Feedback of the electromagnetic environment on current and voltage fluctuations out of equilibrium

We present a theoretical framework for the statistics of low-frequency current and voltage fluctuations of a quantum conductor embedded in a linear electromagnetic environment. It takes the form of a Keldysh field theory with a generic low-frequency limit that allows for a phenomenological understanding and efficient evaluation of the statistics in the saddle-point approximation. This provides an adequate theoretical justification of our earlier calculation that made use of the so-called ``cascaded Langevin approach.'' We show how a feedback from the environment mixes correlators of different orders. This explains the unexpected temperature dependence of the third moment of tunneling noise observed in a recent experiment. At finite temperature, current and voltage correlators of order 3 and higher are no longer linearly related. We show that a Hall bar measures voltage correlators in the longitudinal voltage and current correlators in the Hall voltage. Next, we demonstrate that the quantum high-frequency corrections to the low-frequency limit correspond to the environmental Coulomb blockade. We prove that the leading order Coulomb blockade correction to the nth cumulant of the current fluctuations is proportional to the voltage derivative of the (n+1)-th cumulant. This generalizes to any n earlier results obtained for $n=1,2.$