Amplitude mode oscillations in pump-probe photoemission spectra from a $d$-wave superconductor

Recent developments in the techniques of ultrafast pump-probe photoemission have made possible the search for collective modes in strongly correlated systems out of equilibrium. Including inelastic scattering processes and a retarded interaction, we simulate time- and angle-resolved photoemission spectroscopy (trARPES) to study the amplitude mode of a $d$-wave superconductor, a collective mode excited through the nonlinear light-matter coupling to the pump pulse. We find that the amplitude mode oscillations of the $d$-wave order parameter occur in phase at a single frequency that is twice the quasi-steady-state maximum gap size after pumping. We comment on the necessary conditions for detecting the amplitude mode in trARPES experiments.

Figure 1

trARPES spectra. In all panels, the orange marker indicates the peak position of the EDC at $k_F$, which corresponds to the superconducting gap size, and the red dotted line indicates the energy of the bosonic mode plus the antinodal gap size which corresponds to the gap-shifted position of the kink in the band structure. (a) Nodal ARPES spectrum in equilibrium with superconductivity. (b) Nodal spectrum 25 fs after pump arrives. (c) Off-nodal ARPES spectrum in equilibrium with superconductivity. (d) Pumped off-nodal spectrum shows shift in the kink and a partial melting of the superconducting gap.

Figure 2

Gap and kink dynamics. (a) Tracking the peak position of the EDC at $k_F$ given by the orange marker in Fig. 1 (i.e., superconducting gap size) for an off-nodal cut which is halfway between the node and antinode as indicated by the dotted red line in the inset of Figs. 1 and 1. (b) Tracking the EDC peak position for multiple cuts along the Fermi surface. (c) Tracking the kink position given by the horizontal red dotted line in Fig. 1 based on the inflection point in the peak-dip-hump structure of EDC's at $k_F$ for multiple cuts. Curves for the kink position are offset for clarity. The pump field is $E_{\max }=1.2\text{V}/a_0$ for all plots. The time is measured relative to the center of the pump pulse (which reaches the sample at time $t=0$ fs). (d) The frequency of the oscillations (of the gaps and the kinks) occurs at a single frequency given by the average value of 47 meV, shown as the dotted black line.

Figure 3

Gap dynamics vs fluence. The superconducting gap size (determined by the magnitude of the antinodal EDC peak position) as a function of time for different pump fluences (maximum electric field in $V/a_0$). Solid lines show the fits. (Inset) In the zero fluence limit, the Higgs frequency extrapolates to twice the maximum gap size in equilibrium. The solid line is a quadratic polynomial fit.