Noise correlations in time- and angle-resolved photoemission spectroscopy

In time-resolved photoemission experiments, more than one electron can be emitted from the solid by a single ultrashort pulse. We theoretically demonstrate how correlations between the momenta of outgoing electrons relate to time-dependent two-particle correlations in the solid. This can extend the scope of time- and angle-resolved photoemission spectroscopy to probe superconducting and charge density fluctuations in systems without long-range order, and to reveal their dynamics independent of the electronic gap and thus unrestricted by the energy-time uncertainty. The proposal is illustrated for superconductivity in a BCS model. An impulsive perturbation can quench the gap on ultrafast timescales, while nonequilibrium pairing correlations persist much longer, even when electron-electron scattering beyond mean-field theory is taken into account. There is thus a clear distinction between a dephasing of the Cooper pairs and the thermalization into the normal state. While a measurement of the gap would be blind to such pairing correlations, they can be revealed by the angular correlations in photoemission.

Figure 1

(a) Order parameter $\varphi$ (solid) and pairing correlations $F_{k_f}$ (dashed) at the Fermi surface $\left({\epsilon }_k=0\right)$ after an interaction quench with $\mathrm{\Delta }U=0.0,0.3,...,2.7$ (top to bottom). The bold line corresponds to $\mathrm{\Delta }U=2.4$. (b) Numerical simulation of the noise correlation measurement for $\mathrm{\Delta }U=2.4$: $\mathrm{\Delta }{I}_k\left(t\right)$, as obtained from Eq. (8) with a short pulse $S\left(\tau \right)=\sqrt{100/\pi }{e}^{-100{(\tau -t)}^2}$ centered around time $t$.

Figure 2

(a) Order parameter $\varphi$ (solid lines) and Cooper-pair correlations $F_{k_f}$ (dashed lines) for the ramp protocol with different excitation densities $\mathrm{\Delta }U=1.5,1.8,...,2.7$ (from top to bottom). The bold line shows $\mathrm{\Delta }U=2.4$. The shaded area highlights ramp duration period. Inset: Decay rate $\mathrm{\Gamma }$ obtained by an exponential fit to $\varphi \left(t\right)$ (solid red), the momentum-averaged noise ${\sum }_k\sqrt{\mathrm{\Delta }{I}_k\left(t\right)}$ (dashed red), $F_{k_f}$ (solid, blue), and $\sqrt{\mathrm{\Delta }{I}_{k_F}\left(t\right)}$ (dashed blue) against $\mathrm{\Delta }U$. (b) Numerical simulation of the noise correlation measurement $\mathrm{\Delta }{I}_k\left(t\right)$ for energies around the Fermi edge for $\mathrm{\Delta }U=2.4$.