Fermi-edge problem in the presence of AC electric field

We study in this paper a non-equilibrium Fermi-edge problem where the system under investigation is a single electron reservoir putting under an AC electric field. We show that the electron Green’s function and other correlation functions in the problem can be solved and expressed exactly in terms of a well-defined integral. The qualitative behaviors of the solution is studied and compared with the situation where the impurity is coupled to more than one reservoir at different chemical potentials. The Fermi edge problem in non-equilibrium situation is perhaps one of the most simple but non-trivial problems in non-equilibrium quantum many-body physics. It is the generalization of the Noziéres-De Dominicis X-ray edge problem [1] to non-equilibrium situations where the systems may be (i) connected to N > 1 reservoirs at different chemical potentials [2], (ii) put under a timedependent electromagnetic field, or (iii) put under other non-equilibrium configurations. The problem concerns the reaction of a non-equilibrium electron gas to the sudden turning on of a scattering potential. A simple realization of the problem is a small quantum dot coupled to electron reservoirs as shown in Fig.1. The small quantum dot can be treated as a two-level system, and by changing the gate voltage applied to the quantum dot, the electron occupancy of the dot can be changed, resulting in a change of the local dot potential felt by electrons in the reservoirs. [3–5]. Indeed, Fermi-edge singularity behavior was observed experimentally in a quantum dot system at energy range ∼ meV [6]. The problem is in principle a oneparticle problem where the many-body effects enter only through Fermi statistics. The problem has been studied carefully in those cases where the system is connected to reservoirs at different chemical potentials [3,7–9]. It is now understood that in this case the problem is equivalent to solving an N × N matrix Riemann-Hilbert (RH) boundary value problem [7, 8] of which a general exact solution is unavailable and only the long-time asymptotic behavior of the solution can be extracted [2, 7, 8]. In this paper we consider a different situation where the system has only one electron reservoir putting under an AC electric field generated by, for example a microwave radiation. Going to the center of mass (CM) frame it can be shown that the electric field can be eliminated [10] and the system becomes an “equilibrium” system coupled to a time-dependent (periodic) impurity scattering potential. The corresponding Fermi-edge problem concerns the reaction of this “equilibrium” electron gas to the sudden turning on of a time-dependent scattering potential. The problem provides a rare example of an non-trivial nonequilibrium quantum many-body problem which can be solved exactly and is testable. We find that the equation of motion for the electron Green’s function is a Carleman type integral equation of which the solution can be expressed in terms of a well-defined integral. In this paper, the qualitative behaviors of the solution are analyzed in this paper and compared with the situation where the impurity potential is static but the system is connected to reservoirs at different chemical potentials. The system we consider is described by the timedependent one-particle Hamiltonian H = H0(t) + V (t), where