Photoinduced changes of the chemical potential in superconducting Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$

The chemical potential of a superconductor is of critical importance since, at equilibrium, it is the energy where electrons pair and form the superconducting condensate. However, in non-equilibrium measurements, there may be a difference between the chemical potential of the quasiparticles and that of the pairs. Here we report a systematic time- and angle-resolved photoemission study of the pump-induced change in the chemical potential of an optimally doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi2212) sample in both its normal and superconducting states. The change in chemical potential can be understood by separately considering the change in the valence band energy relative to the vacuum, and the change in chemical potential relative to the valence band energy. We attribute the former effect to a changing potential barrier at the sample surface, and the latter effect to the conservation of charge in an asymmetrical density of states. The results indicate that the pair and quasiparticle chemical potentials follow each other even on picosecond timescales.