Charge transfer statistics and qubit dynamics at the tunneling Fermi-edge singularity

Tunneling of spinless electrons from a single-channel emitter into an empty collector through an interacting resonant level of the quantum dot (QD) is studied, when all Coulomb screening of charge variations on the dot is realized by the emitter channel and the system is mapped onto an exactly solvable model of a dissipative qubit. In this model we describe the qubit density matrix evolution with a generalized Lindblad equation, which permits us to count the tunneling electrons and therefore relate the qubit dynamics to the charge transfer statistics. In particular, the coefficients of its generating function equal to the time-dependent probabilities to have the fixed number of electrons tunneled into the collector are expressed through the parameters of a non-Hermitian Hamiltonian evolution of the qubit pure states in-between the successive electron tunneling events. From the leading asymptotics of the cumulant generating function (CGF) linear in time we calculate the Fano factor and the skewness and establish their relation to the extra average and the second cumulants, respectively, of the charge accumulated during the QD evolution from its empty and stationary states, which are defined by the next-to-leading term of the CGF asymptotics. The relation explains the origin of the sub-Poisson and super-Poisson shot noise in this system and shows that the super-Poisson signals existence of a nonmonotonous oscillating transient current and the qubit coherent dynamics. The mechanism is illustrated with particular examples of the generating functions, one of which coincides in the large time limit with the generating function of the 13 fractional Poisson distribution realized without the fractional charge tunneling