Fermi gas response to time-dependent perturbations
  We describe the Riemann-Hilbert (RH) approach to computing the long-time response of a Fermi gas to a time-dependent perturbation. The approach maps the problem onto a non-commuting RH problem. The method is non-perturbative, quite general and can be used to compute the Fermi gas response in driven (out of equilibrium) as well as equilibrium systems. We illustrate the power of the method by rederiving standard results for the core-hole and open-line Greens functions for the equilibrium Fermi edge singularity (FES) problem. We then show that the case of the non-separable potential can be solved non-perturbatively with no more effort than for the separable case. We compute the corresponding results for a biased (non-equilibrium) model tunneling device, similar to those used in single photon detectors, in which a photon absorption process can significantly change the conductance of the barrier. For times much larger than the inverse bias across the device, the response of the Fermi gases in the two electrodes shows that the equilibrium Fermi edge singularity is smoothed, shifted in frequency and becomes polarity-dependent.These results have a simple interpretation in terms of known results for the equilibrium case but with (in general complex-valued) combinations of elements of the scattering matrix replacing the equilibrium phase shifts. We also consider the shot noise spectrum of a tunnel junction subject to a time-dependent bias and demonstrate that the calculation is essentially the same as for the FES problem. For the case of a periodically driven device we show that the noise spectrum for the Coherent States of Alternating Current can be easily obtained using this approach. 
N. d'Ambrumenil and B. Muzykantskii