that high-energy paramagnons are ubiquitously and strongly coupled to electronic excitations throughout the cuprate phase diagram. Time-domain analysis constrains the build-up of the paramagnon population to $\lesssim 10$ fs, placing a lower bound $λ\gtrsim 0.7$ on the coupling strength. More broadly, our results establish 2DES as a powerful approach for disentangling mode-selective electron-boson interactions and addressing decoherence dynamics, thereby establishing a new avenue for investigating strongly correlated quantum materials. These findings also provide a direct framework for future time-resolved resonant inelastic X-ray scattering experiments aimed at tracking the ultrafast dynamics of magnetic excitations.

Figures (6)

• Figure 1: (a) Schematic of the partially collinear 2DES setup. Two phase-locked broadband pump pulses, separated by a delay $t_{pu}$, excite the sample; after a waiting time $\tau$ a probe pulse measures the response. The reflected probe is spectrally dispersed by a spectrometer to define the detection axis $\hbar\omega_{pr}$. Pump replicas are produced by a common-path wedge interferometer to ensure phase stability and precise control of time delay $t_{pu}$ (see Supplementary Sec. S1). (b) Sketch of a 2D spectrum $S(\hbar\omega_{pr},\hbar\omega_{pu};\tau)$ at fixed $\tau$. The dashed diagonal marks $\hbar\omega_{pu}=\hbar\omega_{pr}$; an off-diagonal cross-peak ($\hbar\omega_c,\hbar\omega_a$) is indicated. (c) Illustration of a boson-mediated three-level scheme. First, pump photons drive a vertical transition to a virtual level at $\hbar\omega_a=\hbar\omega_b+\hbar\Omega_{\mathbf q}$, which relaxes to the real state $\hbar\omega_b$ by emitting a boson (energy $\hbar\Omega_{\mathbf q}$, momentum $\mathbf q$). This results in the creation of a transient non-thermal boson population peaked at $\hbar\Omega_{\mathbf q}$. Second, the probe promotes the system to a virtual intermediate state at $\hbar\omega_c=\hbar\omega_b-\hbar\Omega_{\mathbf q}$, and the transition is completed to the real state by absorbing a non-thermal boson of energy $\hbar\Omega_{\mathbf q}$. The corresponding cross-peak is observed at $(\hbar\omega_c,\hbar\omega_a)$ in $S(\hbar\omega_{pr},\hbar\omega_{pu};\tau)$ in panel (b).
• Figure 2: (a) 2DES map of optimally doped Y-Bi2212 measured at $300$ K with a pump fluence of $400$ µJ/cm$^2$, recorded at a fixed delay time $\tau = 40$ fs. The color map shows the differential reflectivity $\Delta R/R$ as a function of pump photon energy $\hbar\omega_{pu}$ (vertical axis) and probe photon energy $\hbar\omega_{pr}$ (horizontal axis). The white dashed line marks the diagonal direction $\hbar\omega_{pu} = \hbar\omega_{pr}$. Black dashed lines indicate the position of the off-diagonal peak (see Supplementary Sec. S3). The inset shows the position of the measurement in the temperature-doping phase diagram. (b) 2DES map measured under the same experimental conditions as in panel (a) but at a longer delay time, $\tau = 340$ fs. The same color scale as in panel (a) is used; for clarity, the signal amplitude is multiplied by a factor of two. The off-diagonal resonance observed at $\tau = 40$ fs is preserved at long delay.
• Figure 3: Model of paramagnon-assisted charge-transfer pathways. (a) Schematic band structure illustrating the indirect optical transitions mediated by paramagnons. The O-$2p_{\sigma}$ (red) and O-$2p_{\pi}$ (blue) valence bands are shown together with the Cu-$3d_{x^2-y^2}$ conduction band (color scale), with energies referenced to the Fermi level $E_{Fermi}=\mu$. In the pump channel (purple arrows), a photon of energy $\hbar\omega_{pu}$ excites a vertical charge-transfer transition from the O-$2p$ band, followed by emission of a paramagnon with energy $\hbar\Omega_{\mathbf{q}}$. In the probe channel (orange arrows), a photon of energy $\hbar\omega_{pr}$ drives the transition via absorption of a previously generated paramagnon. The green bar indicates the resulting correlation energy $\Delta E = \hbar\omega_{pu} - \hbar\omega_{pr}$. The inset shows the high-symmetry path in the Brillouin zone along which the band dispersions are plotted. (b) Imaginary part of the spin susceptibility $\chi"(\mathbf{q},\hbar\Omega)$ used in the calculations, shown as a color map along a high-symmetry momentum path with $J=105$ meV and $\Gamma=300$ meV. The inset highlights the corresponding momentum direction in the Brillouin zone.
• Figure 4: (a) Momentum-resolved probability $P_{if}(\mathbf{k}_f)$ for the pump process at fixed photon energy $\hbar\omega=1.73$ eV, corresponding to excitation of an O-$2p_\pi$ electron followed by emission of a paramagnon with energy $\hbar\Omega_{\mathbf{q}}$, as sketched in the inset Feynman diagrams. (b) Momentum-resolved probability for the probe process at $\hbar\omega=1.32$ eV, involving absorption of a paramagnon previously generated by the pump, as sketched in the inset. (c) Momentum-integrated transition probabilities for pump (purple) and probe (orange) processes as functions of photon energy, obtained by integrating, for each photon energy, the maps in panels (a) and (b) over the Brillouin zone. (d) Calculated two-dimensional correlation map constructed as the product of the pump and probe energy dependences in panel (c), reproducing the off-diagonal resonance observed experimentally in 2DES.
• Figure 5: Antiferromagnetic (paramagnon) dispersion and momentum-transfer phase space. The semi-transparent surface shows the model paramagnon dispersion $\hbar\Omega_{\mathbf{q}}$ over the full Brillouin zone. Overlaid is the probability $P_{if}$ for transitions mediated by paramagnon emission for incident photon energies between 1 and 2 eV. The darker blue (red) shaded areas highlight the set of momenta $\mathbf{q}$ contributing to this indirect process for the initial O-$2p_{\pi}$ [O-$2p_{\sigma}$] state. For visualization, the $(\pi,\pi)$-centered contribution is translated by a reciprocal-lattice vector and displayed around $(-\pi,-\pi)$ (BZ periodicity). Vertical lines mark the reference initial momentum points at $(-\pi,-\pi)$ (blue) and $(0,0)$ (red).